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'1)(S2)] 'in T whenever 81 < 82 in S. The sets Sand T are order-isomorphic if there tS a bijedwn
'Ii' :
1/; : S
---->
T such that
1.)
and
1/)-1
are isotomc.
Definition 1.1.3 Let (S.:::;) be a partially ordeTed set. Then (S.:::;) is a lattice
tf {.'I, t} has a supTemum and an mfimum for' each Let (S, :::;) be a lattice, and let
8.
t
5,
t
E
S.
E S. Then we set
s V t = sup{s, t}.
.~ 1\
t = inf{s, t}.
For example, let T be a Ilon-empty set. Then (P(T), C) is a lattice: for subsets T1 and T2 of T. T1 V T2 = T1 U T2 and T1 1\ T2 = T1 n T 2 . Similarly, the family of all finite sllh.,ets of T is a lattice. Again. (JR A ,:::;) is a lattice: for j, 9 E JR A . we have
(f V g)(n) = f(ex) V g(o.)
and
(f 1\ g) (0:) = f(a) 1\ g(n)
for
ex EA.
Definition 1.1.4 Let (A,:::;) be a paTfzally oTdered set. Then A tS a directed set tf, fOT each Q. (3 E A. there ensts "Y E A wtth n :::; 1 and [j :::; "y; a tail fa S' if; a subset of the form {Q E A : Q 2:: Qo} for some Qo E A. A netm a set X 'i8 a
function from a dtlw:ted set into X. For example. N is a directed set for the obvious ordering; more generally. each lattice is a directed set. Often a net f : A ~ X is denoted by (,cO' : n E A). or just (,Tt», where the directed set is left unspecified. A sequence (xn) is a net indexed by N. The product of a family of directed sds is a directed set with respect to the product partial order. Let A be a directed set, and let f : A ~ X be a net in a set X. A sub71ct of ~ X. where B is a directed set and l/J : B ~ A is an order-preserving map such that 'Ij!(B) is cofinal in A' We now consider the notion of a projective systenl.
f is a map of the form fOil' : B
Definition 1.1.5 Let (A,:::;) be a d2rccted set, and let {Xc> : 0: E A} be a farmly of non-empty sets. Suppose that, for each Q. (1 E A 'With 0 :::; (3. there 2S a map 'IT"r3 : Xfj ~ X" 81Lch that 'IT"" Z8 the identtty on X"a and 'lTO'd 0 'lTih = 'lTt>;, whenever 0:::; {j :::; "Y. Then {Xl>; 'lTai;; A} is a projective system. Defin(. X
== lim proj{Xa; 1l'a,e; A}
=
{ext»~ E II Xc< : 1l'o.,e(x,e) =
Xc< for all n, j3 E A with
O'EA
Then X is the projective limit of the system {XO'; 1l'a{3; A}.
Q;
S
fJ}
Algebraic foundations
6
Clearly 1ra = 1raf3 0 1rf3 on X whenever (X S [3. In the special case where (with the usual order), we refer to a project we sequence {Xnj7l'mn}. Let (Xn) be a sequence of non-empty sets, and let en : X n+1 ---> Xn be a map for each n E N. For n > m, set 1rmn = ()m 0 ()m+l 0 . . . 0 en-I: Xn ---> X m , and let 1rmrn be the identity on X rn · Then {Xn;1rmn} is a projective sequence. denoted by {Xn; en}; its projective limit is
A
=N
X
== limproj{Xnj ()n}
=
{(xn ) E I1nENXn : ()n(xn+J)
= Xn
(n
E
N)}. (1.1.1)
Definition 1.1.6 Let 5 be a non-empty set. Let'R be a family of subsets of 5 such that Rl n R2 E whenever R J , R2 E 'R. An 'R-filter on 5 25 a non-empty subset F ofP(5) such that: 0 tJ. F; if E, FE F, then En FE F; if FE F and R E 'R with FeR, then R E:F. An 'R-ultrafilter on 5 is an 'R-filter' F on 5 such that either E E For' FE F whenever E, FE 'R wdh E U F = 5. A filter on 5 'i.~ a P(5)-filter; an ultrafilter on 5 is a P(5)-ultrafilter.
n
Let. 'R be as in the above definition. The family of all 'R-filters on 5 is a partially ordered set with respect to inclusion; an 'R-filter is an ultrafilter if and only ifit is maximal in this family. For example, let s E 5. Then {T C 5 : SET} is an ultrafilter OIl 5; these are the fixed ultrafilters on 5, and ultrafilters which are not fixed are free. Suppose that (A, S) is a directed set, and let F be the family of all subsets T of A which contain a tail. Then F is a filter on 5, called the order filter, and each ultrafilter containing F is a free ultrafilter on A. In the case where A = N, the order filter consists of all subsets F of N such that N \ F is finite; it is sometimes called the Frt£chet filter. Let F be a filter on 5, and let f,g E ~s. Then we say that f SF 9 if {8 E 5 : f(8) S g(s)} E:F. Clearly SF is a preorder on ~s. We say that f ""F 9 if f SF 9 and 9 SF f. and so f "'F 9 if and only if {s E 5 : f(s) = g(s)} E :F. In this way we obtain a partially ordered set which we denote by (~S / F, SF)' In the case where F is an ultrafilter, this set is totally ordered. Definition 1.1.7 Let (5, S) be a totally ordered set. Then (5, S) is well-ordered if each non-empty 8'ubsft of 5 conta'ins a minzmum element. Let (sn : n E N) be a sequence in a partially ordered set (5, S). Then (sn) is lnc'1'ea8ing [strictly zncreasing] if 8 n S 8 n +l [sn < Sn+l] (n EN). We define decreasing and 8trzctly decreasing sequences similarly. The sequence (8 n ) has a property eventually if there exists no E N such that (sn : n 2: no) has the property; we also say that 'sn has the property eventually'. In ZFC, a set (5, S) is well-ordered if and only if each decreaHing sequence in 5 is eventually constant. Let 5 be a well-ordered set, let T be a totally ordered set, and take f, gETS with f -I- g. Define
6(1,g) = min{s
E
-I- g(s)}. f(6(1,g)) -I- g(6(1, g));
5 : f(8)
Since 5 iH well-ordered, 6(1,g) exists and we set f -< 9 if f(6(1,g)) < g(o(1, g)) in T. For f,g E TS, we set f :::S 9 if f -< 9 or f = g. Then :::S is a total order on T S : it is called the lexicographic order. In particular, (1'1, .•• , r n) -< (Sl' ... , sn) in the lexicographic order on ]Rn if and only if 1'1 < Sl or there exists k E {2, ... ,n} such that ri = 8i (i E Nk-d and rk < 8k.
Foundatwns and order'ed sets
7
\Ve now give two sentence::; which are equivalent to AC in ZF.
Zorn's lemma is the sentence: each partially ordered set ill which each chain has an upper bound has Ii maximal element. The 'Well-oTdeTing pnnciple is the sentence: on each non-empty set 5, there is a binary relation:::::; such t.hat (5, :::::;) is a well-ordered set.
Theorem 1.1.8 The following sentences aTe eq'uiualeut in ZF: (a) the axiom of choice, AC,. (b) Zorn's lemma; (c) the well-ordering pr'inczple. 0 Thus, as we shall be working in ZFC, we shall be able to apply Zorn's lemma and the well-ordering principl('. Among the many facts that hold in ZFC and which we shall usc in this book, but which cannot be proved ill ZF, are th(, following: (i) each lin('ar space has a basis; (ii) in a unital algebra. each proper ideal is contained in a maximal ideal; (iii) each field is contained in an algebraically closed field; (iv) each R-filter is contained in an R-ultrafilter; (v) each continuous linear functional on a linear subspace of a normed space has a norm-preserving extension to the whole space. A famous model 9J1 of the theory ZF has been constructed by Solovay. In this model, the axiom of choice is false. but a weaker axiom, the 'axiom of dependent choice (DC)'. is true. and it is proved that every subset of each complete. separable metric space has the Bairc property (see Definition A.5.13). It follows rather easily from this (see A.5.24) that, in 9J1, every linear map from an (F)-space into a locally convex space is automatically continuous. Thus one cannot provE' in the theory ZF that discontinuous linear maps exi::;t, and so ZF is not a sufficiently rich theory in which to study the questions of automatic continuity that interest us. Although somewhat weaker axioms would suffice, it seems reasonable for us to assume AC itself, and thus to work in the theory ZFC. This 'We shall do j7'Om. this powt on'WaTds. \Ve now discuss ordinal and cardinal nnmbers.
Definition 1.1.9 Let 5 be a set. Then 5 1,8 transitive if Fach element of 5 is a subset of 5, and 5 ~s an ordinal if 5 2S transitive and (5, E) t8 a totally ordered set, Tegar-dmg 'E' as a stnct paTtwl onteT. Specifically. let 5 be transitive. and, for n.!J E 5. set 0: < (J if 0: E (3. Then 5 is an ordinal if (5,:$) is totally ordered. Suppose that wc are given a property that a set Illay have. The collection of all sets which have this property is the class determined by the propert.y. Certainly, every set is a class, but consideration of Ru::;sell's paradox shows that, for example, the cla::;s of all set::; is not a set. We denote the class of all ordinals by Ord; Ord is not a. set, but it is convenient to extend to Ord the notions of ordering that were previously defined for sets. Thus, let 0, (3 E Ord. Then it can be easily seen that 0 E (3 if and only if 0 0 and x < l/n for each n E N. By taking 51 or 52. respectively, to be 0 in (iii), above, we see that coi 5 > ~o and cof 5 > ~o for an 7]l-set 5; no set is both an aI-set and an 7]l-set. Let U be a free ultrafilter on N. Then it is easy to see that the totally ordered set (JR.N /U, ~u) is an 7]rset; this set will be discussed further later. Let S = {O, 1 }Wl . so that S is the collection of dyadu: sequences of length WI: S is a totally ordered set with respect to the lexicographic order :5. Definition 1.1.17 The set Q is the subset of elements a of S such that {T < WI : aT = I} is non-empty and has a maximum. For each IJ < WI. Qa is the set of elements of Q for which this maximum is less than IJ. Proposition 1.1.18 (i) For each (ii) Q
= U{Qa : IJ < wd
i8
IJ
8. A subset T of a semigroup S generates S if each element of S is a finite product of elements of T.
12
Algebraic foundations Let 8 1 and 8 2 be non-empty o;ubsets of a o;emigroup 8. Then
81
.
82 =
{81S2 : 81 E
81,
S2 E
8d,
and, if 8 is written additively, then
8 1 +82 =
{81 +S2: S1 E
81,
S2 E
8 2 }.
We write s . 8 for {s} ·8 = Ls(8) and s + 8 for {s} + 8, etc. Let 8 be a semigroup. An element e of 8 is a left [right] identity of 8 if es=s
[se=s]
(sE8).
An element e is an identity of 8 if it is both a left identity and a right identity; such an element of 8 is unique. A semigroup which has an identity is a umtal o;emigroup; we o;hall often denote the identity of 8 byes or bye. A semigroup without an identity is non-unital; in this case, take e O}; Q+., iR.+., and II are each subsemigroups of the semigroup (e, + ). Let 8 be a semigroup. An element s E 8 is an idempotent if S2 = 09; left identities and right identities of a semigroup are each idempotents. Let 8 be a unital semigroup, and let s E 8. Then 09 is left [rzght] invertible if there exists t E 8 with ts = es [st = es]. The element s is invertible if it is both left and right invertible. In this case, there is a unique element t E 8 with ts = st = es; t is the mverse of s. written S-1. The set of invertible elements of 8 is denoted by Inv 8, and 8 io; a group if Inv 8 = 8. The set Inv 8 is a group with respect to the operations of 8, and (8t)-1 = C 1 S- l (s, t E Inv 8). A subset T of a unital semi group 8 is a subgroup of 8 if T is a group for the operation of 8 and the identity of T is es. Let G be a group. For 8 c G, we define 8- 1 = {S-1 : s E 8}; 8 is symmetric if 8- 1 = 8. For f E e G , set ](09) = f(8- 1 ) (s E G); f is symmetric if ] = f. Let a E G. Then the left o;hift of f by a il:l defined by (8a f)(s) = f(a- 1 s)
(s
E G).
(1.2.2)
The product [t:>EA 8 a of a family {8a : Q E A} of o;emigroups is a semigroup with respect to the operation (sa)(t a,) = (Sata) (sa, ta E 8 a ). Suppose that 8 is a unital o;emigroup. Then we identify 8<w with 8<w = {(sn) E 8 w : Sn
= es eventually};
clearly 8<w is a subsemigroup of 8 W •
Examples 1.2.2 (i) Let 8 be a non-empty set. The set of bijections from 8 to itself is a group with respect to composition. In the case where 8 = Nn , this group io; the group of permutations on n letters; it is the symmetric group of order n, denoted by en. The group en is a finite group consisting of n! elements. A word with letters from the set 8 is a finite formal product S~l . . . s;;;, where mEN, S1,'" ,Sm E 8 and CI,'" ,Cm E {-I, I}; we also take the formal product with no factors to be a word (denoted bye). A word is reduced if it is e or if
Semigroups
13
whenever Sk+l = Sk· Each word is equivalent to a reduced word in an obvious way. The free group on S, denoted by IF'(S), consists of all reduced words; the product of S~1 •.• s;;; and t~1 ••• t~n iH the reduced word equivalent to S~1 ..• s;;;' t~1 •.• t~n. The identity of IF'( S) is e, and the inverse of s1 1 •.. s;~' is
Ck+l = Ck
s;;,,"m .. '8;-"1
Suppose that lSI = n. Then IF'(S) is denoted by IF'n: it is the free gr01tp on n generators. In particular, IF'2 is the free group on two generators. Abo, if lSI = ~o, then IF'(S), denoted by IF'N o , is the free group on countably many generators.
(ii) Let n E N. Then zn, Qn, ]Rn, and en are groups with reHpect to coordinatewise addition. Similarly, Z<w and ZW, etc., are groups. The former group contains (Z+)<w as a subsemigroup; (Z+)<w is the free abelian semzgroup on countably many generators. For j E N, set X J = (8;,j : i EN). Then each element of (Z+) <w can be written as Xr'1 ... X;;k , where ml, ... ,mk E Z+ and mi + ... +mk > O. (iii) Let n E N. Then §n = N;;:w, the set of finite sequences of elements of N n . The set §n is a semigroup with respect to juxtaposition of sequences; it is the free semigroup on n generators. By replacing N n by N, we obtain §N o , the free semzgroup on countably many generators. In fact, Het aj = (j) for J E N n · Then the generators of §n are a}, ... , an, and each element of §n has the form a'J:.k, where jI,'" ,jk E N n and mI,"" mk EN. Clearly §n is a su bsemigroup of IF' n' 0
aJ:1 ...
Definition 1.2.3 Let S be a semzgroup. Then S zs:
(i) cancellative zf rs
i= rt
and sr
i= tr
i= t; with t n =
whenever r, s, t E Sand s
(ii) divisible if, for each s E Sand n E N, there eX'tsts t E S
s;
(iii) torsion-free if r = s whenever r, s E Sand rn = srt for some n E N; (iv) a cone if S is cancellative, non-umtal, and abelian. Thus S is cancellative if and only if Lr and Rr are injective on S for each rES. Note that an abelian group is torsion-free if r = eo whenever rn = eo for some n E N. Also, if S is a divisible, torsion-free semigroup, then, for each s E S and each n E N, there exists a unique element t E S such that t n = s; this element t is denoted by si/n (or by sin if we are using additive notation). Suppose that S if, a cone and that s E S. Then sm = sn for m, n E N only if
e
m=n. It is clear that a subset T of an abelian group (e, +) is a cone (with respect to the operation of e) if and only if T + T C T and 0 ¢. T. Definition 1.2.4 Let Sand T be semigroups. A map'ljJ : S --> T is a morphism if'ljJ(rs) = 'ljJ(r)'ljJ(s) (r, s E S); 'l/J is an isomorphism if it is also bijective. Definition 1.2.5 Let S be a semigroup, and let K be a subsemigroup of (e, + ) with 1 E K. A K-semigroup in S is the image 'l/J(K) of a morphism 'ljJ ; K --> S; the semigroup 'l/J(K) is non-zero if'l/J(K) c S·.
Algebraic foundations
14
A K-semigroup 'lj;(K) is usually denoted by (s< : ( E K), where 'lj;(1) = s. We refer to rational and real semigroups when K is Q+. or jR+., respectively. Let 5 be a semigroup, let", be an equivalence relation on the set 5, and let 1r: 5 -- 51'" be the quotient map onto the set of equivalence classes of "'. The relation", is compatible with 5 if rs '" rt and sr '" tr whenever 1', s, t E 5 and s '" t. In this case, set ?r(r)1r(s) = ?r(rs) (1', s E 5); we obtain a well-defined operation on 5/", with respect to which 5/", is a semigroup. The quotient map 1r is a morphism. Let G be a group, and let H be a subgroup of G. Then H is normal if s· H = H· s (s E G), and in this case G/ H, the set of cosets of H, is a group. Let '" be an equivalence relation compatible with G, and let H = {s E G : s '" eo}. Then H is a normal subgroup of G, and G I H = G / '" . Let G and H be groups, and let IjJ : G -- H be a morphism. Then the kernel ker~ of '1jJ, defined to be {s E G: ~(s) = eH}, is a normal subgroup of G, and the map s· ker'lj; f--7 'lj;(s). G/ker'lj; -- 'lj;(G), is a bijective morphism.
Definition 1.2.6 Let 5 be a cone, and let G be an abelian group. operates on 5 if there zs a map (a, s) f--7 as, G x 5 -- 5, such that;
Then G
(i) a(st) = (as)t (a E G, s, t E 5); (ii) (a(3)s = a«(3s) (a, (3 E G, s E 5). Gs
In this case, it follows from (i) and (ii) that eos = {as: a E G} for s E 5.
=
s (s E 5). We define
Proposition 1.2.7 Let 5 and T be cones, and let G be a divisible, abelian group that operates on T. 5uppose that 80 E 5 and that X : 5 -- T is such that X(st) E GX(s)X(t) (s, t E 5). Then there zs a morphz8m ~ : 5 -- T such that 'lj;(so) = X(80) and'lj;(s) E GX(8) (s E 5). Proof Let Uo = {sa: n EN}, and define 'Po : sa -- X(so)n, Uo -- T, so that 'Po is a morphism with 'Po(so) = X(so) and 'Po(s) E GX(s) (s E Uo). Denote by F the family of pairs (V, .:= (U, 'lj;) in :F. By the maximality of (U, 'lj;), we have V = U, and so r E U whenever there exist s, t E U with rs = t.
15
Semigroups
For l' E S, set Vr = U u U{ I'm . U : mEN}, so that Vr is a subsemigroup of S with Vr :J U. Assume that there exist l' E S \ U and n :2:: 2 with 1',1'2, . .. ,rn - l ¢. U and rn E U. Then 1/)(1''') E Gx(r n ) = Gx(r)n. Since G is divisible, there exists 0' E G with '1/;(1''') = anx(r)n. Set 'IjJ(rmu) = O'mx(r)m'IjJ(u) for mEN and u E U. Suppose that rmlul = rm2u2 for some mI, m2 E N with mi > m2 and some UI. U2 E U. Then rml-m2111 = 'U2, and so r m, - m2 E U. This shows that m'l = m2 + np for some pEN. We have
'I/;(U2) = 'tf;(rnY'tf;(ud = xnpx(r)"P'Ij)(ut) , and so ami x(r)m l 1j;(ud = O' Tn2 x(r)1n 21/)(U2) , Thus 'I/; is well-defined on the set U{rm . U : mEN}. This extension of'IjJ to \1;. is a morphism such that 'IjJ(v) E GX(v) (v E V). But this contradicts the assumed maximality of (U, '1/). Hence l' E U whenever rn E U for some n E N. Finally, assume that there exists l' E S \ U. Suppose that r1nlul = rm2U2 for some ml,rn2 E N with mi :2:: m2 and some UI, U2 E U. If mi > m2, then rTnl-1n2uI = 112, and so r m, - m2 E U and l' E U, a contradiction. Thus mi = m2. and the map 1/) : I'm f---t x(r)m'IjJ(u), Vr ----- T, is a morphism extending 1/J with 1/J(s) E GX(s) (s E Vr ). Again Vr = U, and so l' E U, a contradiction. Thus U = S, and the result is proved. 0 \Ve now give a construction that will show that each cancellative, abelian semigroup can be embedded in a group. Let S be an abelian semigroup, and let T he a subsemigroup of S. We say that (SI' tt) '" (S2' t2) in S x T if there exists l' E T such that r8 l t 2 = rS2tl' Routine verifications show that the relation", is an equivalence relation which is compatible with S x T.
= (S x T)/"" and let 7r : S x T ----- S[T- I ] be T, set 1/J(s) = 7r«(8t, t») (8 E S).
Definition 1.2.8 Define S[T- I ]
the quotient map. For t
E
The definition of 1/J( s) is independent of the choice of t ill T because it is clear that (Stl,tt) '" (st2,t2) in S x T whenever t I ,t2 E T.
Proposition 1.2.9 Let S be an abelian semigroup with a subsemigroup T.
(i) The semigroup S[T- I ] zs an abelian, unital semigroup; '4): S a morphism: and 1jJ(T) C Inv S[T-I]. (ii) S[T- I ] = {1/J(s)'tf;(t)-1 : S E S, t E T}. (iii) In the case where S is canceliatwc, 'IjJ is an mjection.
f---t
S[T- I ] is
Proof (i) Certainly S[T- 1 ] is an abelian semigrollp. Define e = 7r(t, t» for t E T. Then e is independent of the choice of t in T, and e is the identity of the semi group S[T-I]. The map 1/J is a morphism, and 1/J(T) C Inv S[T- I ] because 7r«rt, t)(t, rt» = e (1', t E T). (ii) Let x E S[T-I]. Then x = 7r(8, t» = 'IjJ(8)'¢(t) -1 for some s E Sand t E T. (iii) Let 81,82 E S with '¢(8d = '1/;(82)' and take t E T. Then there exists l' E T with r81t2 = r82t2. Since S is cancellative, necessarily 81 = 82, and so '¢ ~~~w.
0
Algebraic foundations
16 We now apply 1.2.9 in the case where T notation.
=
S; we continue to use the above
Proposition 1.2.10 Let S be an abelian semigroup.
(i) The semzgroup S[S-1] is an abelian group. (ii) Suppose that S is cancellative. Then 1j; : S
-+
S[S-1] is an mjective
morphism. (iii) Suppose that S is cancellatzvc and tOri,ion-free. Then the gr'oup S[S-1] is torsion-fTee, and IS[S-1]1 = lSI. Proof Set G
=
S[S-l].
(i) Since 1l'((s. t)(t, s)) = eu (8, t E S), each element of G is invertible. (ii) This is immediate. (iii) Suppose that x E G with xn = ee, say x = 1j;(r)ljJ(s)-l. Then rn = 8" in S, and so r = .., because S is torsion-free. Thus x = cu, and G is torsion-free. If S is a singleton, then lSI = IGI = 1. If S is not a singleton, there is an element s E S which is not an identity of S. Since 'ljJ is injective, 1/;(8) i ee. Since G is a torsion-free group, the map n f--+ 1j;(8)", N -+ G, is an injection. Thus lSI = 1'4)(S)1 ;::: No, and so IS x 81 = lSI. Hence IGI ~ IS x SI = lSI ~ IGI, and so IGI = lSI, as required. 0 The group S[S-1] is the group of the scmigTOup S. Henceforth we shall identify S with ljJ(S); each cancellative, abelian semigroup is a subsemigroup of a group. Definition 1.2.11 Let S be a 8emigroup, and let ~ be a binary relation on S. Then (S,~) is an ordered semigroup zf S is a partial order on S and if Ls and Rs are isotonic for each s E S; (S,~) is totally ordered or well-ordered if the underlymg ordered set zs totally oTdered or well-ordered, respectively. Note that a totally ordered semigroup is cancellative and torsion-free, and that each abelian semi group is an ordered semigroup with respect to the binary relation =. A group [cone] which is an ordered semigroup is an ordered group [ordered cone]. For example, let n E N. Then (zn,::5) is a totally ordered group with respect to coordinatewise addition, where ::5 is the lexicographic order. Similarly, (Z<w,::5) and (zw,::5) are totally ordered groups. Now set s --< t in §n if either £(8) < £(t) or £(8) = £(t) and 8 --< t in the lexicographic order on Nr~(s). Then the semigroup (§n,::5) is well-ordered. We now change to additive notation for semigroups, as we wish to identify certain abelian semigroups with subsets of linear spaces. An identity of a semigroup is now denoted by 0, and the inverse of an element 8 is -8. Let (S, +) be an ordered, abelian semigroup, and set G = S[S-1]. Take Xl, X2 E G. By 1.2.9(ii), there exist 81,82, tl, t2 E S with Xl = S1 - t1 and X2 = 82 - t 2 · Define Xl ~ X2 if 81 + t2 ~ 82 + t 1. Then ~ is well-defined on G, and (G, +,:::;) is an ordered group; G is totally ordered if S is totally ordered.
Semzgroups
17
Definition 1.2.12 Let (G,
+,::;)
be an ordered abelzan group. Then
G+={sEG:O::;S},
G-={SEG:8::;O}.
Thus G+· = {s E G : 0 < s} and G-· = {s E G : 8 < O}. In the C8...'le where G 1= {O}, the semigrollps (G+·, +) and (G-·, +) are cones; they are the positive and negative cones of G, respectively. Definition 1.2.13 Let (G, For s E G, define s+ =
8
+,::;)
V 0,
be an ordered abelzan group which is a lattice.
8-
= 81\0,
lsi
=
8
V (-05).
For example, let 5 be a non-empty set. Then (IRS". +. ::;) is an ordered, abelian group which is a lattice, and the definitions of f+, f- and Ifl for f E IRS" coincide with the usual pointwise definitions; clearly we have f = f+ + f- and If I = f+ - f-· Definition 1.2.14 Let T be a cone whzch is a subsemzgroup of an abelzan semigroup 5, and set r < . H be the quotient map, and define 7f«SI,PI/q1» + 7r«S2,P2/q2) (p/Q)7f«Sl,pI/Ql»
= =
7f«PIQ2 S1 + P2Q1 S2. 1/QlQ2), 7r(Sl,ppI/QQl».
These operations are well-defined, and H is a rational-linear space with respect to the operations. The map 1/J : S f-+ 7f((8, 1»). G ---> H, is an injective morphism, and H = {1jJ(s)/n : s E G, n EN}, so that IHI = IGI. Set 7f«s,p/Q» > 0 in H if s > 0 and p/Q > 0 or if s < 0 and p/q < O. Then < is well-defined on H, and (H. :::;) is a totally ordered rational-linear space. The map '1/; : (G,:::;) ---> (H,:::;) is isotonic. 0
Theorem 1.2.19 Let S be a totally ordered conr.. Then there 2S a totally ordered rational-linear space H wzth IHI = lSI and an mjectwe morphism '1/) : S ---> H with 1/'(S) c H+-. Proof Let G be the group of S. Since S is torsion-free, G is torsion-free and IGI = lSI by 1.2.1O(iii). Also, G is an ordered group with respect to the S-order on G. The result now follows from 1.2.18. 0 Let S be an abelian sernigroup. For n E N. set Sn 1/Jn : S
f-+
(n
+ l)s,
S,,+1
--->
S, and define
=
Sn.
Then {Sn; 1/J,,} is a projective sequence of semigroups and morphism;". Define S = lim proj{ Sn; 1/J,,}' so that S is the set
S = {s
=
(sn)
E
SW : Sn
=
(n
+ 1)8n+1
(n E N)}.
In general S may be empty; in the case where S is non-empty, S is clearly a subsemigroup of SW. Note that, in the next theorem, the semigroup S is not assumed to be either cancellative or torsion-free; this will be important for our applications of the result in §4.9 and §5.7.
]9
Semigroups
Theorem 1.2.20 (Esterle) Lft S be an abelian serru,group, and let S be as alioue. Assume that S =I- 0. For each (\' = plq E Qt-. and.'l = (lin) E S, dfjine p((n q )!) ) as = ( q(n!) S,,'!: n EN. Then S, togetheT 71nth thf'. map (n. s)
f--->
0'8, lS
(1.2.4)
a semzgroup 01leT Q-t-•.
Proof Let S = (sn) E Sand () = plq E Q+ •. Since (nq)!/q(n!) E N (n EN). the right-hand side of (l.2.4) belongs to SW. We show that 0'8 is well-defined. First note that. if rn, n E N with In :5 11. then 8 m = (n!/m!)sn because.9 E S. Suppose that jlk = plq in Q, with k ? if in N, say. Since Snq = ((nk)!/(nq)!)8 n b we have
p((nq)!) p((nk)!) j((nk)!) q ( n.I) Snq = q (') n. Snk = k( n.') Snk, and so (plq).'> = (jlk);:;, as required. Also, have
(
n+]
0;8
E
S because, for each n E N, we
) _ p((nq + q)!)((nq)!) _ p((nq)!) . ) ( p((nq + q)!) . )1)( n.') Snq+q ' ) Snq· q (( rz + 1)') . 8 nq + q q ((,nq. q (n.
We nov.. check that the conditions of l.2.16 are satisfied. Clauses (ii) and (iv) clearly hold. Let s = (;;n). and let plq.jlk E Q-t-•. Then
q + 'kj)
(p
.'I
=
((Pk+ qj ) ((nqk)!) qk (n!) .'inqk: n
E
N
)
=
p j qS + 'k o5
and
!!. (Ls) q
k
=
(p((nq)!)j((nqk)!) 8 n k : n EN) = PJ 8. q(n!)k((nq)!) q qk
o
so that clauses (i) and (iii) also hold.
Note that, in multiplicative notation, the image of (0:, s) under the ab a maximuIIl a and B has no minimum, then there exists , s h. We clazm that (5, ::;s) is an 'T7l-COne over Q+ •. This iH a similar argument to that in 1.2.33. For example, let (In) be a strictly increasing sequence in S. and take a strictly increasing sequence (t n ) in lR such that tn ---> ex; as 71 ---> 00 and
fn+l(t)
~
fn(t)
(t
~ t n ),
1
00
Ifn(t)1 d/t < 2- n
(n
E
N).
t"
Set f(t) = 0 (t < td and f(t) = In(t) (t In <s I (71 EN), so cof 5 > No· By 1.2.32, 5 is a universal cone.
E
(tn' tn+1))' Then I E 5 and 0
24
Algfbmic fo'undations
Notes 1.2.35 The construction of S[T- I ] in 1.2.9 is an adaptation to semigroups of a standard COlH;truction (P. ;"'1 Cohn 19H9). The group G of a cancellativt', abelian semigroup S iH sonH'tirrH's called the GT'Othendieck g1'O'Up of S; it is the HOlution of a universal proLlem, in the sense t.hat, if H is a group and X : S ..... H is 11 morphiHm. then there is a group morphism {} : G ..... H such that X = {} 01/) Theorem 1.220 is baRed on part of Theorem I) 1 of (Esteri{' 19S:3a). Ordered groups. including the Hahn group J(K S), are diHcussed in (FuchH 196:3), (Priess-Crampe 1(83). and, especially, (Dales and \Voodin 1996. Chapter 1). On each torsion-free, abelian group G. there is a relation :::5 such that (G.:::5) is a totally ordered group. Propositions 1.2.26. 1.2.28, and 1.2.29 are proved in (DaleR and Woodin 1990, Chapter 1), Proposition 1.2.7 and the example Nil>!/F in 1.2.:~3 are taken from (Esterle 19961', 1(98).
1.:3
ALGEBRAS, IDEALS, AND FIELDS
\Ve present in this section a review of some essentially standard material about. algebras and their ideals; the section concludes with a brief discussion of fields, including ordered fields. Our account is. of course, prejudiced in favour of topics that will form the algel)) aic background to our later work; results are included because they will be required later. ancl our aim is to be efficient. In particular. we shall introduce notation that will be standard throughout the book. We begin hy Iecalling some basic' facts about linear spaces over a tieldlK; it will be assumed throughout that lK is either JR or C. Let E be a linear space over K As in §1.2, we write E· for E \ {a}. Let S be a non-empty subset of E. Then lKS = {(Xx: (l' E lK ..r E S}; we write lKx for lK {.1;}. The l'meaT .'ipan of 8 is lin 8 = lin K8 =
{t
(ti·'Ci :
nl· ... ,(tn
ElK,
'Cl, ... ·.E n
ES, n EN} .
For example, let 8 be a BOil-empty set.. Then coo(8) = lind Ob : S E S} C ([s, so that Coo = coo(N). In the case where 8 is a totally ordered set. coo(8) = ~(())(8). Let E be a linear space over R and set F = Ex E. Then F is a linear space over ([ for the operations:
(Xj·yd + (J'2.Y2) = (Xl +.r2, Yl +Y2) (.(1,:r2,Yl,Y2 E E); (n + i;3)(x, Y) = (nx - j3y, av + j3x) (0, (j E JR, x. Y E E). The space F is the eomplexijieation of E. Let {E-y : '"Y E r} be a family of linear suhspaces of a linear spac(' E. Then the sum L-YEI' E-y = lin(U E-y) is a d'iTect sum if E6 fllin(U-y1'O E-y) = {a} for each 8 E r. In this case, we write O-yEI' E-y for lin(U E-y); an element of (0 E-y)· has the form L~=l X-Yil where X-Yi E E;i (i E N n ) and '"Yi i= '"Yj if i i= j. Let {E-y : '"Y E r} be a family of linear spaces over lK. The Cartesian product TI-YEI' E-y is a linear space with respect to the coordinatewise operations; it is the dZTect pmduct of the family. We identify Eo with {(x-y) : x-y = 0 if ~f i= 8}, and then the direct sum O-YErE-y, also denoted by coo(r,E-y) (ef. A.3.74), is the subspace {(X-y) E TI-YEr E-y : XI'
= 0 for
all but finitely many'Y E
r}.
25
Algebra8, zdeals, and fields
Let E be a linear space. A basis for E is a linearly independent subset 8 of E such that lin 5' = E. A maximal linearly independent subset of E is a hasiti, and each linearly independent suhset is contaiued in a Imf:'is. The cardinality of bases is an invariant of E; it is the dirnension, dim E, of E. The space E is finzte-dzrnrnswnal if dim E is finite, and in.jinde-dzrnenslOnal if dim E is infinite. Let F be a linear subspace of E. and let E / F })(' the quotient space. Then F has codirnen.9wn n in E if dim( E / F) = n, F has Jinzte codirnenswn in E if dim( E / F) is finite, and F has znjinzte corhrnenswn in E if dirn(E / F) is infinite. Let E a.nd F be linear spaces over lK. A map T : E ---7 F is lK-lmea.r (or, usually, lmear) if
T(n:r
+ By)
= nTJ;
+ /3Ty
(0, (J
E lK, :r,
y E E) ,
and T is conJngate-lmear if
T(n:£
+ ~3y) = liT:r + STy
(0,3 E lK, x,y E E).
We often refer to linear maps from E to F as operators. A bijective linear map T : E --> F is a linear ?sornuTplnsrn; in this case, E and Fare lznearly isornorphzc, and we write E ::::: F. The set of linear maps from E into F is denoted by £'(E, F); it is itself a linear space over lK for the usual operations. Let G be a third linear space, and let S E £'(E, F) and T E £.(F. G). Then the composition T 0 S of Sand T belongs to £'(E, G); it is often denoted just by TS. \\Te write £'(E) for £'(E, E). The identity operator on E is often denoted by h. Let T E £'(E). Then z E C is an eigenva17te of T if zh; - T is not injective'. We write EX for £'(E, lK); EX is the algebmzc dual spacr of E. and clements of EX are linear j1J:nctionals on E. \Ve often write (:1',).) for ).(:r), where x E E and), E EX. There is a natural embedding /, : E --> EX x; here I(.l')().) = ).(x). Let E and F be linear spaces, and let T E £'(E, F). For)' E F X, we define TX), by the formula (x, TX),) = (Tx,).) (:1: E E) . (1.3.1) Then TX), E EX and T X E £'(FX, EX); T X is the algebraic dual map to T. The map T X : (Fx. a(FX, F)) ---7 (EX, a(EX, E)) is clearly continuous. An operator T E £( E, F) is rank-one if dim T( E) = 1 and jinite-mnk if dim T(E) is finite; each finite-rank operator is the sum of finitely many rankOlle operators; the ::;et of finite-rank operators is a linear suhspace, denott'd ·by F£(E,F), of £'(E, F). We write F£'(E) for F£'(E,E). Let F be a linear subspace of a linear space E. An element P E £.(E) with P(E) = F and Px = x (.T E F) is a projection onto F. Let (En: n E :2:) be a sequence of linear spaces (each possibly 0), and let Tn E £'(En+l' E,,) (n E :2:). The beqllence '"""' . T,,+l L .... -->
=
Tn
---+
En
n-1 T----> ...
(1.3.2)
0 Tn = 0 (n E :2:). The sequence (1.3.2) is e.Tact at rz E :2: if kerTn - 1; it is exact if it is exact at each n E:2:. Thus a sequence
is a cornplex if T,,-1
Tn(En·n)
E n+1
26
Algebr'aic foundatwns
is a shori exact sequence if S is injective, T is surjective, and S(E) = ker T; in this case, there exists Q E £(G, F) with To Q = la, and F = S(E) (;) Q(G). In the case where 2: is not exact. we measure 'the failure of exactness' by certain 'homology groups'.
2:
Definition 1.3.1 Let
Hn
be a r-07nplex, as in (1.3.2). The linear spaces
(2:)
(n
= kerTrt_I/im T"
E
Z)
are the homology groups of the complex.
Corresponding to a complex denoted by ""' L.....t:
(where we have Tn+!
0
...
E ,,+1 ~ 1~, E ,,~ T n - 1 ...
Tn+l ~
T" = 0 (n E Z)), we obtain the cohomology gT'O'IlpS
H"
(2:) =
(n E Z).
kerT,,+I/imTn
The sequence
L.....t
""'
x
: ...
TX n-1 ---+
x EX Tn EX ,,---+ ,,+1
T
X
n+1 ---+ ...
(1.3.3)
corresponding to the above complex (1.3.2) is also a complex, for clearly T;:: 0 T:_ 1 = (Tn - 1 0 Tn)X for each n E Z; 2: x is called the algebmzc dual complex to 2:. The algebraic dual of a sequence exact at n is also exact at n. Let E J , ••• , En, F be linear spaces over IK. A map T : I1:'=1 Ei - 7 F is 1/,lmeaT if, for each j E N" and each Xi E Ei (i = 1, ... ,j - 1,j + 1, ... , n), the map X f--+ T(Xl, ... , X)-I, X, Xj+l, ... , Xn), E j - 7 F, is linear. The set of n-linear maps from
n:';,,1 Ei into F is denoted by
£"(El"'" E,,; F),
and we write £1'(E, F) in the special case where El = ... = En = E: again. £n(E1 , . .. ,En; F) is itself a linear space over lK. A 2-linear map is said to be bilineaT.
The tensor pT'Oduct of linear spaces E 1 , ... , En is a pair (~, F), where F is a linear space and ~ E £1' (E1' ... , En; F), snch that. the following universal property holds: for each linear space G and each S E £n(E1 • ... ,En: G). there is a unique linear map T : F - 7 G such that S = To I. Such a tensor product always exists, and it is unique (up to linear isomorphism). We write n
® Ei
or
E1 0 ... 0 En
for
L(Xll' .. ,
i=1 for the space F and
Xl
0· .. 0
Xn
x n ), and we regard ®~=J Ei as
the tensor product of E 1 , ... , En; we identify (®~=1 Ei) 0 (®~=k+l Ei) with ®~=1 E i . Each element z of ®7=1 Ei can be written in the form m
Z = LXl,j 0 X2,j 0··· 0 xn.j j=1
Algebras, ideals, and fields
27
for some rn E N, where Xi,} E Ei (i E N n , j E N m ); in the case where z -I- 0, we may suppose that each of the sets {Xi,j : j E Nm } is linearly independent. Let E and F be linear spaces over lK. Then the following identities hold in E®F:
+ y) ® z = X ® z + y ® z r8' (y + z) = X ® y + x ® z
(X X
a(x ® y)
=
(ax) ® y
=x
® (ay)
(x, Y E E, Z E F) ; } (x E E, y, Z E F) ; (x E E, y E F, a E lK) .
(1.3.4)
For each bilinear map S : Ex F --t G, there is a unique linear map T : E®F such that T(x ® y) = Sex, y) (x E E. y E F), and so
--t
G
(E®F)X ",:C(E,F;C) ",:C(E,FX). For example, let Sand T be non-empty sets. Then coo(S) ® coo(T) "': coo(S x T);
here (f ® g)(s, t) = J(s)g(t) (5 E S, t E T) for J E coo(S) and 9 E coo(T). Let E be a linear space, and let ®P E for p E Z+ denote the tensor product of p copies of E (taking ®o E = C and ®l E = E). We define the linear space Q9E=O{®PE:pEZ+} , and denote a generic element of ® E by U = (up), where up E ®P E (p E Z+). Let U E ®P E and l' E ®q E, where p, q E Z+. Then U ® v E ®p+q E. For each a E 6 p, there is a linear map (j : ®P E --t ®P E such that (j(Xl ® ... ® .Tp) = Xa(l) ® ... ® Xa(p)
(Xl, ... , Xp
E E).
(1.3.5)
An element U E ®P E is symmetric if (j(u) = u (a E 6 p); we write VP E for the linear subspace of ®P E consisting of the symmetric elements. The linear map 1 Sp : 11!---> I ' 2)a(u) : a E 6 p.
(1.3.6)
p}
is a projection from ®P E onto VP E; Sp is called the symmet'r'iz'ing map. We define VE=O{VPE:PEZ+} . Note that Sp+q(u ® v) = Sp+q(Spll ® Sqv) (u E ®P E, v E
® qE).
Definition 1.3.2 An algebra over lK is a lK-linear space (A. +) which has an additwnal binary operatwn . such that (A, .) is a semzgroup, the di8tributwe law8 hold, and a(a· b) = (aa) . b = a· (ab) (a E lK, a, bE A). The algebra A i8 commutative zJ (A, . ) is abelian. Henceforth, we shall usually write ab for a . b, but sometimes we shall retain the '; (A, . ) is the multiplicative 8ernigroup of A. Note that the map rnA : (a, b)
!--->
ab,
A x A
--t
A,
is a bilinear map, called the product map. There is a linear map 7rA : A ® A --t A such that 7rA(a 0 b) = ab (a, bE A); this map 7rA is called the induced product map.
28
Algebraic fo'undations
Let K be a subsemigroup of (c, + ) with 1 E K. Then a K -semigroup in A is a K-semigroup (ar: : ( E K) in (A, . ). An algebra over C is a complex algebra. These algebras are the main objects of study in this book, and in Chapters 2-5 a complex algebra will just be called an algebra. A complex algebra is also a real algebra when scalar multiplication is restricted to R The set {O} is an algebra: it is the zero algebra, and it is denoted by O. Let A be an algebra. Then AOP is the algebra formed by reversing the order of the product in A; AOP is the opposite algebra to A. Note that, even when we are considering AOP, ab always denotes the product of a and b in the original algebra A. A subset of an algebra A is a .mbalgebra of A if it is an algebra with respect to the operations of A. The intersection of the subalgebras of A containing a subset S of A is the subalgebra of A generated by S; it is denoted by algAS
or
Let A be an algebra over JR, and let B containing A described above. Set (Xl, Yl) (X2' Y2)
=
alg S.
=A
(XIX2 - YIY2, :CIY2
x A be the complex linear space
+ X2yd
(Xl, :1'2, Yl, Y2 E A) .
Then B is an algebra over C containing A as a (real) subalgebra; B is the complexification of A. An element eA of an algebra A is an identdy of A if fA i- 0 and eA is the identity of the semigroup (A, . ); similarly, we define left and right identities of A. An algebra A with an identity is a unital algebra, and in this case we identify the field lK with lKCA. A subalgebra B of a unital algebra A is a unital subalgebra if eA E B. We sometimes write (eA - a)A and A(eA - a) for {b - ab : b E A} and {b - ba : b E A}, respectively, even when A does not have an identity. Let A be a unital algebra. An element of A is left mvertible [nght mvertible], [invertl,ble] if it is left invertible [right invertible], [invertible] in the semigroup (A, .). The set of invertible elements in A is denoted by Inv A. A unital subalgebra B of A is inverse-closed if B n IllV A = Inv B. Let B be a unital subalgebra of a commutative, unital algebra A. Clearly {be -
J :
b E B, c E B
n Inv A}
is a unital subalgebra of A, and it is the smallest inverse-closed subalgebra of A containing B. It is called the znver8c-clos'ure of B in A. Definition 1.3.3 Let A be an algebra over lK. Then AD i8 the linear space lK x A together wzth the product (a, a)({3, b) = (a{3, ab +;Ja + ab) Define A #
=
A if A is umtal and A #
=
((X, (3
E
lK, a, bE A).
AD if A is not unital.
Certainly A' is a unital algebra over lK; the element (1,0) is the identity of A', and we regard A as a subalgebra of AI>. Note that, if A is unital, then the element (1,0) is different from the identity eA of A. The algebra A# is the algebra formed by adjoining an identity to A; we shall usually be concerned with
Algebras, ideals, and fields
29
this 'conditional unitization'. We often denote the identity of A# by eA, even when A is non-unital; thus, in the non-unital case, we write O:eA + a for the element (a,a) of A# = AO. However, eO always denotes the identity of AO.
Definition 1.3.4 Ld A be an algebra, and let a, bE A. The quasi-product of a and b i8 a b, where a b = a + b - ab . An element a of A is left [right] quasi-invertible if there exists b E A such that b a = 0 [a b = 0]. The element a 1,1, quasi-invertible tf it is both left and right quasl,-mverttble, and a subset of A 1,8 left quasi-invertible if each of tts elements is left quasi-inver·tible, etc. The quasi-product is an associative operation with identity O. Suppose that a b = (' a = O. Then clearly b = c, and so, if a is quasi-invertible, there is a unique dement b E A such that a b = b a = 0; b is the quasi-mverse of a, and is denoted bya'l. \Ve write q-Inv A for the set of quasi-invertible elements of A. Clearly 0, b = 0 if and only if (eA - a)(cA - b) = eA in A#, and so Inv A# = eA + q-Inv A. Let A be an algebra. and let Sand T be non-empty subsets of A. As in §1.2, S . T = {ab : a E S, Ii E T}. We define ST = lin S . T, so that ST =
{t
aiaibi : 0'1, ... , an E
1K, aI, .... an E S, b], ... ,bn E T, n E N} ;
t=1
we write aT for {a}T, etc. Note that aA = a . A, for example, but that, in general, A . a . A i= AaA. Also, A#T, calculated in A#, is a subset of A. There is a special notation which we introduce because we wish to distinguish between two sets: for a nOll-empty subset S of A and n E N, set s[n]
= {al ... an
: ai, ... ,an E S},
sn
= lin s[n]
(whereas s(n) denotes the n-fold Cartesian product of S). The set S is mulhplicative if S[2] c S.
Definition 1.3.5 Let A be an algebra. Then: an element 0, E A factors if a E A[2], and A factors if A = A[21; A factors weakly tf A = A2; a pazr {al,a2} in A has a common factor b E A if {ai, 0'2} C bA, and A has factorization of' pairs zf each pair in A has a common factor. Of course, a unital algebra factors and has factorization of pairs; later we shall be concerned with non-unital algebras that either factor or factor weakly.
Definition 1.3.6 Let A be an algebra. A linear functional T on A is a trace if T(ab) = T(ba) (a, bE A). Examples 1.3.7 (i) The linear space IK n is a complex, unital, commutative algebra over IK with respect to the coordinatewise operations; IKn is taken to have this product, unless otherwise stated. More generally, let A')' be algebras for 'Y E r. Then I1-yH A')' is an algebra with respect to operations defined
Algebraic foundatwns
30
coordinatewise, and O-YEr AI' is a subalgebra thereof. (ii) Let S be a non-empty set. The pomtwise opemtzons on S are defined as follows. For f, 9 E OC S and a E OC, define
(af)(s) = af(s), (f (fg)(s) = f(s)g(s),
+ g)(8) = 1(8) =
f(s) 1,
+ g(S)'}
(s E S).
(1.3.7)
Then OC S is a commutative algebra with identity 1, and coo(S) is a subalgebra of CS. For f E OC S , we also define
(3tf)(s) = 3t(f(s)),
(:sf)(s) = :s(f(s)),
If I (8) = If(s)1
(s
E
S).
(1.3.8)
Let A be a subalgebra of OC S , and let U be a non-empty subset of S. Then A I U is an algebra; it is the restrictzon algebra of A to U. (iii) Let S be a semigroup; the linear space J(O)(S) = coo(S) was defined in 1.2.24. Clearly there is a unique product * on coo(S) such that
8..
* 8t = 8st
(s,t E S);
the operation * is called the convolutwn product on S, and (c 00 (S), *) is the algebrazc semigroup algebra. In the cru:;e where S hru:; an identity e, 8e is the identity of (coo(S), *). We refer to the algebraic gmup algebra when S is a group. Of course, properties of the algebra coo(S) correspond to properties of S; for example, coo(S) is commutative if and only if S is abelian. Now take S = +) X (Q+, Then the non-unital algebra coo(S) factors weakly. but it does not factor. (iv) Let qX] be the algebra of polynomials in one indeterminate, with coefficients from Co (See §1.6, below.) Then qX] is a commutative, unital algebra. We shall use the fact that each p E qXJ- can be written as
+»-.
w:.)!+,
p = no(X - al)" . (X - on), where aI, ... ,an E C and aD E C-. This algebra will be studied in §1.6. (v) Let E, F, and G be linear spaces, and let T : E x F ~ G be a bilinear map. Set A = E x F x G, as a linear space. The map
«.rl, Yl,
zd,
(X2' Y2, Z2»
f->
(0,0, T(Xl' Y2)),
A x A ~ A,
is bilinear and defines a product with respect to which A is an algebra with A:3 = O. Clearly A2 = {O} X {O} x lin T(E, F); in general, A[2j =I- A2. (vi) Let E be a linear space over K Then £(E) is a unital algebra over OC (with composition as product), and T E £(E) is invertible if and only if T is a bijection. Suppose that E is a finite-dimensional space. Then the following are equivalent: (a) T E Inv £(E); (b) T is injective; (c) T is surjective; (d) det T =I- 0, where det T is the determinant of T. Let E and F be linear spaces, and take Yo E F and Ao E EX. We define
Yo ® Ao : x
f-+
(x, Ao)Yo,
E
~ F.
The map Yo ® AO is either a rank-one operator on E or 0, and we may identify F ® EX with F£(E, F). Note that, given Xo E E- and Yo E E, there exists AO E EX with (xo, Ao) = 1, and then Txo = Yo, where T = Yo ® Ao.
Algebras, ideals, and fields In particular, E product
(9
EX
(Xl
(9
31
= F£(E); AI)
the product in F£(E) corresponds to the
(X2 (9 A2) = (X2'
0
Al)Xl 0
(1.3.9)
A2,
and the product::! of Xo 0 Ao E F£(E) with T E £(E) are
To (.1:0 ® Ao) = Txo ® AD,
(TO 0 AD) 0 T = Xo ® T X(AD) ;
(1.3.10)
we have (xo 0 AoV = AD ® ~(XO) E F£(EX). There is a unique linear functional Tr : E 0 EX --> C such that
Tr(x0A)
=
(x, A)
(x
E
E, A E EX).
(1.3.11 )
This map is called the trace map; clearly Tr is a trace on E 0 EX. Indeed it follows from (1.3.9) that each trace on E 0 EX is a scalar multiple of Tr. (vii) Let A
= C4
with the product given by
(Zl, Z2, Z3, Z4)( WI, W2, W3, W4)
=
(Zl WI, Zl W2, Z3W3, Z4W3)
for Zl, ... , Z4, WI, ... , W4 E C. Then A is a non-commutative algebra which factors, but the pair {CO, 1,0, 1), (1,0, I,D)} does not have a common factor. We shall note in 2.2.51, 2.9,45, and 2.9,48 that there are various Banach algebras that factor weakly, but do not factor, and in 4.3.19(i) that there is a commutative Banach algebra that factors, but does not have factorization of pairs. 0 Definition 1.3.8 Let E be a linear space, and let m, n E No Then MIm.n(E) zs the linear space of m x n matrices with coefficients mE.
An element of MIrn.n(E) is denoted generically by
(Xij) = (Xij : i E N m , j E Nn ). The transpose of (xi]) E MIm,n(E) is the matrix (Xij)t = (Xji) E MIn,m(E). An element (Xij) E MIm.n(E) can be identified in the mmal way with an element T E £ (E(rn),E(n»): for (Xl, ... ,Xn ) E E(n), T(XI"" ,Xn ) = (YI,'" ,Yrn), where Yi = L7=1 XijX) (i E N m ). Let n E N. We write MIn(E) for MIn,n(E). A matrix (Xi) E MIn (E) is upper-trzangular if Xij = (i > j); strictly upper-trzangular if Xij = 0 (i ::::: j); lower-triangular if Xij = (i < j); strzctly lower-triangular if Xlj = (z ::::: j); . and dzagonal if Xi) = (i =I- j). Let A be an algebra. Then MIn(A) is an algebra for the product given by (ai])(bij ) = (L~=l aikbkj): it is the full matrix algebra of order n over A. We write MIn for MIn(C), so that MIn is a complex algebra of dimension 11. 2 . \Ve also write G L( 11.) for Inv MIn, the general {meaT group of order n. Let A be a commutative algebra, and take (ai]) E MIn(A). Then the determinant of (aij) is
° ° °
det(aij) =
2) (-1)10'Ia1,O'(l) ... an.O'(n) : IY E Sn} ,
where IIYI is the sign of IY. The following result is standard.
°
Algebrazc fo'undatwns
32
Proposition 1.3.9 Let A be a commutative algebra, and let n EN,
(i) deteST) = (det S)(det T) (S, T E Mn(A», (ii) Let T E Mn(A). Then T E Inv M,,(A)
'tf and
only 'if det T E IllV A.
(iii) Suppose that (ai]) E Mn(A) and (b 1 , ... ,b,,) E A(n) are such that I:~=l aijbi = 0 (j E N n ). Then (det(aij»bi = 0 (i E Nn ). 0 Definition 1.3.10 Let A be an algebm. A set (eij : i,j E N n ) of elements of A is a system of matrix units (of size n) for A if CijekC=6j.kCi£
(i,j,k,fEN n );
in the case where A is umtal, the system
28 unital
2f, further, I:~1 eii =
CA.
In particular, let n E N, and consider the full matrix algebra Mn. \Ve sometimes write En for the identity matrix (6i,j) ill Mn. For i, j E N n , take Eij to be the matrix with 1 in the (i,])th position and 0 elsewhere. Then (Ei] : i,j E N n ) is a unital system of matrix units for Mn; it iH the standani system of matrix units. Proposition 1.3.11 Let AI, .... An be algebras. and let A be the linear- space ®:~1 Ai' Then there is a unique pmduct on A wzth respect to which A is an algebm and such that (0.10'" ® a n )(lJ1 @ ... 0 b7l ) = a l b1 @ ... 0 anb n
(a,. bi E Ai, i E N n ).
If A!, ... ,An are commutatwe, then A 2S commutative; if Ai has the zdentity (i E N n ), then C1 ® ... 0 en zs the 2dentity of A.
ei
Proof Take
(!
= (al,"" an) E
rr=l Ai'
The IIlap n
(b i
, ... ,
bn )
1--7
a) bl 0 ... 0 an bn .
II Ai
is n-linear, and so there is a unique linear map /La : A /La(b 1 0 ... 0 bn )
= a 1 bl
-->
/,=1
-->
A,
A such that
(hi E Ai, i E N n ).
® ... 0 anbn
The map a 1--7 f.La, I1:~ 1 Ai --> £: (A), is n-linear. and so there is a unique linear map f.L: A --> C(A) such that f.L(aI0·· ·®an ) = f.La (a = (al, ... ,an) E I1~~I Ai). The map (x, y) 1--7 fL(X)(Y), A x A --> A, is the required product. 0 The algebra A is the tensor- product of the algebras AI, ... ,An. For example, the space A# 0 A#op is an algebra for a product which satisfies (1.3.12)
This algebra is the algebmic enveloping algebra of A. Let E be a linear space. For u u0v=(up)0(Vq)=
=
(up) and v
(L
p+q=r
=
(vq) in
® E,
Up0vq:rEZ+).
we define (1.3.13)
33
Algebras, ideals. and fields
Then ® E is an algebra with respect to the product (u, v) >----+ u C>9 v; the sequence (1,0,0, ... ) is the identity of ® E. In the case where dim E > 1, ® E is noncommutative. For u E VP E and v E Vq E, define
uVv
=
Sp+q (U C>9 v) E
where Sp+q is the symmetrizing map. For U
Vv =
(lIp)
U
L
V (v q ) = (
Vp+q E ,
= (up) and v = (uq) in IIp
V 11q : r E Z+) .
(1.3.14)
VE. define (1.3.15)
p+q=r If 1I E Ep and v E Eq, then U C>9 v = a(v C>9 u) for a certain a E 6 p+q, and so U V v = 11 V 11. Thus V E is a commutative, unital algebra with respect to the product (u, v) >----+ U V 11. (However, VE is not a subalgebra of the algebra ® E.) Definition 1.3.12 Let E be a linear space. Then (®E,C>9) and (V E. V) are the tensor algebra and the symmetric algebra over E, respectzvely. Definition 1.3.13 Let A be an algebm. A subset S of A zs commutative if ab = ba (a, bE S). The commutant of S is
Se
=
{b E A : ab
= ba
(a E S)} .
The centre of A zs A e, and zt zs denoted by 3(A).
For each subset S of A, S e is a subalgebra of A, and S (' is a unital subalgebra in the case where A is a unital algebra. Clearly S iH commutative if and only if SeSe. The serond commutant, sec, of S is defined to be (se)c; always S c sec and se = secc. Let S be a commutative subset of A. Then it follows from Zorn 'H lemma that S is contained in a maximal (with respect to inclusion) subset AI of A. Let a E Ale. Then AI U {a} is commutative, and so a E AI. Thus AI = AIC and AI iH a subalgebra. Since se ~ MC and see c Mce = lvI, the set sec is a commutative subalgebra. The centre 3(,4) of A is a commutative subalgebra of A. Definition 1.3.14 Let A be an algebra. A subset M of A which is maximal (with respect to inclusion) in the fam'ily of commutatwe s'ubsets of A is a maximal commutative Hubalgebra of A.
For example, 3(Mn(lK») = IKEn. For let T = (O~rB) E 3(Mn(IK»). For each = TEij , and so ari = 0 (r i= i), a JB = 0 (s i= j), and (tii = Q jj. Thus T E IKEn- Also. we ::;ee ::;imilarly that lin {E 11 • ... , Enn} is a maximal commutative subalgebra of Mn(IK).
i,j EN", we have EijT
Definition 1.3.15 Let A be an algebra. Then a E A is nilpotent zf an = 0 for some n E N; a set SeA is nil if each a E S zs nilpotent, and S is nilpotent if s[n] = {O} for some n E N.
We write SJ1(A) for the set of nilpotent element::; of A, so that A is nil if and only if A = IJ1(A). The index of a E IJ1(A) is the minimum n E N such that
34
Algebrazc foundatwns
an = 0, and the mdex of a nilpotent set S is the minimum n E N such that s[n] = {O}. Let a, b E ')'teA). Then in general neither a + b nor ab belong to ')'teA), but both a + b E ')'teA) and ab E ')'teA) in the ca..(ab) = A(ba). It now follows from (1.3.18) that A(ab) = >.(a)>.(b), and so AECPA. D
Definition 1.3.30 Let A be an algebra for whzch CPA#- 0. For a E A, define 0:('1') = cp(a) (cp E CPA), and define
9 :a
f-+
0:, A
--+ ]K
n (n EN). The sets of infinitesimals and of finite elements in K are denoted by
KO
and
KO#,
respectively. Clearly KO and K O # are subalgebras of K, and KO is an ideal in KO#. Let f K be the value set of the ordered group (K, +, ::;), with the archimedean valuation v, as in 1.2.23, and define + on fK by the formula v(a)
+ v(b) = v(ab) (a, bE
Ke).
+ is well-defined and that (f K, group; it is called the value group of K. We have v(l)
It is easy to check that
KO = {a E K : v(a) > a}, so that KO#
=
R1
+, ::;) is a totally ordered = a and
KO# = {a E K: v(a) ~ O},
+ KO.
Definition 1.3.62 Let K be an ordered field. Then K is: (i) an Cil-fieid [1]l-field] if (K, :s:) is an Cil-set [1]l-set]; (ii) a i3I-field if K Cil -subfields of K.
= U{Kv :
/.I
G (G E F). Thus F is the maximum member of F. 0 Let A be an algebra, let E be a left A-module, and let (an) be a sequence in A. Then
lima1" +--
.a
n .
E
=
{x
E
E: t~~e exists (X1~ in E ,such that }. (1.4.16) ], Xl and Xn - an . Xn +1 (n E N)
Thw;, in the the notation of §l.l,
~
a] ... an . E is
1l'1 (lim
proj{E; en}). where
n:=l
en(x) = an . X (x E E). Note that ~al" ·an . E = a1' ··an . E in the case where E is torsion-free. Let a E A. Then clearly ~ an . E is the maximum a-divisible submodule of E. Let A be a unital integral domain, and let E be a unital A-module. If E is injective, then E is divisible. For let a E A·, and take x E E. Define T : ab I---> b . x, aA --> E. Then T E AC(aA, E), and so T hru; an extension T E AC(A, E). We have x = T(aeA) = a . T(eA) Ea· E, and so E = a . E, as required. We shall show that the converse of this result is true in the case where A is a principal ideal domain. Lemma 1.4.18 Let A be a unital algebm, and let E be a unztallejt A-modnle. Suppose that each A-module homomorphism from a left zdeal of A mto E has an extension to an A-modnle homomorphi8m from A into E. Then E i8 an injective modnle.
Proof Let G be a left A-module, let Fo be a submodule of G, and let To E AC(Fo, E). Let F be the family of pairs (F, T), where F is a submodule of G with F ::> Fo , T E A£(F, E), and T I Fo = To. Set (Fl. Td ::5 (F2' T 2 ) in F if F1 c F2 and T2 I F1 = T 1. Then (F,::5) is a partially ordered set, and each chain in (F,::5) has an upper bound. It follows from Zorn's lemma that (F,::5) . has a maximal element, say (F, T).
58
Algebraic foundatzons
Assume towards a contradiction that F i= G, and take :1: E G \ I = {a E A: a . x E F}, so that I is a left ideal in A and define ():a ..... T(a.x),
F.
Set
I-+E.
Then () E A£(I, E), and so, by hypothesis, (I has an extension B E A£(A, E). Define H = F + A . x and S : y + a . x ~ Ty + a . B(eA)' H -+ E. If YI + al . x = Y2 + (L2 . x in H, then T(YI - Y2) = {}(a2 - ad = (a2 - al) . B(eA), so that S is well-defi~ed. Clearly S E A£(H, E) and (H, S) ~ (F, T) in F, a o contradiction. Thus F = G, and E is injective. Theorem 1.4.19 Let A be a prmcipal ideal domam, and let E be a umtal left A-module. Then E is injective 2f and only 2f it is dwisible. Proof We must show that E is injective in the case where it is divisible. Let I be a non-zero ideal in A, and take T E A£(I, E). Since A is a principal ideal domain, there exists ao E A· such that I = aoA. Since E is ao-divisible, there exists x E E with ao . x = Tao. Define T : a ~ a . x, A -+ E. Then T E A£(A, E) and T I I = T. It follows from 1.4.18 that E is injective. 0 Corollary 1.4.20 Let A be a princ2pal ideal domam, and let E be a umtal left A-module. Suppose that E 2S dwisible. Then there is a torsion-free, divisible, unital left A-module F such that E = F 8 E t . Proof Since E is divisible, E t is divisible, and so, by 1.4.19, E t is injective. Thus there is a left A-module F such that E = F8Et . Clearly F is torsion-free, divisible, and unital. 0 Definition 1.4.21 Let E be a lmear space, and let T E £(E). For each subset W of C, the algebraic spectral space ET(~V) is the maximum linear ,mbspace F of E such that (zIe - T)(F) = F for each z E C \ W. As in 1.4.8, E is a left £(E)-module, and so the existence of ET(W) is a special case of 1.4.17. Clearly ET(W1 ) C ET(WZ ) whenever WI C W 2 in C. Proposition 1.4.22 Let E be a linear space, and let T E £(E).
(i) Suppose that F is a linear subspare of E such that T(F) = F and such that, for each x E F, there exists n E N with Tnx = O. Then F C ET(0). (ii) Suppose that W C C, z E W, and x E E with (zIE - T)x E Br(W). Then x E ET(W),
n
(iii) For each W C C, ET(W) = {ET(C \ {z}) : Z E C \ W}. (iv) For each family {Wv } of subsets ofC, ET Wv) = ET(Wv ),
(nv
Proof (i) Take x E F and z E C·. There exists n S = 2.:::~=1 z-kT k - 1 and y = Sx. Then y E F and (zIE - T)(F) = F for each z E M(A), is an isomorphism. Proof The map e is an embedding. Take (L, R) E M(A). Then there exists bE B with L = Lb. For a, c E A, we have Rb(a)c = abc = aL(c) = R(a)c, and so Rb(a) = R(a) because A-L = O. Thus Rb = R, and e is an epimorphism. 0 Proposition 1.4.21 Let A be a fmthful algebra. Then, for each rp E exzsts a unique '15 E (j)M(A) such that '15 I A = 'P.
(j) A,
there
Proof Let (L, R) E M(A). For a, bE A, we have
rp(a)rp(Lb) = rp(aL(b)) = rp(R(a)b) = rp(Ra)rp(b).
(1.4.19)
Now take a E A with rp(a) = 1, and set '15«L, R)) = rp(La). By (1.4.19), rp(Ra) = 'P(La). Suppose also that b E A with rp(b) = 1. Then, again by (1.4.19), rp(Lb) = rp(La), and so '15«L, R)) is well-defined. Certainly '15 E M(AV and '15 I A = rp. Let (LI, Rd, (L2' R 2) E M(A), and take a E A with 'P(a) = 1. Then
'15( (L1' R 1)(L2, R 2)) = rp(Ll (L2a)) = '15( (Ll> Rd )'15 ((L2' R 2)) , and so '15 E (j) M(A)' Clearly '15 is uniquely specified.
o
Now suppose that A is a commutative, faithful algebra. Then each left [right] multiplier is a right [left] multiplier. Let (L1' Rd, (L2' R 2) E M(A). For a, bE A,
b· (L1L2)(a) = L1(bL2(a)) = Ll(R2(b)a) = R2(b)L1(a) = b . (L2Ld(a) , and so L1L2 = L 2L 1; similarly RIR2 = R 2R 1 . Thus M(A) is commutative; also Me(A) e:! M(A), and so we can regard M(A) as a commutative, unital subalgebra of .c(A) and A as an ideal in M(A).
61
Modules and primitive ideals
Definition 1.4.28 Let A be an algebra, and let E be a left A-module. Then E zs non-trivial zf A . ~rJ:-a:naE'is' simple -if it {,9non-trzvw,[ cma Tj 0 -rind E are the only submodules of E. The 'module E zs decomposable if E = F 8 G for some non-zero left A-modules F and G, and E zs indecomposable zf zt is not decomposable. A representation is simple [decomposable], [indecomposable] if the corresponding module is simple [decomposable], [indecomposable].
Similar definitions apply to right A-modules and A-bimodules. A simple left A-module is indecomposable, but the converse is not necessarily true. For each maximal left ideal M in A, A/AI is a simple left A-module. Proposition 1.4.29 Let A be an algebra, let E be a simple left A-module, and let Xo E E-. Then:
(i) A . Xo = E; (ii) the map a + x~
I-->
a . Xo, A/x~
-+
E" is a left A-mod~lle isomorphism;
(iii) x~ is a manmal mod~Llar left ideal in A. Proof (i) Let x E E. Then A . x is a submodule of E, and so A . x = 0 or A . x = E. Let F = {x E E: A . x = o}. Then F is a submodule of E; F i= E because A . E i= 0, and hence F = O. Thus Xo rJ. F and A . Xo = E.
(ii) Define T : a + x~ I--> a . xc, A/x~ -+ E. Then T E AL(A/x~, E) and T is an injection. By (i), T is a surjection, and so A/x~ 3:! E. (iii) Since E is a simple module, x~ is a maximal left ideal. By (i), there exists u E A with u . Xo = xc, and so A(eA - u) . Xo = 0 and A(eA - u) C x~. Thus u is a right modular identity for x~. D 9orollary 1.4.30 Let A be an algebra, let M be a maximal left zdeal in A, let bE A, and set J = {ao E A : ab E AI}. Then ezther J = A or J is a maximal modular left ideal in A. Proof Set E = A/M. Then either Ab eM. and hence J = A, or E is a simple left A-module and b + M E E-. In the latter case, J = (b + M)l. is a maximal modular left ideal in A by 1.4.29(iii). D
There are two standard theorems about simple modules, each to be used several times, which we shall now establish. Theorem 1.4.31 Let A be an algebra, and let E be a simple left A-module. Then AL(E) is a dzvision algebra. Proof We have remarked that AL(E) is a unital sub algebra of L(E). Take T E AL(E)-. Then kerT and T(E) are submodules of E. Since T f 0, we have ker T i= E and T(E) f 0, and so, since E is simple, ker T = 0 and T(E) = E. Thus T is a bijection, with inverse S, say. Certainly S E AL(E), and so T is invertible in AL(E). 0
Algebraic foundations
62
Theorem 1.4.32 (Jacobson's density theorem) Let A be an algebra over lK, and let E be a simple left A-module such that AC(E) = lKIe. Let {Xl, ... ,xn } be a linearly independent set zn E, and let YI,' .. , Yn E E. Then there exzsts a E A with a . Xj = Yj (j E N n ). Proof We first claim that, for each finite-dimensional subspace F of E and each X E E \ F, we have Fl. . X = E. The proof of this claim is by induction on the dimension of F. The case dim F = 0 is 1.4.29(i): if x E E-, then A . x = E. Now suppose that dim F 2': 1, and assume that the claim holds for each subspace G of E such that dim G < dim F. Let Yo E E be such that Fl. . Yo = O. Take Zo E F-, and take a linear subspace G of E with F = G 8lKzo. By the inductive hypothesis, Gl. . Zo = E, and so, for each x E E, there exists a E Gl. with a . Zo = x; set Tx = a . Yo. If also b E Gl. with b . Zo = x, then a - b E FJ.. and (a - b) . Yo E Fl. . Yo = 0, so that a . Yo = b . Yo. Hence T is well-defined. It b easily checked that T E AC(E). By hypothesis, AC(E) = lKls, and so there exists 0:' E lK such that T = O:'lE. For each c E GJ.., we have T(c . zo) = c . Yo, and so c . (Yo - O:'zo) = O. Thus GJ.. . (Yo - O:'zo) = 0, and hence, by the inductive hypothesis, Yo - O:'Zo E G and Yo E G + lKzo = F. Let x E E \ F. It has been shown that Fl. . x =1= 0, and so FJ.. . X = E because E is simple. Hence the claim holds for this F, and the induction continues. Now let Xl, .. " Xn, YI, ... , Yn be as specified in the statement. It follows from the claim that, for each j E N n , there exists aj E A with aj . Xj = Yj and aj . Xi = 0 (i =1= j). Set a = al + ... + an. Then a . Xj = Yj (j E N n ). 0 Definition 1.4.33 Let I be a left zdeal in an algebra A. The quotient of I is I :A
=
{a E A : aA C l} .
The quotient of a maximal modular left ideal is a primitive ideal. The algebra A is primitive if 0 is a primitwe ideal. Each quotient I : A of a left ideal I of A is an ideal in A. Clearly the quotient of a modular left ideal I is maximal in the set of ideals in A which are contained in I. The algebra A itself is not a primitive ideal.
Proposition 1.4.34 Let A be an algebra. (i) An ideal in A is a primitive ideal if and only if it is the kernel of a simple representation of A. (ii) Each primitive ideal in A is the intersection of the maximal modular left ideals which contain it. (iii) Each primitive ideal in A is a prime ideal. (iv) Each maximal modular ideal in A is a primitive ideal, and each modular ideal is contained in a primitive ideal.
Proof (i) Let 1= M : A, where M is a maximal modular left ideal in A. Then AIM is a simple module, and I is the kernel of the left regular representation of A on AIM.
Modules and primitzve ideals
63
Conversely, let p be a simple representation of A on E, take Xo E E·, and set M = X6-. By 1.4.29, A . Xo = E and M is a maximal modular left ideal in A; we have a E ker p if and only if a . (A . xo) = 0, and this holds if and only if aA eM. Hence kerp = lvI: A. (ii) and (iii) Let I be a primitive ideal, say I = E1-, where E is a simple module. Then I = n{x1- : x E E e }, giving (ii). Take ideals J, K in A with JK c I and K ¢. I, say. Then K . E = E, and so J . E = JK . E = 0, whence J C I, giving (iii). (iv) Let lvI be a maximal modular ideal. There is a maximal modular left ideal L with MeL. Clearly lvI C L : A, and so lvI = L : A is primitive. Each modular ideal is contained in a maximal modular ideal. 0 It follows from (i), above, that, in the case where I is a proper ideal in A, the following are equivalent: (a) I is a primitive ideal; (b) AI I is a primitive algebra; (c) there is a simple left A-module E such that I = E1-.
Example 1.4.35 Let E be a non-zero linear space. Then E is a simple left C(E)-module for the map (T, x) f--? Tx, C(E) x E -+ E, and C(E) i::; a primitive algebra. In the case where E is infinite-dimensional, FC(E) is a non-zero, proper ideal in C(E), and so the primitive ideal is not maximal. 0
°
Proposition 1.4.36 Let I be an ideal in a commutative algebra A. Then the following conditions on I are equivalent: (a) I is a primitive ideal; (b) I is a maximal modular ideal; (c) AI I is a field. Proof Suppose that (a) holds, say I = lvI : A, where M is a maximal modular ideal. Then lvI : A = lvI, and so (b) holds. The remainder is immediate. 0 Proposition 1.4.31 Let A be a complex algebra, and let E be a finite-dimensional, simple left A-module wzth representation p. Then p : A -+ £(E) is an epimorphism. Proof By 1.4.31, AC(E) is a division algebra. Since the linear space E is finitedimensional, AC(E) is finite-dimensional, and so, by 1.3.56, AC(E) = Cle. Let {Xl, .. " Xn} be a basis for E, and let T E C(E). By Jacobson's density theorem 1.4.32, there exists a E A with a . x J = TX J (j E N n ). Clearly pea) = T, and so p : A -+ C(E) is a surjection. 0 Corollary 1.4.38 Let I be an ideal of finite codzmenswn in a complex algebra A. Then the following conditions on I are equivalent: (a) I is a prime ideal; (b) I is a maximal modular ideal; (c) I is a primitive ideal; Cd) All ~ Mn for some n E N.
64
Algebrazc foundations
Proof Suppose that I is primitive, and let P be a simple representation of A on a finite-dimensional space E with ker P = I, say dimE = n. By 1.4.37, AI I ~ C(E), and so AI I ~ Mn. By 1.3.51, Mn is a simple algebra, and so I is a maximal modular ideal. Thus (c)=*(d)=*(b)=*(c). By 1.4.34(iii), (c)=*(a), and so it remains to prove that (a)=*(c); it suffices to do this in the case where I = 0 (and A is finite-dimensional). Let J be a non-zero left ideal in A of minimum dimension. Since A is a prime algebra, aAb:j:. 0 (a, bE Ae), and so Ab :j:. 0 (b E Ae). For each b E J, we have Ab C J, and so Ab = J (b E r). Thus J is a simple left A-module, and aJ:j:. 0 (a E Ae). Hence 0 = J 1- is a primitive ideal. 0 Corollary 1.4.39 Let A be a complex algebra, and let PI,"" Pn be szmple, finite-dimensional representations of A on spaces E 1, ... , En, respectzvely, such that ker Pi :j:. ker Pj (i:j:. j). Then the map n
a
f--->
(PI (a), .. . , p,,(a)), A ----
0
C(E;) ,
i=l
zs an epimorphism.
Proof Set M j = ker Pj (j EN,,), so that, by 1.4.38, each M j is a maximal modular ideal in A. Take kEN", and let 1= M1
n··· n M k- 1 n Mk+! n··· n Mn.
Assume that I C Nh. Then Ah ... M k - 1M k+1'" Mn C M k ; by 1.4.34(iii), M j cAlk for some j :j:. k, whence Alj = Mk, a contradiction. Thus I ct- Mk and I . Ek :j:. O. By 1.4.37, for each T E C(Ek), there exists ak E I with Pk(ak) = T. The result follows. 0 Corollary 1.4.40 Let A be a complex, umtal algebra, let E be a szmple, finitedimensional A-bimodule, and let F be a subspace of E such that F is a simple left A-module. Then there exist a1>"" am E A with a1 = eA and such that F ~ F . OJ (j E N m ), E = O~l F . aj, and EJ. = FJ. is a maximal ideal in A. Proof Let a E A. By 1.4.31, either F . a = 0 or the map x f---> x . a, F ---- F . a. is an isomorphism of left A-modules. Since E is finite-dimensional, there is a maximal finite set {F . a1,"" F . am} with each F . aj :j:. 0 and with a1 = eA. Let i E Nm and a E A. Then either F . aia = 0 or F . aia = F . aj for some j E Nm · Thus 2:;"=1 F . aj is an A-bimodule, and so 2:.7=1 F . aJ = E. Since F . ai n F . aj = 0 for i :j:. j, the sum is direct. Clearly EJ. = F1- is an ideal of finite codimension in A; by 1.4.38, this ideal is maximal. D Let A, E, F, and a1, .. ·,am be as in the above corollary, with dimF = k. Then dimE = km. Set M = EJ., so that AIM ~ M k . Let {rll,'" ,rkd be a basis of F, and set riJ = ril . aj (i E N k , j E Nm ). Then we obtain a basis of E that can be written as a (k x m)-array (rij); a generic element x of E is x = 2:.:=12:7=1 Xijrij' Let a E A. Then a + M corresponds to a k x k matrix
Modules and primitive zdeals
65
(aij) E M k , and the element a . x of E is represented (with respect to the basis (rij) of E) by the matrix product
X1m)
X~rn
.
We can also regard the left action of a E A on x E E as being specified by a km x km matrix
(1.4.20) where Em is the identity m x m matrix, and this matrix acts on the element x = (X11' X12,'" ,X1rn, X21,'" ,X2m,··· ,Xk1,.· ., Xkm) of E. Now consider the product x . b for x E E and b E A. This product can be represented by a km x km matrix acting on the left on the km x 1 matrix x. But this matrix commutes with all matrices of the form (1.4.20), and so it has the form
0
rr
B (
0)
1,
where B is an m x m matrix and there are k copies of B. Write the transpose Bt of B as (f3ij), so that the product x . b corresponds to the product given by
In particular, AI ET ~ M m , and the simple right A-modules in E are spanned by the rows of the array (riJ)' We obtain the following result.
Proposition 1.4.41 Let A be a complex, unital algebra, let E be a szmple, finite-dzmensional A-bimod71,le, and let F be a subspace of E such that F is a simple left A -module. Suppose that dim F = k and dim E = km, and let {riJ : i E Nk, j E Nm } be a basis for E, as above. Then there exist families {Pst: s, t E N k } and {quv : U, v E Nm } in A such that: (i) (Pst + EJ.. : s, t E N k ) and (quv + ET : u, v E Nrn) are sets of unital matrix units in AI EJ.. and AI E T, respectively, with L:=l Pss = 2:::'=1 quu = eA;
(ii) for a = Ls t astPst + EJ.., b = Lu v f3uvquv + E T , and x = where (ast) E M k , (f3uv) E M m , and (XiJ) Eo Mk,m, we have
a .x =
L: (2..: aikXkj) rij, 2,J
k
2::i j
Xijrij,
'
o
66
Algebraic foundations
Definition 1.4.42 Let A be an algebra. Then ITA is the set of primitive ideals of A. Let I be a subset of A, and let S be a subset of ITA· Then
~A(I)
= ~(I) = {P E ITA
: p:J I}
and
tA(S)
= t(S) = n{p: PES}
are the hull of I and the kernel of S, respectively (taking t(0) = A).
Let I and J be subsets of A with Ie J. Then ~(J) :J ~(I). :For each S C ITA, t(S) is an ideal in A, and t(S) :J t(T) whenever SeT c ITA. We write ~t(S) for ry(t(S»; clearly S c ~t(S) and t(~t(S)) = t(S) whenever S c ITA, so that ryt(~t(S» = ~t(S) and ryt(0) = 0. For each subset I of A, we have I C try(I) and ~t(ry(I) = ry(I). Let Sand T be subsets of ITA. Then t(SUT) = t(S)nt(T), and so we see that ~t(S U T) :J ryt(S) U ~t(T). For each P E ITA with t(S U T) C P, we have t(S)t(T) C t(S)
n t(T) = t(S U T)
C
P,
and so either t(S) c P or t(T) C P because, by 1.4.34(iii), P is a prime ideal. It follows that ~t(S U T) C ryt(S) U ~t(S), and so ~t(S U
T) = ryt(S) U ~t(T)
(S, T
c
ITA) .
We have shown that the map S 1-+ ~t(S) is a closure operation on P(ITA)' Definition 1.4.43 Let A be an algebra. The hull-kernel topology on ITA is the topology defined by the closure operation S 1-+ ~t(S) on P(ITA); the space ITA with the hull-kernel topology is the structure space of A.
It is certainly possible to have distinct primitive ideals P and Q in A with Pc Q, and so points are not necessarily clol:led in ITA; in general, the hull-kernel topology is only a To-topology. Proposition 1.4.44 Let I be an ideal in an algebra A. Then: (i) the map R : P (ii) the map Q: P
1-+ 1-+
P
n I,
ITA \
PII, ry(I)
~(I) ---->
---->
IT I
,
is a homeomorphism;
ITA/I, is a homeomorphism.
Proof (i) Let p: I ----> L(E) be a simple representation of Ion a space E, and set J = ker p E ITI. Fix Xo E E-. By 1.4.29(i), for each x E E, there exists bEl with b . Xo = x; we define p(a)(x)
=
p(ab)(xo)
(a E A).
For a E A and b, c E I, we have p(c)p(ab) = p(cab) = p(ca)p(b). Thus, if b . Xo = 0, then p(c)p(ab)(xo) = 0 (c E I), and so, by 1.4.29(i), p(ab)(xo) = O. It follows that p(a)(x) is well-defined in E. It is now easily checked that p is a simple representation of A on E and that p I I = p. Set P = kerp. Then PEllA \ 1)(1) and R(P) = J, and so R is a surjection. Suppose that PI, P2 E IIA \fJ(I) with PIn1 = P2 n1. Then PI1 C Pln1 c P2 . But Pz is a prime ideal by 1.4.34(iii) and I ct P2 , and so PI C P2 • Similarly P2 C PI, and so n is an injection.
Modules and primitwe ideals
67
Take Po E ITA \ ~(I) and S C ITA \ ~(I), and set K = £I( {Q n I : Q E S}). Clearly, if £(S) c Po, then K c Po n I, and, if K C Po n I. then £(S)I C Po and £(S) C Po. Thus £(S) c Po if and only if K c Po n I. This proves that n is a homeomorphism. (ii) Clearly Q is an injection. Let p be a simple representation of AI I on a space E. Then p 0 7r is a simple representation of A on E, where 7r : A --+ AI I is the quotient map. Hence Q is a surjection. For each S C ~(I), necessarily tA/I(Q(S)) = Q(t(S)), and so, for Po E ITA, we have £(S) C Po if and only if tA/I(Q(S)) c Q(Po). This proves that Q is a 0 homeomorphism. It follows from 1.4.44(ii) that co dimension in an algebra.
~(I)
is finite whenever I is an ideal of finite
Proposition 1.4.45 Let A be an algebra. and let p E J(A). n: P t-> pPp, ITA \ ~(pAp) --+ ITpAp , is a homeomorphism.
Then the map
Proof Let P E ITA \ b(pAp), say P = EJ.. for a simple left A-module E. Then p ~ P, and so p . E =f O. Suppose that x E (p. E)·. Then A . x = E by 1.4.29(i), and so pAp· x = p . (A . x) = p . E. Thus p . E is a simple left pAp-module; the kernel of the corresponding representation of pAp is pAp n P = pPp. Thus n has range in ITpAp. Suppose that PI, P 2 E ITA \ ~(pAp) with pP1p = pP2P. Then (Ap)PI(Ap) C ApPIP = ApP2P C P 2
.
rt
rt
By 1.4.34(iii), P 2 is a prime ideal in A. Since pAp P2 , certainly Ap P 2 . By 1.3.42. necessarily PI C P 2 . Similarly P 2 C PI, and so n is injective. Let Q E IT pAp . Then there is a maximal modular left ideal 1'v! in pAp such that Q = M : pAp. Consider the left ideal I = M + AM + A( e A - p) in A. Then p is a right modular identity for I, and so there is a maximal modular left ideal J in A with I c J and p ~ J. Define P = J : A; we have P C J and p ~ P, and so P E ITA \ b(pAp). Clearly QAp = QpAp c M c I and QA(eA - p) C I, and so QA C J. Thus Q c pPp. On the other hand, M + pPp c I + Pc J and p E pAp \.1, and so M + pPp =f pAp. Since M is a maximal modular left ideal, this shows that pPp C M. Thus pPp c Q. We have shown that Q = pPp, and so is surjective. Suppose that F is a dosed subset ofITpAp, and set S = n-1(F). To show that S is closed in ITA \ lJ(pAp), we must show that Q E S whenever Q E ITA \ ~(pAp) and Q :::l £(S). But this is immediate. Thus is continuous. Conversely. suppose that S is closed in ITA \ ~(pAp), and set F = n(s). To show that F is closed in ITpAp, we must show that Q E F whenever Q E ITpAp and Q :::l t(F). Set P = n-I(Q). Then pt(S)p = t(S) n pAp = n{pIp : IE S} = t(F) c Q c P, and so (Ap)t(S)(Ap) C P. Again this implies that £(S) C P. Since S is closed, this implies that PES and hence that Q E F. Thus n- l is continuous. 0
n
n
Definition 1.4.46 Let A be an algebra, and take S Jo(S) = {a E A : ua = a for some and J(S) = A# Jo(S)A#.
C
ITA. Then
U E
P(S)} ,
Algebraic foundatwns
68
Thus J(S) is the ideal in A generated by Jo(S). We have J(S) so ~(J(S)) -:) ~e(s).
c reS), and
Proposition 1.4.41 Let A be an algebra, and take S C ITA. Then J(S) for each idPaI I of A wzth ~(I) c S.
c I
Proof Let I be an ideal with ~(I) c S, and take a E .1o(S), i:iay ua = a, where E r(S). Define K = {b E A : ba E I}. Then K is a left ideal of A with K -:) I. Assume that a ~ I. Then u ~ K and u is a right modular identity for K. and so there is a maximal modular left ideal 111 of A with M -:) K and u ~ M: set P = 11.1 : A E IT.4, so that P -:) I and PES. Then 11 E £(S) c P c M, a contradiction. Hence a E I. o It follows that Jo(S) c I, and so J(S) c I. U
Notes 1.4.48 Again, almost all the results in this section arf' contained in standard texts such as (P. M. Cohn 1989), (Hungerford 1980), (Jacobson 1974, 1980), (McCoy 1973), and (Pierce 19i52). Definition 1.4.12 introduces a new notation in an attempt to clarify concepts which are often confused. Our concept of 'split' in 1.4.15 is sometimes referred to as 'splitexact'. There is an extensive theory of injective and divisible modules (and the related 'projective' modules); see (Jacobson 1980, Chapter 3), for example. Every left Amodule can be embedded in an injective left A-module (ibid., 3.18). The left and right multipliers of 1.4.25 are sometimes called centralizers; see (Palmer 1994), where M(A) is called the double centralizer algebra. Our 'primitive ideals' are more accurately termed left primitive ideals; it is not true that each right primitive ideal is left primitive (Bergman 1964). The terminology and early results on primitive ideals in an algebra A are due to Jacobson (1956): the hull-kernel topology on the structure space I1A is also called the Jacobson topology. Other structure spaces are considered in (Palmer 1994, Chapter 7).
1.5
RADICALS AND SPECTRA
In the structure theory of algebras, several different radicals are considered. In the theory of Banach algebras, it is the Jacohson radical which is by far the most important, and so we shall just refer to this radical as 'the radical' for general algebras. There are several different characterizations of this radical; we shall take the notion of primitive ideal a::; the ba::;ic one. We shall also de::;cribe the strong and prime radica]::; of an algebra, and we shall include some cla::;sical structure theorems for finite-dimensional algebras. The section will conclude with the definition of the ::;pectrum of an element in a complex algebra and of the joint spectrum of an n-tuple in a commutative algebra. To simplify the statement of some definitions and results, we adopt throughout the convention that, if the class of subsets of an algebra A satisfying a certain property be empty, then the intersection of all the sets in the class is A itself. Definition 1.5.1 Let A and B be algebras. The (Jacobson) radical, radA, of A is the intersection of the primitive ideals of A. The algebra A is semisimple zf rad A = 0 and radical if rad A = A. A homomorphism 8 : A -+ B is radical if 8(A) c rad B.
69
Radicals and spectra
Thus rad A = t(TIA)' and A is radical if and only if TIA = 0. Let P be a primitive ideal in A. By 1.4.44(ii). AlP is a primitive algebra, and so AlP is semisimple. For example, let E be a non-zero linear space. Then the algebra L(E) is primitive, and hence semisimple. Let R be a commutative, radical algebra. Then R# is a local algebra with maximal ideal R; the unique character on R# with kernel R is denoted by 'PR, so that Inv R# = {a E R# : 'PR(a) of- O}. Theorem 1.5.2 Let A be an algebra. Then:
(i) rad A '/,s the intcrsectzon of the kernels of the simple representatzons of A: (ii) rad A zs the mtersectwn of the maxzmal modular- left zdeals of A; (iii) rad A zs a quasz-inver-hble zdeal, and mel A contams each left quasiinver-tzble left zdeal of A; (iv) radA
=
{a E A : A#a C q-Inv A}.
Proof (i) and (ii) These arc immediate from 1.4.34, (i), (ii), and (iv).
(iii) Let a E rad A. and set I = A(a - e A)' Then I is a modular left ideal with right modular identity a. Assume that a ~ I. Then there is a maximal modular left ideal AI with modular identity a; since a ~ M, we have a ~ rad A by (ii), and this is a contradiction. Thus a E I, and so there exists b E A with b a = O. Also, b = ba - a E rad A, and so there exists c E A with c b = O. Necessarily c = a, and so a E q- Inv A. Thus rad A is a quasi-invertible ideal. Let I be a left quasi-invertible left ideal of A, let E be a simple left A-module, and take x E E e . AHsume that I . E of- O. Then, by 1.4.29(i), there exists a E I with a . x = x. Take b E A with boa = O. Then x = (b + a - ba) . :E = 0, a contradiction. Thus I . E = 0, and so, by (i), I c rad A. (iv) Suppose that a E radA. Then A#a C radA, and so A#a C q-InvA by (iii). Conversely, suppose that a E A and that A#a C q-InvA. Assume that a ~ rad A. Then there is a simple left A-module E and x E E with a . x of- O. By 1.4.29(i), there exists b E A with ba ..'T = x. Since ba E q-InvA, necessarily x = 0, a contradiction. Thus a E rad A, as required. 0
= radA. rad A = {a E
It follows that radA#
and, in particular, ert
+ radA C
In the case where A iH unital, A : eA
+ Aa C
(1.5.1)
Inv A} ,
Inv A.
Corollary 1.5.3 Let A and B be algebras.
(i) Suppose that a
E
A, b E rad A, and a
(ii) Suppose that B : A
---+
= abo
Then a = O.
B is an epimorphzsm. Then B(rad A)
C
rad B.
Proof (i) By 1.5.2(iii), there exists c E A with b 0 c = 0, and now we have a = a - a(b 0 c) = a - ab - (a - ab)c = O. (ii) Since O(a 0 b) = O(a) oO(b) (a, bE A) and B(A) = B, the result follows 0 from 1.5.2(iv).
Algebraic foundations
70
We have characterized primitive ideals, and hence rad A, in terms of left Amodules. Let J be the ideal defined analogously in terms of right A-modules. Then J contains each right quasi-invertible right ideal of A, and so, by 1.5.2(iii), rad A c J. Similarly J C rad A, and so J = fad A. Thus fad A is independent of the choice of left or right in its definition. In particular, radA
=
{a
E
A: aA#
C
q-Inv A}.
(1.5.2)
The equivalence between the characterizations of rad A in 1.5.2(iv) and in (1.5.2) can also be seen directly: if b E A and if c E A is such that c J1(A) = rad A and A=
~(A) J1(A) = lin {PI , ... ,Pk} J1(A).
In the case where A is umtal, eA = PI
+ ... + Pk.
(ii) Each cP E ell A has the form CPj : L::=l O::iPi
+ >J1(A)
f---t
O::j for a umquf
j E Nk .
(iii) For each cP E J1(A) = radA, and, by 1.5.7(iv), ~(A) n radA = {O}. The result now follows from 1.3.23(iii); in the unital case, we use 1.5.7(ii) to see that eA = PI + ... + Pk·
(ii) Let cP E J1(A) = 0 and cp(pi) E {O, I} (i E N k ). There exists j E Nk such that cp(Pj) =I- 0; also, CP(Pi)CP(pj) = 0 for z =I- j, and so there is a unique j E Nk such that cp(pi) = 6i,J (i E Nk), and then cP = CPJ' (iii) Suppose that cP = CPj in the above notation. If pj>J1(A) = 0, take ao = P)' If pj>J1(A) =I- 0, there exists b E (p/l1(A))- with bpj>J1(A) = 0, and then we take ao = bpj. In each case, ao has the required property. 0 Theorem 1.5.9 (Wedderburn structure theorem) Let A be a non-ze7'O, finitedimensional, semisimple, complex algebra. Then A has an identzty e A and there exist kEN, an orthogonal set {PI, ... ,pd of central idempotents in A, and nl,"" nk E N such that eA = PI + '" + Pk, such that {pjApj : j E Nd zs the family of minimal ideals in A, such that pjApj ~ Mnj (j E Nk), and such that
A
k
k
j=1
j=1
= (0pjApj ~ OMnj
.
73
Radicals and spectra
Proof Since A is semisimple, 0 is an intersection of primitive ideals. Indeed Pj , for there exist finitely many primitive ideals PI, ... , Pk such that = otherwise there would exist a sequence (Qn) of primitive ideals with
° n1=1
QI
n ... n Qn+1
~
QI
n ... n Qn
(n E N) ,
and this is not possible in a finite-dimensional algebra. We may suppose that n'#i Pj i- 0 (i E Nk)' J Let j E N k . By 1.4.38, Pj is a maximal modular ideal and there exists nj E N such that AI Pj ~ M nj ; let Pj : A -+ Mnj be the natural epimorphism. By 1.4.39, the map k
p: a
f--t
(PI (a), ... , Pk(a)),
A -+
0
Mnj ,
j=1
n1=1
is an epimorphism, and ker P = Pj = 0, so that P is an isomorphism. For j E Nk , take Uj to be the identity of M nj , and set Pj = p-I(UJ ). Then {PI, ... ,Pk} is the required set of central idempotents. 0
Corollary 1.5.10 Let A be a non-zero, finite-dimenswnal, semisimple complex algebra. (i) For each non-zero ideal I in A, there is a central idempotent p in I such that p is the identity of I and I = pAp. (ii) Let {PI, .. ' ,pe} be an orthogonal set of minimal idempotents such that eA = PI + ... + Pf. Then dim A :s; £2. Proof (if By 1.5.4(i), I is semisimple, and so, by 1.5.9, 1 is unital; set p = e[. Certainly pAp = 1. For a E A, we have pa E I and ap E I, and so pa = pap = ap, whence p E 3(A).
07=1
(ii) Suppose that A ~ M nJ , a!:l in 1.5.9. Then £ = 1.3.19, and dim A = ni + ... +n~:S; £2.
n1
+ ... + nk by 0
There is a structure theorem related to 1.5.9, which we shall now establish by elementary arguments. The result will be placed in a more general !:letting in §1.9. In the next few pages we write a for a + R in AIR.
Definition 1.5.11 Let A be an algebra with radical R. Then: (i) A is an SBI algebra if, for each x E R, there exzsts Y E R with yo Y = x and {x}C = {yV; (ii) a set U of orthogonal zdempotents in AI R can be lifted to A zf there exists an orthogonal set P in J(A) such that {p : pEP} = U; (iii) orthogonal idempotents can be lifted zf each finite set of orthogonal zdempotents in AIR can be lifted to A. For each x E R, the element Y arising in 1.5.11(i) is unique. For suppose that Y1, Y2 E R, that YI 0 YI = Y2 0 Y2, and that {yd C = {Y2}c. Then we have (YI - Y2)(YI + Y2 - 2eA) = 0 in A#; by (1.5.1), YI + Y2 - 2eA E Inv A#, and so Yl = Y2· We also see that, if ax = 0, then ay = 0, and that, if x belongs to a left or right ideal I, then also y E I.
74
Algebrazc foundatwns
Lemma 1.5.12 Let A be an algebra whose radical zs a nzl ideal. Then A is an
SBI algebra. Proof Let 1-2:j:1 cxJZ j be the Taylor expansion of(1-Z)1/2. Take a E radA. and set b = 2:;'=1 cxja j , where n is the index of a, so that b E rad A. Then bob = (1 and {a}C = {by, and so A is an sm algebra. 0 Lemma 1.5.13 Let A be an SBI algebra 'Unth radical R.
(i) Suppose that 'U E J(A/ R) and (1 E A is such that P E J(A) with p = u and {ay c {p}c. (ii) Orthogonal zdempotents can be lifted.
a=
u. Then there exists
Proof Write e for the identity of A # .
(i) Set x = (12 - a, so that x E R. Since A is an SBI algebra, there exists y E R with (e + y) 2 = e + 4x and {x y = {y y. Set
p=(a-~e)(e+y)-l+~e. Then pEA,
P=
u, and {a}C
C
{py. We calculate:
4p2 = (2a - e)2(e + 4X)-1 + 2(2a - e)(e + y)-l + e = (4a 2 - 4(1 + e)(e + 4X)-1 + 4p - e = (e + 4.c)(e + 4X)-1
= p2
(1.5.3)
+ 4p -
e
= 4p,
J(A). (ii) Take an orthogonal set {U1' ... ,un} in J(A/ R). First, by (i), there exists P1 E J(A) with P1 = U1· Now assume that there exist P1,'" ,Pk E J(A) with Pi = Ui (i E Nk) and Pi 1. Pj (i =I- j), and set P = P1 + ... + Pk E J(A). Choose a E A with a = Uk+1. and set b = (e - p)a(e - p). Then b = Uk+1 because PUk+1 = Uk+1P = 0, and pb = bp = O. By (i), there exists q E J(A) with q = Uk+1 and {by C {qy. Set Pk+1 = (e - p)q(e - p). Then p E {q}C, and so Pk+1 E J(A). Also Pk+1 = Uk+1 and PiPk+1 = Pk+1Pi = 0 (i E Nk)' The result follows by induction. 0 and so p
E
Suppose that R2 = 0 in the above lemma. Then the formula (1.5.3) for p becomes P = 3a 2 - 2a 3 . Lemma 1.5.14 Let A be an algebra. Suppose that (Uij : i, j E Nn ) is a system of matrix units in A/rad A such that the set {Uii : i E N,,} of orthogonal idempotents
can be lifted. Then there is a system (eij : i,j E N n ) of matrzx units in A such that ei] = Uij (z,j E Nn ). Proof By the hypothesis, there exists an orthogonal set {ell, ...• enn } in J(A) such that eii = Uii (i E Nn ). Fix i E {2, ... , n}, and choose a, b E A with a = U1i and b = Uil. Since U11U1iUii = U1i, we may suppose that a E ellAeii, and similarly that b E eiiAell' Since ab = ell, we have ell - ab E rad A, and so there exists r E rad A with r
+ ell
- ab - (ell - ab)r = 0 .
Radicals and spectra
75
Since ella = a, this implies that ab(eA - 1') = ell. Define eli = a and eil = b(eA - r)eu, so that eli = 'Uli, eil = 'Un, eli E ellAeii, and eil E eiiAel1' We see that elieil = ab(eA -1')e11 = ell' Next, set x = eii - eileli. Then x 2 = x and x = Uii - 'lLilUli = 0, and so, by 1.5.7(i), x = O. Thus eileli = eii. Now take i,j E {2, ... ,n}, and define eij = eilelj' Then eij = UilUIJ = Uij' Suppm;e that i, j, k, £ E Nn . If j #- k, then eijeke E AeJjekkA = O. Also, eijej£ = eileljejlel£ = eile11elf = eilelf = eif· Thus we have verified that eijekl = {)j,keil. Hence (eiJ : i,j E Nn ) is the required system of matrix units. 0 Definition 1.5.15 Let A be an algebra with radical R. Then A has a Wedderburn decomposition, or A is decomposable, if there is a subalgebra B of A such that A = B 8 R; in thts case B 8 R lS a (Wedderburn) decomposition of A.
Of course, if A = B 8 R, then B ~ A/ R. The algebra A is decomposable A/ R -; A with 1[' 0 = fAIR' if and only if there is a homomorphism where 1[' : A -; A/ R is the quotient map; the homomorphism e is a splitting homomorphism.
e:
e
Proposition 1.5.16 Let A be an algebra such that A2 n rad A = O. Then A is decomposable. Proof Let E be a linear subspace of A such that A = (A2 8 radA) 8 E, and set B = A2 8 E. Then B is a subalgebra of A, and A = B 8 rad A. 0 Proposition 1.5.17 Let A be a commutative algebra. Suppose that A = B 8 I, where B is a subalgebra of A and I is an ideal with f C rad A. Then ~(A) c B. Proof Take P E J(A), and set P = b + r, where b E Band rEf. Then b + r = b2 + 2br
+ r2 ,
and so b = b2 and r3 = (p - b)3 = p3 - 3p2b + 3pb2 - b3 r2 E rad A, there exists s E R with r2 0 s = O. But now
+ s - r 2 8) = (r - r 3 ) - (r - r3)8 = 0, b E B. Thus J(A) c B, and so ~(A) C B. r
and so p =
= P- b=
=r
r. Since
- r(r 2
0
Theorem 1.5.18 (Wedderburn's principal theorem) Let A be a complex algebra such that A/rad A is the algebra 07=1 M n " where nl,.'" nk EN. Suppose that the standard set of minimal idempotents can be ltfted. Then A is decomposable. Proof Set R = rad A, and let (EW : r, s E Nn j ) be the standard system of matrix units in Mnj for j E N k • By hypothesis, there is an orthogonal set Ro.T· Ro.T}· (j) + R = E(j) J: "'T d' "'T { e(j) r r : r E 1'lnj') E l'lk In A sueh t h a t err rr LOr r E l'lnj an ) E l'lk'
eW,
For j E N k, set Pj = 2:;~ 1 so that {PI, ... ,Pk} is an orthogonal set of idempotents in A. Take j E Nk' By 1.5.7(iii), rad (pJApj) = pJRpj = (pjApj) n R and so pjApj/rad (pjApJ) ~ M nj . By 1.5.14, there is a system (eW : r, S E NnJ of
76
Algebrazc foundations
matrix units in A such that eW + R = EW (r, S E Nnj)j we may suppose that (J) : r, s E 1M} (i)...i. (j) wh enever Z. ,. ...i. • Th l' { e rs !'ilnj C Pj APj, so t h at ers ,. e uv J. e mear map () : A/ R -+ A such that ()(EW) = eW is a splitting homomorphism, and so A is decomposable. 0 Corollary 1.5.19 Let A be a jinite-dimenswnal, non-nilpotent complex algebra. Then A is decomposable, A contains a non-zero idempotent, and there exist kEN and nl, ... ,nk E N such that A ~ 0~=1 Mnj 0 rad A. Proof By 1.5.6(iv), rad A is nilpotent, and so, by 1.5.12. A is an SBI algebra. Thus, by 1.5.13(ii), orthogonal idempotents can be lifted. By 1.5.9, A/ R has the form specified in the theorem, and so the result follows. 0
Let B be a complex, unital algebra with a maximal ideal R such that R2 = O. Then R = rad B. Suppose, further, that R has finite codimension in B. By 1.4.38, there exists n E N such that B/R ~ M n , and we have B ~ Mn 0 R. Let (Eij : i,j E N n ) be the standard system of matrix units for Mn. For i,j E N n , set Bij = EiiBEjJ = {b E B : EiibEjj = b} and Rij = EiiREjj, so that B = L~j=l B ij , R = L~j=l ~j, and Bij = CEiJ + Rij . For each i,j E N n , the map x 1--+ EilxElj , B11 -+ B ij , is a linear isomorphism. Let V be a linear subspace of R 11 , and define W = L~j=l eil Velj' Then W is a subspace of R which is an ideal in Bj if V has codimension k in Rll, then W has co dimension kn 2 in R. Now let A be a complex, unital algebra with a maximal ideal M of finite codimension in A, and set B = A/M2. Then the unique maximal ideal of B is M/M2, and M/M 2 = rad B = R, say. Let V C Rll and W be as above. Then W + M2 is an ideal of A with M2 C W + M2 C Mj W + M2 has finite codimension in A in the case where V has finite co dimension in Ru. We briefly mention a second radical of an algebra. Definition 1.5.20 Let A be an algebra. The strong radical 9l(A) of A is the intersection of the maximal modular ideals of A. The algebra A is strongly semisimple if 9l(A) = O.
It follows from 1.4.34(iv) that rad A C 9l(A). A simple, unital algebra is strongly semisimple. Let E be an infinite-dimensional linear space. Then £( E) is semisimple, but it is not strongly semisimple because, by 1.4.35, 9l(£(E» contains :F£(E). The following result is similar to 1.5.4, and the proof will be omitted. Theorem 1.5.21 Let A be an algebra. (i) Let I be an ideal in A. Then 9l(I) = In 9l(A). In particular, if A zs strongly semiszmple, then I is strongly semisimple. (ii) Let I be an ideal in A with I c 9l(A). Then 9l(A/I) = 9l(A)/I. In particular, A/9l(A) is strongly semisimple. (iii) Let I and J be ideals in A with I c J c 9l(A), and let 7r : A/I -+ AIJ be the natural surjection. Then 7r(9l( AI I» = 9l( AI J). 0
Radicals and spectra
77
The third radical of an algebra that we shall consider in this section is the prime radicaL Definition 1.5.22 Let A be an algebra. The prime radical intersection of the prirne ideals of A.
~(A)
of A is the
Clearly ~(A/~(A)) = O. Since each prime ideal contains a minimal prime ideal, ~(A) is equal to the intersection of the minimal prime ideals of A. Proposition 1.5.23 Let I be an ideal in an algebra A. Then
~(I)
=
In~(A).
Proof Let P be a prime ideal in A, and take a, bEl with alb c In P. Then aAlb c P, and so, by 1.3.42, either a E P or Ib c P. In the latter case, lAb C P, and so either I c P or b E P. In each case, either a E P or b E P, and so In P is a prime ideal in I. Thus ~(I) c I n ~(A). Let a E I \ ~(I), and take a prime ideal Q in I with a f/: Q. Then 1\ Q is an m-system in A, and so, by 1.3.44(i), there is a prime ideal P in A with pn(I\Q) = 0. Since a f/: P, we have a f/: ~(A), and so In~(A) c ~(I). 0 Theorem 1.5.24 Let A be an algebra. Then A is a sernzprzrne algebra if and only if ~(A) = O. Proof Suppose that ~(A) = O. Take a E A with aAa = O. Then a E P for each prime ideal P of A, and so a = O. Thus A is a ::;emiprime algebra. Conversely, suppose that A is a semi prime algebra, so that, by 1.3.43, we have aAa # 0 (a E Ae). First take al E A-, and then successively choose a2, a3,'" in Ae so that a n +1 E (anAa n )- (n EN). For m ::::; n, we have an+l E amAa n , and so {an: n E N} is an m-system. By 1.3.44(i), there is a prime ideal P with al f/: P, and so ~(A) = O. 0 It follows that an ideal is semiprime if and only if it i::; an intersection of prime ideals.
Proposition 1.5.25 The following conditions on an algebra A are eq'uivalent:
(a) A is serniprirne; (b) A contains no non-zero, nilpotent left or right ideal; (c) A contains no non-zero, nilpotent ideal. Proof (a)::::}(b) Assume that I is a non-zero, nilpotent left or right ideal in A, say of index n + 1. Take a E l[n] \ {O}. Then aAa c l[n+l] = 0, and so a = 0, a contradiction. (b)::::}(c) This is trivial. (c)::::}(a) Set] = A.L, an ideal in A with ]2 = O. By (c). ] = O. Thus, if bE A and bA = 0, then b = 0; similarly, if Ab = 0, then b = O. Now take a E A with aAa = O. Then, successively, (AaA)2 = 0, AaA = 0, Aa = 0, and a = O. Hence A is semiprime. 0
Algebrazc foundations
78
Proposition 1.5.26 Let A be an algebra. (i) The prime radical of A is a nil ideal contained in rad A. (ii) SfJ(A) contains each nilpotent left or right ideal of A. (iii) Suppose that A is commutative. Then SfJ(A) = SJt(A). (iv) Suppose that A is commutative and complex, and that IJ1(A) has finite codimension in A. Then SfJ(A) = rad A. (v) Suppose that rad A is finite-dimensional. Then SfJ(A) = rad A and SfJ(A) is the maximum nilpotent ideal in A.
Proof (i) Take a E A \ SJt(A). By 1.3.44(ii), there is a prime ideal P with a tj. P, and so a tj. SfJ(A). By 1.4.34(iii), SfJ(A) c rad A. (ii) Let I be a nilpotent left or right ideal in A. Then I + SfJ(A) is a nilpotent left or right ideal in AjSfJ(A). By 1.5.24, AjSfJ(A) is a semiprime algebra, and so, by 1.5.25, I C SfJ(A). (iii) Let a E IJ1(A). Then A#a C SJt(A) because A is commutative, and so a E A#a C SfJ(A) by (ii). (iv) For each prime ideal P in A, we have P :::) SJt(A) and dim(Aj P) < oc, and so, by 1.3.57, P is a maximal modular ideal. Thus radA C P, and so rad A C SfJ(A). Always SfJ(A) C rad A. (v) By 1.5.6(iv), radA is a nilpotent ideal, and so, by (ii), radA C SfJ(A). Thus SfJ(A) = rad A is the maximum nilpotent ideal. 0
Definition 1.5.27 Let A be a unital, complex algebra, and let a E A. resolvent set of a in A is
PA(a) = {z
E
C : ze A - a
E
The
Inv A} ,
and the resolvent function is the map Ra : z
f-+
The spectrum of a is 0" A(a)
(zeA - a)-I, =
vA(a)
PA(a)
-+
Inv A.
C \ PA (a), and the spectral radius of a is =
sup{lzl : z E O"A(a)}.
We take vA(a) = 0 if O"A(a) = 0, and vA(a) = 00 if O"A(a) is an unbounded set in C. We usually write pea) and O"(a) for PA(a) and O"A(a), respectively. Now suppose that A is a complex, non-unital, algebra, and that a E A. Then we define PA, R a , 0" A (a), and v A (a) by regarding a as an element of A #; in this case, necessarily 0 E 0"( a). If A is either unital or non-unital, then
O"A(a) U {O} = {z E e
: ajz
is not quasi-invertible in A} U {O}
(a E A).
Note that, in the case where A is commutative and unital, z E O"(a) if and only if (zeA - a)A i= A. For example, let E be a finite-dimensional space, and let T E £(E). Then O"(T) consists of the eigenvalues of T. We shall use the following easily checked identity. For each a E A, Ra(w) - Ra(z) = (z - w)Ra\z)Ra(w)
(z,w
E
p(a)).
(1.5.4)
79
Radicals and spectra
~ B be a O"A(a)U{O}, vB(O(a» :::; vA(a),
proposition 1.5.28 Let A and B be complex algebras, let () : A
homomorph2sm, and let a E A. Then O"B(O(a» anda(~A) C O"A(a) U {a}.
C
Proof Take z E PA(a) \ {O}. Then a/z E q-Inv A, and so O(a)/z E q-Inv B, whence z E PB(B(a». For 'P E ~A, 'P(a) =1= z, and so a(A) C O"A(a) U{O}. 0 Proposition 1.5.29 Let A be a complex algebra with radical R, and let a, b E A.
(i) 0" A/R(a + R) U {a} = O"A(a) U {a} and VA/Rea + R) = VA(a). (ii) O"A(ab) U {O} = O"A(ba) U {O} and vA(ab) = vA(ba). (iii) (TB(a) = o"A(a) for each maximal commutative subalgebra B of A wzth aEB. (iv) Suppose that A is unital. Then (J,C(A) (La) = (JC(A) (Ra) = (JA(a). (v) Suppose that E is a left A-module and 0: E C e 2S such that a . x = o:x for some x E Ee. Then 0: E (JA(a). (vi) Suppose that a = 2::7=1 O:jPj, where {PI, ... ,Pn} is an or·thogonal set m J(A) \ {O} and 0:1>"', O:n E e. Then {0:1' ... , O:n} C
(vii) O"A(a)
= U{(J A/p(a + P)
0" A
:P
(a)
E
C
{O, 0:1, ... , O:n} .
ITA}.
Proof (i) By 1.5.28, (JA/R(a + R) C (JA(a) U {a}. Now take z E PA/R(a + R) with z =1= O. Then there exists c E A with co (a/z), (a/z) 0 c E R. By 1.5.2(iii), co (a/z) E q-Inv A, and so a/z E q-Inv A. Thus PA/R(a + R) \ {a} C PA(a). (ii) We have noted that ab E q-Inv A if and only if ba E q-Inv A. (iii) Let B be a maximal commutative sub algebra of A with a E B. Certainly o"A(a) C O"B(a). Let z E PA(a). Then there exists c E A# with
(zeA - a)c = c(zeA - a) = eA . For b E B, (zeA - a)b
= b(zeA - a),
and so bc
= cb.
Thus c E B# and z E PB(a).
(iv) By 1.5.28, O"c(A)(L a ) C O"A(a). Now suppose that La E Inv A, and take T E C(A) with LaT = TLa = lA. Set c = T(eA)' Then ac = (LaT)(eA) = eA. and so LaLc = IA. Thus Lc = T, and hence ca = LaLc(eA) = eA, showing that a E Inv A. It follows that (Jc(A)(L a ) = (JA(a). Similarly O"c(A)(Ra) = O"A(a).
a+
(v) Assume that 0: EPA (a). Since 0: =1= 0, there exists c E A such that o:c - ca = 0, and then o:x = (a + o:c - ca) . x = 0, a contradiction. (vi) By (v), {0:1,'" ,O:n} C (JA(a). For z E c=
t
)=1
e \ {0:1,""
(~)P)' z 0:)
Then zc + a - ca = zc + a - ac = 0, and so z E PA(a). (vii) We may suppose that A is unital. By 1.5.28, U{O"A/p(a+P): P E ITA} C (JA(a).
O:n}, set
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Algebraic foundatwns
We claim that, if bE A and b+ P has a left inverse in AlP for each P E ITA, then b has a left inverse. For assume that Ab -I- A. Then there is a maximal left ideal M in A with Ab c M. Set P = M : A E ITA. There exists c E A with cb + P = eA + P, and so eA E A1, a contradiction. The claim follows. Now assume that z E aA(a), but that z tf. aA/p(a + P) (P E ITA)' By the claim, zeA - a has a left inverse, say b. Since z tf. aA/p(a + P) (P E ITA), the element b + P has a left inverse in AlP for each P E ITA. By the claim again, b has a left inverse, and so zeA - a = b- l E Inv A, a contradiction. It follows that aA(a) C U{aA/p(a + P) : P E H A }, giving the result. 0 The following result will be crucial for the proof of the Gel'fand-Mazur theorem for topological algebras. Theorem 1.5.30 Let A be a complex dimsion algebra such that a(a) each a E A. Then A = CeA.
-I-
0 for
Proof Let a E A. Since a(a) -I- 0, there exists z E C with zeA - a tf. Inv A. Since A is a division algebra, it follows that zeA -a = 0, and so a E CeA. 0 Definition 1.5.31 Let A be a complex algebra. Then a E A is quasi-nilpotent if vA(a) = O. We write D(A) for the set of quasi-nilpotent elements of A. Clearly a if and only if Ca C q-Inv A.
E
D(A)
Proposition 1.5.32 Let A be a complex algebra. Then:
(i) meA)
c
(ii) radA
=
(iii) I
D(A); {a E A : A#a C D(A)}
c
D(A);
c rad A for each left zdeal I with I c D( A);
(iv) A is radical if and only zf D(A) = A. Proof (i)-(iii) These are immediate from 1.5.6(i), 1.5.2(iv), and 1.5.2(iii), respectively. (iv) This follows from (ii) and (iii).
0
Even if A is commutative it is not necessarily the case that radA = D(A). For let A be a complex field with A -I- CeA. Then D(A) = A \ C·eA, but radA = O. We shall require a generalization for commutative algebras of the notion of the spectrum of an element. Definition 1.5.33 Let A be a commutative, unital, complex algebra, and let a = (aI, ... ,an) E A (n). The joint spectrum of a is aA(a)
= aA(al,'"
,an) = {z = (Zb.'·' zn) E en: (zleA - adA + ... + (ZneA - a)A =1= A}.
polynomwls and formal power senes
81
Let A be a unital algebra, and let a, b E A (k). Then a . b = L7=1 ajb j • The element a is unimodular if there exists bE A(k) with a . b = eA. Clearly, if A is commutative and a E A(k), then
a(a) = {z We define a
=
E
C k : zeA - a is not unimodular in A}.
(a1' ... ,an) E A(n) for a E A (n). Clearly we have
a([>A)
C
aA(a)
(a
(1.5.5)
E A(n).
It is clear that the new definition of a(a) coincides with the old definition in the case where n = 1, and that we always have a(a1, ... , an) C 0;=1 a(aj). Notes 1.5.34 Let A be an algebra. A left A-module is said to be semisimple if it is a direct sum of simple modules. An algebra A is semisimple (in the sense of 1.5.1) if and only if A is semisimple as a left A-module. For the radical of an algebra, see (Hungerford 1980, §IX.2), (Jacobson 1980, Chapter 4), and, in particular, (Jacobson 1956); the calculations on SBI algebras are from (Jacobson 1956, IlL8). (The letters 'SBI' stand for 'suitable for building idempotents'.) A form of Wedderburn's principal theorem is given in the early work (Albert 1939). The strong radical of an algebra is sometimes called the Brown-McCoy radical; it is characterized in (Palmer 1994, §4.5). The prime radical of an algebra is called the Baer radical in (Divinsky 1965) and (Palmer 1994, §4.4); it is a semiprime nil ideal which is included in every semi prime ideal. See also (McCoy 1973). An example of a simple, radical algebra is given in (Sasaida and Cohn 1967), and an example to show that .;p(A) need not contain every nil ideal of A is in (Baer 1943). Several other 'radicals' are considered in the literature; see (Divinsky 1965), for example. A 'history of radicals' is given in (Palmer 1994,4.8.1).
1.6
POLYNOMIALS AND FORMAL POWER SERIES
We shall continue in this section our accumulation of algebraic results that will be required in the later chapters of this book. We shall consider the polynomial algebra A[X] and the formal power series algebra A[[X]] over an algebra A, and their n-variable analogues; this leads to a study of algebraic and transcendental elements of an algebra with respect to a subalgebra, and, in Theorem 1.6.31, to a condition for the extension of a homomorphism defined on a subalgebra of a given algebra. Recall that the ground field OC of an algebra A is always IR or C. Definition 1.6.1 Let A be a unital algebra over K Then:
A[[X]] = {a = (an: n A[X] = {a = (an: n
A}; A[[X]] : an = 0 eventually}.
E Z+) : an E E
Z+) E
For a = (an) and b = (b n ) in A[[X]] and a E lK, set a + b = (an aa = (aa n : n E Z+), and ab = (tarbn-r : n r=O
E
Z+) .
+ bn : n
E Z+),
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82
With respect to these operations, A[[X]] is an algebra over lK, and A[X] is a subalgebra of A[[X]]. The algebra A[[X]] is the algebra of formal power series over A, and A[X] is the polynomial algebra over A; its elements are polynomzals. Clearly IA[X]I = IAI. We identify A with {(a,O,O, ... ): a E A}, so that eA is the identity of A[[X]J, and we write X for (0, eA, 0, 0, ... ); X is the zndeterminate of the algebras A[[X]] and A[X]. Each formal power series a = (an) E A[[X]] can be formally expressed as 00
where XO
= eA,
and each polynomial p
= (an)
E
A[X] can be expressed as
n
p = p(X)
=
2:arXr. r=O
-I- 0, and then the above expression for p is unique; in this case, the degree of p, written op, is n, and the leadzng coefficient of p is an. We set 00 = -00, with the conventions that -00 + (-00) = -00 and that -00 < nand -00 + n = -00 for n E Z+. A non-zero polynomial is monic if its leading coefficient is eA. In particular, we have now formally defined the algebras qX] and q[X]]; the identity of qX] is denoted by 1. We define Ao[X] = {(an: n E Z+) E A[X] : ao = O},
If Pi- 0, then we may suppose that an
and, for n EN,
A(n) [X]
=
{p
E
A[X] : op:S; n}.
(1.6.1)
Let p, q E A[X]. Then o(p + q) :s; max{ op, oq}, and so A(n) [X] is a linear subspace of A[X]. Also, o(pq):S; op+oq, and o(pq) = op+oq in the case where A is a domain. Suppose that A is commutative. Then A[[X]] and A[X] are also commutative; if A is an integral domain, then A[[X]] and A[X] are integral domains. In the latter case, the quotient fields of A[[X]] and A[X] are denoted by A«X)) and A(X), respectively, so that
A(X) = {pjq: p
E
A[XJ, q E A[X]-};
for example, IC(X) is the field of rational functions. Let p = E~=o arXr E A[X], and let bE A. Then p(b) = E~=o arb r (where bO = eA). In the case where A is commutative, the map p 1-+ p(b), A[X] ~ A, is a unital homomorphism. Let A be a commutative, unital algebra, let p be a monic polynomial in A[X] with op = n + 1, where n ~ 1, and let Ap = A[X]jpA[X]. Then each element of Ap corresponds uniquely to an element E~=o aiX i , where ao, al, ... ,an E A. We regard Ap as an algebra containing A as a subalgebra. Now suppose that B is a commutative, unital subalgebra of A and that a E Be. Then:
B[a]
=
{p(a) : p E B[X]};
B(a) = {p(a)q(a)-l : p, q E B[X], q(a) E Inv A}.
Polynomial8 and formal power series Thus B[a] = algA(B U {a}) and B(a) is the smallest inverse-clo~ed subalgebra of A containing B U {a}. Let a E A. Then OCo [a] = {pC a) : p E OCo [Xl}, so that a E OCo[a] and OCo[a] is a commutative subalgebra of A. Let A be a commutative, unital algebra, let p E A[X], and let a E A. Then a is a root of p if pea) = o. Let q E A[X]-. Then the dwi8ion algorithm asserts that there exist b E A- and r, s E A[X] with bp = qr + s and as < aq; if tht' leading coefficient of q belongs to IllV A (and in particular if A is afield), then we may take b = eA, and the representation p = qr + s is then unique. Thus, if a is a root of p, there exist mEN and q E A[X] with p = (X - a)m q and q(a) "10: a is then a root of mult1phcity m. If A is an integral domain and p E A[X]-, then p has at most ap roots (including mUltiplicities). It is immediate from the division algorithm that OC[X] is a principal ideal domain. Let p = L~=o arxr E A[X]. Then the formal denvative of p is the polynomial p', where p' = a1 + 2a2X + ... + na n X n - 1 . The kth formal derivative p(k) is defined by induction: p(j+l) Clearly we have
p
= (p(J)' (j EN).
(n)(o)
= p(O) + p'(O)X + ... + P_ _ xn;
(1.6.2)
n!
this is Taylor'8 formula for polynomials.
Definition 1.6.2 Let A be a commutative, unital complex algebra. A polynomial p E A[X] of the form
(1.6.3) where n ;::- 2 and ao, a2, ... ,an E rad A 1S a Henselian polynomial. The algebra A is Henselian if each Hen8elian polynomial has a root in A. Each root of a Henselian polynomial is necessarily in rad A. Let p be a Henselian polynomial. Then, for each a E A, p' (a) E e A + rad A c Inv A and p(k)(a) E radA (k = 2, ... , n), and also
pCb) - pea) E (b - a) . (eA
+ rad A) c
(b - a) . lnv A
(a, bE A),
(1.6.4)
so that p has at most one root in A.
Proposition 1.6.3 Let A be a Henselian algebra, and let
p = aD
+ alX + a2X2 + ... + anXn ,
where n ;::- 2, ao E rad A, al E Inv A, and a2, ... ,an E A. Then p has a root in radA. Proof We may suppose that a1 = eA. Set q(X) = p(aoX - aD) E A[X]. Then q(O) = p( -ao) E a6A, and q(k) (0) = a~p(k) (-ao) E a6A for k = 2, ... ,n. On the other hand, q'(O) = aop'( -aD) E ao . lnv A, and so q = hr, where bE radA and r is a Henselian polynomial in A[X]. By hypothesis, r has a root, say a, in A, and then aoa - ao is a root of p in rad A. 0
84
Algebraic foundatwns
Proposition 1.6.4 Let A be a local, Henselian algebra, and take a E A and n ~ 2. Suppose that there exists b E A with bn Ea' Inv A. Then there exists c E b . Inv A with cn = a. Proof Take 0: E C· and Cl E MA with bn = a(aeA + cd. Choose (3 E C· with (3n = a, and set p = ((3eA + x)n - (aeA + cd. Then p(O) = -Cl E MA and p'(O) = nBn-leA E Inv A. By 1.6.3, there exists C2 E MA with P(C2) = O. Set c = b((3cA + C2)-1. Then c has the required properties. 0 For P E qX], define Z(p)
= {( E C: p(() = O}.
Proposition 1.6.5 (i) Each character on qX] has the form some (E C.
C( :
P
f-+
p(() for
(ii) Each non-zero ideal m qX] has fimte codimension m qX]. (iii) Each maxzmal ideal m qX] zs the kernel of a character. (iv) Each non-zero, prime ideal in qX] is a maximal ideal. (v) The algebra qX] is semisimple.
(vi) Let PI, ... ,Pn E qX] be such that n?=l Z(Pj) q1, ... , qn E qX] such that 2:.]=1 Pjqj = l.
= 0.
Then there exist
Proof (i) For cp E qX], set (= cp(X). Then cp(p) = p(() = C«(p) (p E qX]). (ii) Let I be a non-zero ideal in qX]. Since qX] is a principal ideal domain, 1= pqX] for some P E qX]·, say P = TI~=l(X - (j)"i, where (1, ... ,(k are distinct points in C and nl, ... , rLk EN. Clearly
I = {q
E
qX] : q((j) = ... = q(nr 1)((j) = 0 (j E Nk)},
and so I has finite codimensioIl in qX]. (iii), (iv) The ideal I in (ii) is maximal (equivalently, prime) if and only if P has the form X - ( for some ( E C, and then I = ker C(. (v) This is now immediate. (vi) Let I = 2:..7=1 pjqX] be the ideal generated by {PI, .. . ,Pn}. Assume that I ii:l a proper ideal. Then I is contained in a maximal ideal of q X]. and so, by (iii), I c ker C( for some ( E C. But this is a contradiction of the fact that n;=l Z(Pj) = 0. Thus I = qX], and the result follows. 0 Definition 1.6.6 Let A be an algebra. An element a of A is algebraic if OCo[a] is finite-dimensional, and A zs algebraic if each element of A is algebraic. Clearly each finite-dimensional algebra is algebraic. Proposition 1.6.7 Let A be a complex algebra, and let a E A. Then the following conditions on a are equivalent: (a) the epimorphzsm p f-+ p(a), Co[X] - 7 Co [a], is an isomorphism; (b) a is not algebraic; (c) the set {an: n E N} is linearly independent.
Polynomials and Jormal power series
85
Proof By lo6.5(ii), the kernel of the epimorphism in (a) is either 0 or has finite codimension Co[X]. The result follows. 0 Corollary 1.6.8 Let A be a complex algebra, and suppose that a E A is not algebraic. Then Co [a] is semism~ple. Proof By lo6.5(v), Co[X] is semisimple. By the implication (b)=>(a), above, Co[X] ~ Cora], and so Coral is semisimple. 0 Proposition 1.6.9 Let A be a commutatwe, complex algebra, and let a E A. Then the Jollowing condztions on a are equzvalent:
(a) (b) (c) (d)
a is algebrazc; a E (E(A) 891(A); a E CeA
+ Q Jor
each przme ideal Q in A#.-
Jor each prime zdeal P in A, there exzsts p E Co[X]- with pea) E P.
Proof (a)=>(b) By lo5.8(i), Coral
=
(E(Ca[a]) 8 91(C o [a]) C (E(A) 891(A).
(b)=>(c) By 1.5.26(iii), we have 91(A) c Q, and so a E (E(A#) 1.3.46(ii), dim ((E(A#)/(~(A#) n Q)) :s; 1, and so a E CeA + Q.
+ Q.
By
(c)=>(d) By 1.3.54(i), there is a prime ideal Q of A# with Q n A = P. Take o E C with a - oeA E Q, and set p = X(X -0). Then p E Ca[X]- and pea) E P. (d)=>(a) Define U = {pea) : p E Ca[X]-}. Then U[2j c U. Assume that U. Then, by 1.3.44(i). there is a prime ideal P in A with P n U = 0, a contradiction of (d). So 0 E U, and (a) follows. 0
o 1.
Proposition 1.6.10 Let A be a commutatwe, complex algebra. Then the Jollowing condztzons on A are equivalent:
(a) A is algebraic; (b) A
=
~(A)
891(A);
(c) each prime zdeal in A zs the kernel oj a character on A; (d) each prime ideal in A has jinite codimenszon. IJ the condl,tzons are satzsjied, then rad A = 91( A).
Proof (a)q(b) This followi:i from the equivalence of (a) and (b) in 1.6.9.
(a)=>(c) Let P be a prime ideal in A. By 1.3.54(i), there is a prime ideal Q in A# with Q n A = P. By 1.6.9, A c CeA + Q, and so, by 1.3.37(ii), there exists
A# with MA· (c)=>(d) This is trivial. (d)=>(a) Let a EA. For each prime ideal P in A, the set {an + P : n E N} is linearly dependent, and i:iO there exii:iti:i p E Ca[X]- such that pea) E P. By the implication (d)=>(a) in 1.6.9, a is algebraic. Thus A is algebraic. Now suppose that A = (E(A) 891(A). By 1.5.6(iii), 'Jt(A) C rad A, and, by 0 1.5.7(iv), \C(A) nradA = {O}. Thus radA = 91(A).
Algebrazc foundatwns
Proposition 1.6.11 Let A be a unital, complex algebra, and let a E A. (i) For each p E qX], a(p(a»
= p(a(a».
(li) Suppose that a is algebraic. Then a(a) is fimte. Proof (i) We may suppose that op ~ 1. Take z E C. Then there exist 0:0 E C· and (Xl ....• (1:" E C such that z - p = ao(X - ad", (X - an), and HO it is dear that ZPJ\ - pea) E Inv A if and only if a - {tjeA E Inv A (J E N n ). Thus ::; E a(p(a») if and only if a 7 E a(a) for some j. The result follows.
o
(ii) This is immediate from (i).
Definition 1.6.12 An algebm A is spectrally finite ~f the spectrum a(a) is finite for each a E A. It follows from 1.6.11(ii) that each algebraic algebra is spectrally finite.
Proposition 1.6.13 Let A be an algebraic, commutative, unital, complex algebra. SUppOi'Ie that \)"leA) is nilpotent, that 1> A is fimte, and that A is mfinitedimensional. Then there zs a maximal ideal M in A such that M / M2 is mfimtedimensional. Proof Set N = \)"leA). By 1.5.26(iii), N = !.p(A), and so, since A is algebraic, it follows from 1.6.10 that A = (E(A) 8 N and that N = n{M
k.
Polynomwls and Jormal power senes Definition 1.6.18 Let a
= (a r : r
E
89 z+n) E In (OC)·. Then the order of a is
o(a) = min{lrl E Z+ : ar =I O}; also, 0(0) =
00.
Further, JOT r E Z+, Mr = {a E In(OC) : o(a) ::::: r}.
It is clear that o(a a, bE In(OC).
+ b) 2
min{o(a),o(b)} and that o(ab) = o(a)
+ o(b)
for
Proposition 1.6.19 Let n E N.
(i) Let a
E
In(OC). Then a is mvert'ible zJ and only zJ o(a)
(ii) The algebra
J n (OC)
= 0.
is a local algebra with rnaximal zdeal M 1 .
(iii) For kEN, Mf = Mk, Jvh has finite codzmension m In(OC), Mk is generated by {xr E In(lK) : Irl = k}, and each zdeal contaming Mk 1.S finitely generated.
°
Proof (i) Certainly o( a) = for a E Inv J n (lK). Now take a = (a r : r E Z+) with o(a) = 0, say ao = 1. Set bo = 1, and inductively define br, for r E N by setting br, = - (a1 br - 1 + ... + arb o). Then each br is a homogeneous polynomial of degree r, and it is easily checked that ab = 1, where b = (b r : r E Z+).
o
Oi) and (iii) These are clear.
Proposition 1.6.20 Let I be a non-zero ideal of J(OC). Then there exzsts r E Z+
such that I
=
Mr = xrJ(OC).
Proof Set r = min{o(a) : a E I}, and take a E I with o(a) = r. Then a = Xrb, where o(b) = 0. By 1.6.19(i), bE InvJ(OC), and so x r E I and I = Mr. 0
Let n E N with n 2 2. For each subset S of z+n such that 7' E S whenever r E z+n and r ~ s for some s E S, the family {a E In : as = (s E S)} is an ideal in In. However, there are many more ideals not of this form. Let a, b E J with 1l'o(b) = O. Then formal composition a 0 b can be defined: for each n E Z+, 1l'n(b T ) = for r > n, and so 1l'o(a 0 b) = 1l'o(a) and 1l'n(a 0 b) = L;'=o 1l'r(a)1l'nW) (n EN).
°
°
Proposition 1.6.21 Let b E J w1.th 1l'o(b)
= 0, and set Ab(a) = a 0 b (a
E
J).
(i) Ab is an endomorphism of J with Ab(Mn) C Mn (n E Z+).
(ii) Suppose that 1l'l(b) =I 0. Then there exists c E J with 1l'o(c) = 0, 1l'l(C) =I 0, and Ac = Ab 1 . In particular, Ab is an automorphism of J. Proof (i) This is clear.
(ii) Define 1'1 = 1l'l(b)-l, and successively define I'n for n 2 2 so that 2:~=1 I'r1l'n(b r ) = O. Then c = 2:::1 I'rxr has the required properties. Take a E J. Then Ab(Ac(a)) automorphism.
=
a, and so Ab is a surjection, and hence an 0
90
Algebraic foundations
Definition 1.6.22 Let £(OC) be the set of sequences a = (an: n E Z) such that an E OC (n E Z) and such that ther-e exists no E Z with an = 0 for n < no. For a = (an) and b = (b n ) in £(OC) and for a E lK, set a + b = (an + bn : n E Z), aa = (WIn: n E Z), and
For each nEZ, a, bn - r -I- 0 for only finitely many values of r, and so ab is well-defined. It is now ellliily checked that £(OC) is an algebra over OC: it is the algebra of Laurent series (in one indeterminate). We write £ for £(q. Each non-zero element of £(OC) can be uniquely expressed in the form L~=no anXn, where no E Z and ano -I- O. The following result is clear. Proposition 1.6.23 The algebra £(OC) is the quotient field of ~(OC).
0
Thus £(OC) = OC«X)) in our previous notation. Note that OC(X) is a subfield of £(OC), and that £(OC) = ~(OC, Z) in the notation of 1.2.24 and 1.3.63; the product in £(OC) coincides with *. The order 0 extends to £(OC). We define the following linear subspace of £: X
=
{t
U
x_jx-j : Xo, X-l,""
X-k E
C, k E Z+} .
(1.6.8)
J=O
Then £ = X + ~ and X n ~ = Cl; the projection L:=no anXn f----t L~=no anXn of £ onto X is denoted by P x . The space X is a unital, torsion ~-module with respect to the map (a,x)
f----t
a . x
=
Px(ax),
~ x
X ---+ X.
(1.6.9)
There is one further subfield of £ that we shall refer to later. introduce C{ X}, the algebra of absolutely convergent power senes:
C{X} =
{~anxn : ~ lanl En
First we
o} .
Then qX) is defined to be the quotient field of C{ X}; qX) is the field of meromorphic functions at O. We have qX) c qX) c q(X)) = £. Definition 1.6.24 A subalgebra A of ~ is ordinary if, faT each a E A-, there exists kEN with Xk E aA.
By 1.6.20, J' itself is ordinary; also, the subalgebra C{ X} is ordinary. Proposition 1.6.25 Let A be an ordinary subalgebra of J', and let E be an A-module. Then E is divisible if and only if E is X -divisible. Proof Suppose that E is X-divisible, and take a E A-. Then there exists kEN and bE A with X k = abo For each x E E, there exists y E E with X k . y = x, and then x = ab . yEa· E, and so E is a-divisible. Thus E is divisible. 0
Polynomials and formal power series
91
It follows that X is a divisible ~-module.
Let A and B be commutative, unital algebras, and let () : A --t B be a unital homomorphism. Take p E A[X}, ... , Xn], say p = 2: arxr. Then the element ()p E B[X 1, ... ,XnJ is defined by
(l.6.10) clearly, the map () : p morphism, and
f--->
()p, A[X 1, ... , XnJ ....... B[X 1, ... , XnJ, is a unital homo-
(()p)((}(ad , ... ,(}(an )) = ()(p(al"" ,an))
(a1,'" ,an E A).
Definition 1.6.26 Let B be a unital subalgebra of a commutative, unital algebra A, and let a E A. Then: (i) a is algebraic over B tf there exists p E B[XJe with p(a) = 0; (ii) a is integral over B if there exists a momc polynomwl p E B[XJ with p(a) = 0; (iii) a is transcendental over B if a is not algebratc over B. For example, let B = qXJ and A =~. Then the element exp X E ~ is transcendental over B; here, exp X is defined as 1 L,xn. n. 00
expX=
n=O
A subset S of A is algebraic over B if each element is algebraic over B. The set of elements of A which are algebraic over B is the algebraic closure of B in A, denoted by (l.6.11) B is algebraically closed in A if Alg B = B, and A is an algebratc extension of B if Alg B = A. Clearly IAlgABI = IBI in the case where A is an integral domain. The set of elements of A which are integral over B is the integral closure of B in A, denoted by IntAB or lnt B, and B is integrally closed in A if lnt B = B.
Lemma 1.6.27 Let a E A. Then a tS integral over B ~f and only if there is a finitely generated B-submodule E of A with eA E E and aE c E. Proof Suppose that a is integral over B. Then there exists n E N such that an E E, where E = BeA + Ba + ... + Ba n - 1. Then E is a finitely generated B-submodule, eA E E, and aE C E. Conversely, suppose that E = Bb 1 + ... + Bb n is a B-submodule with eA E E and aE c E. For i E N n , we have abi E E, say abi = 2:;=1 bijbj , and (b 1, ... , bn ) is a solution of the set of simultaneous linear equations n
i)bi] - oi,]a)tj
=
0
(i
E
N n ).
j=1
Let d = det(bij - Oi,ja). Then db i = 0 (i E Nn ) by 1.3.9(iii), and so d . E = O. Since eA E E, necessarily d = deA = O. This shows that a is a root of the monic polynomial (_l)n det(bij - Oi,jX) in B[X], and so a is integral over B. 0
92
Algebrazc foundations
Theorem 1.6.28 Let B be a unital subalgebra of a commutative, umtal algebra A. Then lnt B is an integrally closed s1Lbalgebra of A containing B. Proof Set C = lnt B, and take a, bE C. By 1.6.27, there are finitely generated B-submodules E and F of A such that eA E En F, aE C E, and bF c F. Let G = EF. Then G is a finitely generated B-submodule of A, eA E G, and (a + b)G c G and abG c G. Thus a + band ab are integral over B, and so C is a subalgebra of A. Certainly C ::) B. Let a E A be integral over C. Then there is a finitely generated C-submodule M with eA EM and aM c M, say M = Cal + .. ·+Can . We may suppose that al = eA. For i E N n , there exist Cij E C such that aai = 2:,7=1 CiJaj. Let Eij be a finitely generated B-submodule such that eA E Eij and cijEij C E ij , and let E = fI E ij . Then E is a finitely generated B-submodule, eA E E, and cijE C E. Set F = Ea1 + ... + Ea n , so that F is a finitely generated·B-submodule, eA E F, and aP C 2:,i,J CiJajE C 2:,j Eaj = F. By 1.6.27, a is integral over B, and so a E C. Thus C is integrally closed in A. 0 Corollary 1.6.29 Let B be a unital subalgebra of a umtal integral domain A. Then Alg B is an algebraically closed subalgebra of A. Proof First note that, if a E Alg B, then there exists b E B- such that abc is integral over B for each C E B. For suppose that p( a) = 0, where p = 2:,~0 biXi. Take b = bn , and, for c E B, set q = 2:,;:01 bibn-1-icn-ixi + xn. Then q is a monic polynomial, and q(abc) = O. Now suppose that a, bE Alg B. Then there exists c E B- with ac, bc E lnt B. By the theorem, (a + b)c and abc 2 belong to lnt B, and so a + b, ab E Alg B. Thus AIg B is a subalgebra of A. Finally take a E Alg (AIg B). Then there exist co, Cll ... , Ck E AIg B with Ck -I- 0 and Co + C1a+· .. + Ckak = O. We may suppose that Co, Cl,'" ,Ck E lnt B. Then, as above, Cka E lnt (lnt B), and so, by the theorem, Cka E lntB. Hence a E AIg B, proving that Alg B is algebraically closed. 0 Definition 1.6.30 Let B be a unital subalgebra of a commutative, umtal algebra A, and let a E (AIg B) \ B. Set aa = min{ap: p E B[Xt, pea) = O}. Then a is algebraic of degree aa over B; a polynomial p with p( a) ap = aa is a minimal polynomial for a.
o and
Theorem 1.6.31 Let A and C be commutative, umtal algebras, let B be a unital subalgebra of A, and let () : B -+ C be a unital homomorphism. Suppose that a E Alg B, that q is a minimal polynomial for a, and that Co E C is a root of (jq. Suppose further that either
(i) ()(b)c -I- 0 (b E B-, c E C-); or (ii) A is a integral domain and q(O) E Inv B. Then (j : pea) ~ (()p)(co), B[a] -+ C, is a homomorphism which extends ().
93
Polynomials and formal power senes
Proof We first show that (j is well-defined. Suppose that p E B[Xl e with pea) = o. By the division algorithm, there exist b E Be and r,8 E B[Xl with bp = rq + 8 and < oq. Since sea) = 0, necessarily 8 = O. Thus
as
B(b)(Bp)(co)
=
(Br)(co)(Bq)(co)
= O.
Under hypothesis (i), it follows immediately that (Bp)(co) = O. Now suppose that hypothesis (ii) holds, and set r = L~=o rjX) E B[Xl. Since q(O) E Inv B, it follows by induction on j that there exist 80, ... ,8k E B with rj = b8j (j E zt). Set 8 = L~=o SjXj. Then bp = bq8. Since A is an integral domain, p = q8, and again (Op)(co) = O. We have shown in both cases that (j is well-defined. It is now easily seen that (j is a homomorphism and that (j extends O. 0 The following result is immediate. Theorem 1.6.32 Let A and C be commutative, umtal algebras, let B be a unital subalgebra of A, and let 8 : B ~ C be a unital homomorphism. Suppose that a E A zs transcendental over B, and that c E C. Then the map (j : pea)
f-+
(8p)(c), B[a]
---7
C,
o
is a homomorphism which extends B.
Definition 1.6.33 Let B be a unital subalgebra of a commutatzve, umtal algebra A. A subset S of A zs algebraically independent over B if P(SI, ... , sn) =I 0 for each polynomzal p E B[X1 , ••. , Xn]e and each set {81, ... , sn} of distinct element8 of S. A transcendence basis for A over B is a maximal element in the family of subsets of A which are algebraically independent over B. It follows from Zorn's lemma that each algebraically independent set over B is contained in a transcendence basis for A over B. The cardinality of transcendence bases is an invariant of the algebras A and B. Let A be a commutative, unital, complex algebra. A transcendence basis for A over CeA is a transcendence basis for A; the dimension of such a basis is the transcendence degree of A. Since each algebraically independent set is linearly independent, this degree is at most dim A. Proposition 1.6.34 Let A be a commutative, umtal algebra, let B be a unital subalgebra of A, and let r be a transcendence basis for A over B. Then each a E B [r]- can be written m a unique way in the form a = p( 71, ... , 7 n), where n E N, p E B[X1 , ... , Xnl-, and 71, .. " 7 n are dzstmct elements of f. Further, A is an algebraic extension of B[f]. Proof Set C = B [fl. Certainly each element a E C- has an expression of the given form. Suppose that a = p( 71, ... ,7m ) = q( 7{, ... ,7:,), where {71,' .. ,7m } and {7i, ... , 7~} are sets of distinct elements of r. By relabelling, we may suppose that 71 = T{, . .. ,Tk = T~ and that {Tl, ..• , Tm, T~+I" .. , v(b - a). Also r(y) = q(x) and r(k)(O) = (b - a)kq(k) (b) (k E Z;t;.), and so
v(r(k)(O» = v«x - a)kq(k) (x» < v«x - a)k+l q(k+1) (x» = v(r(k+l) (0» for each k E N'm. Finally,
v(r(O»
= v«b -
x)q'(x» > v«b - a)q'(b»
= v(r'(O)).
Thus v(r'(O» < v(r(k)(O» (k = 0,2, ... ,m). Since V is a valuation algebra, there exist be, b2, ... , bm E Mv such that r(k)(O) = k!r'(O)bk (k = 0,2, ... , m).
Valuation algebras
103
Set Po = bo+X +b2X 2+ .. ·+bmXm E V[X]. Then Po is a Henselian polynomial in V[X], apo = m S n. and r'(O)po = r. We now show that the polynomial Po has no root in V. Take e E V. Since v(r'(O)) = v«b - a)q'(b)) = v«b - a)q'(x)), we have
v(r'(O))
+ v(y -
e)
=
v«x - b - eb + ae)q'(x)) ,
and so, by (1.7.3), v(r'(O)) +v(y - e) < v(q(x)). Since q(x) = r(y) = r'(O)po(y), we have v(y - e) < v(Po(y)). By (1.6.4), v(Po(y) - po(e)) = v(y - c), so that v(Po(y) - po(e)) < v(Po(y)). Thus po(e) =f 0, as required. We have shown that V is not Henselian. 0 Lemma 1.7.22 Suppose that W is an immedzate extension of V and that V zs Hen.~elzan.
Then, for each p E V[X]· and each x such that v(P(x)) < v«x - y)p'(x)).
E
W \ V, there exists y
E
V
Proof Since W is an immediate extension of V, condition (i) of the above lemma holds. Since V is Henselian, condition (ii) must fail. 0 Lemma 1.7.23 Suppose that x E W \ V, that x zs algebraic oj degree n over V,
and that, for each q E V[X]- with Oq < n, there exists an element a E V such that v(q(x)) < v«x - a)q'(x)). Then V(V[x]) zs an zrnmediate extension oj V. Proof By 1.7.13, V(V[x]) is a valuation subalgebra of W, and so, by 1.7.6(ii), it suffices to show that v(V(V[x])) c v(V). Take q E V[x]· with aq = m, where m < n. Then there exists a E V with
v(q(k)(x)) < v«x - a)q(k+l) (x))
(k E Z~_l)'
and so v(q(x)) = v(q(a)) E v(V). Now fix p E V[X]· with 8p = n such that p(x) = O. For each q E V[X]·, there exist b E V· and r, s E V[X] with bq = pr + s and as < n. Then bq(x) = sex), and so v(q(x)) E v(V). Thus v(V[x]) C v(V). Take y = al/b l E V(V[x]), where al E V[x], b1 E V[x]·, and Veal) ~ v(bI). Then there exist a2 E V and b2 E V· such that v(a2) = v(ad and v(b2) = v(bI). Also, there exists z E V such that a2 = b2z, and then v(y) = v(z) E v(V). Thus v(V(V[x])) C v(V), as required. 0 Lemma 1.7.24 The valuation algebra V has a maximal immediate extension V
in W. Moreover, zJ W is Henselzan, then
V zs also Henselian.
Proof Let F be the family of all valuation subalgebras U of W such that U ::> V and v(U) = v(V). Clearly (F, c) is a partially ordered set and each chain in F has an upper bound. By Zorn's lemma, F has a maximal element, say V. By 1.7.6(ii), V is a maximal immediate extension of V. Now suppose that W is Henselian. Assume that V is not Henselian, and take a Henselian polynomial p E V[X] with no root in Vj we may suppose that p is of minimal degree among such polynomials. Since W is Henselian, there exists
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Algebraic foundations
x E W \ V such that p(x) = o. We have vex - a) = v(p(a» (a E V), and so vex + V) c v(V). Take q E V[X]- with aq < ap. If it were the case that v(q(x)) ~ v«x - a)q'(x))
(a E V),
then, by 1.7.21, there would exist a Henselian polynomial Po E V[X] with no root in V and with apo < ap. But this contradicts the assumed minimality of ap. Thus there exists a E V with v(q(x» < v«x - a)q'(x». By 1.7.23, V(V[x]) is an immediate extens~on of V. By the maximal~y of V in (F, C), V(V[x]) = V, and in particular x E V, a contradiction. Thus V is Henselian. 0 Theorem 1.7.25 Each complex valuatwn algebra has an tmmedwte, algebraic extension whzch is a Henselian algebra. Proof Let V be a valuation algebra. By 1.7.10, there is an algebraically closed valuation algebra W which is an algebraic extension of V. By 1.7.11, W is Henselian. By 1.7.24, there is a Henselian algebra V C W such that V is an immediate extension of V. Certainly V is an algebraic extension of V. 0 Corollary 1.7.26 Let G be a totally ordered abelian group. Then J(G+) is a Henselian algebra. Proof By 1.7.7, J(G+) is a maximal valuation algebra. By the theorem, J(G+) is always Henselian. 0 In fact, it is easy to see directly that J(G+) is always Henselian. We now define a 'framework map'. Recall that the positive cone rt; of the value group rw of a valuation algebra W is denoted by Pw , and that the unique maximal ideal of a local algebra A is M A . Definition 1.7.27 Let W be a valuatwn algebra with Mw i= 0, and let A be a local algebra. A map 'lj; : P w ........ MA zs a framework map zf:
(i) 'lj;(s + t) = 'lj;(s)'lj;(t) (.'.I, t E P w ); (ii) 'lj;(s).L = 'lj;(t).L (s,t E Pw). The framework map 'lj;
'U~
freely acting zf:
(ii)' 1f;(s).L = 0 (s E Pw). Let W be a valuation algebra with Mw i= 0, and let 'lj; : Pw ........ framework map. We systematically extend 'lj; to a morphism 'lj; : rt setting 'lj;(0) = eA. Throughout we write
W = 'lj; 0 v,
be a by
W ........ A,
where v is the valuation on W, taking w(O) = w(xy) = w(x)w(y) (x, yEW)
MA
. . . . A-
o.
and
Clearly W(Inv W) = {eA} .
Suppose that U is an immediate extension of W. Then may regard W as a map from U into A.
ru
=
fw, and so we
Valuation algebras
105
Let x, yEW. Then it is clear that lI1(x) E lI1(y)A whenever vex) 2: v(y) and that lI1(x) E lI1(y)MA whenever vex) > v(y). Assume that s < t in Pw and that 1/)(s) = tjJ(t). Then t/J(s)(eA + a) = 0 for some a E MA, and so t/J(s) = 0, a contradiction. Thus t/J is an injection. We shall prove our extension theorems 1.7.42 and 1.7.44, first in the case where t/J is freely acting, and then in the general case; note that a framework map is certainly freely acting in the case where A is an integral domain, and many of our later results will apply in this context. Definition 1.7.28 Let S be a subset of W. A map () : S with a framework map t/J if ()(x) E lI1(x) . Inv A (x E S).
-+
A
tS
compatible
Let () be such a map. Then ()(x) = 0 for xES if and only if x = O. If x, yES with vex) 2: v(y), then ()(x) E ()(y)A. Now suppose that 'l/J is freely acting, that a E A·, and that xES·. We claim that ()(x)a =I O. For, if x E M then vex) E Pw and lI1(x)a =I 0 because t/J is freely acting, and, if x E Inv W, then lI1(x)a = a =I O. In either case, it follows that ()(x)a =I 0, as claimed.
w,
We shall now give a sequence of results which will culminate in our main algebraic extension theorem 1.7.42. Until we state otherwise, Wand A are as in 1.7.27, 1/) is a freely actmg framework map, Uo is a fixed subalgebra of W, and ()o : Uo -+ A is a homomorphism which is compatible with t/J. We denote by F the family of all pairs (V, ()), where V is a subalgebra of W with Uo c V and () : V -+ A is a homomorphism extending ()o such that () is compatible with t/J. Set (VI,()t) =;< (V2,()2) in F if VI c V2 and ()2 I VI = ()l. It is clear that (F, =;r (V, 0) in F. 0
Al.'Jebrmc foundatzons
106
Let (V, (9) E F with Vi- W, and let x E W \ V. Then a E A is 19-compat~ble with x if the map x + y ~ a + 19(y), x + V ~ A, is compatible with 1/1. Recall from (1.6.1) that, for each n E N, V{n)[x] = {p E v[X] : 8p ::; n}; if x E V, then V{n)[x] = {p(x) : p E V{n)[X]},
Lemma 1. 7.30 Let (V, (9) E F, let x E W \ V, and let n EN. Suppose that a is 19-compatible with x and that, for each p E V{n) [X]-, there e~sts y E V such
that v(p(x)) < v«x - y)p'(x)). Then the map p(x) ~ (19p)(a), V{n)[x] a lmear map which extends 19 and which is compatible with 1/1.
~
A, is
Proof Set J-L(p(x» = (19p)(a) (p E V(n)[X]), Fin;t note that v(p(x» < 00, and so p(x) i- 0, for each p E V(n) [X]-. Thus J-L : V{n) [x] ~ A is a well-defined linear map which extends 19. Take p E V{nl [X]-. It follows from the hypothesis that there exists y E V such that 'I.'(p(k)(x)) < v«x - y)p(k+l) (x» (k E p ). Let k E p • From Taylor's formula (1.6.2), v(p(k)(y)) = v(pCkl(x)), and so
zt
zt
v(p(k)(y)) < v«x _ y)pCk+l)(y». Thus w«x_y)pCk+ 1 )(y)) E W(p(k)(y))MA' Since a-Bey) E lI1(x-y) . Inv A, this implies that (a-B(y))B(pCk+l)(y)) E B(pCk)(y»MA. Again from Taylor's formula, we have (Bp)(a) E B(p(y)) . Inv A, and so J-L(P(x)) E w(P(y)) . Inv A. SincE' v(p(x)) = v(p(y», it follows that w(p(x)) 111 (p(y)), and so IL is compatible with 1/1. 0
Corollary 1.7.31 Let (V, B) E F, and let x E W \ V. Suppose that V is Henselian, that W zs an immediate extension of V, and that a ~s an element
of A wh~ch ~s B-compat~ble with x. Then there is a homomorphism J-L : V[x] such that J-L(x) = a and (V[xJ, J-L) >,;= (V, B) in F.
~
A
Proof It follows from 1.7.22 that, for each p E V[X]-, there exists y E V such that v(P(x)) < v«x - y)p'(x)). Set J-L(P(x)) = (Bp)(a) (p E V[X]). By 1.7.30, J-L : V[x] ~ A is a linear map which is compatible with 1/1. Clearly J-L is a homomorphism satisfying the required conditions. 0 Lemma 1.7.32 Let (V, B) E F, and let x E W \ V be algebm~c of degree n over V. Let q be a minimal polynomial for x, and suppose that a E A is a mot of Bq and that the map ()' : p(x) ~ (Bp)(a), V{n-l)[X] ~ A, is compatible with 1/1. Then there is a homomorphism J-L : V[x] ~ A which extends B' and which is such that (V [x], J-L) >,;= (V, B) in:F. Proof Since 1/1 is freely acting, the homomorphism B : V ~ A satisfies hypothesis (i) of 1.6.31, and so J-L : p(x) ~ (Bp)(a), V[x] ~ A, is a homomorphism which extends B. Certainly J-L extends B'. Take p E V[X]. By the division algorithm, there exist y E V-, r E V[X], and S E V{n-l) [X) with yp = qr+s. We have B(y) (Op) (a) = (Os)(a) E lI1(s(x)) . Inv A and yp(x) = sex), and so (Oy)(Op)(a) E O(y)lI1(p(x)) . Inv A. Thus
J-L(P(x» and so 1-£ is compatible with
E
w(P(x» . Inv A,
1/1. Clearly (V[x),I-£)
>,;=
(V, 0) in F.
o
Valuation algebras
107
We now return to consideration of Henselian polynomials, defined in 1.6.2. Note that, if V is a valuation subalgebra of W, if (V,O) E F, and if p is a Henselian polynomial in V[X], then Op is Henselian in A[X] because 8(ev) = eA and 8(Mv) C MA. Lemma 1. 7.33 Let (V, 0) E F, where V is a valuatzon subalgebra of W, and let E W\ V. (i) Suppose that p is a Henselzan polynomzal m V[X] with p(x) = 0, and that
x
a E A is a root of 8p. Then a is O-compatible with x. (ii) Let n E N. Suppose that a E w(x) . Inv A, and that kv(x) fj. rt (k E N n ). Then the map p(x) t-+ (8p)(a), V(n)[x] ---+ A, is a Imear map which extends 0 and which is compatible with 'Ij;. Proof (i) Let y E V. It follows from (1.6.4) that x - y E -p(y) . (ev and a - 8(y) E -8(P(y)) . (eA + MA), and so
a - 8(y)
E
+ Mv)
w(p(y)) . Inv A = w(x - y) . Inv A.
x + V ---+ A, is compatible with 'Ij;. (ii) Set f.L(p(x)) = (8p)(a) (pE V(n)[X]). By 1.7.16(i), p(x) =1=0 (p E V(n)[Xj-), and so f.L : V(n)[x] ---+ A is a well-defined linear map which extends 8. Take y, z E V- and k E N n . Since kv(x) fj. rt, necessarily v(yxk) =1= v(z). If v(yxk) < v(z), then w(z) E w(Y)W(X)k MA, and so 8(z) E O(y)a k MA. Similarly, if v(yxk) > v(z), then O(y)a k E 8(z)MA. Now take p E V(n) [X]-, say Thus the map x
+ y t-+ a + 8(y),
p(X) =
xm1Xml
+ ... + Xm,.Xmk,
where 0 ::; m! < m2 < ... < mk ::; nand X m1 , ... , x mk E V-. There exists j E Nk such that v(xm,x mj ) < v(xmix m.) (z =1= j). Then v(p(x)) = v(xmjx mj ), and so w(p(x)) = w(xmJx mj ). If i < j, then v(xmjXmj-m;) < v(xmJ, and so
O(amJ E O(xm,)amj-mi M A . Hence O(xmJa mi E 8(xmJa m'MA. This also holds if i », and so J.L(p(x)) E O(xmj)amj . (eA + MA) C w(xmjx m;) . Inv A. Thus f.L is compatible with W. 0 Lemma 1.7.34 Suppose that A and W are both Henselian. Then:
(i)
V is a valuation algebra;
(ii) W is an immediate extension of V; (iii) V is algebraically closed in W; (iv) V is Henselzan. Proof We write V and 0 for V and 8, respectively. (i) By 1.7.29, V = V(V), and so V is a valuation subalgebra of W. (ii) Assume that there exists x E W \ V such that kv(x) fj. rt (k EN), and set a = 1I1(x). By 1.7.33(ii), the map p(x) t-+ (8p)(a), V{n)[x] ---+ A, is a linear map which extends 8 and which is compatible with 'Ij; for each n E N. Define f.L(P(x)) = (8p)(a) (p E V[X]). As in 1.7.31, (V[x], f.L) ~ (V,8) in F. But this
Algebraic foundations
108
contradicts the assumed maximality of (V, 0). Thus, for each x E W \ V, there exists n E N with nv(x) E rt. Next, assume that there exists x E W \ V and n ~ 2 such that nv(x) E rt, but kv(x) fj. rt (k E Nn-d. Then xn E y . Inv V for some y E V. Since W is Henselian, it follows from 1.6.4 that there exists z E W with v(z) = vex) and zn = y. We have w(z)n E w(zn) . Inv A = O(y) . Inv A, and so, since A is also Henselian, there exists a E w(z) . Inv A such that an = O(y). By 1.7.16(i), z is not algebraic of degree less than n over V, and so z is algebraic of degree n. By 1.7.33(ii), the map p(z) ...... (Op)(a), V{n-l)[Z) -+ A, is a linear map which is compatible with t/J. Thus, by 1.7.32, there exists I-' such that (V[z),I-') >r (V,O) in F. By the maximality of (V,O), necessarily z E V, and so vex) E rt, a contradiction. Thus vex) E rt (x E W), and hence W is an immediate extension of V. (iii) Assume towards a contradiction that V is not algebraically closed in W. Then there exist x E W \ V and p E V(X)· with p(x) = O. Certainly
v(p(x» ~ v«x - y)p'(x»
(y
E
V).
(1.7.4)
We may suppose that p and x are chosen so that op is minimal among polynomials satisfying (1.7.4) for any x E W \ V, say op = n. By 1.7.21 there is a Henselian polynomial Po E V(X) with oPo ~ n such that Po has no root in V. Since W is Henselian, Po has a root, say Xo. in W \ v. By the choice of p and x, Xo is algebraic of degree n over V. Since A is Henselian, Opo has a unique root, say a, in A, and, by 1.7.33(i), a is O-compatible with Xo. By 1.7.30, the map q(xo) ...... (Oq)(a), V{n-l)(XO) -+ A, is compatible with t/J. By 1.7.32, there exists I-' such that (V(xoJ, 1-') ~ (V, 0) in:T. But this contradicts the maximality of (V, 0). Thus V is algebraically closed in W. (iv) Since W is Henselian and V is algebraically closed in W, the subalgebra V is Henselian. 0 The above lemma allows us to deduce an embedding theorem for a valuation algebra W into the algebra A = ~(rt). For s E Pw, define 1jJ(s) = as. Then t/J : Pw -+ MA is a freely acting framework map. Let U be a subset of W. Then 0 : U -+ A is compatible with t/J if and only if v(O(x» = vex) (x E U); such a map is said to be valuatwn-preservzng. Let V be a subalgebra of W, and let 0 : V -+ A be a homomorphism which is compatible with t/J. Suppose that V =f. W, that x E W \ V, and that f E A. Then f is O-compatible with x if and only if v(l - O(y» = vex - y) (y E V). Proposition 1.7.35 Let Uo be a subalgebm of a valuatwn algebra W, and let 00 : Uo -+ ~(rt) be a valuation-preserving embeddzng. Then there is a valuationpreserving embedding 0 : W -+ ~(rt) which extends 00 , Proof Set A = ~(rt), and let :F be as above. By 1.7.26, A is Henselian. By 1.7.25, we may suppose that W is Henselian. Let (V, 0) be a maximal element of :F. By 1.7.34, V is Henselian and W is an immediate extension of V. Assume that there exists x E W\ V. Then, essentially as in 1.7.7(i), there exists f E A such that v(l - O(y» = vex - y) (y E V); f is O-compatible with x. By 1.7.31, there exists I-' such that (V[x],I-') ~ (V, 0) in:F,
Valuation algebras
109
a contradiction of the maximality of (V, (J). Thus V = W, and (J is the required m~.
0
Corollary 1.7.36 (Kaplansky's isomorphism theorem) Let W be a maximal, complex valuation algebra. Then there is a valuation-preservzng isomorphism from W onto ~(rtv)· Proof By the above proposition, there is a valuation-preserving embedding (J : W ---+ ~(rtv). Since W is maximal and ~(rtv) is an immediate extension of (J(W), (J is a surjection. 0 We can now show that the algebra M# is unzversal in the class of /31-valuation algebras.
Theorem 1.7.37 (Esterle) Let V be a /31 -valuation algebra. Then there is an embedding of V into M# . Proof Set V = U{Vu : a < wd, where {Vu : a < wd is a chain of aI-valuation subalgebras of V; we regard each rv" as an aI-subgroup of rv. By a transfinite recursion, there is a family {(JT : 7 < WI} such that each (JT : VT ---+ ~(rtJ is a valuation-preserving embedding and (JT I Vu = (Ju whenever a < 7 < Wl. Indeed, given OTl where 7 < WI, it follows from 1.7.35 that there exists (JT+l such that (JT+I I VT = (JT. Now suppose that 7 < Wl is a limit ordinal and that (Ju has been defined for each a < 7; define (J~ on U{Vu : a < 7} by requiring that (J~ I Vu = (Ju for each a < 7, and then extend (J~ to VT by using 1.7.35 again. Finally, define (J : V ---+ ~(I)(rt) by requiring that (J I VT = (JT (7 < wt). Then (J is a well-defined, valuation-preserving embedding. By 1.2.29(i), rv can be identified as an ordered subgroup of G, and so we may regard ~(1)(rt) as a valuation subalgebra of M#. 0 Theorem 1.7.38 Let B be an zntegral domain with a transcendence degree at most ~I. (i) Suppose that B is non-unital. Then there zs an embedding of B in M. (ii) Suppose that B zs unital and has a character. Then there is a unital embedding of B in M# . Proof (i) By 1.7.9, there is a valuation algebra V in the quotient field of B with B c Mv. Clearly the transcendence basis of B is a transcendence basis for V, and so, by 1.7.17, V is a /31-valuation algebra. By 1.7.37, there is an embedding (J : V ---+ M#, and (J(Mv) C M. o (ii) This follows in a similar way; we use 1.7.8 instead of 1.7.9. Let B be an integral domain of cardinality c. Then, with CH, B certainly has transcendence degree at most ~l. The following result will allow us to insert a 'twist' in certain homomorphisms; recall that a pair {a, b} of elements of a commutative algebra is algebraically independent if p{a, b) # 0 whenever p E qx, Yj-.
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Proposition 1.7.39 Let {I, g} be an algebraically independent set m M#. Then there zs an automorphism J.t on M# with J.t(f) = f and J.t(g) =I- g. Proof Choose ao < WI such that f,g E M~o' and then choose hEM- with v(h) » Gto ' say h E M~. For each p E qx, Yj-, we have p(f,g) =I- 0 and p(f,g + h) - p(f,g) E hM#, and so v(p(f,g + h» = v(p(f, g». Thus the map
p(f, g)
()o :
t-+
p(f, 9 + h),
qf, gj
--->
M1a '
is a well-defined, valuation-preserving embedding. By 1. 7.35, there is a valuationpreserving embedding J.tTo : M~ ---> M~ which extends ()o. By a transfinite recursion using 1.7.35, there is a family {J.tT : T < WI} such that each J.tT : M~ ---> M~ is a valuation-preserving embedding and such that J.tq I Mf' = J.tT whenever TO S T < a < Wl· Since each M~ is a maximal valuation algebra, each J.tT is an epimorphism. Define J.t : M# ---> M# by the formulae /1 I M~ = J.tT (TO S T < wd. The map J.t is well-defined, and is the required automorphism. 0 We have given the preliminary results necessary for the proof of our algebraic extension theorem in the case where the framework map 't/J is freely acting. We now explain the modifications that will give the general case. Thus we suppose that 't/J : W ---> A is a framework map; we may suppose that Pw =I- 0. Define I = 't/J(Pw)J... Then I is a proper ideal in A. Write 7r : A ---> AI I for the quotient map, and set B = CeA
+ w(Mw)
. A,
so that W(W) c B. Since A is local, we have 1I'(Inv A) = Inv (AI I) and rad (AI I) = (rad A)I I. It follows that, if A is Henselian, then so is AI I. Note that the only point at which the fact that a framework map satisfies 1.7.27(ii) is used in our theory is in clause (ii) of the following result. Proposition 1.7.40 (i) B is a subalgebra of A. (ii)
11'
IB
:B
--->
AI I is an injection.
(iii) 1I'(B) = CeA/I
(iv)
11'
0
't/J : Pw
--->
+ (11' 0 W)(Mw)
. (AI I). All zs a freely actmg framework map.
Proof (i) Take a,b E A and x,y E M w , say with vex) ~ v(y), so that w(x) E w(y)A. Then w(x)a + w(y)b E w(y)A c w(Mw) A, and hence w(Mw)A = w(Mw) . A. It follows easily that B is a subalgebra of A. (ii) Take b = aeA + W(x)a E B, and suppose that 1I'(b) = 0, so that bEl. Since eA ¢ I and w(x) E M A , necessarily a = O. If x = 0, then b = O. If x =I- 0, then w(x) E w(M and so 0 = bW(x) = W(x)2a = w(x 2)a. By 1.7.27(ii), W(x)a = 0, and so b = O. This shows that 11' I B is an injection. (iii) This is immediate.
w),
(iv) Certainly 11' 0 't/J is a framework map. If a E A and (7r 0 W)(x)1I'(a) = 0, then w(x) = 0 by (ii), and so a E I. Thus 7r 0 'I/J is freely acting. 0
111
Valuatwn algebras
We write 11"-1 : 1I"(B) ---> B for the inverse of the injection 11" I B. Let V be a subset of W. Suppose that a map B : V ---> A is compatible with 'l/J. Then clearly 11" 0 B : V ---> AI I is compatible with 11" 0 'l/J, and B(V) C B, so that 11"-1 011"0 B = B. Proposition 1.7.41 Let () : W ---> AI I be a homomorphzsm which zs compatible with 11" 0 'l/J. Then (}(W) C 1I"(B) and 11"-1 0 () : W ---> B is a homomorphism which is compatible with 'l/J. Proof Let x E W, say x = aew + y, where a E C and y E Mw. Since () is compatible with 11" 0 'l/J. (}(y) = (11" 0 w)(y)1I"(a) for some a E Inv A, and so B(W) C 1I"(B). Since aew + w(y) E B, we have (11"-1 0 (})(x) = aew + W(y)a. If a -1= 0, then we;]:) = eA, and, if 0: = 0, then w(x) = w(y). Thus, in either case, (11"-1 0 B) E w(x) . Inv A, and so 11"-1 0 () is compatible with W. 0 We can now give our algebraic extension theorem. Theorem 1.7.42 Let Uo be a subalgebra of a complex valuation algebra W with Mw -1= 0, let A be a local Henselian algebra, let'l/J : Pw ---> MA be a framework map, and let (}o : Uo ---> A be a homomorphism whzch is compatzble with '1/" Suppose that W is algebraic over Uo. Then there is an embedding of W into A which extends Bo and whzch zs compatzble with 'l/J. Proof By 1. 7.25, W has an immediate, algebraic extension which is a Henselian algebra, and so we can suppose that W is Henselian. Suppose first that the framework map 'l/J is freely acting, and let F be as above. Let (\7,0) be a maximal element of F. By 1.7..:34(iii), V is algebraically closed in W. Since W is algebraic over Uo, we have V = W, and so the result follows in this case. Now consider the general case, and take I, 11", and B as above. Then AI I is a Henselian algebra, 11" 0 'l/J : W ---> AI I is a freely acting framework map, and 11" 0 0 : U ---> AI I is a homomorphism which is compatible with 11" 0 'l/J. Thus there is a homomorphism J.L : W ---> AI I which extends 11" 0 0 and which is compatible with 11" 0 'l/J. By 1.7.41, 11"-1 0 Jt : W ---> B is a homomorphism which extends () and which is compatible with 'I/J; this implies that B is a monomorphism. 0 The above results are also sufficient to cover certain non-algebraic extensions; for this, we require that W be an aI-valuation algebra (see 1.7.15(i» and that a certain subset of A be a Mittag-Leffler set (see 1.3.40). Let W, A, 'l/J, and F be as above, with '1/' a freely acting framework map. The key lemma that we require is the following. Lemma 1.7.43 Let (V, 0) E F, and let x E W\ V. Suppose that v(x+ V) has no maximum element, that W is an aI-valuation algebra, and that 'l/J(Pw) . Inv A is a Mittag-Leffler set. Then there exists a E A which is f)-compatible with x. Proof First suppose that x
E
Mw, so that vex) E Pw.
Algebraic foundations
112
Since v(x + V) has no maximum element and W is an aI-valuation algebra, there exists a sequence (Yn) in V such that (v (x - Yn» is a strictly increasing, cofinal sequence in v(x + V). We have
= v(x -
V(Yn - Yn+l)
Yn) < v(x - YnH)
= V(YnH
- Yn+2)
and we may suppose that v(x - yr) > v(x). Then V(Yl) V(Y2 - yr) > v(yr). Set Zl = Yl, Z2 = (Y2 - Yl)/Yl, and Zn
= (Yn - Yn-d/(Yn-l - Yn-2) (n
~
(n
E
N),
= v(x) > 0 and also 3).
w,
Then (Zn) C M Zl'" Zn+l = Yn+l - Yn, and 'Lj=l Zl'" Zj = Yn (n EN). Set an = (J(zn) (n EN). Then (an) C 1jJ(Pw) . Inv A, a Mittag-LefHer set in MA, and so there exists a E MA such that (1.3.22) holds. We have a - (J(Yn) E (J(YnH - Yn) . Inv A = W(Yn+l - Yn) . Inv A
(n E N).
(1.7.5)
We now show that a is (J-compatible with x. Take Y E V, and choose n E N such that v(x - Yn) > v(x - y). Then V(YnH - Yn) > v(x - Y), and so W(Yn+l - Yn) E w(x - y)MA
.
By (1.7.5), a - (J(Yn) E w(x - y)MA . Also, V(Yn - y) = v(x - Y), and hence (}(Yn) - (J(y) E w(x - y) . Inv A. Thus a - (}(y) E w(x - y) . Inv A, as required. For the general case, let aew + x E W, where a E C and x E Mw. Take a E MA to be (}-compatible with x. Then oeA + a is clearly (J-compatible with oew +x. 0
a
a-
We remark that, if E A is such that a E n{W(Yn+l - Yn)A : n E N} for a sequence (Yn) such that (v(x - Yn» is cofinal in v(x + V), then is clearly also (J-compatible with x.
a
We can now establish the main extension theorem of this section. Theorem 1.7.44 Let Uo be a subalgebra of an 01 -valuation algebra W with Mw =f 0, let A be a local Henselian algebra, let 1jJ : Pw -+ MA be a framework map, and let (}o : Uo -+ A be a homomorphism which is compatible with W. Suppose that 1jJ(Pw) . Inv A ~s a M~ttag-LejJler set in A. Then there ~s an embedding of W into A which extends (Jo and which ~s compatible with 1jJ. Proof As in 1.7.42, we may suppose that W is Henselian. Suppose first that 1jJ is freely acting, and define F as above. Let (V, 8) be a maximal element of:F. By 1.7.34, V is Henselian and W is an immediate extension of V. Assume that there exists x E W \ V; by 1.7.6, v(x + V) has no maximum element, and so, by 1.7.43, there exists a E A which is (}-compatible with x. By 1.7.31, there exists v such that (V[x],v»);= (V,8) in F. But this contradicts the maximality of (V, 8), and hence V = W. The result follows in this case. Now consider the general case, and take I, 71", and B as before. Clearly (71" o'I/J)(Pw) . Inv(A/I) is a Mittag-LefHer set in A/I, and so there is a homomorphism J.L : W -+ A/ I which extends 71" 0 (J and which is compatible with 71" 0 'I/J. The result follows from this, as in 1.7.42. 0
Valuation algebras
113
The above theorem does not cover the case where W = M# because M# is not an aI-valuation algebra. However, M# is a ,a1-valuation algebra, and so we can easily extend 1.7.44 to obtain such an embedding. Theorem 1.7.45 Let A be a local Henselian algebra. Suppose that there is a morphzsm 1j; : (P, +) ---+ (M.A, .) such that al. = bl. (a, bE 1j;(P)) and such that 1j;(P) . Inv A zs a Mittag-Leffier set m A. Let fo E Me and ao E 1j;(P) . Inv A. Then there zs an embedding () of M# into A such that ()(Me) c 1j;(P) . Inv A and ()(fo) = ao· Proof The map 1j; : P
---+
()o :
M.A is a framework map, and p(fo)
1-+
p(ao),
Co[fo]
---+
A,
is a well-defined homomorphism which is compatible with 1j; . Take 7"0 < WI such that fo E Mfa. By a routine transfinite recursion using the main extension theorem 1.7.44, we see that there is a family {()" : 7"0 ~ a < WI} such that each ()" : M~ ---+ A is a homomorphism which is compatible with 1j;, such that (}r I M~ = ()" whenever 7"0 ~ a < 7" < WI. and such that ()" I Co[fo] = (}o for 7"0 ~ a < WI· Define (} : M# ---+ A by requiring that () I M~ = (}" (7"0 ~ a < wt). Then (} is a well-defined homomorphism with the required properties (where we recall that M# = U{M~: a < wd). 0 Notes 1.7.46 For the general theory of valuations, see (Jacobson 1980, Chapter 9). Some texts do not require a valuation algebra to be an integral domain. The connection between valuation algebras and abstractly defined valuations is explained in (Dales and Woodin 1996, Chapter 2); the fundamental theorem for the extension of places 1.7.8 is exactly (ibid., 2.13), where other references are given. Theorem 1.7.17 is essentially the same as (ibid., 2.28). Further properties of the fields Rand C can be found in (tbid.). It is a theorem of J. R. Esterle that each real-closed .81 -1/1-field is isomorphic to R; see (ibid.,2.33). Thus, in the theory ZFC + CH, each ultrapower ~N /U, where U is a free ultrafilter on N, is isomorphic to R. Theorem 1.7.19 was first proved in (Mac Lane 1939); see also (Priess-Crampe 1983, 11.5.6) and (Dales and Woodin 1996, 2.15). Most of the results on the extensions of homomorphisms compatible with a given framework map are abstract versions of calculations given in (Esterle 1977, 1978b, 1979a) and (Dales 1979a); 1.7.21 and 1.7.37 were explicitly given in (Esterle 1984b). Kaplansky's isomorphism theorem 1.7.36 was first proved in (Kaplansky 1942); for a different approach to 1.7.36 and 1.7.37, see (Dales and Woodin 1996, Chapter 2). In fact, each algebraically closed .81-1/1-Valuation algebra V is isomorphic to M#; this follows by a development of 1.7.37, and also follows from the fact that each realclosed .81-1/1-field is isomorphic to R. It follows from 1.7.38 that, with CH, each complex, unital integral domain of cardinality c is isomorphic to a sub algebra of (~N /U)O#. The following related, stronger result, which holds in the theory ZFC itself, is proved in (Dales and Woodin 1996, 5.25). Theorem Each complex, non-unital integral domain A of cardinality c is isomorphic to a subalgebra of co/U for some free ultrafilter U (depending on A) on N. 0 Thus the class of integral domains of the form non-unital integral domains of cardinality c.
eo/U
is universal in the class of all
Algebmic foundations
114 1.8 DERIVATIONS
In this section, we shall develop the algebraic theory of derivations from an algebra A into an A-bimodule E and, in particular, into A itself. The role of point derivations, defined in 1.8.7, will be significant later. We shall see that higher point derivations of infinite order on A correspond to homomorphisms from A into q[X]). We shall establish in Theorem 1.8.14 an important connection between homomorphisms and derivations, and this will lead in Theorem 1.8.15 from 1.6.31 to a condition for the extension of a derivation defined on a subalgebra of a given algebraj we shall also construct derivations from the algebra 3' into its algebraic dual space X. These results will be used in §5.6 to construct discontinuous derivations from certain Banach algebras.
Definition 1.8.1 Let A be an algebm, and let E be an A-bimodule. A linear map D : A ----+ E is a derivation zf
D(ab) = a . Db + Da . b (a, bE A) .
(1.8.1)
Equation (1.8.1) is the derivation identity. The set of derivations from A into E is denoted by Zl(A, E)j it is linear subspace of .c(A, E). For example, take x E E, and set 8x (a) = a . x - x . a (a E A) . (1.8.2) Then, for a, b E A, we have 8x (ab)
=a
. (b . x - x . b)
+ (a
. x - x . a) . b = a . 8x (b)
+ 8x (a)
. b,
and so 8x is a derivation. Derivations of this form are termed inner denvations, and an inner derivation 8x is implemented by Xj derivations which are not inner derivations are outer denvations. The set of inner derivations from A to E is a linear subspace Nl (A, E) of Zl (A, E). Clearly the kernel of a derivation from A to E is a subalgebra of A. Let E and F be A-bimodules, and let D E Zl(A, E) and R E A.cA(E, F). Then RoD E Zl(A,F). Proposition 1.8.2 Let A be an algebm, let E be an A-bzmodule, and let D : A ----+ E be a derivation. (i) Suppose that p E J(A). Then p . Dp . p = o. (ii) Suppose that p E J(A) and p . Dp = Dp . p. Then Dp =
o.
(iii) Suppose that a E A and a . Da = Da . a. Then
D(an ) = nan-I. Da
(n E N) .
(iv) Suppose that A and E are unital and that a E Inv A with a· Da = Da· a. Then D(an ) = nan-I. Da (n E Z) . (v) Suppose that (a" : a E Q+.) is a mtional semigroup in A and that a· Da=Da· a. Then
D(a Ct ) = aaCt -
1 •
Da
(a E Q n (1,00)).
Der'lVations
115
Proof (i) Since p = p2, we have Dp = p . Dp + Dp . p, and so immediately p . Dp = p . Dp + P . Dp . p and p . Dp . p = O. (ii) Now Dp
= 2p
. Dp
= 4p
. Dp, and so p . Dp
= 0 and hence
Dp
= O.
(iii) This is an immediate induction. recalling that aD . x = x (x E E).
(iv) By (iii). the result holds for n E N. By (ii), it holds for n = 0, and it now follows from the derivation identity that it also holds for -n E N. (v) Suppose that a = p/q, where p,q E Nand p/q > 1. By (iii), D(aP/ q ) = pa(p-l)/q . D(a 1 / q ) and Da = qa(q-l)/q . D(a l / q ), and so D(aP/ q)
= pa(p/q)-l
.
al-(l/q) .
D(a l / q )
=
(p/q)a(P/q)-l . Da,
o
as required.
The module operations on £(1, E) in the next result were defined in (1.4.14). Proposition 1.8.3 Let A be an algebm, let E be an A-bzmodule, let I be a left ideal in A, and let D : I --+ E be a der'lVatzon. For each a E A, the map Da : x
1--+
D(ax) - a . Dx,
I
--+
is a right I-module homomorphism, and the map a derivation.
1--+
E, Da. A
--+
£(I,E), is a
Proof Note that Dab(X) = (a . Db + Da x b)(x) (a, bE A, x E I). The results are now obtained by immediate calculations. 0 Theorem 1.8.4 (Gr!/lnbrek) Let I be an ideal in a commutative algebm A, let E be an A-module, and let D : I --+ E be a derivation. Then the map
jj : (a, b)
1--+
D(ab) - b . Da,
I x A
--+
E,
is a bilinear map such that: (i) D(a, b)
=a
. Db (a, bE I);
(ii) for each a E 12, the map b 1--+ D(a, b), A
--+
E, zs a derivation.
Proof Certainly D : I x A --+ E is bilinear, and (i) holds. Take aI, a2 E I and bl , b2 E A. Then D(ala2, bl b2)
= D(ala2blb2) - bl b2 . D(ala2) = alb l . D(a2 b2) + a2b2 . D(alb l ) = bl . (al . D(a 2b2) - b2al . Da2)
bl b2 . D(ala2)
+ b2 . = bl
(blal . Da2 + a2 . D(alb l ) - bl . D(ala2)) (D(ala2 b2) - b2 . D(ala2))
.
+ b2 . b1 and this implies (ii).
.
(D(ala2bl) - b1
.
D(ala2))
D(ala2, b2 ) + b2 . D(ala2, bd
, o
Algebraic foundations
116
A particular case of a derivation arises when E is equal to A; we refer to a derivation on A. For example, the map 8b : a
J---+
ab - ba, A
~
(1.8.3)
A ,
is an inner derivation on A for each b E A. Note that, by induction, 8i:(a)
= ~(_l)k(~)bkabn-k
(a,b E A, n E N).
(1.8.4)
The formula in part (i), below, is Leibnzz's identity. (Our convention is that DO is the identity map.) Theorem 1.8.5 Let D be a derivation on an algebra A, and let n E N. (i) For each a, b E A, we have Dn(ab) =
t (~)Dka
. Dn-kb.
k=O
(ii) Suppose that I is an ideal in A. Then D(I) I I is an ideal in AII and {a E I : Dna E I (n E N)} is an ideal in A. (iii) Suppose that a E A with D 2 a = o. Then Dn(a n ) = n! (Da)n. (iv) Suppose that I is an ideal m A, and that al, ... , an E I. Then Dn(al ... an) - n! (Dal) ... (Dan) E I.
(v) Suppose that r.p E q,A and a E M A, the cocycle identity for JJ, on M
--+ (Tt)* is an automorphism on M n , and so there is an invertible matrix P such that (Tt)* = P- 1TP (T E Mn). Set Q = (p*)-l, so that Tt = Q-1T*Q and T = Ttt = (PQ)-lT(PQ) for all T E Mn. We have PQ E 3(M n ), and so, by 1.3.51, P = aP* for some a E C; necessarily, lal = 1, and so, by replacing P by (3P throughout, where (3 -=1= and a(3 = /3, we may suppose that P = P*, and hence that Q* = Q. The eigenvalues of Q are real. Assume that some are positive and some are negative. Then there is a non-zero vector x E cn with [Qx, xJ = (where [ ., .J is the usual inner product in C n ). Let R be the orthogonal projection of C n onto Then RQR = 0, and so Rt R = Q-l RQR = 0, a contradiction because t is proper and R -=1= O. Thus the eigenvalues of Q are all positive, say, and there is an invertible matrix 8 with Q = 8*8. We have Tt = 8- 1(8*)-lT*8*8 and 8Tt 8- 1 = (8T8- 1 )* for T E M n , giving the result. 0
°
°
ex.
Proposition 1.10.9 Let (A, *) be a non-zero, jinzte-dimenszonal, proper *algebra. Then A zs *-zsomorphzc to Mnl 0 ... 0 Mnk for some n1, ... ,nk E N. Proof We first note that the algebra A is semisimple. For take a E rad A. By 1.5.6(iv), (a*a)2m = for some mEN, and so a = 0 because * is a proper involution. Let J be a minimal ideal in A. Then J* is also a minimal ideal in A. Assume that J* -=1= J. Then J* nJ = {O}, and so a*a = 0 (a E J), whence J = 0 because the involution is proper, a contradiction. Thus J* = J. The result now follows from 1.5.9 and 1.10.8. 0
°
We also define Apos
= {a
E
Asa : a(a)
C
JR.+}.
(1.10.6)
In general, little can be said about Apos (save that aa E Apos when a E JR.+ and a E Apos); its relation to A+ in the case where A is a Banach algebra will be clarified in Chapter 3.
146
Algebraic foundations
Definition 1.10.10 Let A be a *-algebra. A projection in A is a self-adjoint zdempotent; the set of projections in A is denoted by Jsa(A). An element zn Jsa(A) n 3(A) is a central projection.
Thus Jsa(A) = J(A) nAsa = {p E A : p2 = p* = p}j we regret using the same word 'projection' as that used in §1.3 for an element of .c(E), but no confusion should arise. We have already (in 1.3.18(ii» defined a partial order j on J(A). Clearly p :5 q in Jsa(A) if and only if qp = p. Proposition 1.10.11 Let A be a *-algebra. (i) Suppose that a ::; b m Asa and that c E A. Then c*ac ::; c*be in Asa.
(ii) S-uppose that p j q in Jsa(A). Then p ::; q in A~a' Now suppose that A is very proper. (iii) Suppose that b*b ::; amAsa and that a
= ac.
Then b = bc.
(iv) Suppose that p, q E J sa (A). Then p j q if and only if p ::; q. Proof (i) This is immediate.
(ii) Since p j q, we have qp = p, and so q - p = (q - p)*(q - p) E A+. Now suppose that A is very proper. Take e to be the identity of A#. (iii) By (i), (e - e)*b*b(e - e) ::; (e - e)*a(e - c) A is very proper.
= 0,
and so b = be because
(iv) Suppose that p ::; q. Then e - q ::; e - p, and so, by (i), (p - pq)(p - pq)* = pee - q)2p::; pee - p)p = O.
Thus p = pq hecause A is very proper. The result now follows from (ii).
0
Definition 1.10.12 Let A and B be *-algebras. A linear map () : A ~ B is positive if ()(A+) C B+; a Imear functzonal A on A is a positive functional if A(A+) C jR+. The set of positive functzonals on A zs denoted by PA. Let A EPA. Then A is a positive trace if (aa*, A) = (a*a, A)
(a E A).
A positive functional A on a unital *-algebra A is a state if (eA, A) = 1: the state space of A, denoted by SA, zs the set of states on A; a state whic~ is also a trace is a tradal state.
Let A and B be *-algebras, and let () : A ~ B be a *-homomorphism. Then () is positive, and () : (Asa, ::;) ~ (Bsa, ::;) is order-preserving. Clearly (PA,+) is a unital subsemigroup of (AX)sa, and (PA,+) is a scmigroup over jR+.. Let::; be the PA-order on PA, so that A ::; J1- if and only if Jl. - A EPA. Then (PA I A2,::;) is a partially ordered set, and (PA,::;) is a partially ordered set if and only if A = A2. In the case where A is unital, the state space SA is a convex subset of AX . Let A EPA. Then clearly Ab : a ~ (b*ab, A) is also in PA for each b E A.
Involutions
147
It follows from (1.10.3) that (ab, A) = (ba, A) (a, b E A) for each positive trace A on A, and so A is a trace in the sense of 1.3.6. For example, the trace map Tr : (O'.ij) --+ L}=l fl'.jj of (1.3.16) satisfies n
Tr(AA*) = Tr(A* A) =
L
(A = (Cl'ii) E Mn),
/nij/2 2: 0
i,j=l and so Tr is a positive trace on the *-algebra Mn in the above sense. Proposition 1.10.13 Let A be a *-algebra. and let A EPA. Then:
(i) (a*,A) = (a,X) (a E A2); (ii) (a*b, A) = (b*a, A) (a, bE A); (iii) (Cauchy Schwarz inequality) /(a*b,A)/2::; (a*a,A)(b*b,A) (a,b E A). Proof (i) Let a E A2. By (1.10.3), a = L}=l O'.jajaj for some 0'.1, ... 'O'. n E 0 such that
m IIxlll
~
IIxl12
~ M
Ilxlll
(x
E
E).
In the case where (E, 11·111) and (E, 11.11 2 ) are Banach spaces, it follows from the closed graph theorem A.3.25 that the norms 11·111 and 11.11 2 are equivalent if and only if the identity map £ : (E, 11.11 1) -+ (E, 11.11 2 ) is continuous.
Introductzon to normed algebras
155
We shall not be concerned with the geometric properties of norms on Banach algebras, but only with the topologies which they define. Thus our interpretation of 'uniqueness-of-norm' is the following. Definition 2.1.6 Let (A, 11·11) be a Banach algebra. Then A has a unique complete norm [unique norm] if each norm with respect to which A is a Banach algebra [normed algebra] is equivalent to the given norm 11·11.
Thus, in the case where A is a Banach algebra with a unique complete norm, the topological and algebraic structures of A are intimately linked. We shall prove in Corollary 2.3.4 that each commutative, semisimple Banach algebra has a unique complete norm, and we shall later describe several classes of Banach algebras with this property. In particular, we shall give three proofs (5.1.6, 5.1.9(iii), and 5.2.28(iv)) of the famous theorem of Johnson that each semisimple Banach algebra (not necessarily commutative) has a unique complete norm. Other Banach algebras with a unique complete norm are certain convolution algebras Ll(W) (see 5.2.18(i)) and Banach algebras of power series (see 5.2.20(ii)). On the other hand, one may easily form Banach algebras which do not have a unique complete norm. For example, let A be a linear space which is a Banach space with respect to two inequivalent norms, and set ab = 0 (a, bE A). Then A is a Banach algebra with respect to each of the two norms. Less trivial examples will be given in §5.1, which is devoted to the question of the uniqueness of complete norms. However, we do not have any tractable characterization of exactly which Banach algebras have a unique complete norm. For example, it is not known whether or not each commutative Banach algebra which is an integral domain has a unique complete norm; see Question 5.1.B and 5.3.13. The property of having a unique norm is a stringent condition. Of course, finite-dimensional algebras satisfy this condition; infinite-dimensional examples, such as the Banach algebras B(E) and K(E) for certain Banach spaces E, will be presented in 5.4 12. The following remark will be useful in determining when a Banach algebra has a unique norm. Proposition 2.1.7 Let (A, 11·11) be a Banach algebra such that, for each algebra norm 111·111 on A, there exists C > 0 with lIall ::; C IlIalil (a E A). Then A has a umque norm if and only if each homomorph%sm from A mto a Banach algebra is contmuous. Proof Suppose that () is a discontinuous homomorphism from A into a Banach algebra, and set IIlalil = max{lIall, 1I()(a)lI} (a E A). Then 111·111 is an algebra norm on A, and III· III is not equivalent to II . II· Conversely, suppose that III· III is an algebra norm on A not equivalent to 11·11. By hypothesis, the identity map from A into the completion of (A, III· liD is a discontinuous homomorphism. 0
The automatic continuity problem for derivations is the following. Let A be a Banach algebra. Under what conditions on A is it true that all derivations from A into some or all Banach A-bimodules (see 2.6.1) are automatically continuous? For example, all derivations from a C* -algebra A into a Banach A-bimodule
Banach and topological algebras
156
are continuous; see 5.3.7. In fact, we shall consider more general maps than derivations; these are the intertwining maps of Definition 2.7.1. We shall see in 2.7.5 that, if all homomorphisms from a Banach algebra A are continuous, then all derivations (and, indeed, by 2.7.7, all intertwining maps) from A are also continuous. The converse is not true; for example, it will be shown in 5.3.7 that all intertwining maps from each C(O) are continuous, but, with CR, there are discontinuous homomorphisms from C(O) whenever 0 is infinite. Definition 2.1.8 A normed algebra (A, II· II) is unital if A has an identity eA and IleAIi = 1.
Let (A,p) be a seminormed algebra. Then the formula
p(ael>
+ a)
= lal
+ pea)
(a E C, a E A)
defines an algebra seminorm on AP; it is a [complete] norm if and only if p is a [complete) norm on A. Thus, in the case where A is a Banach algebra, so are AI> and A#. Proposition 2.1.9 Let A be an algebra, let 11·11 be a norm such that (A, 11·11) is a Banach space and multzphcation on A is separately continuous, and let S be a bounded semigroup in (A, . ). Then there zs a norm III . III equivalent to II . II on A such that (A, III· liD is a Banach algebra and IIlslll :s; 1 (s E S). Proof By A.3.39, there exists C ~ 1 such that lIabli :s; C Ilalillbli (a, b E A). Also there exists M ~ 1 such that IIsli :s; M (s E S). Set pea) = sup{lIall, IIsali : s E S} (a E A), so that p is a norm on A. Then lIall :s; pea) :s; CM lIall (a E A), and so p is equivalent to 11·11. Clearly:
p(sa) p(ab) p( s)
:s; pea) :s; Cp(a)p(b) :s; M
(a E A, s E S); (a, bE A); (s E S) .
Define IIlalll
= sup{p(aa. + ab) : Cl'
E
C, bE A, M lal
+ pCb) :s; I}
(a E A).
Then 111·111 is a norm on A and p(a)/M :s; IIlalll :s; Cp(a) , and so 111·111 is equivalent to p, and hence to II . II. For a, b E A, we have IIlablli
:s; sup{p(aab + abe) : a E C, c E A, p(ab + be) :s; IIlblll} :s; sup{p(ad) : d E A, p(d) :s; I} IIlblll :s; Illallllllblil ,
and so (A, 111·111) is a Banach algebra. Finally, take s E S. For a E C and b E A with M lal + pCb) :s; 1, we have peas + sb) S; 1, and so IIlslll s; 1. 0 In particular, we may always suppose that a Banach algebra with an identity is a unital Banach algebra.
Corollary 2.1.10 Let A be a unital Banach algebra, and let G be a bounded subgroup of (Inv A, . ). Then there is a Banach algebra norm III . Ilion A equivalent to the given norm such that IIlalil = 1 (a E G). 0
Introduction to norrned algebras
157
There is a further renorming result that is sometimes useful. Let (A, 11·11) be a commutative, unital normed algebra, and let 8 be a unital subsemigroup of (A, . ) with 8 c A[k], where k > O. Define Illalll
= inf {~lla.11 : a =
~aisi'
Sl, ... ,Sm
E 8, mEN} .
(2.1.3)
Then (A, 111·111) is a unital normed algebra, and Iiall jk ::::; Illalll ::::; Iiall (a Also IIlslll ::::; 1 (s E 8).
E
A).
Definition 2.1.11 Let A be a norrned algebra, let a E A-, and let K be a subsemigroup oj (re, +) with 1 E K. A K -semzgroup (a( : ( E K) zs continuous [bounded] zJ the map ( f---+ a(, K ---+ A, is continuous [bounded]. In the case where K zs an open set in re, the semzgroup zs analytic zJ thzs map is analytic onK. Proposition 2.1.12 Let A be a normed algebra, let K be an open semzgroup m (!C. +) with 1 E K, and let (a,( : ( E K) be an analytic semigroup in A. Then:
(i) a(A = aA (ii) (a( + I: (
«( E K); E
K) is an analytzc semigroup in A/I Jor each closed ideal I
in A. Proof (i) Fix (0 E K, and take A E A' such that A I a(o A = O. For each b E A, the function Fb : ( f---+ (a(b, A), K ---+ re, is analytic. For each ( E (o+K, we have a(b E a(oA, and so Fb«() = O. Hence Fb = 0 on K, and so A I a(A = 0 E K). It follows from the Hahn-Banach theorem A.3.18(i) that a("A C a(oA E K). The result follows. (ii) This follows immediately from A.3.76. 0
«( «(
Many specific examples of Banach algebras will be extensively discussed in this book. At this stage, we wish to list a few well-known examples to establish notation and to allow us to make remarks as we progress. All the examples will be considered more fully later. Examples 2.1.13 (i) Let 8 be a non-empty set, and let £00(8) denote the Het of complex-valued, bounded functions on 8. Then £00(8) is a unital sub algebra of res (with the pointwise operations of (1.3.7». We define
IJls
= sup{IJ(s)1 : s E 8} (f E £'>0(8».
(2.1.4)
Then I· Is is an algebra norm on loo(8), called the unzJorm norm on 8, and (£00(8), I· Is) is a commutative, unital Banach algebra. Let X be a non-empty topological space. Then C(X), Cb(X), and CoCX) are subalgebras of rex. Also Cb(X) = £oo(X) n C(X) and Co(X) are commutative Banach algebras which are closed subalgebras of (£oo(X), I· Ix ); Coo(X) is a dense ideal in Co(X). The algebra Co(X) is unital if and only if X is compact. As a special case of the algebras Co(X), note that (eo, I· IN) is a commutative Banach algebra. The subspace C(X, 1R) of C(X) is a closed, real subalgebra.
158
Banach and topological algebms
For a study of the Banach algebras Cb(X) for a completely regular topological space X and of Co(O) for a non-empty, locally compact space 0, see §4.2. We shall see that algebraic properties of Co(O) correspond to topological properties of O. For example, it is clear that J(Co(O)) consists of the functions XK for K a compact and open subset of O.
(ii) Let A (iD) = {J E C (iD)
: f IIDl is analytic} ,
e : Izl < I} is the open unit disc, as in Appendix 2. Then is a commutative, unital Banach algebra; it is called the disc algebm (see 4.3.11). Each f E A(iD) has a Taylor expansion:
where
IDl
= {z E
(A(iD) , I· Ii»)
f(z) =
~
L.J
n=O
f(n) (0) zn n!
(z
E
iD) .
However, this series need not converge in the uniform norm on iD. Let f E A(iD). For each r < 1, set fr(z) = f(rz) (z E iD). Then fr --> fin A(iD) as r --> 1-, and each such fr has a uniformly convergent Taylor series. Thus the restrictions to iD of the polynomials are dense in A (iD) . We denote by A+ (iD) the set of functions f = 2::=0 QnZ n in A(iD) which have an absolutely convergent (and hence uniformly convergent) Taylor expansion on iD. It is easy to see that A+ (iD) is a commutative, unital Banach algebra with respect to the pointwise operations and the norm II . 111' where
it is called the algebra of absolutely convergent Taylor series. We exhibit some specific functions in A+(iD). Let (3 E jR+. \ N. We claim that 00
(1 - z)13 = 1 +
L Q13,nZn
(z E iD)
,
(2.1.5)
n=l
where
IQ13,nl '" n- 1 -{3 as n --> 00.
Indeed, we have
Q{3,n = (-I)n(3«(3 - 1) ... «(3 - n
+ 1)/n!
= (n - 1 - (3)(n - 2 - (3) ... (1 - (3)( -(3)/n!
= r(n - (3)/f(n + l)f(-(3) = f(n - (3)/nf(n)f( -(3), where f is the gamma function. By (A.2.3), f(n - (3)(n - (3)13 /f(n) --> 1 as n --> 00, and so Q13,nn1+13 --> f( _(3)-1 as n --> 00. Thus (1- Z)13 E A + (iD). Note that, in the case where 0 < (3 < 1, we have Q13,n < 0 (n EN). Clearly «1- Z){3 : (3 E JR+.) is a continuous real semigroup in A+(iD). (iii) Let E be a non-zero Banach space, and let 8(E) be the algebra of all bounded linear operators on E. Then 8(E) is an algebra with identity IE, and it is a Banach algebra with respect to the operator norm II . II. An idempotent P in 8(E) is a projection onto the closed subspace P(E) of E. In the case where E = en, we have 8(E) = C(E) ~ Mn.
Introduction to normed algebms
159
Let A be a finite-dimensional algebra. Then there is a norm II . II on the finitedimensional space A#, and the map a 1-+ La, A -+ B(A#), is an embedding. Set Illalil = IILall (a E A). Then (A, 111·111) is a Banach algebra. (iv) Let 8 be a non-empty set, let w : 8 -+ R+ e be a function, and define £1(8, w) = {I =
L l(s)08 E C
S :
II/lIw =
sES
L I/(s)1 w(s) < oo}.
(2.1.6)
sES
Then (P1(8, w), II·IU is a Banach space; we write £1(8) and 11·111 in the case where w = 1, and we write £1 for £l(N), as in Appendix 3. The space £1(8, w) is separable if and only if 8 is countable. If the function w is bounded below, then ((1(8, w), ., 1I·lI w ) is a commutative Banach algebra with respect to the pointwise product . on 8; this algebra is unital if and only if 8 is finite. (v) Now suppose that 8 is a semi group, not necessarily abelian. A weight on 8 is a map w : 8 -+ R+e such that
west) ::; w(s)w(t)
(s, t E 8) j
we also require that w(e 8 ) = 1 in the case where 8 is unital. It follows from A.1.26(iii) that limn----+oow(sn)l/n exists for each s E 8. A weight on Z+ is a weight sequence. For example, w : s 1-+ exp(-s2) is a weight on (R+,+), and w : p/q 1-+ q, where p and q are coprime in N, is a weight on (tQ+e, +). Here is a recipe for constructing rather strange weights on tQ+.. Let U be a subset of tQ+e. Then a function 1j; : U -+ R is subadditzve if 1j;(u) ::; E;=l1j;(Uj) whenever u, U1, ... ,Un E U and U = Ul + ... + Un. Let 1j; : U -+ R be subadditive, and take v E Q+. \ U. Then there is a subadditive function ;p : U U {v} -+ R such that 7jj I U = 1j;. Now let {Pit P2, ... } be an enumeration of the prime numbers (with Pn < Pn+1 (n EN)), and let Uo = {1/Pn : n EN}. Suppose that u, UIt ... , Un E Uo and U = U1 + ... + Un. Then necessarily n = 1, and so every function 1j; : Uo -+ R is subadditive. Each such 1j; has an extension to a subadditive function :;j; on Q+e j exp:;j; is then a weight on tQ+e. By starting with the function which takes the value -n at I/Pn, we obtain a weight w on Q+e such that liminfs----+o+w(s) = O. Let w be weight on a semigroup 8. We define a product *, called convolution, in £1 (8, w) such that Os * Ot = Ost (s, t E 8). First note that, for I, 9 E £ 1(8, w), we have
~ L~t I/(r)llg(s)l} wet) ::; ~ I/(r)1 w(r) ~ Ig(s)1 w(s) = II/l1w IIgllw ' and so £1(8, w) is a Banach algebra with respect to the product defined by (f
* g)(t) =
L
I(r)g(s)
(t
E
8).
(2.1.7)
rs=t
(If there are no elements r, s E 8 with rs = t, set (f * g)(t) = 0.) The algebra (£1(8, w), *, 1I·ll w ) is the weighted semigroup algebm for the weight w on 8j in the case where 8 has an identity e, Oe is the identity of the algebra. The algebra £1(8, w) is commutative if and only if S is abelian.
Banach and topological algebras
160
The algebraic semigroup algebra coo(S), described in 1.3.7(iii), is a dense subalgebra of each weighted semigroup algebra £l(S, w). For example, by taking S = z+n, we obtain subalgebras of the algebra ~n of formal power series in n indeterminates by identifying the element O(Tl, .,Tn) with the monomial X[l '" X~n for (r1,"" rn) E z+n; these algebras will be studied in §4.6. In particular, the constant function 1 is a weight on S; the corresponding algebra, denoted by £ 1 (S) = (£ 1(S), *, II . 111)' is the (discrete) semzgroup algebra of S. It is easy to see, for example, that (£l(Z+), *, 11'//1) is isometrically isomorphic to (A + (ii)) , . , 1/ . "1)' We refer to the weighted group algebra £ 1(G, w) in the case where G is a group; in this case, the formula for the convolution product of f,g E fl(G,w) is (J
* g)(t) =
L
f(s- l t)g(s)
(t E S).
(2.1.8)
sEG
In particular, £ 1 (G) is the group algebra of the group G.
o
We begin by showing that certain algebras are not normable. Proposition 2.1.14 (i) Let X be a non-empty topological space. Then C(X) is normable if and only if X is pseudocompact. (ii) Let E be a non-zero linear space. Then C(E) is seminormable if and only if E zs finite-dimensional. (iii) Let A be a unital algebra containing elements a, b such that ab - ba = eA. Then A is not seminormable. Proof (i) Suppose that X is pseudocompact. Then C(X) = Cb(X) is a Banach algebra with respect to the uniform norm / . /x on X. Conversely, suppose that X is not pseudocompact, and take f E C(X) with f(X) unbounded; we may suppose that f(X) C ~+. Inductively choose a sequence (xn) in X such that f(xn+l)
> f(xn) + 2 (n E N),
and set In = [J(Xn) - 1, f(xn) + 1] (n EN). Then the pairwise disjoint, closed intervals In are such that there exist h E C(JR) with h(t) = n (t E In) for each n E N, and (gn) C C(JR) with 9n(J(Xn» = 1 and supp gn C In for each n E N. Set a = h 0 f and bn = gn 0 f (n EN). Then a E C(X), {bn : n E N} c C(X)-, and abn = nbn (n EN). By 2.1.3(i), C(X) is not normable. (ii) Set 2l = C(E). Suppose that E is finite-dimensional. Then 2l = B(E) is a Banach algebra with respect to the operator norm. Conversely, suppose that E is infinite-dimensional. Then there is a sequence (En) of subspaces of E such that E = O{En : n E N} and En ~ E (n EN). For n E N, take Pn : E --+ En to be a projection and take Qn E 2l such that Qn / En : En --+ E is a linear isomorphism and Qn / Em = 0 (m =1= n). Then the map QnPn : E --+ E is an epimorphism, and hence there exists Un E 2l with QnPnUn = IE· Thus we have !2lPn !2l =!2l. Define A E !2l by requiring that A / En = nPn (n EN). Then APn = nPn (n EN), and so, by 2.1.3(i), !2l is not seminormable.
Introduction to normed algebras
161
(iii) By an immediate induction, anb - ban = na n -
(n E N) .
1
(2.1.9)
Assume that p is a non-zero algebra seminorm on A. It cannot be that peak) = 0 for any kEN, for, by (2.1.9), this would imply that p(eA) = 0 and hence that p = O. From (2.1.9), np(a n - 1 ) ~ 2p(a)p(b)p(a n - 1 ) (n EN), and so n ~ 2p(a)p(b) for each n E N, a contradiction. 0 It is clear that each algebra A with A2 = 0 is normable: every norm on A is an algebra norm. However, we now !ihow that this result does not extend to the case of algebras A such that A3 = o. Proposition 2.1.15 (Dales) (i) There is a commutative algebra A with A3 = 0 such that A is not normable.
(ii) There zs a commutatwe algebra A with a countable basis such that A is not normable. Proof (i) Take A to be the algebra which, as a linear space, has as a basis the set {e s : S E [0, I)}, and for which the product is defined by the formula: _ { eo/ Is - tl eset 0
(s, t E (0,1), s (otherWlse .) .
=I t) ,
It is clear that the product of any three elements of A is 0, and so A is associative and A3 = O. Assume that there is an algebra norm /I . /I on A. Then
0< IIeoil ~ Is - tl lies II IIetil
(s,t E (0,1), s
=I t).
For n E N, set Un = {s E (0,1) : IIesil > n}. If s E (0,1) and t E (0,1) with Is - tl < IIeo/i /n IIesll, then t E Un, and so each Un is open and den!ie in (0,1). By the category theorem A.1.21, Un =10. Take s E Un. Then IIesil > n for each n E N, a contradiction. Thus A is not normable. (ii) Let S = Ul:+)<w be the free abelian semigroup on countably many generators, as in 1.2.2(ii), and let A = coo(S), the algebraic semigroup algebra, as in 1.3.7(iii). We regard the elements of A as polynomials with zero constant term in the countable family {Xn : n EN}. Let I be the ideal in A generated by the elements X 1 X n - nXn for n 2 2, and set 2{ = A/I and bn = Xn + I (n EN), so that 2{ is a commutative algebra with the countable basis ibn : n EN}. We claim that bn =I 0 for each n 2 2. For assume that Xn E I for some n 2 2. Then there exist kEN and PI, ... , Pk E A with Xn = L~=2(XIXj - jXj)Pj. Evaluate this expression at the point «(J) where (1 = n, (n = 1, and (j = 0 otherwise; we obtain a contradiction, and so the claim holds. Clearly b1bn = nbn (n EN), and so, by 2.1.3(i), 2{ is not normable. 0
n:=1
n:=1
Let A be a unital normed algebra. Then the map a ....... La, A ~ B(A), is an isometric, unital homomorphism which identifies A as a closed subalgebra of 8(A) such that fA EA.
162
Banach and topological algebras
Proposition 2.1.16 Let A be a unital normed algebra. Then eA is an extreme point of A[l]' Proof It suffices to show that IE E ex8(E)[1] for each Banach space E. Set a = a(E',E) and K = Efl]' so that K is convex and compact in (E',a)j
by the KreIn-Mil'man theorem A.3.30(i), (exK) (0-) = K. Take T E 8(E)[1] with IIIE ± Til ~ 1. Then IIIEI ± T'" ~ 1 in 8(E'). Let A E exK, and set Al = A + T'(A) and A2 = A - T'(A). Then AI. A2 E Efl] and --(0-)
2A = Al + A2, and so Al = A2 = A. Thus T'(A) = O. Since (exK) = K and T' is continuous on (E', a), it follows that T' = O. Thus T = O. Now suppose that 2IE = R + S, where R, S E 8(E)[1], and set T = IE - R. Then IIIE ± Til ~ 1, and so T = 0, R = S = IE, and IE E exB(E)[l]' 0 Definition 2.1.17 Let (A, II "D be a normed algebra. Then a Banach algebra extension of A is a Banach algebra (B, ",, III) such that there is an isometric embedding of (A, 11·11) in (B, ",, III).
In this case we regard (A, II· II) as a closed subalgebra of (B, ",, "'). We now give some ways of building new Banach algebras from old ones. Examples 2.1.18 (i) Let A and B be Banach algebras, let I be a closed ideal in B, and let () : A - B be a continuous homomorphism. Set 21 = A Eel I as a Banach space, and define
(aI, bl )(a2, b2) = (ala2, bl b2 + ()(al)b 2 + bl ()(a2)
(aI, a2 E A, bI, b2 E I). (2.1.10) Then 21 is a Banach algebra extension of Aj 21 is commutative if both A and B are commutative. (ii) Let A be a unital Banach algebra, and set 21 = MIn(A), the algebra of n x n matrices over A. Then 21 is easily checked to be a unital Banach algebra with respect to the norm given by
II(aij)'1 = max{IIailil
+ ... + IIainil
: i E N n}
and identity" = (oi,jeA); 21 is a Banach algebra extension of A with respect to the map a 1-+ (oi,ja), A - 21. In particular, the full matrix algebra MIn is a Banach algebra with respect to the norm //(ai)// = max{laill + ... + lainl : i E Nn }. The algebra MIn is also a Banach algebra with respect to the norm "'lI p ' formed by regarding MIn as an algebra of (bounded, linear) operators on (en, "./1 ) for p E [1, oo)j the case p = 2 gives the C*-norm on MIn, to be discussed in §3.~. (iii) Let {A-y : I E r} be a family of Banach algebras, and take p E [1,00). Then the Banach spaces fP(r,A-y), co(r,A-y), and fOO(r,A-y) (c/. Appendix 3) are Banach algebras for the coordinatewise product. Clearly co(r, A-y) is a closed ideal in .e 00 (r, A-y ) j the quotient .e 00 (r, A-y ) / Co (r , A-y) is sometimes useful.
163
Introduction to normed algebras
(iv) Let A be a Banach algebra, and let n be a non-empty, locally compact space. Then the Banach space (eb(n, A), II ·11) is a Banach algebra with respect to the pointwise product, and eo(n, A) is a closed ideal therein; here,
IIFII = sup{IIF(x) II : x E n} (F E eben, A)). (v) Let (A, II· II) be a unital Banach algebra, let n E N, and let W
be a weight
on z+n. Set !!w
=
{a
=
LarX r E A[[X1, ... ,Xn]] : IIall w = L
IIarilwr
k)}. Then I is a closed ideal in !!w, and !!w/ I is a Banach algebra that we identify with {~lrl:5k arX r }; !!w/ I is also a Banach algebra extension of A with respect to the norm given by (vi) Let (A,
\\~Irl~k arX r \\ = ~Irl~k IIarll·
11·11)
be commutative, unital Banach algebra, and let p = ao
+ a1X + ... + anX n + X n+ 1
be a monic polynomial in A[X]. Set Ap = A[Xl/pA[X], so that Ap is a commutative, unital algebra. Each elemcnt of Ap has a unique expression in the form ~~=o biXi, where bo, b1 , ... , bn E A. Choose t > 0 such that t n +1 ;::: ~;=o Ilajll tj, and then consider
!!t = {a E LarX r E A[[X]] : lIali t = L
liar II t r < oc} ,
so that, as in (v), (!!t, II . lit) is a Banach algebra. It is easy to check that IIrp + slit;::: IIsli t (r, s E !!t), and so p!!t is a closed ideal in !!t. Also, wc have Ap 9:! !!t/p!!t. and so Ap is a commutative, unital Banach algebra which is a Banach algebra extension of A; Ap is called the Arens-Hoffman extension of A by p. In the case where p = Xk+1, we identify Ap with
{~~=o arxr }.
0
Definition 2.1.19 Let A be a normed algebra wzth unzt sphere S, and let a E A. Then a zs a topological divisor of 0 ifinf{lIabll + llball : bE S} = o. Theorem 2.1.20 (Arens) Let A be a unital, commutative Banach algebra, and let a E A. Then there is a Banach algebra extension!! of A such that a E Inv!! if and only if a is not a topological divisor of o. Proof The unit sphere of A is denoted by S. Suppose that a is a topological divisor of O. Then there exists a sequence (b n ) in S such that abn ---+ 0 as n ---+ 00. Assume that!! is a Banach algebra extension of A such that there exists c E !! with ac = e21. Then 1 = IIbnll ::; Ilclillabnll ---+ 0 as n ---+ 00, a contradiction. Now suppose that a is not a topological divisor of 0, say lIabll 2': 1 (b E S). Set B = {b = E'f=oa,Xj E A[[X]] : IIbll = E'f=o lIajll < oo}, as in 2.1.18(v), so that B is a unital Banach algebra which is an extension of A. Set J = (eA - aX)B,
Banach and topological algebras
164
a closed ideal in B, and set 21 = B/J. We claim that the natural embedding c 1---+ c + J, A --+ 21, is an isometry. Indeed, take p = Ej=o ajXj E B. Then (eA - aX)pl/ = I/a - aol/ + Iial - aaoll + ... + Ilan - aan-lll + Ilanll I/all-Ilaol/ + Ilaaoll-llalll + ... + lIaan-III-llanll + Ilaanil ~ lIall because Ilaaj II ~ Ilaj II (j E Z;i) The claim follows from this inequality. Clearly a + J E Inv 21, and so we may regard 21 as the required extension. 0
I/a -
~
Proposition 2.1.21 Let kEN, and let 11·11 be an algebra norm on M k • Then there is an automorphi,sm r : Mk --+ Mk such that (2.1.11) Proof Let the linear subspace
E={(~:~·::~): QI, ... ,QkEC} ak
0 ···0
of Mk have the relative norm 11·11, and let 111·111 denote the corresponding operator norm on £(E). Clearly Mk acts on E by left multiplication, and we have an isomorphism Mk ~ £(E) in this way; certainly IIITIII :::; IITII (T E Mk)' By A.3.5, there is a linear homeomorphism L : (C k ,II'II) --+ (C k ,II'112) with l v'k. Define an isomorphism
IILlillc l1 : :;
r :T
1---+
LoT
0
L-l,
£(E)
--+
£((C k , 11·lb)).
By identifying Mk with £(E) and £((C k , 11·lb)), respectively. we obtain a.n automorphism r : Mk --+ Mk which satisfies (2.1.11). 0 Let AI, ... , An be algebras. The tensor product ®~l Ai was defined in 1.3.11; the projective tensor norm II· 1/11" and the projective tensor product are defined in A.3.65 and A.3.67, respectively. Theorem 2.1.22 Let AI"'" An be normed algebras. The norm
algebra
on the
®~l Ai zs an algebra norm, and (®~=l Ai, 11·1111") is a Banach algebra.
Proof Let a = ®~=l Ai' Then r
Ila bll
1/ ·1I 7r
7r :::;
E;=l al,j I8l ... 18l an,j and b= E~=l bl,k I8l ... I8l bn,k belong to 8
r
8
L L al,Jbl,k I8l ... I8l an,Jbn,k : :; L L j=lk=l
: ; (t
I/al,jll" 'lI an ,jl/)
I/al,jbl,kli" 'lIan,jbn,kll
j=lk=l
(1; I/b1,kll" 'lI bn,kll),
where we are using A.3.66, and so
lIabll 7r :::; Ilall7r IIbll ....
The result follows.
0
Introduction to normed algebms
The Banach algebra
165
(®~=1 Ai, II . lin) is the proJectzve~ensor product of the
normed algebras AI,"" An; it is sometimes denoted by ®nAi' A particularly important example is given in the next definition. Definition 2.1.23 Let A be a normed algebr·a. The enveloping algebra of A is the Banach algebm Examples 2.1.24 Let 8 and T be non-empty sets, and consider the Banach space [1(8) 0£1(T). The map (8 s , 8 t ) f-+ 8(s.t) defines a continuous bilinear map from £1(8) x £l(T) into £1(8 x T), and, by A.3.69, there is a unique continuous linear map W : £1(8) 0£1(T) -+ £1(8 x T) with w(os ® 8t ) = 8(s.t) (s, t E 8). Clearly W is an isometric isomorphism. (i) Let £1(8) have the pointwise product· of 2.1.13(iv). Then the algebra [1(8) 0[1(8) is the Banach space ([1(8 x 8), 11·111) with the pointwise product. (ii) Now suppose that 8 is a unital semigroup: (£1(8), *,11,111) is the semigroup algebra of 8. as in 2.1.13(v). Then the enveloping algebra of £1(8) is the semigroup algebra of 8 x 8, where the semigroup operation on 8 x 8 is given by (s, t)(u. v)
= (su, Vi) (.'I. t. U. v
E
8).
o
Let (A. II· II) be a normed algebra, and let It, ... .In be idealr:; in A. Then It ... In is an ideal in A; we transfer the projective norm from It ® ... ® In to 11 ... In. so that, for a E II ... In, we have
lIall.
~ in!
{t,
Ila,.,II·· ·1Ia"., II ' a
~
t,
a,,'" anJ with a;J E I; } .
(2.1.12) Clearly II· lin is an algebra norm on It· .. In with Iiall :::; Iialin (a EIt ... In); the norm II· lin is again called the projective norm. In particular. we may consider the projective norm II· lin on A2; note that
lIablln :::; lIalln Ilbll
E A 2 , bE A). related to II· lin' First (a
There is a norm which is closely 'Banach' version of the induced product map 7fA of 1.9.19.
(2.1.13)
we introduce a
Definition 2.1.25 Let A be a Banach algebm. Then the contm'ILOllS lmear map 7rA: A0A -+ A surh that 7fA(a®b) = mA(a,b) = ab (a,b E A) is the projective induced product map. and In = ker7fA. The existence of 7fA follows from A.3.69; In is a closed linear subspace of A0A, and I" is a left ideal in the algebra A0Aop, (~alled the projectzve diagonal ideal. The quotient norm on the image 7fA(A0A) e:: (A0A)jln is denoted by 111·lIl n, so that
IIlalil. ~ in!
{t,
lIa, II 11M '
a~ t, a,b,} (a ~A(A®A)) E
.
(2.1.14)
Banach and topologzcal algebras
166 We have
(2.1.15) Suppose that A is unital. Then clearly 7rA is an admissible epimorphism, is a norm on A, and
111" = lin {a ® b - e A
(g
111·11111"
(2.1.16)
ab : a, b E A} .
Definition 2.1.26 A normed algebra A has the S-property [7r-property] if there is a constant C > 0 8uch that Ilal 11" ::; C lIall [lllalll11" ::; C Iiall] (a E A2).
Clearly, if A has the S-property, then A has the 7r-property, and a Banach algebra A has the 7r-property if and only if 7rA(A0A) = A2, so that A has the 7r-property whenever A factors weakly. An example of a commutative, separable Banach algebra which has the 7r-property, but which does not have the S-property, will be given in 4.1.43. Proposition 2.1.27 Let A be a Banach algebra such that A is a finitely generated left ideal in A #. Then A has the S -property. Proof Take all"" an E A[1] with A = L;=l A#aj, and set E = A(n). Define T : (b i , ... , bn ) ~ L~~=l bjaj, E --+ A, and set 1 = T(E) = Lj=l Aaj. Then T E 8(E,A), I has finite codimension in A, and I C A2. By A.3.24, I is closed in A, and so, by the open mapping theorem A.3.23, there exists C > 0 such that, for each eEl, there exist bl, ... ,bn E A with L;=lllbjil ::; Cllell and e = Ej=l bjaj. Thus lIell11" ::; C llell (e E 1). By A.3.42(i), the identity map (A2,1I'11) --+ (A2, 11·1111") is continuous, and so A has the S-property. 0 Proposition 2.1.28 Let (A, 11·11) be a norm,ed algebra. Then there is an algebra norm 111·111 on A such that IIlalil = II a 1111" (a E A2) and lIall ::; IlIalil (a E A). Proof There is a linear subspace E of A such that A = A 2 0 E. For an element a = (b,x) E A2 0 E, define IIlalil = IIbll11" + IIxli. Then it is immediately checked that III . III has the required properties. 0
We conclude this first section with a result that is the foundation stone of automatic continuity theory for Banach algebras. Let A be a Harmed algebra, and let a E A. Since lIam+nll ::; lIa"'lIlIanll for m, n E N, it follows from A.1.26(iii) that lim
n-+oo
lIanll l /n
= inf {lIanlll/n : n E
N} .
(2.1.17)
Theorem 2.1.29 Let A be a Banach algebra. (i) Suppose that a E A with infnEN
lIanll l / n < 1.
Then a E q-Inv A wzth
00
a = - La q
k .
k=l
FUrther, o-(a) C ])(0; IIall)· In the case where A is unital and (1
+ lIall)-1 ::;
lIall < 1,
lI(eA - a)-III ::; (1- 110.11)-1.
we have (2.1.18)
Topologzcal algebms
167
(ii) Let r.p be a chamcter on a Banach algebm A. Then cp is continuous. and In the case wher-e A zs unital, IIcpll = r.p(eA) = 1. (iii) Let a E A. Then Ip(z)1 ::; IIp(a) II (z E O'(a), p E qX]).
IIcpll ::; 1.
Proof (i) Since limn-+oc.lla n Il 1 / n < 1, the series E:=l ak converges, say to -b. Clearly aob = boa = 0, and so b = aq • For Izl > lIall, we have zeA -a E Inv A#, and so v(a) < 14 Thus O'(a) C Jl)l(0; Ilall). The estimate (2.1.18) is immediate. (ii) Assume towards a contradiction that there exists a E A with lIall < 1 and Ir.p(a) I 2: 1, and set b = a/cp(a). Then IIbll < 1, and so, by (i), there exists c E A with b + c = be. But cp(b) = 1, and so 1 + cp(e) = r.p(c), a contradiction. Thus IIcpll ::; 1. If A is unital, then r.p(eA) = 1 = IlcAII, and so Ilcpll = 1. (iii) This follows from 1.6.11(i) and (i). 0 Notes 2.1.30 The elementary theory of Banach algebras can be found in many texts; for example, (Bonsall and Duncan 1973), (Gel'fand et al. 1964), (Helemskii 1993), (Naimark 1964). (Palmer 1994), (Rickart 1960), (Rudin 1973), and (Zelazko 1973). Questions of automatic continuity were stressed at It very early stage in the history of the subject; for example, Rickart focused on whether a semisimple Banach algebra has a unique complete norm in (1960, II, §5). An account of the historical origins of Banach algebra theory is given in (Palmer 1994). The semigroup algebra 1 (8) (for 8 abelian) is discussed in (Hewitt and Zuck('rmann 1956) and (Barnes and Duncan 1975). Propositions 2.1.14. (i) and (ii), are from (Meyer 1995). ThE' Weyl algebra is the algebra of polynomials in two variables X, Y such that XY - YX = 1; by 2.1.14(iii), the Weyl algebra is not normable. Proposition 2.1.15 is from (Dales 1981a). where it is also proved that each commutative, radical algebra with a countable basis is normable. The Arens-Hoffman extensions of 2.1.18(vi) were introduced in (Arens and Hoffman 1956). It is shown in (ibid.) that
e
0 such that p(ab) ~ Cq(a)q(b) (a, bE A). Let A be a topological algebra. Then AOP is a topological algebra; if A belongs to a class of these algebras that we have defined, then AOP belongs to the same class. Clearly the closure of a sub algebra of A is a subalgebra, and the closure of an ideal is an ideal. Further, the closure of a commutative subset of A is commutative, so that maximal commutative subalgebras of A are closed. For each subset S of A, the commutant se is closed; in particular, the centre 3(A) of A is closed. The subalgebra pAp is closed in A for each p E J(A). Finally, let I be a closed ideal in A. Then AI I is a topological algebra; it is an (F)-algebra, a locally convex algebra, a normed algebra or a Banach algebra in the case where A has the corresponding property.
Banach and topological algebms
170
The following result is a special case of A.3.39.
Proposition 2.2.6 Let A be an algebm which Z8 a~m an (F)-8pace 8uch that multzplicatwn i8 8epamtely continuou8. Then A is an (F)-algebm. 0 Let A be a topological algebra. Then Ab is a topological algebra for the product. topology, and Ab is complete or locally convex or metrizahle if and only if A has the corresponding property.
Definition 2.2.7 Let A be a topological algebm, and let S be a subset of A. Then: (i) the .subalgebm of A polynomially generated by S is Co[S]: (ii) the subalgebm of A# unit ally polYllomially generated by S is qS]; (iii) the subalgebm of A# rationally generated by S is qS). An element a E A is a polynomial generator of A [rational generator of A#] if Co [a] = A [q a) = A #]. The algebm A is singly generated if it has a polynomial genemtor. Let A be a topological algebra and I be a closed ideal in A. Then, as in 2.2.3. WE' have
OA(I) = {a E I : aA =
I}
and
O(A) = {a E A : aA = A} .
Recall from (1.3.20) that Ia = n{ an A : n E N} for an elE'ment a of an algebra A. Now let A be a topological algebra. and take a E O(A). Then we define
ia=n{an.O(A):nEN}. Clearly ia is a subsemigroup of la, and
(2.2.1)
fa . O(A) cia.
Theorem 2.2.8 Let A be a commutative (F)-algebm, and let a E O(A). (i) Then: Ia is dense in A.; the map Lb I aA is inJectwe for each b E O(A); La I Ia E Inv.e(Ia)' (ii) Suppose that A is non-umtal and that O(A) = A. Then: ia is dense in A; is a cone; La I E Inv.e(L).
(Ia, .)
Ia
Proof (i) That Ia is dense in A is a special case of 2.2.4(ii). Let b E O(A). Suppose that x E aA and that bx = 0, say x = ay, where yEA. Then (abA) y = 0 and so x = 0 because a E abA. Thus Lb I aA is injective. Now suppose that x E Ia. For each n E N, there exists Xn E A with x = anxn . Take n ~ 3. Then a(ax2 - anxn+d = 0, and so aX2 = an Xn+l because La I aA is injective. Thus aX2 E la, and hence x E ala, which shows that La : Ia -+ Ia is a surjection. Since it is an injection, La I Ia E Inv.e(1a). (ii) By 2.2.4(iii), ia is dense in A. Certainly ia is an abelian semigroup, and it is non-unital because A is non-unital. Take b E n(A). By (i), Lb is injective on i a , and so fa is cancellative. As in (i), La is a bijection on i a. 0 Let A be a local algebra which is a topological algebra. Then Inv A is dense in A. For take a E MA· Then a = limn--+oo(eA/n + a), so that a E Inv A.
171
Topological algebms Corollary 2.2.9 Let R be a commutative, mdzcal (F)-algebm.
(i) Let I be a closed ideal in R such that nRC!) =I- 0. Then nRC!) is a dense G8-subset of I. La
(ii) Suppose that a E nCR). E Inv £(1:).
Then: fa zs dense m R; (la, .) is a cone;
I fa
Proof (i) By 2.2.4(i), nRC!) is a G.s-set. Take a E nR(I). Then, by 2.2.4(i), a . Inv R# C nRC!)' and so I = aR = a . Inv R# = nR(I).
(ii) By (i), nCR)
o
= R, and so this follows from 2.2.8(ii).
We now introduce the important notion of an element of finite closed descent. Definition 2.2.10 Let A be a commutative topologzcal algebm, and let a E A. Then a has finite closed descent if there exists n E N such that
(2.2.2) a has finite closed descent k if a has finite closed descent and k is the mmimum element n E N such that (2.2.2) holds. We write 8A(a) = k if a has finite closed descent k and 8A(a) = 00 if a does not have finite closed descent. Note that, if A is unital and a E Inv A, then 8A (a) = 00, and that, if A is radical, then an E a n+1A only if an = O. Proposition 2.2.11 Let A be a commutative (F)-algebm, and let a EA.
(i) Suppose that kEN is such that a k E ak+1A, and set b = a k and I = bA. Then bEl = Ia = anA (n ~ k) and an . Inv A# C n(!) (n ~ k). (ii) Suppose that a has finite closed descent. Then am f/. a m+1A (m EN), and the map p t-> pea) + la, qX] ---+ AIla, is an embedding. (iii) Suppose that 8A (a) = k. Then k = min {n EN: an E a n+1A}. (iv) Suppose that A zs mdical and that a E n(A). Then 8A(a) = 1. Proof (i) For each j E Z+, we have aJ+1 I = ak+H1 A = I. For n ~ k, an E I, and so an . Inv A# C n(I). Also bI = a 2k A = I. By 2.2.8(i), bn I is dense in I, and so Ia = I. (ii) Set k = 8A(a), and assume that am E a m+1A. Then clearly m > k. There exists c E A with am(eA - ac) = OJ we have amA(eA - ac) = 0, and so, by (i), ak(eA - ac) = 0, whence a k E a k+! A, a contradiction. The map p t-> pea) + Ia is a homomorphism. Assume that p E qx]e with pea) E la, say 8p = n. Then pea) E an+! I, a contradiction. Hence the map is an embedding.
n:'=l
(iii) This is immediate from (ii). (iv) We have a E aA = a 2A, and a 8A (a) = 1.
rt a 2A because A is radical. Thus 0
172
Banach and topological algebms
Proposition 2.2.12 Let A be a commutatzve (F)-algebm. (i) Let B be a closed subalgebm of A, and suppose that a E B that ~B(a) < 00. Then ~A(a) ::; 8B (a).
(ii) Let I be a closed ideal in A, and let a
n rad A
and
E I. Then 8r (a) = ~A(a).
Proof (i) Set n = ~B(a). Then an E a n+ 1 Be a n+1 A, and an ¢ a n+1 A because a E radA and an -# o. So 8A(a) ::; n. (ii) Suppose that ~A(a) = kEN. Then a k E ak+l A = ak+2 A c ak+l I and k a ¢ ak+l I, and so ~r(a) ::; k. Suppose that ~I(a) = kEN. Then a k E ak+lI C a k+1A. Assume towards a contradiction that a k E a k+1 A. Then a k E a k+2A C a k+ l I, a contradiction. So ~A(a) ::; k. Thus ~r(a) = ~A(a), as required. 0 Definition 2.2.13 Let A be a topological algebm. Then A is essential if A2 is dense in A. Theorem 2.2.14 Let A be a commutative (F)-algebm.
(i) Suppose that A zs essential. Then n{An : n E N} is dense in A. (ii) Suppose that A = alA + ... + akA for some aI, . . . ,ak E A. Then
n{{L:
a? ... a~k A: r}, ... ,rk E Z+, rl
+ . " + rk =
n} : n EN}
is dense in A. Proof (i) Let ~ be the given metric on A. For each mEN, the Cartesian product space A (m) is also a complete metric space with respect to the product metric, which we denote by ~m' Choose a E A and c > O. We shall make a certain inductive choice involving an increasing sequence (k n ) in N. Having defined (kn ), set Xn = A(k n ) (n EN). Then each (Xn, 8k n ) is a complete metric space. We shall specify for each n E N a complete metric dn on Xn defining the same topology as ~kn' a point Xn E X n , and a map On : X n +l - - Xn such that
(2.2.3) and
(2.2.4) To start the inductive construction, take kl = 1, Xl = a, and d l =~. Now take n E N, and assume that k n E N, Xn E X n , and d n have been specified. Temporarily, we write k for kn' and set Xn = (al,"" ak). Since A2 = A, for each j E Nk there exist mj E Nand aJ,I, ... , aj,mj , bJ,I, ... ,bj,mj E A such that dn(xn,Yn) < c/2n , where
Topologzcal algebms
173
Set kn+l = E;=l mj and Xn+l = (al.l,"" al,ml"'" ak,l,"" ak,mk)' and then define On: X n+1 -+ Xn by the formula
Since addition and multiplication in A are continuous, On is continuous. Set dn+1(x, y) = 8k n + 1 (x, y)
+ dn(On(x), On(Y))
(x, y E Xn+d.
Then d n +l is a complete metric defining the topology of X n +l, and (2.2.3) is satisfied. Since On(xn +1) = Yn, (2.2.4) is also satisfied. Thus the inductive construction continues. It follows from (2.2.3) and (2.2.4) that the hypotheses of the Mittag-Leffler theorem A.1.24 are satisfied, and so there exists (zn) E limproj{Xn ; On} with 8(a, zd :::; E~=l c/2n = c. Clearly Zl E n~=l An. It follows that n~=l An is dense in A, as required. 0 (ii) This is a similar argument. A related result applies just to sepamble (F)-algebras; it follows from the theory of analytic spaces in Appendix 5.
Theorem 2.2.15 Let A be a sepamble (F)-algebm. (i) Suppose that A2 has countable codzmension in A. Then A2 is closed and has finite codimension in A, and there exzsts mEN such that each a E A2 can be written as a = Ej:l ajbj for some al,"" am, bl , ... , bm E A. Further, An is closed and has fimte codzmension in A for each n E No (ii) Suppose that each element of A zs the sum of fimtely many squares of elements of A. Then there exzsts mEN such that each element of A is the sum of m squares. Proof The space A2 is analytic. Thus (i) follows from A.5.19(ii) and A.5.20(i), and (ii) follows by applying A.5.20(i) with 'P : a ~ a 2 , A -+ A. 0 Theorem 2.2.16 (Loy) Let A be a sepamble Banach algebm. (i) Suppose that A2 has countable codimension in A. Then: An is closed and has finite codimenswn in A for n EN; A has the S -property; there exist mEN and M > 0 such that, for each a E A 2 , there exzst al, ... ,am,bl, ... ,bm E A with m m a=
L ajb
J
j=l
and
L Ilaj 1IIIb II :::; 111 Iiall . j
j=l
(ii) Let h, .. . , In be close.d ideals in A. Suppose that h ... In has countable codimenswn in A. Then h··· In is closed and has finite codimension in A, and • 11·11 and 11·11.". are equivalent on h ... In.
Proof (i) By 2.2.15(i), A2 has finite co dimension in A, and so An has finite codimension for each n E N by 1.5.6(vii). By A.5.19(ii), each An is closed in A. The remainder is now a special case of A.5.21. (ii) This also follows from A.5.19(ii) and A.5.21. 0
174
Banach and topological algebms
Proposition 2.2.17 Let A be a sepamble Banach algebm which factors. Then there exists 111 > 0 such that, for each a E A, there exzst b1 , b2, Cl, C2 E A with a = b1c]
+ b2C2
IIb1 11ilclil + IIb21111c211 S M lIall . = {a E A : lIall < 1}; S is an analytic set
and
Proof Set S = U!21, where U in A, and S is absorbing because A factors. Thus S is non-meagre in A, and so, by Pettis's lemma A.5.15, S - S is a neighbourhood of O. The result follows. 0
Recall that Mittag-Leffler sets and Mittag-Leffler algebras were defined in 1.3.40; we give some examples related to these notions. Proposition 2.2.18 Let M be a commutatwe (F)-algebm, and let I be a proper, dense ideal in M such that Mil is a mdical algebm. Then (M11)# is a MittagLeffter algebm. Proof Set A = (MI1)#, so that A is a local algebra and MA = Mil. Let d denote an invariant metric defining the topology of 111. Take (an) in MA. We inductively choose a sequence (xn) in M such that, for each n E N, we have Xn + I = an and max{ d(Xl ... Xn , 0), d(X2 ... Xn , 0), ... , d(Xn-IX n , 0). d(xn, O)} < 1/2n .
For n E Z+, define Yn = Z=~l Xn+l '" Xn+k; each series converges in M, and Yo - Z=~=l Xl' .. Xk = Xl' .. XnYn' Set a = Yo + I. Then a satisfies condition 0 (1.3.21), and so MA is a Mittag-Leffler set. Proposition 2.2.19 Let A be a commutative (F)-algebm such that n(A) =I- 0. Then n(A) is a Mittag-Leffter set. Proof Take (an) in n(A). By A.3.28, applied with Tn : X ~ anx, A -+ A, and Un = an+!, there exists b E A with b - Z=~=l al'" ak+! E al '" a1l +IA (n EN). Set a = b + al. Then a - Z=~:~ al ... ak E al ... an+IA (n EN), and so n(A) is a Mittag-Leffler set. 0 Proposition 2.2.20 Let R be a commutative, mdical (F)-algebm, and let a E R have finite closed descent k. Set A = R# a,nd
60
= {b E la : bak A =
akA} ,
6
= 6 0 u U {an.
Inv A: n E N} .
Then: (i) 6 is a subsemigroup of (A, . ) with a E 6 and 6 . Inv A = 6; (ii) 6 0 =I- 0, and n{al" ·an . 6: n EN} =I- 0 for each sequence (an) in 6: (iii) 6 is a Mittag-Leffler set in A; (iv) if {an}.L = 0 (n EN), then {b}.L = 0 for each b E 6.
Proof Set 1= akA. By 2.2.11(i), we have anA = I (n ~ k), and so
60U
U{an. Inv A: n ~ k} c n(I).
(i) Clearly 6 0 • 6 0 c 6 0 and a E 6. Also a . 6 c 6, and so 6 is a subsemigroup of (A, .), with 6 . Inv A = 6.
Topological algebras
175
(ii) Since fl(1) =I 0, fl(1) = I by 2.2.9(i). Take a sequence (an) in ~, and define bn = a(n-I)k+l ... ank (n EN). Since ~ c aA, we have (b n ) C akA C I, and so (b n ) C fl(1). By 2.2.4(iii), there exists (cn ) C fl(1) with Cn = bnCn+1 (n EN). Clearly (c n ) C la, and so (c n ) C ~o, which is thus not empty. Also E b bn . ~ = n~=1 al ... an . ~, and so n~=1 al ... an . ~ =I 0. (iii) Take (an) in~. By A.3.28, there exists bEl with
CI n::1 l...
n
b - ~)al ... aj ) (aj+l ... aj+k) E al ... an+lI J=l
(n E N) .
Set a = b + L:;=I al ... aj. Then, for each n E N, we have a-
Thus
~
n
k+n
j=l
j=n+l
L al ... aj = L
al'" aJ
+ al ... an+lI C
al ... anA.
is a Mittag-Leffler set.
(iv) Take C E A with bc = O. If b E ~o, then b E fl(1), and so ak+ 1 c C = O. If bEam . Inv A, then amc = 0, and so c = O.
= 0 and 0
We noW' introduce a rather strange condition which will nevertheless be seen later to be important in automatic continuity theory.
Definition 2.2.21 A topological algebra A is pliable if, for each closed
~deal
J
of mfinite codimension in A, there are sequences (an) and (b n ) in A such that (bnal ... an) C J and (bn+1al ... an) C A \ J.
Proposition 2.2.22 Let A be an (F)-algebra. Suppose that, for each closed ideal J of infimte codimension m A, there exu;ts a sequence (bn ) ~n A such that at least one of the followmg holds:
(i) bmbn E J (m
=I n)
and b~
f/- J
(n EN);
(ii) bmbn - bm/\n E J (m, n E N) an1 bn+ 1
-
bn
f/- J
(n EN).
Then A is pliable.
Proof In both cases, choose a sequence (Qk) C 1R+- such that L:~l Qk < 00 and such that (L:~=l Qkbk) is a Cauchy, and hence convergent, sequence in A, say a = L:~l Qkbk· Set f3n = L:~n+l Qk (n EN). In case (i), set an = L:~n+l Qkbk (n EN). Let n E N. For m ?: n + 1, bma n E Qmb~ + J, and, for m ~ n, bma n E J, and so (bnal ... an) C J and bn+1al ... an E Q~+lb~t} + J, so that (bn+lal ... an) C A \ J. In case (ii), set an = a - abn (n EN). For each n E N, abn - bna E J, and so (bnal ... an) C J. Since (bn + 1 - bn)a E f3n(b n+ 1 - b.,) + J, we have b"+lal ... an E f3;:+I(b n +l - bn ) + J, and so (bn+Ial ... an) C A \ J. 0 Let A be a unital (F)-algebra with a continued bisection of the identity {(Pn), (qn)}. Then A is pliable: (qn) satisfies the conditions on (bn ) in 2.2.22(i).
Banach and topologzcal algebras
176
Definition 2.2.23 Let A be a topological algebra. Then A zs the directed union of a family {Aa} of closed subalgebras if: (i) for each Q,{3, there exists'Y wzth Aa U A,6 CAy;
(ii)
Ua
Aa zs dense in A.
Theorem 2.2.24 (Willis) Let A be a unital topological algebra which is a dzrected union of a famzly {Aa} of pliable subalgebras. Suppose that Aa/ J is semisimple for each Q and each closed ideal J with finite codimension in Aa. Then A is pliable. Proof We may suppose that eA E Aa for each Q. Let I be a closed ideal of infinite co dimension in A. Suppose that I n Aa has infinite codimension in Aa for some Q. Then the appropriate sequences exist in A a , and hence in A. Thus we may suppose that In Aa has finite codimension in ACt for each Q. Since In (U Aa) has infinite co dimension in U A a , there is a sequence (O:n) such that dim(Acl!n/(I n Aan )) -+ 00 as n -+ 00 and Ae>n C Aan+l (n EN). Write An for A a ", and set In = I n An; the embeddings are 7rm ,n :
Am/1m
-+
An/In
(m ~ n).
Choose an orthogonal set Sl = {P1,1, ... ,P1,k]} of minimal idempotents in Ad11 with L7~1 P1,j = eA + 11, Then {7r1.2(P1,1), . .. ,7r1.2(P1.k t )} is an orthogonal set of idempotents in Ad12. Each element 7r1,2 (P1,J) in this set can be expressed as a finite sum of an orthogonal set of minimal idempotents in Adh the children of 7r1,2(P1.J)' The family of all children is an orthogonal set S2 = {P2,b ... ,P2.k2} of minimal idempotents in A2/I2 with L7~1 P2,j = eA +h We continue in this way, obtaining an orthogonal set Sn = {Pn,1, ... , Pn,k" } of minimal idempotents in An/In with L~~l Pn,j = eA + In for each n E N. Since An/In is semisimple and dim(An/ In) -+ x, it follows from 1.5.1O(ii) that kn -+ 00 as n -+ 00. The collection of idempotents obtained from a given idempotent in one set Sn by taking its children, their children, etc., are the descendants of the given idempotent; an idempotent P ha..., dn(P) descendants in the nth subsequent generation, and P is said to be infinzte if dn(p) -+ 00 as n -+ 00. Clearly there is an infinite idempotent in Sl, and each infinite idempotent has an infinite child. Thus we may choose (nr) in N such that, for each r E N, we have nr+1 > n r . Snr contains two distinct elements Pr and qr with Pr infinite, and Pr+1 and qr+ 1 are descendants of Pr. Note that two elements which are descendants of distinct idempotents in an earlier generation are orthogonal, and so 7rnr ,n r+l (qr) 1. qr+]' For each r E N, choose br E Anr with br + Ir = qr' Then brb s E I (r #- s) and b~ ¢ I (r EN). By 2.2.22(i), A is pliable. 0 We shall shortly discuss inversion in topological algebras. However, as a preliminary, we note the following result. Let kEN. Then InvMk = {T E Mk : detT
#- O},
Topolog~cal
177
algebras
and so InvMk is an open subset of (M k , 11.11 2 ), and the map T f-t T-l is continuous on (Inv M k , 11·112) because det is a continuous function on (Ck 2 • Also the trace map Tr is continuous on (M k , 11·112). Proposition 2.2.25 Let A be an (F)-algebra, let {7rn : A ---+ (Mk n , 11·11 2 )} be a of contznuous epimorphisms for some sequence (k n ) zn N, and let no EN. Suppose that ker 7rm -=I ker 7rn when m, n E N with m -=I n. Then there exists ao E A such that 7rn (ao) = 0 (n E N no ), 7rn (ao) E InvMkn (n > no), and Tr7rm (ao) -=I Tr7rn (ao) (m,n > no, m -=I n).
fam~ly
Proof Let B = {a E A : 7rn (a) = 0 (n E Nno)}' a closed subalgebra of A. For n > no, set Un = {a E B : 7rn (a) E InvM kn }, and, for m,n > no with m -=I n, set Um,n = {a E B : Tr7rm (a) -=I Tr7rn (a)}. Each Un and each Um,n is an open subset of B, and clearly each Un is dense in B. For m -=I n, the ideal 7r n (B n ker 7rm ) of the simple algebra Mk n is non-zero, and so it follows that 7rn (B n ker 7rm ) = Mk n • Consequently B \ Um •n is a proper linear subspace of B, and hence Um,n is also dense in B. Let U = n{Un : n > no}nn{Um,n: m,n > no, m -=I n}. By the category theorem A.1.21, U is dense in B. Choose any ao E U.
0
Corollary 2.2.26 Let {'Pk : kEN} be a family of d~stinct contznuous characters on an (F)-algebra A. For each no E N, there exists ao E A such that 'Pk(ao) = 0 (k E Nno ), 'Pk(ao) -=I 0 (k > no), and 'Pl(aO) -=I 'Pk(ao)
(j, k ~ no, j -=I k) .
o
For each infinite set S of continuous characters on A, there exists ao E A such that (lo(S) is infinite. In particular, if A is spectrally finite, then q, A is finite. Definition 2.2.27 Let A be a topological algebra. q-InvA zs open in A.
Then A is a Q-algebra zf
For example, suppose that A is a topological algebra which is a division algebra. Then Inv A = A· is open, and so A is a Q-algebra. It is easy to see that A is a Q-algebra if and only if q-InvA has a non-empty interior in A. For suppose that U is a non-empty, open set with U c q-Inv A. For a E A, define 'Pa : b f-t a b and 1/Ja : b f-t b a, so that 'Pa,1/Ja : A ---+ A are continuous. If bE q-InvA and 'Pa(C) = b, then (b q a) c = bq b = 0, and so c has a left quasi-inverse. Thus 'P;;l (U) is left quasi-invertible. Similarly, 1/J;;1 (U) is right quasi-invertible. Now take a E q-InvA and bE U. Then 'Pboa.(a) = 1/Ja Qob(a) = b,
and so q- Inv A is open because a E 'P~~. (U) n 1/J-;q~b(U) .
Let A be a Banach algebra. Then it follows from 2.1.29(i) that q-Inv A contains the open ball {a E A: lIall < I}, and so q-Inv A has non-empty interior;
178
Banach and
topolog~cal
algebras
by the above remark, this shows that A is a Q-algebra. Thus Banach algebras are the main examples of Q-algebras, but we ::;hall not.e some other examples later (see 4.4.9, for example). Some incomplete normed. algebras are Q-algebras: the algebra (Coo(X), !·!x) is a Q-algebra for each completely regular ::;pace X, but it is only complete for special spaces X. Indeed. it is clear that a normed algebra (A, II·!D is a Q-algebra if and only if :E:'=1 an converges in A for each a E A with lIall < 1. The next result shows that certain ideal::; in a Q-algebra are necessarily closed. As in §1.4, ITA is the structure space of an algebra A; ITA con::;ists of the primitive ideals of A and has the hull-kernel topology; for the definition of J(0), see 1.4.46.
Theorem 2.2.28 Let A be a Q-algebm. (i) Each maximal modular left or right ideal and each maX'tmal modular ideal in A is closed. (ii) Each chamcter on A ~s continuous.
(iii) Each primitive ~deal 2n A is closed. (iv) The (Jacobson) mdical radA and the strong ideals in A. (v) Suppose that I is an ideal zn A. Then ~(I) = ~
md~cal
(7)
9t(A) are closed
in ITA, and
rad (A/1) = {a + I : a E A, a + 1 E rad (A/I)} "J 1/1. (vi) Suppose that I is a dense ideal in A. Then A/ I is a md~cal algebm and J(0) C I.
Proof (i) Let M be a maximal modular left ideal with right modular identity u. Assume towards a contradiction that u E 1M. Since q-Inv A is a neighbourhood of 0, there exists a E M with u - a E q-Inv A, say b + u - a - b(u - a) = O. But then u = a - ba - (b - bu) E M, a contradiction. Hence u rt. M, and so M = M. A ::;imilar argument applies to both right ideals and ideals.
(ii) Let r.p E q, A . Then M'f' = ker r.p is a maximal modular ideal, and hence closed.. Thus r.p is continuous. (iii) By 1.4.34(ii), each primitive ideal is the intersection of the maximal modular left ideals which contain it. Thus each primitive ideal in A is closed. (iv) By 1.5.1, rad A is the intersection of the primitive ideals in A; by 1.5.20, 9t(A) is the intersection of the maximal modular ideals of A. Thus both radA and 9t(A) are closed. in A. (v) By (iii), each primitive ideal in A containing I also contain::; 1. (vi) It is immediate from (v) that A/ I is a radical algebra. Since ~(I) = 0. it follows from 1.4.47 that J(0) C I. 0 It follows from 2.2.28(ii) that the space q, A U {O} for a Q-algebra A can be regarded as a subset of the dual space A'; unless otherwise stated, it is supposed to have the relative weak* topology 0"( A' , A), called the Gel 'fand topology. Let A be a Banach algebra. Then A/radA is a semisimple Banach algebra. For each S C ITA, f(S) is a closed. ideal in A. A closed subset S of IIA is a set
Topological algebras
179
of 8yntheszs if e(s) is the only closed ideal of A with hull equal to S; 8pectral analysts holds for A if each proper closed ideal of A is containd in a primitive ideal. The algebra A is weakly Wzener if spectral analysis holds for A: ill the case where A is not weakly Wiener, there is a proper closed ideal [ in A with ~(I) = 0, and then AI[ is a radical Banach algebra. These notions will be explored for commutativc Banach algebras in §4.1. 'We have explained that we shall be concerned with the question when all homomorphisms from a topological algebra are automatically continuous. \Ve give a condition for this in the cru:;e where the range algebra is finite-dimensional.
Lemma 2.2.29 Let A be a unital Q-algebra, and let AI be a maximal ideal of fimte codzmenswn in A such that M2 has infinite codimenswn. Then there 't~ an zdeal [ in A of finite codimension such that M2 C [ ~ M and 7 = M. Proof Let B = AIM2. as in the remarks after 1.5.19, and let R, E iJ • R'J' V, and IV be as in those remarks. Since 1\,12 has infinite codimension in ,M. the radical R is infinite-dimensional, and so Rl1 is infinite-dimcnsional. Thus V + E11/o.12 Ell is a dense subspace of codimension 1 in the closed subspace Ell JvIE u . The ideal [ = W + M2 has the required properties. 0 Theorem 2.2.30 (Dales and Willis) Let A be a unital Q-algebra. following conditions on A are equivalent:
Then the
(a) each homomorphism from A into a fimte-dzmensional Banach algebra zs continuous;
(b) each zdeal in A of fimte codimenszon is closed; (c) [2 is closed and of fimte codzmension m A for each closed ideal [ of finite codimension in A.
Suppose, further, that A zs either a separable (F) -algebra or 1,S such that AI[ is semi.mnple for each closed ideal [ of finite codimension in A. Then the above conditwns are also equivalent to: (d) M2 is of fimte codimenswn in A for each maximal zdeal M of finite codimension in A.
Proof (b)::::}(a) This follows immediately from A.3.8(i). (a)::::}(b) Assume that (b) fails, and take [to be an ideal of finite codimension such that [ is not closed. The finite-dimen.."ional algebra AI [ is a Banch algebra, and the quotient map from A onto AI[ is discontinuous, a contradiction of (a). (c)::::}(b) Let [ be an ideal of finite codimension, and set R = rad (AI1) and J = 1f- 1 (R), where 1f : A ---> AI [ is the quotient map. Then J is an intersection of maximal left ideal:; in A, and so, by 2.2.28(i), J is closed in A. By (c), J2 n is closed in A for each n E N. Since AI[ is finite-dimensional, R m = 0 eventually, and so J2 n C [ eventually. Thus [ is closed. (c)::::}(d) This holds because each maximal ideal in A is closed. (b)::::}(c) To obtain a contradiction, assume that [ is a closed ideal of finite codimension such that the conclusion of (c) fails. By (b), [2 has infinite codimension in A; we may suppose that I has minimum codimension among the
180
Banach and topolog'tcal algebras
ideals with this property. As above, set J = 7r- 1 (rad (Aj I). Then Jk C [2 eventually; if J2 has finite codimension, then so does J k , and hence so does [2. Thus J = I by the minimality condition on I, and Aj [ is semisimple. By the Wedderburn structure theorem 1.5.9, Aj [ ~ 0~=1 Mnj for some kEN. Assume that k ? 2, and set L1 = 7r- 1 (M nl ) and L2 = 7r- 1 (0;=2 M n , ). Then L1 and L2 are ideals of finite codimension in A. each properly containing [, and L1 n L2 = L1L2 = L2L1 = I. Choose pEA to be such that p + [ is the identity of MIn" and set q = eA - p. Next define K = L~ n L~. Then K = (p + q)3K c L~L~ c K, and so K = L~L~ and 12 = L1L2L1L2 = LiL~ = Li n L~.
If both of L~ and L~ were of finite codimension, then the same would be true of [2, a contradiction. Thus k = 1 and [ is a maximal ideal. By 2.2.29, this is a contradiction of (b). Now suppose that A satisfies either of the additional conditions; we shall prove that (d)===>(c) . Assume that there is a closed ideal [ in A of finite co dimension such that [2 is either not closed or not of finite codimension; again, suppose that [ has minimum co dimension among the ideals with this property, and set J = 7r- 1 (rad(Ajl). If J i=- [, then J2 is of finite codimension, and so, in the case where A is a separable (F)-algebra, Jk is closed and has finite codimension for each kEN by 2.2.15(i), a contradiction because Jk C 12 eventually. Thus under either of the additional hypotheses. J = [ and AjI is semisimple. As before, it follows that I is a maximal ideal in A, and so we have a contradiction of (d). 0 Proposition 2.2.31 Let A be a Banach algebra, let kEN, and let {In : n E N} be a family of distinct ideals such that Ajln ~ Mk (n EN). Then there are a famtly {7r n : n E N} of eptmorphisms from A onto Mk and ao E A such that: (i) In = ker 7rn (n EN);
(ii) l17rn (a)1I2 ~ v'k lIall (n E N, a E ~); (iii) 7rn (ao) E Mk has upper-trzangular form for each n E N; (iv) there exists j E Nk such that {(7r n (ao»j,j : n E N} c C is mfinite. Proof By 2.2.28(i), each ideal In is closed. For n E N, choose an epimorphism ?Tn : A -> Mk with ker ?Tn = In, and let II· lin denote the quotient norm on Mk induced by 7rn . Then, by 2.1.21, there is an automorphism f n : Mk -> Mk with
(2.2.5) By 2.2.25, there exists ao E A such that Tr Vm i=- Tr Vn (m, n E N, m i=- n), where Vn = (fn 0 7Tn )(ao) (n EN). For each n E N, there is a unitary matrix Un in Mk such that the matrix Un VnU~ E Mk has upper-triangular form; define 7rn : a t-+ Un(f n 0 7Tn)(a)U~, A -> M k • Then it is immediate that the family {7r n : n E N} satisfies (i) and (iii), and (2.2.5) implies that (ii) holds as well. Furthermore, since the trace of a matrix remains unchanged under conjugation with unitaries, it follows that Tr7rm (ao) i=- Tr7rn (ao) (m,n E N, m i=- n), and so the set {Tr 7rn (ao) : n E N} is infinite; from this (iv) follows. 0
Topologzcal algebras
181
We now turn to a very mysterious class of topological algebras.
Definition 2.2.32 Let A be a topologzcal algebra. Then A zs topologically simple zf A2 i= 0 and if 0 and A are the only closed idmls in A. For example,
-Ra(z)2
as w --;
Z
in p(a) \ {z},
and so Ra is analytic at z. Since Ra(z) = z-l(eA - a/z)-1 (z E C e ). we have Ra(z) -~ 0 as Z --> 00 in p(a). 0
Theorem 2.2.41 Let A be a locally convex algebra with identzty Jor which inversion Z8 continuous. Then 0"(0.) zs non-empty Jor each a E A. Proof Assume towards a contradiction that there exists a E A with 0-( a) = 0. Since A is a locally convex space, there exists ..\ E A' with (a -1, ..\) =I- O. Set
J(z) = (Ra(z),..\)
(z E
q.
By 2.2.40(ii), J is an entire function with J(z) --> 0 as z --> 00, and so J = 0 by Liouville's theorem. In particular, J(O) = _(a- 1 ,..\) = 0, a contradiction. Thus O"(a) =I- 0 (0. E A). 0 A further theme of this book is that topological algebras whose topologies satisfy certain algebraic conditions must have a special form. We now give our first example of a result of this type: it is a general version of the celebrated GelJand-Mazur theorem.
Theorem 2.2.42 Let A be a dwzs'tOn algebra. Suppose that either: (i) A zs a locally convex (F)-algebra; or (ii) A is serninorrnable. Then A = CeA. Proof Suppose that A satif:,fies (i). By 2.2.39, inversion is continuous for A. Suppose that A satisfies (ii), and let p be a non-zero algebra seminorm on A. ThC'n kerp = {O}, so that (A,p) is a normed algebra, and inversion is continuous for (A,p) by 2.2.36. In either case, 2.2.41 shows that O"(a) =I- 0 (a E A), and so the result follows from 1.5.30. 0 Corollary 2.2.43 Let A be a topologically simple, commutative, locally convex (F)-algebra with A '1- C. Then A zs radical and an integml domain. Proof By 1.4.36. a primitive commutative algebra is a field, and the only field which is a locally convex (F)-algebra is C. Thus A is radical; by 2.2.33(i), A is an integral domain. 0 Consider the following three fields, which were all introduced in §1.6: the field .c = C((X)) of Laurent series; the field C(X) ofrational functions; the field C (X) of meromorphic functions at O. These fields are not isomorphic to C, and so the theorem shows that there is no topology with respect to which they are locally convex (F)-algebras, and that they are not seminormable. In §4.1O. we shall give various examples of commutative topological algebras which show that the conditions imposed in the above theorem cannot be removed completely.
Topological algebras
185
We now introduce a special class of topological algebras. Definition 2.2.44 Let A be a topological algebra. Then A 2S a locally multiplicatively convex algebra or an LMC algebra if there is a base of nezghbourhoods of the orzgin cons2sting of sets which are absolutely convex and multiplicative. A complete, metrzzable LMC algebra zs a Frechet algebra. Note that our terminology is such that a Frechet space which is a topological algebra is not necessarily a Frechet algebra. Clearly a subalgebra of an LMC algebra is LMC, and the quotient of an LMC algebra by a closed ideal is also LMC; closed subalgebras of Frechet algebras and quotients of Frechet algebras by closed ideals are Frechet algebras. Let A be an algebra. The Minkowski functional of an absolutely convex, multiplicative neighbourhood of 0 in A is clearly an algebra semi norm on A, and so the topology of an LMC algebra can be specified by a separating, saturated family of algebra seminorms. In particular, the Gel'fand-Mazur theorem 2.2.42(ii) shows that each LMC algebra which is a division algebra is isomorphic to .) = I};
(2.3.1)
we have shown in 2.1.29(ii) that CPA c KA. The set KA is non-empty and convex, and it is closed, and hence compact, in (Ahl' a(A', A». By the KreIn-Mil'man theorem A.3.30(i), KA = (exKA). Let a E A. Then V(a) = {(a, >.) : >. E KA}
is the numerical range of a; clearly, ~T(a) is a non-empty, compact, convex subset ofC, and V(a) C lIJ)(O; Ilall). Let z E a(a). Then (z+1] E a((a+17eA) ((,1] E q, and so, by 2.1.29(i), I(z + 1]1 ~ II(a + 1]eAIi. Thus
>'0 : (a + 1]eA 1-+ (z + 1], lin{ a, eA} -+ C, is a well-defined, continuous linear functional with (a, >'0) = z, (eA' >'0) = 1, and 11>'011 = 1. Take>. E A'to be a norm-preserving extension of >'0. Then>. E KA and (a, >.) = z. This shows that a(a) C V(a). Theorem 2.3.1-f.~ _~ be a oo lIanll l / n = inf {lIanlll/n : n EN};
(iv) v(a) = limt->oo lIatll l / t whenever (at: t semigroup in A.
E
R+ e ) is a continuous real
Banach and topological algebms
194
Proof We may suppose that A is unital. (i) By 2.2.36, inversion is continuous for A; by 2.2.40(i), O"(a) is compact; by 2.2.41, O"(a) is not empty.
(ii) For r > "all, the series L~oak/(k+l is uniformly convergent on 'Jl"(O:r) to «(eA - a)-l E A, and so (2.3.2) holds. By 2.2.40(ii), Ra is analytic on p(a), and so, by A.3.77(i), (2.3.2) holds whenever r > v(a). (iii) By 2.1.29(i), v(b) ::; I/bl/ (b E A). Let z E O"(a) and n E N. By 1.6.1l(i), zn E O"(a n ), and so Iznl ::; Ilanli. Hence v(a) ::; inf I/anl/ l/n . Take r > v(a), and set Mr = sup{IIRa(z)11 : Izl = r}. Then it follows from (ii) that I/anll ::; rn+1Mr (n EN), and so limsuPn-+oo I/anl/ l/n ::; r. The result follows. (iv) Set M = sup{I/ati/ : t E [1,2]} For each c > 0, there exists no EN such that I/anl/ < (v(a) + c)n (n;::: no). For t > no + 1, there exists n ;::: no with t - n E [1,2), and then lIa t i/ ::; M(v(a) + c)n. The result follows. 0 Let a E A. Then a E .Q(A) if and only if lim n--+ oo "anI/lin = 0, and an ~ 0 in A if and only if v(a) < 1. By 1.5.32(iv), A is radical if and only if lim Ilanli l/n = 0
n-+oo
(a E A);
A is semisimple if v(a) > 0 (a E Ae). In fact, each non-empty, compact subset n of C is the spectrum of some element of some commutative Banach algebra, namely n = O"C(!l) (Z). The result of 2. 1. 14(iii) for a Banach algebra A also follows from 2.3.8(i). For assume that A contains elements a and b such that ab - ba = eA. Then O"(ab) = {z
+ 1: z E O"(ba)}
and O"(ab) U {OJ
= O"(ba) U {O}.
These two statements are incompatible with the fact that O"(ab) is compact and non-empty. Proposition 2.3.9 Let A = lim proj Ap be a .complete LMC algebm, and let
a E A. Then O"A(a) = UO"A p (7l'p(a»
and
p
v(a) = sup lim p(a n )lln. p
n--+oo
Proof This follows from 2.2.49(ii) and the spectral radius formula.
0
Let (a z E II) be a continuous semigroup in a Banach algebra A. We shall be concerned with the quantity Z :
""a = Note that
11
-
7f
00
-00
log+
d Ila1+ iY II ~. 1+y
(2.3.3)
""a < 00 in the case where (a< : ( E VI) is bounded.
Proposition 2.3.10 Let A be a Banach algebm, let (a Z : z E II) be a continuous semigroup in A such that K-a < 00, and let ( E II. Then there is a constant C< > 0
such that
Spectra and Gel 'fand theory
195
Proof Set U = {t E ~: 210g+ IlaHitl1 > 1 +t 2 }. Then U is an open set in~, and U ha..'l Lebesgue measure at most 211"Ka because 1I"Ka ~ J~J(1/2) dt. Take Y E ~. Then there exists t E ~ \ U with Iy - tl ~ 1I"Ka , and so Ila«(/2)+ Hiy ll
~ Il aHit lllla((f2)+i(y-t)11
~ exp (1 ~ t 2 )
Ila«/2)+i(V- t ) II
~ Cl exp(y2) ,
where Cl = exp(1/2 + (1I"Ka)2)suPlsl:S::71"l v(a)
we have
Ila«/2l+Hi Y 1IIIa«/2)+X II
~ C( exp(x2 + y2) = C< exp(lzI2) o
> 0, as required.
Definition 2.3.11 Lct (A, II·ID be a Banach algebra. Then A is uniformly radical if sup {llanlll/n : a E A[l]} --+ 0 as n --+ 00. Certainly a nilpotent Banach algebra is uniformly radical. Proposition 2.3.12 (Dixon) Let (A, 11·11) be a commutatwe, non-zero, umformly radical Banach algebra. Then sup {ila l ... anll 1 / n : at. ... , an E A[l]}
--+
0
as n
(2.3.4)
--+ 00,
and A does not factor weakly. Proof Take J.ln = sup {ila n Il l / n : a E A[l]}' so that J.ln Let n E N, and let al, ... ,an E A[l]' Define p((l, ... , (n) = ((tal
+ ... + (nan)n
--+
0 as n
((t. ... , (n E
--+ 00.
q.
Then n! al ... an is the coefficient of (1 ... (n in the expansion of p( (1, ... , (n), and so lIal" ·anll
=
2 2 n!(;1I")n 111 71" .. ,1 71" p(ei0 1 ,
~~ sup{llp((l, ... , (n)1I n.
•••
,eiOn)e-i(lh+··+On) dOl'" dOnl1
: (1, .. " (n E 1l'}
~ ~(nJ.ln)n ~ (eJ.ln)n. n.
The result (2.3.4) now follows. Assume towards a contradiction that A factors weakly. Then 11"A : A ®A --+ A is a surjection, and so there exists C > 0 such that each a E A[l] has the form a = ~';:l ajbj , where aj, bj E A and ~';:lllajllllbjil :::; C. Let a E A[l]' By an
Banach and topological algebras
196
immediate induction, for each n E N, there exist flj,l, .... aj.n+1 E A such that a = Lj:l aj.1 ... aj,n+1 and Lj:1 Ilaj,1 II ... IIaj.n+l" s en. But now
IIail
00
00
3=1
j=l
s :L IIaj.l"· a3 •n+l11 S :L(eJ.Ln+1)n+1 IIaj,dl'" IIa3,n+lll S en(eJ.Ln+l)n+l -40 as n -4
00.
Hence a = 0, and A = 0, a contradiction. Thus A does not factor weakly.
0
Examples 2.3.13 (i) Let A = ®wE be the (non-commutative) Banach algebra which is a weighted tensor algebra defined in 2.2.46(ii) for a weight sequence w. Suppose that infw(n)l/n > 0, and take u = (up) E A·, say Uk =I- 0 and Uj = 0 (j < k). Then
Ilu®nll ~ lIu~nllnk w(nk) =
"ukll~ w(nk)
(n E N),
and so v(n) > O. Thus radA = D(A) = {O}, and hence A is semisimple. (ii) Let S be a semigroup, let w be a weight on S. and set A = e1(s.w). For s E S, define Suppose that A is radical. Then 8s E D(A) (s E S), and so v .. = 0 (s E S). Now suppose that Vs = 0 (s E S) and west) = wets) (s, t E S). We claim that A is a radical algebra. For take s E S. For each I E A and n E N, we have
11(158
* f)* "1Iw =
:L {:L {1/{rl) ... l(rn)1 w(srl ... srn) : srI" . srn = r} } . rES
But
w(S7'lSr2'" srn) S w(srlS7'2'" rn-lS)w(rn) = w(s2r18r2 ... rn-dw(rn) S ... S w(Sn)w(rl) ... w(rn) . and so II(Js * f)*nIL S W(8 n) 11/11:. Since Vs = 0, it follows that VA(J B * f) = O. Thus J s E rad A. Since lin {88 : s E S} is dense in A, the claim follows. Even in the case where S is abelian and w = 1. it is not true that A is semisimple whenever VB > 0 (8 E S). For example, take S = {a,b,p}, and make S into an abelian semigroup by defining all products to be p. Then clearly Vs = 1 (8 E S). but (Sa - Sb)2 = 0, and so Da - Db E radA by 1.5.6(iii). (iii) Let (A k,,,· Ilk) be a sequence of Banach algebras, and let!.2l = eOO(N, Ak)' as in 2.1.18(iii). Take a = (ak) E!.2l. Then v2I(a) ~ sUPkEN VAk (ak), and so!.2l is semisimple in the case where each Ak is semisimple. Now take each Ak to be a radical Banach algebra R. Suppose first that R is uniformly radical, and take a E !.2l[lj. Then IIanlll/n S sup {IIanlll/n : a E R[lj} -40 as n -4
00,
and so !.2l is radical. On the other hand, suppose that R contains a non-zero. bounded, rational semigroup (aO! : a E Q+-), and set a = (a l / k ) E!.2l. Then we have IIanil ;::: IIail (n EN), and so a fI. rad!.2l. Hence 2l is not radical. 0
197
8pectm and Gel'fand theory Let (8,~) be a well-ordered semigroup, and let 1 E C S with 1 define a(f) = min supp 1 .
-# O.
Theorem 2.3.14 Let (8,~) be a well-ordered semzgroup, and let on 8.
W
(i) The Banach algebm £1(8,w)
Then we (2.3.5)
be a weight
a domam. (ii) 8uppose that 1 E £l(S,w)- zs quasi-nzlpotent. Then 2S
lim w(a(f)1!)l/1! n--+oo (iii) For the Banach algebm A
= £1(8),
= O.
= D(A) = {O}. 8 = a(f) and t = a(g). radA
Proof (i) Take 1.g E f l (8,w)-, and set Then clearly (f * g)(st) = 1(8)g(t) -# 0, and so 1 * 9 -# O. (ii) Set 8 = a(f). Clearly IIJ*nll w ~ 11(8)l n w(8 n ) (n EN), and so
limsupw(8 71 )1/71 11(8)1 ~ I/(f)
= O.
Hence limn--+00 w(, 0 and hence that j ¢ rad B, so that B is semisimple. The algebra B contains elements ao with v(ao) > 0; each such element is a limit of nilpotent elements, and so neither v nor a is continuous at ao. Now define w(s) = n(s)n(s) (s E S).
Clearly w is a weight on the semigroup S, and so Aw = [l(S,w) is a Banach algebra which is continuously embedded in A as a dense subalgebra. Define B", = Aw/(I n A w), a Banach algebra which is continuously embedded in A as a dense subalgebra; the quotient norm in Bw is denoted by 1I·llw' Let kEN, and consider a product s = S1'" S2k of 2k elements of S. First, suppose that n(si) < k for i E N 2 k. Then s E I. Second, suppose that n(si) ~ k for exactly one i E N2k' If i E Nk, then Si+1 •.• S2k E I, and, if i E {k+ 1, ... ,2k},
199
Spectm and Gel'fand theory
then 81'" 8k-1 E I. In both cases, s E I. Third, suppose that least two values of i E N 2k . Then
n(8i) ::::
k for at (2.3.6)
Thus (2.3.6) holds whenever 81, ... . 82k E S. It follows that l/ a 1
•.•
a2k IIw
::; k-k Iiali/ w... IIa2k Ilw
(aI, ... ,a2k E Bw) .
Since (k-k)1/2k ----+ 0 as k ----+ 00, it follows that (Bw, 11·1Iw) is a radical algebra; indeed, it is uniformly radical. In summary, the Banach algebra B is semisimple, lJ1(B) C nCB)
0, there exists b E R with Ilbll = 1 and Ilball < E. (ii) Suppose that R 1.S a domain. For' each a E R-, aR 1.S not closed m R. (iii) Suppose that R is a pnncipalleft ideal m R#. Then R is finite-dimenswnal.
Proof (i) The result is immediate if a E lJ1(R), and so we may suppose that a ¢ lJ1(R). Since a is quasi-nilpotent, lim infn-->oo Ila n+1 11 / Ila n II = 0 by A.1.26(ii); set b = a k / Ilakll, where kEN is such that Il a k+111 / Ilakll < E. (ii) Let a E R-, and assume towards a contradiction that aR is closed. The map b 1--+ ab, R ----+ aR, is a continuous surjection between Banach spaces, and it is an injection because R is a domain. By A.3.23, there exists m > 0 such that Ilbll ::; m Ilball (b E R). But this is a contradiction of (i). (iii) Suppose that R = R#a, where a E R. By A.3.23, there exists m > 0 such that, for each x E R, there exist a E C and y E R with lal + I/yll ::; m Ilxl/ and x = aa + ya. Let x E R. By an immediate induction, there exist (an) in C and (xn ) in R with lanl + Ilxnll ::; mn I/xll and x = L~=l akak + xna n for each n E N. Since limn-->oo Ila n 1/1/n = 0, we have limn-->oo xnan = 0, and hence ex;,
x= Lakak.
(2.3.7)
k=l
Take (13k) in C such that L~=l 13kak = 0 and 13k = O(rnk) as k ----+ 00. First assume that necessarily 13k = 0 (k EN). Then the representation (2.3.7) of each x E R is unique, and so, for each Z E 1Dl(0; 11m), the map x 1--+ L~l O!kzk is a character on R, a contradiction of the fact that R is radical. Hence there exists kEN such that 13k =I- 0; let ko be the minimum such k. Then clearly (3ko + L%"=ko+1 13k ak - ko E lnv R#, a ko = 0, and R is finite-dimensional. 0 In §5.7, we shall see that the question of the normability of algebras KO#, where K is an ordered field, is important. In fact, if K O # is normable, then a cardinality condition on the value group r K must hold.
Banach and topologzcal algebras
200
Theorem 2.3.17 (Esterle) Let K be an ordered field wzth value group that KO# is normable. Then irK I ~ c.
rK
such
Proof Let II· II be an algebra norm on KO#, and denote the archimooean valuation on K by v. First, take a, b E KO. with v(a) > v(b). Then there exists c E KO with a = bc. We have IIanil ~ Ilbnll IIcnil (n EN). Since e E O(KO#), IIenil ~ 0 as n ~ 00, and so IIanil < IIbnll eventually. Define ¢ : a 1----+ (IIanll), KO. ~ ]RI'
v(a + b)
~
v(a)
+ v(b)
and
v(ab)
~
v(a)v(b).
(2.3.8)
Thus Q(A)n3(A) c radA, for, if a E Q(A)n3(A), then A#a C Q(A) by (2.3.8), and so A#a C q-Inv A and a E radA by 1.5.2(iv). The above equation leads to a second, somewhat different, proof of 2.3.4. For suppose that an -+ 0 in (A, 11·11) and an -+ a in (A. III· III). Then, by (2.3.8),
v(a) ~ v(an) + v(a - an) ~ lIanll
+ lila -
anlll-+ 0,
and so v(a) = O. By (2.3.8) again, v(ba) = 0 (b E A#), and so, by 2.1.29(i), A#a C q-Inv A. By 1.5.2(iv), a E radA. and so a = 0 because A is semisimple. Thus I . I and III· III are equivalent. As in Appendix 1, the frontier of a non-empty, compact set K c C is denoted by oK, and the union of K and the bounded components of C \ K is K. Theorem 2.3.21 Let A be a unztal Banach algebm. (i) Suppose that (an) zs a sequence in Inv A and that an -+ a E A \ Inv A. Then limn-+oo = 00. (ii) Suppose that a E A and that ( E oO"(a). Then there exzsts a sequence (an) in IllV A with lIanll = 1 and aan = ana for· n EN, and such that
Ila;;lll
lim «(eA - a)a n
n--+:::x>
= n---+oo lim Qn«eA - a) = O.
(iii) Suppose that B is a closed, unital subalgebm of A and that a E B. Then
oO"B(a) C O"A(a) C O"B(a)
and
O"A(a) C O"B(a) C
;;W
(2.3.9)
and vB(a) = vA(a). (iv) Let a E A, and set B = C[a]. Then O"B(a) =~. Proof (i) Assume towards a contradiction that (a nk ) is a subsequence of (an) such that a;;; is bounded. Then a;;; (a - ank ) < 1 for some kEN j since a = ank(eA + a;;; (a - ank )), this implies that a E Inv A, a contradiction. Thus lim = 00. (ii) Take «n) in p(a) with (n -+ (, and set
(II
II)
I
I
Ila;;lll
an = «(neA - a)-l / 11«(neA - a)-lll
(n
E
N),
so that an E Inv A, lIanll = 1, and aan = ana for n E N. It follows from (i) that 11«(neA - a)-lll-+ 00 as n -+ 00, and so
I «(eA -
a) an I = lIan«(eA - a)1I ~ 11«(neA - a)-l
r l + I( - (nl -+ O.
(iii) Certainly O"A(a) C O"B(a), and oO"B(a) C O"A(a) by (ii)j (2.3.9) follows. (iv) By 2.1.29(iii) and the maximum modulus theorem A.2.12.
Ip(z)J ~ JJp(a)JJ
(z E~, P E C[Xl) .
(2.3.10)
Assume that there exists z E ~ \ O"B(a). Then there exists (Pn) in qXl such that (zeA - a)Pn(a) -+ eA as n -+ 00. Set qn(X) = (zl - X)Pn(X) - 1.
Banach and topologzcal algebras
202 Then qn(a) --+ 0, and so, by (2.3.10), qn(z) contradiction. The result follows.
--+
O. But qn(z) = -1 (n EN), a 0
Clause (ii), above, shows that (eA - a is a topological divisor of 0 in A. Let B be a closed subalgebra of A with a E B. If PA (a) is connected and. in particular, in the cases where O"A (a) C IR or O"A (a) is countable, necessarily aB(a) = aA(a).
Proposition 2.3.22 Let E be a non-zero Banach space, and let T E B(E) be an UiOmetry. Then either aCT) = ~ or aCT) c 11'. Proof Since IITnll = 1 (n EN). we have veT) = 1 and aCT) c~. Assume that there exists ( E Da(T) n lJ)J. Since T is an isometry,
(1 -1(1) IIxll :S 1I«(1E - T)(x)11
(x E E) .
(2.3.11)
By 2.3.21(ii), there exists a sequence (8n ) in B(E) such that 118n ll = 1 and «(1E-T)8n --+ O. For each n E N, take Xn E E with IIxnll = 1 and 118nxnll 2: 1/2. Then, by (2.3.11),
(1 -1(1)/2 :S (1 - 1(1) 118nxnll :S 1I«(1E - T)(8nxn) II :S 1I«(1E - T)8nll
--+
O.
o
a contradiction. So 00'(1') n lJ)J = 0, and the result follows.
Proposition 2.3.23 Let E be a non-zero Banach space, let T E B(E), and let Z E Ga(T). Then: (i) (zh - T)(E)
-I- E;
(ii) zf (z1E - T)(E) zs closed in E, then z is an eigenvalue of T.
Proof Set F = (zlE - T)(E). By 2.3.21(ii), there exists (Tn) in B(E) such that IITnll = 1 and TT.. = TnT for n E N and such that Tn(zIl'J - 1') --+ 0 as n --+ 00; take (xn) in E with IIxnll = 1 and IITnxnll 2: 1/2 for n E N. . (i) Assume that F = E. By the open mapping theorem A.3.23, there exists (Yn) E [OO(N, E) with (zIE - 1') (Yn) = Xn (n EN). But then
1/2:S /lTnxnll :S IITn(z1E - T)IIIIYnll a contradiction. Hence F
--+
0
as
n
--+
x,
-I- E.
(ii) Assume that z is not an eigenvalue of T. Then zIE - T : E --+ F is a bijection onto the Banach space F, and so there exists 8 E B(F, E) with (zIE-T)8 = IF. But then 1 = IITnll :S IITn(z/e - T)1I1I811 --+ 0, a contradiction. Thus z is an eigenvalue of T. 0
Proposition 2.3.24 Let A be a Banach algebra. (i) For each open set U in C, {a (ii) The map v : A
E
A : a(a)
C
U} is open in A.
IR+ is upper semi-continuous. (iii) The set .Q(A) is a Go-subset of A. --+
Spectm and Gel'land theory
203
Proof We may suppose that A is unital.
(i) Take a E A with 0, the set {a E A : lI(a) < r} is open. (iii) This follows because .o(A) =
n:=l {a
E A : lI(a)
max{lIa(lI. I}. By applying this result with a replaced by rna for rn sufficiently large in JR (and noting that /'i,ma :::; logrn + /'i,a), we obtain (2.3.12). 0 Corollary 2.3.34 Let A be a Banach algebm, and let (a( : ( E ll) be an analytu: semigroup in A such that a E Q(A) and /'i,a < 00. Then a = o. Proof We have v(a) = 0, so that a( = 0
As in A.l.18, diammK denotes the subset K of C.
«( E II
mth
2 ).
It follows that a =
o.
0
diameter of a non-empty, compact
Theorem 2.3.35 (Slodkowski) Let A be a Banach algebm, let U be a nonempty, open subset of C, and let f : U -> A be an analytic function. Set
u(z) = logdiammu(f(z)) Then u is subharmonic on U.
(z
E
U).
Spectra and Gel 'fand theory
207
Proof Fix z E U. We apply 2.3.31 in the case where n = m + 1, where each fez), and where the corresponding ai is h(z): there is a maximal
ai is equal to
commutative subalgebra B of such that
®:=1 Ai containing the set {h (z), ... , inez) }
-) a-B ( ft(z), ... , fn(z) = (a-(f(Z))) (n) .
Define i<j
and set b(z) = (h(z), ... ,im+1(z») E n(m+l). It follows from the spectral mapping theorem 2.3.30(ii) that a-B(p(b(z))) = p(a-B(b(z))), and so
a-B(p(b(z))) = P (a-(f(z)))(m+1)) . Thus vB(P(b(z») = (diam ma-(f(z)))m(m+1)/2. -n The map z f-+ p(b(z», U ~ ®i=lAi' is analytic, and so, by Vesentini's theorem 2.3.32(ii), the function 10g(vB 0 p 0 b) is subharmonic on U. Thus u is subharmonic on U. 0 For much of §5.7, we shall be seeking to construct discontinuous homomorphisms from various Banach and normed algebras, including the algebra C(O). As a gentle preliminary to that work, we conclude this section by proving as a counterpoint some weak 'automatic continuity' theorems; at some points we use the spectral radius formula. For related results, see §5.4. Theorem 2.3.36 Let 0 be a homomorphism from Co into a Banach algebra, and let a E Co. Then there is a constant M such that IIO(an)1I ::; M lanh~ (n EN). Proof We may suppose that lah", = 1. Then there is a non-empty, orthogonal set {PI, ... ,pd of idempotents in Co such that a = L:;=1 ZjPJ + b, where Zl, • .. ,Zk E '][', pjb = 0 (J E N k ), and v(b) = Ibl N < 1. Clearly k
an
= LzjPj +bn
(n
E
N),
j=1
and so IIO(an)1I ::; L:;=1110(pj)1I + IIO(bn)1I (n EN). We have v(O(b» < 1 by 0 1.5.28, and hence 1I0(bn )1I ~ 0 as n ~ oc. The result follows. Proposition 2.3.37 There is a norm III· III on qX] such that every homomorphism from (qX], 111·111) into each Banach algebra 'tS continuous. Proof Define n
= Lj!lajl j=O
so that 111·111 is a norm on qX]. Let 0 be a homomorphism from qX] into a Banach algebra (A, 11·11). There exists G > 0 such that 118(X)lI j ~ Gj! (j EN), and then 118(p)1I ~ G IIlplll (p E qX]). Thus 8 is continuous. 0
Banach and topological algebras
208
Proposition 2.3.38 Let A and B be commutatwe, unital Banach algebras, let a E A with qa] = A, and let £I : A --+ B be a unital homomorphism. Suppose that £I I Coral is continuous. Then £I I C(a) is continuous. Proof We may suppose that A, and hence cora], is infinite-dimensional; by 1.6.7, the map p t-+ pea), qX] --+ A, is an embedding. Set b = B(a). By hypothesis, there exists k > 0 such that IIp(b)1I ~ k IIp(a) II (p E Co[X]). Now take p E qX]. If a E Inv A, then
IIp(b)1I ~
lib-III IIbp(b) II ~ k Ilb-Ilillallllp(a)11
.
If a f/:. Inv A, then, by 2.3.1, there exists 'P E A with 'P( a) = 0; we have Ip(O)1 = Icp(p(a))1 ~ IIp(a)ll· Set q = p - p(O). Then
IIp(b)11 ~ Ip(O)1
+ Ilq(b)11
~
(k
+ 1) Ip(O)1 + k IIp(a) II
~
(2k
+ 1) IIp(a) II
.
Thus £I I qa] is continuous. say IIp(b)1I ~ K lip (a) II (p E qa]). Take p, q E qX] with q(a) E Inv A. Since qa] = A, there exists (qn) C QX] with qn(a) --+ q(a)-l as n --+ 00. Thus
IIp(b)q(b)-lll = lim II(Pqn)(b)1I ~ Klimsup II (pqn)(a)1I = K IIp(a)q(a)-lli . n-+oo
n-+oo
Thus £I I q a) is continuous.
o
Proposition 2.3.39 (Esterle) Let A be a unital Banach algebra, let R be a commutatwe, radical Banach algebra, let £I : A --+ R# be a unital homomorphism, and let a EA. Suppose that (CPR 0 B) (a) E int~. Then £I I qa] zs continuo'us. Proof Set B = qa] and K = uH(a), so that ('PR 0 B)(a) E intK by 2.3.21(iv). For convenience, we suppose that ('PR 0 8)(a) = O. There exists r > 0 such that JI)(O; r) C K, and so, by A.2.4, each p E qX] h&oo IIB(a)J1I1/i = 0 by the spectral radius formula 2.3.8(iii). The result follows. 0 Corollary 2.3.40 Let R be a commutative, radical Banach algebra, and let £I : A (iij) --+ R# be a unital homomorphism.
(i) Suppose that (a 2 +', oX) is of exponential type. Theorem 2.3.35 is from (Slodkowski 1981); 2.3.39 and 2.3.40 are from (Esterle 1979b).
Banach and topological algebms
210
2.4 THE FUNCTIONAL CALCULUS
We turn in this section to the functional calculus theorems for Banach algebras. Let A be a Banach algebra, and let a E A. Then p( a) is already defined in A for each polynomial p in Co[X). Now suppose that A is unital and that p, q E qX) with 0 tj. q(O'(a). Then q(a) E Inv A, and there is a natural definition of (pjq) (a), namely p(a)q(a)-l. Finally, suppose that I = 2::'=0 anZ n is a power series with radius of convergence R > vA(a). Then 2::'oO:'nan converges in A, and its sum is naturally denoted by I(a). The single-variable analytic functional calc'1J.11S extends these ideas by defining I(a) for each I which is analytic on a neighbcurhood of 0'( a). We shall give a full proof of this single-variable theorem. Now suppose that A is commutative and that al, ... , an EA. Then the element p(at, ... , an) is already defined in A for each p in CO[Xl , ... , X"j. The several-variable analytic functional calculus defines an element I(at, ... , an) in A for each I which is analytic on a neighbourhood of the joint spectrum 0'( at, ... , an) in cn. We shall state general forms of the several-variable thearerns, but we shall omit the proofs, because these depend on a substantial theory of analytic functions of several variables; references will be given in the notes. We shall give some immediate applications of the functIOnal calculus; these involve idempotents and the set of exponentials in a Banach algebra. Before discussing the analytic fundional calculi, we define a closely related functional calculus; the Banach algebra A+ (K») was defined in 2.1.13(ii). Theorem 2.4.1 Let A be a umtal Banach algebm, and let a be a power-bounded element in A. Then the map 00
00
Ela ; I = L anZ n n=O
f---+
Lanan , n=O
A+(iij) ~ A,
is a continuous, unital homomorphism with Ela(A) C qa]. Suppose that I i..~ a closed ideal zn A, that eA - a E I, and that 1(1) = O. Then Ela(f) E I.
Proof Since 11/111 = 2::'=olan l < 00 and M = sup{lIanll ; n E N} < 00, the series 2::'=0 anan converges in A, say to Ela(f), where II Ela (f) II ~ M 11/111· Clearly Ela is a unital homomorphism with IIElall ~ M anel Ela(A) C qa). Let 7r ; A ~ Aj I be the quotient map. Then 00
00
7r(El a (f)) = L O:'n7r(a n ) = L an(eA + 1) = 0 n=O n=O because 2::'=0 an = 0, and so a (f) E I.
e
0
Let A be a Banach algebra, and suppose that a E A is power-bounded and that f E A+(iij). Throughout, we shall write I(a) for Ela(f) in A#. Let (J E jR+-. As in 2.1.13(ii), (1 - Z)f' E A+ (iij), and so we have defined (eA - a)f' in A#; clearly «eA - a)f' ; (J E JR+-) is a continuous semigroup in (A#, .). Suppose that lIall < 1 and < (J < 1, and set (eA - a)f' = eA - b. Then
°
IIbll ~
00
00
n=l
n=l
L lap,nliialin = - L ap,n lIalin = 1 -
(1 -
lIall)P < 1,
(2.4.1)
The functional calc'Ul1tS
211
where (a/3,n) is as in (2.1.5). Now let a E radA. so that a is power-bounded. Then there exists bE Coral with bob = a and {bY = {aY: this Rhows that A is an SBI algebra in the sense of 1.5.11(i). Proposition 2.4.2 Let A be a Banach algebra. Then orthogonal zdempotmts can be lifted in A.
o
Proof Since A is an SBI algebra, this follows from 1.5.13(ii). The function sin- 1 is defined on ii) by the formula 00
sin- 1 z
=
La
(z
n Z 2n + 1
E
where
ii)),
an
= 2n + 21 ( 2n + 1
n=O
1/2 ) 1 n +1
(2.4.2)
The function sin -1 belongs to A + (ii)), and so sin -J a is defined in A for eadl power-bounded element a of A. Let A be a unital Banach algebra, let a E A, let U E /va(a), and let r be a contour in U surrounding o-(a) (see Appendix 2). For each ( E r, we have (eA - a E Inv A. Further, for each f E O(U), the map (I--> f«)«eA - a)-l,
r
~
A,
is continuous. As in Appendix 3, we define the A-valued integral
e~ (f) =
1. -2 7fl
rf«)«eA - a)-1
ir
d(.
(2.4.3)
It follows from Cauchy's theorem for A-valued functions (see A.3.77(i)) that e~ (f) does not depend on the choice of the contour r. Clearly the map e~ is linear and continuous. Consider the special case where f = Z. We take the contour r in (2.4.3) to be the circle ']['(0; r), whf're r > v(a). By (2.3.2), e~(Z) = a, and, similarly, e~(1)=eA. Let f, 9 E O( U), and choose a contour r 1 surrounding 0- (a) in U to specify e~ (f). Let V be the open set bounded by f1' and choose a contour f2 surrounding o-(a) in V to specify e~ (g). We have
1.) e~ (f)e~ (g) = (-2
2
7fl
By (1.5.4) and the fact that
r r f«)g('TJ)«eA - a)-1 ('TJeA - a)-l d'TJ d(.
ir2 ir1
Jr2 f«()('TJ -
e~ (f)e~ (g) = (~) 2 r nl ir2 =
1. -2 7fl
()-1 d( = 0 for each 'TJ E f1' we have
f«)«eA _ a)-1
r f«)g«()«eA - a)-l
ir2
(rir g('TJ)('TJ -
()-1 d'TJ) d(
1
d(
= e~ (fg) .
Thus e~ : O(U) ~ A is a homomorphism. It follows that e~ (.f) = f(a) whenever f E Ro(U), the algebra of rational functions with poles off U. Now suppose that 0 : O(U) ~ A is any unital homomorphism with O(Z) = a. Then O(.f) = e~ (.f) (.f E Ro(U)). Suppose further that 0 is continuous on O(U).
212
Banach and topological algebras
Then O(f) = 8~ (f) (f E O(U)) because Ro(U) is dense in O(U) by Runge's theorem A.2.1O. We have thus proved that 8~ : O(U) --> A is the unique continuous, unital homomorphism with 8~ (Z) = a. For each O(V) is the restriction map, and so 8~ 0 rv,u = 8~. For f E Oa(a), define 8 a (J) = 8~(J) whenever f E O(U). Then 8 a (J) is well-defined. Definition 2.4.3 Let A be a umtal Banach algebra, and let a E A. A functional calculus for a is a unital homomorphism 8: Oa(a) --> A such that 8(Z) = a. The following important theorem is now clear. Theorem 2.4.4 (Single-variable analytic functional calculus) Let A be a unital Banach algebra. and let a E A. Then the map 8 a : Oa(a) --> A is the unique continuous functional calculus for a. Further: (i) 8 a (J) = f(a) for each rational function f with poles off O'(a); (ii) the range of 8 a is contained in the commutative subalgebra A# with e(Z) = a; in the case where f E Oa(a) and f(O) = 0, there exists 9 E Oa(a) with f = Zg, and then f(a) = ag(a) E A. Let A be a unital Banach algebra, and let a E A. Suppose that there is a partition {K 1 , ... , Kn} of O'(a) into non-empty, compact subsets (where n 2:: 2). For each j E N n , there exists XJ E Oa(a) such that Xj I Ki = 0 (i =1= j) and Xi I Ki = 1; clearly {Xl> ... , Xn} is an orthogonal family of non-zero idempotents in Oa(a) with '£";=1 Xi = 1. Define Pj = Xj(a) (j E Nn ). Then {PI, ... ,Pn} is an orthogonal family of non-zero idempotents in A with '£";=l Pj = eA. Also, for each j E N n , Pj E C(a), so that Pj E {a}ee and Pia = aPJ = Pjapj. In the following corollary, we maintain this notation; we also set B j = pJApj, a Banach algebra with identity Pj.
213
The functional calculus
= K j U {O} and UBj(apj) = K). and so oA(apj) = f(uA(a)) = K j U{O}.
Corollary 2.4.5 Let j E N n . Then oA(apj)
Proof Set f = ZXj' Then f(a) = apj, Take i E N n and (E C\Ki . Then there exists 9 E Ou(a) with ((l-Z)g = Xi. Set b = g(a). Then ((eA - a)b = Pi, and so ((Pi - api)Pibpi = Pi; similarly, Pibpi((Pi - api) = Pi, and so Pibpi is the inverse of (Pi - api in B i . Thus oB;(api) C K i . Assume towards a contradiction that there exists ( E K) \OBj (ap)). For each i E Nn , there exists bi E Bi with ((Pi - api)b i = Pi. Set b = L~=l bi E A. Then L~=l Pibi = b because Prbs = 0 when r i: s, and so n
((eA - a)b
n
= ~)(eA
- a)pibi
=
i=l
a contradiction of the fact that ( E
0A
LPi i=l
(a). Thus
= eA = b((eA 013 j (apj) =
a),
o
KJ •
Corollary 2.4.6 Let E be a Banach space, and let T E B(E). Suppose that o(T) = KUL, where K and L are disjoint, non-empty, compact subsets of o(T). Then there exist non-zero, orthgonal idempotents P, Q E B(E) with P + Q = IE and PT = TP and such that o(T I P(E)) = K and o(T I Q(E)) = L. 0
A related argument can be used to prove a result about the continuity of the spectrum. Theorem 2.4.7 Let A be a unital Banach algebra, let a E A, and suppose that is a non-empty. open and closed subset of o( a). Then, for each neighbourhood U of 01 in C, there exists 8 > 0 such that O'(a + b) n U i: 0 for each b E A with Ilbll < 8.
01
Proof Let
r
be a contour in U surrounding Ar
=
{b E A : O'(b)
0'1
with o(a)
nr
= 0, and define
n r = 0},
so that, by 2.3.24(i), Ar is an open neighbourhood of a in A. Define F(b)
= ~ { ((eA 2m
Jr
- b)-l d(
(b
E
Ad.
Then F is continuous on A r . By A.3.77(i), F(b) = 0 whenever o(b) n U = 0. Now let B be a maximal commutative subalgebra of A containing a, so that 0 B (a) = 0 A (a). Then there exists c.p E
0 such that a + bEAr and
IIF(a + b) - F(a)11 < IIF(a)11 whenever b E A with O'(a + b) n U i: 0.
Ilbll < 8.
For each such b, necessarily F(a
+ b) i: 0,
and so 0
Corollary 2.4.8 Suppose that an ---4 a in A and that IIA(a n ) is a connected subset of C containing the origin, and {( E
C : (eA - a
ri- Inv A} =
0.
---4
O. Then u(a)
214
Banach and topologic-al algebm8
Proof Th(' first statement is immediate from the theorem. Set U = {( E C ; (eA - a ¢ Inv A}. Assume that there exists ( E U. Then «(!A - an E A \ Inv A eventually, and so ( E a(an ) eventually. Thus ( = O. Since U iH open, this is a contradiction, and so U = 0. 0
Tn the following theorem, the family Kc \ {0} of non-empty, compact subsets of C has the Hausdorff metric ~ (see A.1.16). Theorem 2.4.9 (Newburgh) Let A be a Banach algebm. and let a: a b(- the
sp(~ctrum
r-+
(A,
a(a},
II ,11)
~
(Ke \ {0},~),
function.
(i) The map a is upper semz-continuous. (ii) Suppose that ao E A and a(ao) is totally disconnected. Then a zs continuous at ao.
Proof (i) This is immediate from 2.4.7. (ii) Take c: > 0, and let {U1,"" Un} be an open cover of a(ao) by discs of radius c:: we may suppose that Uj n a(ao) =1= 0 (j E N n ). Since a(oo) is totally disconnected, each Uj n a(ao} contains a non-empty. open and closed subset of a(ao). By 2.3.24(i) and 2.4.7. there exists 6 > 0 ~mch that a(ao + b) C U7=1 Uj and a(oo+b)nUj =1= 0 (J E Nn ) whenever b E A with IIbll < 6, and, in this case, ~(a(ao + b), a(ao}} < 2c:. Thus a is continuous at ao. 0
Example 2.3.15 shows that the spectrum function a is not always continuous. Theorem 2.4.10 Let A be a unital Banach algebm. (i) Let a E A. f E
(ii) Let an
~ a in
OU(ll) ,
and 9 E Ou(J(a»' Then (g
A, and let f E
O.,.(a).
0
f)(a) = g(l(o.».
Then f(a n } ~ f(a} in A.
(iii) Let B be a unital Banach algebm, let 0 : A ~ B be a continuous, unital homomorphism, and let a E A. Then O(f(a» = f(O(a» (I E Ou(a»' Proof (i) Set w(h) = 8 a (h 0 f) (h E OU(J(ll»)' Then W : Ou(J(a» ~ A is a continuous, unital homomorphism with W(Z} = f(a}, and so W = 8 f (a). Hence (g 0 f)(a) = 8 a (g 0 f) = 8f(a) (g) = g(l(a». (ii) Take U E Nu(a) such that f E O(U), and let r be a contour in U surrounding a(a). Since r is compact and Ra is continuous on pea), there exists M > 0 such that 11«eA - a)-l/1 :5 M « E r). Take c > O. Then there exists no E N such that Ila n - all < min{c:,1/2M} (n ~ no). For n ~ no, we have II«(eA - an) - «eA - a)II :5 1/2 «eA - a)-III « E r), and so, by 2.2.36,
II
1i«eA - a.. )-1 - «eA - a)-III :5 2M 2c:.
Thus «eA - a .. )-1 ~ «eA - a)-1 uniformly on
r, and hence the result follows.
(iii) This follows from the definition (2.4.3) and A.4.17.
0
The functwnal calculus
215
Let A b(l a unital Banach algebra, and let a E A. By using the above functional calculus, we can define many specific clements of A, and we now discuss some of these dements. First, we define: k
:xl
expo. =
L :,: k=O
•
:xl
sma=L
(_1)k a 2k+1 (2k+1)! ;
DC
cos a =
k=O
L k=O
(_1)ka 2k (2k)!
(2.4.5)
Clearly we have 2isina
= exp(ia) + exp(-ia). (2.4.6) (-rr/2.rr/2). Then O'(sina) C (-1,1) c ]lJ) by 2.4.4(iv),
= exp(ia) - exp(-ia)
and
2cosa
Suppose that 0'(0.) C and so it follows from 2.4.1O(i) that, in the notation of (2.4.2), :xl
a = sin-1(sina) = Lan sin2n + 1 a.
(2.4.7)
n=O
Now let a, bE A. Then exp a, exp b, and exp(a + b) belong to A. In general, exp(a+b) =I- (exp a) (exp b). However, in the special case where ab = ba, we have (a + b)n = 2:~=o (~)akbn-k (n EN), and so it follows from (2.4.5) that exp(a+b) = (expa)(expb)
(a,bEA,ab=ba).
(2.4.8)
In particular, (expa)-1
= exp( -a) (a E A).
(2.4.9)
Proposition 2.4.11 Let A be a commutatzve, unital Banach algebra, and let a, b E A. Suppose that a = band exp a = exp b. Then a = b. Proof Set r = b - a. By (2.4.8), (expa)(expr) = expb, and so expr = eA and r(eA + 2:;:'1 rk/(k + I)!) = O. Since r E radA, it follows from 2.1.29(i) that eA + 2:;:'1 rk/(k + I)! E Inv A, and so r = O. Thus a = b. 0 Proposition 2.4.12 Let A be a unital Banach algebra, and let a E A. Then O'(a) C JR if and only if v(exp (ita» = 1 (t E JR). Proof By 2.4.4(iv), O'(exp(ib)) = {eiz : z E O'(b)} for bE A. First, suppose that O'(a) C JR and t E JR. Then O'(exp(ita)) C 'll', and so v(exp(ita» = 1. Second, suppose that v(exp(ia» = v(exp( -ia» = 1. Then O'(exp(ia» C 'll', and so O'(a) C R 0 Definition 2.4.13 Let (A, 11·11) be a umtal Banach algebra. hermitian if Ilexp(ita)II = 1 (t E JR).
Then a E A is
By 2.4.12, O'(a) C JR for each hermitian element a E A. Theorem 2.4.14 (Sinclair) Let (A,II'II) be a unital Banach algebra, and let a E A be hermztian. Then vA(a) = 110.1\. Proof We may suppose that v(a) < rr/2, so that 0'(0.) C (-rr/2,rr/2). By (2.4.6), 1\ sin all ~ (lIexp(ia)II + lIexp(-ia)ID/2 ~ 1, and so it follows from (2.4.7) that lIall ~ E:'=o an = sin- 1 I = rr/2, where (an) was specified in (2.4.2). The result follows. 0
Bana('h and topological algebras
216
Definition 2.4.15 Let A bf' a umtal Banach algebra. Then the set of exponentials in A is exp A = {exp a : a E A} . Clearly exp A c Inv A; in the ca.'>e where A is commutative. exp A is a subgroup of (Inv A, . ). Let n E A be such that 0 belongs to the unbounded component in C of the resolvent set pea). Then there is a region U in C· with C \ U connected such that £T(a) C U. Let log E O(U) be such that exp(Iog z) = Z (z E U). Then log a E A. By 2.4.1O(i), a = exp(Iog a). so that a E exp A. For each n E N. we have (exp((Iog a)jn))n = a, and hence a is divisible in (Inv A. .). This shows. for example, that expMn = InvM n (n E N). (2.4.10) In general. the definition of log a depends on the choice of U and the function log. Howev('r. suppose that £T(a) C C\IR-. Then we set U = C\IR-. and require that log 1 = O. so that log is uniquely specified as an element of Gu(a). and log a is uniquely specified in A. In this case. set
a"
= exp(zloga)
(z
E
q.
(2.4.11)
Then a Z E ClaJ and (a Z : z E q is an a.nalytic group in (Inv A, . ) with a 1 = a and aD = eA. In the case where 0 < n ::; 1. we have
£T(a Q ) C 8 mr = {rei'P E C : r > 0, l.pl < Q7r} .
(2.4.12)
Now suppose that a E A with v(eA - a) < 1. Then log a is uniquely defined in A, and OC (eA a)n log a = (2.4.13) n n=1
L
-
Thus exp A contains the neighbourhood {a E A : ileA - all < 1} of eA in A. Let b E radA. Then, by (2.4.13), 10g(eA +b) = L:~=1(-1)n+lbnjn E radA.
Proposition 2.4.16 Let A be a 'umtal Banach alg(~bra wzth radzcal R such that R is uniformly radical. Then, for each c > 0, there eXtsts C > 0 such that Illog(eA +r)lI::; C wheneverr E R[c]. Proof Take (IJ.n) E Co such that :mp{lIanIl 1 / n : a E R[1]} The series L:~=1 JL~Cn jn converges; set C = L::=11J.~Cn In. II]og(eA + r)1I ::; C whenever r E R[c].
::;
lin (n EN). From (2.4.13). 0
Proposition 2.4.17 Let R be a radical Banach algebra. Then: (i) InvR# = exp R# = {oeR +a E R#:
0
E C·, a E R};
(ii) (lnv R#, . ) is a divisible group.
Proof For b = oeR + a E R#, we have £T(b) b E exp R#. The remainder is clear.
= {o},
and so, if
0
i= 0, then 0
The functwnal calculus
217
Proposition 2.4.18 Let A be a umtal Banach algebra, and let a E A with (T(a) C IC \ ~ -. Then ther'e ex?'st::; a umque b E A with b2 = a and a(b) C IT. Further, b E {a}C, Proof Set b = a I/2 , so that bE {ale. Then b E lC[a]. b2 = a, and a(b) C II by (2.4.12). Now suppose that c E A with c2 = a and aCe) C II. Then ea = c3 = ar, and so B = lC[a. r.J is a commutative. unital Banach algebra with b, e E B. We have (b - e)(b + e) = O. For each 'P E '11 exp(lzlllIallI) S Cc 11>'11 exp«v(a)
+ c) Izl)
(z E C),
o
and this gives the result.
Theorem 2.4.21 Let A be a nnital Banach algebra. Suppose that>. E A x is such that >.(eA) = 1 and >.(expa) =f. 0 (a E A). Then>. E '11 = 1. Let a E A. By 2.4.20, >. 0 Fa is an entire function. There is an entire function G such that>. 0 Fa = expG because (>. 0 Fa){z) =f. 0 (z E C); we may suppose that G(O) = O. For z E C, 1(>' 0 Fa)(z)1 S exp (Iziliall) by (2.4.16), and so ~G(z) S Iziliali. By A.2.29, there exists a E C with G = aZ. For z E C, we have 1 (>. 0 Fa)(z) = 1 + >.(a)z + "2>.(a 2)z2 + ...
(expG)(z) = 1 + az + ~a2z2 and so >.(a2)
= a 2 = >.(a)2.
+ ... ,
It follows from 1.3.29 that). E oo
Ink = o.
o.
(ii) Suppose that IIa n il
= a(lnl) as Inl -> 00. Then a = eA.
Proof (i) Since (T(a) = {I}, there exists b E A with expb = a and v(b) = o. Set Fb(Z) = exp(zb) (z E q, so that Fb(n) = an (n E Z), and take A E A'. By 2.4.20, A 0 Fb is an entire function of minimal exponential type. Set M = sup{llFb(t)II : It I :::; I}. For x E JR, there exist n E Z and t E [-1,1] with x = n + t and Inl :::; Ixl. We have I(A
0
Fb)(x)1 :::; IIAII IlFb(x)1I :::; II All lIexp(nb) II lIexp(tb) II :::; M IIAlllla n ll = O(lxl i ) as Ixl -> 00.
Also, liminflzl->oo I(A 0 Fb)(Z) I I Izlk = 0, and so, by A.2.28, A 0 Fb is a polynomial of degree at most k - 1. It follows that (A 0 Fb)(k)(O) = 0 (A E A'), and hence bk = O. Since a - eA E 00. Proof We may suppose that A is unital. Define
pCb)
= lim n __sup ao Ilbanil
/ JLn
(b E A).
Then p is a continuous seminorm on A. and p(bl b2) ~ IIb11l p(bz) (b l , b2 E A). Set I = ker p, so that I is a closed ideal in A. Suppose that p(eA) = O. Then lIanll = o(JLn) as n ---> 00, and so the result is immediate. Suppose that p(eA) =I- O. Then E = AI I is a non-zero Banach space for the quotient norm. For b E A, define
O(b)(x+I)=xb+I
(xEA).
222
Banach and topological algebms
Then O(b) E B(E) and 0 : A ~ B(E) is a unital homomorphism. By 1.5.28, a(O(a)) C Jl)) U {I}. Also, for each x E A, we have 1I0(a)(x + 1)11 = p(xa) = lim sup Ilxan+111 / JLn = p(x) = !Ix + III n-+oo
because limf.Ln+t!f.Ln = 1, and so O(a) is an isometry in B(E). By 2.3.22, either a(O(a)) = iij or a(O(a)) cT. It follows that a(O(a)) = {I}. Since 1I0(a)1I = 1, we have 1I0(a)nll :::; 1 (n EN), and so O(a) satisfies the conditions on a in 2.4.22(ii). Thus O(a) = IE, and so eA - a E I, i.e., limn->oo Ila n+1 - anll IJLn = o. D Let A be a Banach algebra, and let a E A. Suppose that as n ~ 00. Then clearly a(a) C IDlU {I}. Also,
lIonll
~ 110+ ~(Ok+l - Ok}11 ~ o(n)
as
n_
Ilan+1 -
I
an ~ 0
00.
The theorem shows that, in the case where a(a) C Jl)) U {I} and a is powerbounded, we have Ila n+1 - anll ~ 0 as n ~ 00. Let A be a unital Banach algebra. Then Inv A is an open subset of the locally connected space A, and so the components of Inv A are open and closed subsets of Inv A; the component containing eA is the pnncipal component, denoted by InvoA. The following result shows that InvoA is the subgroup of Inv A generated byexpA. Theorem 2.4.26 Let A be a unital Banach algebm. Then
InvoA = {(expar) ... (expa n ) : al, ... , an E A}.
(2.4.17)
Proof Let Go be the right-hand side of (2.4.17); certainly eA EGo C Inv A. For b = (expal) ... (expa n) E Go. set F(t) = (exptal) ... (expta n) (t ElI). By 2.4.1O(ii), F : 1I ~ A is continuous, and so F is a path in G connecting eA to b. Thus Go C InvoA. 1 . Then IleA - a-Ibll < 1, and Let a E Go and b E A with lib - all < so a-1b E expA and b = a(a-Ib) E Go. Thus Go is open in Inv A. Now take a sequence (b n) in Go with bn ~ b in InvA. Then b;;1 ~ b- I by 2.2.36, and so lib - bnll < IIb;;1 l eventually. Thus b E Go, and so Go is closed in Inv A. Since InvoA is connected, Go = InvoA. D
Ila-111-
r
In particular, Go = InvoA is a subgroup of G = Inv A. For a E G, the subgroup a . Go· a-I is a connected subset of G containing eA, and so we have a . Go . a-I = Go. Thus Go is a normal subgroup of G, and GIGo is a group, called the index group for A. The cosets a . Go of Go are the components of G. For example, it is easy to check that the index group of the Banach algebra C(O), for a non-empty, compact space 0, is the group of homotopy classes of maps from 0 to C·, and so it is naturally isomorphic to HI(O,Z), the first Cech cohomology group of 0 with integer coefficients. Corollary 2.4.27 Let A be a commutative, unital Banach algebm. Then
InvoA = expA, and the index group is tOTsion-free.
The functional calculus
223
Proof Since exp A is a group, InvoA = exp A. Suppose that a E Inv A and that an E expA for some n E N, sayan = expb. Set c = aexp(-bjn). Then en = eA, and so aCe) C {z E C : zn = I}. In particular, 0 belongs to the unbounded component of p(e), and this implies that c E expA. Hence a E expA, and so Inv AjlnvoA is torsion-free. 0 We now consider the several-variable analytic functional calculus for commutative Banach algebras; the main theorems are stated without proof.
Definition 2.4.28 Let A be a commutative, unital Banach algebra, and let a = (al' ... ' an) E A(n). A functional calculus for a is a unital homomorphism S: Ou(a) --+ A such that S(Zj) = aj (j E N n ). Theorem 2.4.29 (SHov, Arens and Calderon, Waelbroeck, Zame) Let A be a commutative, unital Banach algebra, and let a E A(n). Then there 'tS a contmuous functional calculus Sa for a with Sa(f) = f 0 a (f E Ou(a), and Sa is uniquely specified by these conditions. 0
--
Again we shall usually write f(a) for Sa(f). Theorem 2.4.30 Let A be a commutative, unital Banach algebra, and let a = (al, ... ,an) E A(n).
(i) For f E Ou(a), a(f(a» = f(a(a». (ii) Suppose that h, ... , fm E Ou(a) and b = (h(a), ... , fm(a». Then g(b) = (g
0
(h, ... , fm»(a)
(g E Ou(b).
(iii) (Implicit function theorem) Let h E C(A), and set K= (h,al, ... ,an)(A).
Suppose that F E OK, that F I K = 0, and that (8Fj8z l )«() =1= 0 Then there exists a unique b E A wzth F(b, al, ... , an) = 0 and b = h.
«( E K). 0
Corollary 2.4.31 Let A be a commutative, unztal Banach algebra. (i) Let a E A, and suppose that h E C(A) wzth exph = a. Then there exists a unique b E A such that exp b = a and b = h. (ii) Let aO,al, ... ,an E A, and suppose that there exists hE C(A) with E7=0 ~hj = 0 and with
(t.fo;hH) Then there exists a unique b E A wzth
( 0 with 28(1 + kn 2) < 1. Since ao E rad (A/I), it follows from 1. 5.4 (iii) that eo E rad (A/ J), and so lI(co) = O. By 2.3.20, we may suppose that the norm 11·11 on A/J is such that lleoll < 8 and IIcjll :S k (j = 2, ... , n). By 2.4.38, there exists z E eo . Inv (A/ J) c rad (A/ J) such that ('Tr2p)(Z) = O. Take yEA with 'Tr2(Y) = z. Then p(y) E J. Also p'(y) E eA + VA, and so
'Tr2(P'(y»
E
eA/J +z(A/J) c eA/J +rad(AjJ) c Inv(A/J).
Hence there exist b E A and u E J such that bp'(y) = eA + u. Now define q(X) = p(y + bX) E A[X]. Then q(X) = do + (eA + u)X + d2X 2 + ... + dnXn , where do = p(y) and dJ = bjp(j)(y)/j! (j = 2, ... , n). Since I is dense in J, there exists v E I with Ilu + vII < 1, and then, by 2.1.29(i), eA +u+v E Inv A, say w = (eA +U+V)-I. Since dow E J, there exists rEI with 2(1 + Kn 2 ) IIdow - rll < 1, where K = max{llwd 2 11,··., Ilwdnll}; by 2.4.38, the polynomial (dow - r) + X + wd2X 2 + ... + wdnxn has a root, say 8, in (doW - r) . Inv A c J. We have
+ u + v)r + (eA + u + v)s + d2s2 + ... + dns n = and so p(y + bs) = q(s) = (eA + u + v)r - vs E I. do - (eA
0,
Set x = 'Trl(y+bs) E A/I. Then ('TrlP)(X) = 0, and so x is a root of the given polynomial. 0 Notes 2.4.40 The single-variable analytic functional calculus for Banach algebras has a long history (see (Palmer 1994, §3.3»; key early papers are (A. E. Taylor 1938) and (Beurling 1938). Accounts are given in many of the texts we have cited, including (Bonsall and Duncan 1973) and (Rudin 1973). Sometimes, Runge's theorem A.2.1O is used to show that e~ is a homomorphism. Newburgh's theorem 2.4.9 is from (1951). For many examples and further results on the continuity of a and v, see (Aupetit 1979) and (Burlando 1994). For the concept of a hermitian element, see (Bonsall and Duncan 1973, §1O) and (Palmer 1994, 2.6.5). Let H(A) denote the set of hermitian elements in a Banach algebra A. Then H(A) is a real-linear subspace of A, but it is not necessarily a subalgebra.
228
Banach and
topolog~cal
algebras
The following conditions are equivalent for an f'lement a of a unital Banach algebra: (a) a E H(A); (b) V(a) C lR; (c) (IleA - itall - 1)/ It I -+ 0 as t ....... 0 in lR. Theorem 2.4.14 is from (Sinclair 1971); see (Bonsall and Duncan 1973, 11.17). The proof of Theorem 2.4.19 is based on one in (Dixrnier 1981, IlL9 4) Theorem 2.4.21, from (Browder 1969, 1.4.4), g'o E E®E' with the continuous rank-one operator x 1-+ (x, >'0) :ro, E ---- E, and we identify E(l)E' with :F(E), the algebra of continuous. finite-rank operators on E, so that :F(E) = :FC(E) n B(E). The products in B(E) of Xo ® >'0 E E 0 E' with T E B(E) are given by
To (xo 0 >'0)
= T:ro (l) >'0,
(xo ~ >'0)
0
T
= .EO 0 T'(>.o),
(2.5.1)
so that :F(E) is an ideal in B(E). Now suppose that E and Far£' normed spaces. Then B(E. F) is itself regarded as a normed space with respect to the opemtor norm II . II. We shall also consider: A(E, F), the space of approximable operators; N(E, F), the space of nuclear operators (with the nuclear norm II·II,J; K(E, F), the space of compact operators; WeE, F), the space of weakly compact operators; SeE. F), the space of strictly singular operators; I(£, F), the space of integral operators (with the norm II· liz)· These spaces are defined in Appendix 3.
Definition 2.5.1 Let E be a Banach space. An operator algebra [operator ideal] on E is a. subalgebra [an ideal] of B(E) containing :F(E). An opemtor algebm [opemtor ideal] 2l on E is a Banach operator algebra [Banach operator ideal] if it is a Banach algebm with respect to some norm. A closed operator algebra [closed operator ideal] is an opemtor algebm [opemtor ideal] which is closed in
(B(E),II·II)·
Banach and topological algebras
230
The following result is an immediate consequence of results from Appendix 3, where properties such as AP are defined.
Theorem 2.5.2 Let E be a Banach space. Then:
(i) :F(E) is an ideal in B(E); (ii) A(E), K.(E), WeE), and SeE) ar-e closed operator ideals on E; (iii) A(E) = K.(E) m the case where E has AP; (iv) (N(E), 11·llv) and (I(E), II· III) are Banach operator ideals on E; (v) :F(E) c N(E) c A(E) c qE) c (S(E) n W(E»; (vi) A(E) c I(E) c WeE); (vii) WeE) = B(E) if and only if E zs reflexive.
o
The quotient algebra B(E)jK.(E) is called the Calkin algebra of E. As in Appendix 3, the completions of E®E' with respect to the injective norm II '11 eand the projective norm II ,1171" are the injective tensor product (E0E', II . lie) and the projective tensor product (E0E', 11·1171")' respectively. Using A.3.60 and A.3.71, the following result is an easy verification.
Theorem 2.5.3 Let E be a Banach space.
(i) The algebras (E®E', 0, II· lie) and (E®E', 0,11·1171") are normed algebras with completions the Banach algebras E 0 E' and E 0 E', respectively. (ii) The identification of E ® E' with :F(E) extends to an isometric zsomorphzsm (E ®E', II· lie) --t (A(E). 11·11) and to an epimorphism
11·1171") --t (N(E), II· IIv>
R: (E0E', with
IIRII = 1.
The epimorphism R is injective if and only if E has AP.
Definition 2.5.4 The Banach algebra (E~ E', of the Banach space E. Proposition 2.5.5 Let space E.
(2l, III·IID
(i) The embedding of (2l,
2l,
is the nuclear algebra
be a Banach operator algebra on a Banach
III·IID
(ii) The bzlinear map (x, A) (iii) N(E) c
0,11·1171")
0
~
mto (B(E), 11·11) is continuous. x ® A, E x E' --t 2l, Z8 continuous.
and the embedding oj(N(E),II·II.J into
(2l,III'111)
is con-
tinuous. Proof (i) Let x E E and A E E'. By (1.3.9), (x ® A)2 = (x. A):r 0 A, and so I(x, A)I ::; Ilix ® Alii. By (2.5.1), I(Sx, A)I ::; IIISx ® Alii::;
IIISlIllIlx ® Alii (x E E, A E E', S E 2l). (2.5.2) Let (Sn) beasequencein21withSn --t Oin (2l, 111·111> and Sn --t Sin (B(E), 11·11). By (2.5.2), I(Sx, A)I = 0 (x E E, A E E'), and so S = O. The result follows from the closed graph theorem A.3.25.
Banach algebras of operator's
231
(ii) It follows easily from (i) and the closed graph theorem that the specified map is separately continuous; by A.3.39, it is continuous. (iii) It follows from (ii) and A.3.69 that there is a continuous linear map S: (E0E', 11'11,,) ~ (21,111,111) with S(U) = U (U E :F(E)). By A.3.1O(i), S extends to a continuous embedding of (N(E), 11·11..,) into (21,111'111). 0
Let (21, III· liD be a Banach operator algebra on E. It follows from (ii), above, that {X 0 A: x E E[IJ. A E E lIJ } is a bounded semigroup in (21,111,111): by 2.1.9, we may suppose that (x E E, A E E').
III x 0 Alii = IIxllllAl1
(2.5.3)
In the case where (21, III . III) is a Banach operator ideal, we also suppose that max{IIISTIII, IIITSIII} ~ IIISIIIIITII
(S E 21, T E B(E)).
(2.5.4)
The following theorem shows that the algebra B(E) determines the Banach space E. In fact, it will follow from 5.1.14 that an isomorphism from B(E) to B(F) is automatically continuous. Lemma 2.5.6 Let E be a non-zero Banach space. Then To E B(E)- is a rankone operator' if and only if (STO)2 E CSTo (S E B(E)). Proof Suppose that To is rank-one and that S E B(E). Then (STO)2 E CSTo by (1.3.10). Conversely, suppose that To sat.isfies the given condition, and assume that To is not rank-one. Then there exist Xl' X2, YI, Y2 E E with Toxj = Yj (j = 1,2) and {VI, Y2} linearly independent. Take A!, A2 E E' with (YI, A2) = (Y2, AI) = 1 and (YI,AI) = (Y2,A2) = 0, and set S = Xl ®AI +x20A2' Then (STO)(X2) = Xl and (STO)2(X2) = X2, so that X2 E CXI and hence Y2 E CYI, a contradiction. Thus To is rank-one. 0 Theorem 2.5.7 (Eidelheit) Let E and F be non-zero Banach spaces. Then there is a continuous isomorphism from B(E) onto B(F) 1/ and only if E and F are linearly homeomorphzc. Proof Suppose that
1] :
E
~
F is a linear homeomorphism. Then
TI---t1]oTo17-1,
B(E)~B(F),
is a continuous isomorphism. Conversely, suppose that cI> : B(E) ~ B(F) is a continuous isomorphism. Choose Xo E E- and Ao E E' with (xo, Ao) = 1. and set Po = Xo 0 Ao and Qo = cI>(Po). It follows from 2.5.6 that Qo is a non-zero, rank-one operator, say Qo = Yo 01Lo, where Yo E F- and /1-0 E F'·. For each x E E, choose U E B(E) with Ux'o = x, and set 1](x) = cI>(U)yo. We claim that 1](x) is well-defined. For suppose that UIXo = U2Xo. Then U1PO = U2Po, and so cI>(U1)Qo = ct>(U2 )Qo and cI>(Ut}yO = cI>(U2)Yo, as required. Clearly",: E ~ F is a linear map. Suppose that ",(x) = 0. Then, successively, cI>(U)Qo = 0, U Po = 0, and X = UXo = 0, so that", is injective. Now take Y E F, and choose V E B(F) with VyO = y. Set U = cI>-l(V) and x = Uxo. Then cI>(U)Yo = y = ",(x), and so
Banach and topological algebras
232
Tf is surjective. Finally, let (x n ) E co(N, E), and set Un = Xn ®>.o (n EN). Then (Un) E co(N,13(E» and Unxo = Xn (n EN), so that 7J(xn) = cp(Un)Yo -+ 0 as n -+ 00, and hence 7J is continuous. It follows that Tl : E -+ F is a linear
homeomorphism.
0
We now give some standard properties of Banach operator algebras. As in Appendix 3, Lat T is the lattice of invariant subspaces of T E 13(E) for a Banach space E.
Theorem 2.5.8 Let E be a non-zero Banach space. (i) Let Qt be an operator algebra on E. Then: E is a szmple left Qt-module; Qt 1,S a primitive, and hence semzsimple, algebra; Qtc = Ch; 13(E) zs left and right faithful over Qt; F(E) is the mmimum ideal in ~. (ii) Let Qt be a closed operator algebra on E. Then A(E) is the minimum non-zero, closed ideal in ~.
(iii) Suppose that E zs znfinite-dimensional and that Qt is a closed operator algebra on E. Then the strong radzcal9t(Qt) of~ contams A(E); Qt is a semiszmple Banach algebra whzch is not strongly semisimple. (iv) The algebra K,(E)jA(E) is a radical Banach algebra.
Proof (i) Take Xo, Yo E E e . Then there exists S E F(E) c Qt such that SXo = Yo. It follows that E is a simple left Qt-module, and so Qt is a primitive algebra. Let S E F(E)c. For each x E Ee, take>. E E' with (x. >.) = 1. Then it follows from (2.5.1) that Sx ® >. = x ® S'(>'); evaluating both sides of this equation at x. we see that Sx = (Sx, >,)x E Cx. It follows that S E CIE , and so Qtc = ClEo For x E Ee, choose>. E E' with (x, >.) = 1 and set S = x ® >. E Qt. Take T E 13(E) with TQt = O. Then Tx = TSx = 0, and so T = O. Thus B(E) is left faithful over Qt. Now take T E 13(E) with QtT = O. Then ST = x:8l T'(>') = 0, and so T' = 0 and T = O. Thus B(E) is right faithful over Qt. The argument that F( E) is the minimum ideal in Qt is exactly that in Example 1.3.36: now we choose Xo, Yo E E e with SXo = Yo, and then choose >'0 E E' with (Yo, >'0) = 1. (ii) and (iii) These are immediate from (i). (iv) Take T E K(E). By the standard properties of compact operators, for each z E u(T) \ {OJ there exist a finite-dimensional subspace F in Lat T and G in LatTsuch that E = FEBG and (zIE-T) I G E Inv 13(G). Thus zIE-T+A(E) is invertible in (K,(E)j A(E»#. and so T + A(E) is quasi-nilpotent in K(E)j A(E). By 1.5.32(iv), K,(E)jA(E) is a radical algebra. 0
Theorem 2.5.9 (i) Let E be a Banach space wzth AP, and let Qt be a closed operator algebra on E. Then K{E) is the minimum non-zero, closed ideal in Qt. (ii) Let E = Co or E = £P for p E [1, (0). Then K(E) is the unique nontrwial, closed ideal in 13(E), and K{E) is the strong radical of B{E). Proof (i) Since E has AP, IC(E) = A{E); the result follows from 2.5.8{ii).
Banach algebras of operators
233
(ii) By (i), IC(E) is the minimum non-zero, closed ideal in B(E). Suppose that J is an ideal of B(E) with J IC(E), and take T E J \ IC(E). By A.3.58(ii), T tf- See), and so there is a closed, infinite-dimensional subspace F of E such that T IF: F ~ T(F) is a linear homeomorphism. By A.3.58(i). there is a closed, complemented subspace G of E with G c T(F) and G ~ E. We have E ~ T-1(G), say A: E ~ T-l(G) is a linear homeomorphism. Also let P E 3(B(E)) with P(E) = G, and set S = (TA)-1 : G ~ E. Then SPT A = IE and SP, A E B(E). so that h; E J and J = B(E). Thus IC(E) is the maximum ideal in B(E). It is now clear that IC(E) is the strong radical of B(E). 0
rt
By 2.5.8(ii). the algebra A(E) is topologically simple for each non-zero Banach space E, and, by 2.5.9(ii), the Calkin algebra B(E)jIC(E) is a topologically simple (and hence simple) Banach algebra in the cases where E = Co or E = f,P for p E [1,00). Proposition 2.5.10 Let E be a Banach space. Then A(E) zs plzable. Proof We may suppose that E is infinite-dimensional. Since A(E) if> topologically simple, the only closed ideal of infinite co dimension is O. Let (Xk) be a basic sequence in E. and set F = lin {Xk : kEN}. For each x = 2:%"=IO'.kXk E F, define Pn(x) = 2:~=1 O:/.:X", (n EN), and extend each P n to be an element of B(E) with P,,(E) = lin{xl, ... ,Xn }. Then (Pn ) C F(E), PmPn = P mAn (m, n EN), and Pn+l - p .. =f. 0 (n EN). Thus 2.2.22(ii) applies to show that A(E) is pliable. 0
We next consider when the Banach algebra B(E) has a continued bisection of the identity (see 1.3.24); the result will be seen to be significant in 5.4.12. Theorem 2.5.11 Let E be a Banach space which zs lineady homeomorphzc to E EB E. Then B(E) has a contmued bisection of the identity. and B(E) ha.~ no
proper ideal of jinzte codzmenszon. Proof We have E ~ E EB E ~ (E@E) ffi E ~ «E EB E) EB E) EB E ~ ... ;
at the nth stage E is linearly homeomorphic to the direct sum of n + 1 copies of itself. Let Pn and Qn be the projections of E onto the first and second of these components, respectively. Clearly IE = PI +Ql and Pn = Pn+ 1 +Qn+l (n EN). Let Un be the operator on E which exchanges the first two of the n+l components at the nth stage of the decomposition. Then P,~ = UnQnUn and Qn = UnPnUn . Thus {(Pn ) , (Qn)} is a continued bisection of the identity. The last clause follows from 1.3.34. 0 A collection of examples of Banach spaces E with E ~ E ED E is given in the notes. We next consider the multiplier algebra of a Banach algebra. The multiplier algebras Ml(A) and M(A) of an algebra A were described as subalgebras of £(A) and £(A) x £(A)OP, respectively, after 1.4.25; in the case where A is faithful,
Banach and topological algebras
234
A is rpgaroed as an ideal in M(A). In the next results, we take the norm on B(A) x B(A) to be given by II(S, T)II
= max{IISII, IITII} (S, T
E
B(A».
Proposition 2.5.12 Let A be a Banach algebra which is left and right faithful.
Then: (i) M(A) is a closed, unital subalgebra of B(A) x B(A)OP;
(ii) the embeddmg of A m M(A) zs continuous: (iii) the product maps from A x M(A) into A ar-e contmuous. Proof (i) Let (L. R) E M(A). Suppose that (an) E co(N, A) and L(a n ) --+ b in A. For each c E A, we have cb = limn -+ CXl cL(an) = limn ..... CXl R(c)a n = 0, and so b = because A is right faithful. By the closed graph theorem A.3.25, L is continuous. Similarly R is continuous. Thus M(A) C B(A) x B(A); clearly M(A) is a closed subspace of B(A) x B(A).
°
(ii) For each a E A, II (La, Ra) II ::; Iiali.
(iii) We have lIa . (L, R)II = IlRaIl ::; II(L, R)lllIall (a E A, (L, R) E M(A». Thus the product on the left is continuous; similarly, the product on the right is continuous. 0 In the case where A is commutative and faithful, we regard M(A) as a closed subalgebra of B(A). Theorem 2.5.13 (Johnson) Let E be a Banach space, and let (Q1, III·IID be a
Banach opemtor zdeal on E. Then: (i) T I-t LT I Q1, B(E) --+ M£(Q1) c B(Q1), zs an isometric isomorphzsm; (ii) T I-t (LT I Q1, RT I Q1), B(E) --+ M(Q1) c B(Q1) x B(Q1)OP, zs an tsometr"ic isomorphism. Proof (i) For T E B(E), LT I Q1 E B(Q1) by 2.5.5(i), and liLT I Q111 ::; IITII; the map () : T I-t LT I Q1, B(E) --+ Me(Q1) n B(Q1) , is a monomorphism because, by 2.5.8{i), B(E) is left faithful over Q1. Let L E Me(Q1), and suppose that Yt,Y2 E E and A l ,A2 E Q1 are such that A l Y1 = A2Y2. If Yl = Y2 = 0, then L(AdYl = L(A2)Y2. If Yl =I- 0, take>. E E' with (YI. >.) = 1, and set Sl = Yl ® >. and S2 = Y2 ® >.. Then S1, S2 E Q1, SlYl = Yl, S2Yl = Y2, and AlSl = A2S2, so that
L(Al)Yl = L(AdSlYI = L(AISdYl = L(A2S2)Yl = L(A2)Y2 . We can thus define TL : E --+ E by the formula TL(x) = L(A)y for any A E Q1 and Y E E such that Ay = x. Let al,Q2 E C and XI,X2 E E, and take P to be a projection of E onto lin {Xl, X2}, so that P E Q1. Then
TL(QIXl + Q2X 2) = L(P)(QlXl + Q2X2) = QIL(P)(Xl) + Q2L(P)(X2) = QITL(Xl) and so TL is linear.
+ Q2TL(X2) ,
Banach algebras of operators
235
We show that TL E B(E). Assume towards a contradiction that there exists {xn : n E N} in E with IIxnll < 2- n (n E N) and II Tdxn) II --+ 00. Take a biorthogonal system in E consisting of sequences (Yn) E E[l] and (>'n) in E' with sUPnEN II>'nli < 00 such that (Ym,>'n) = c)m,n (m,n EN). Define 8 = L::'=l Xn ® >'n, so that, by (2.5.3), 8 E Qt. For each n E N, we have 8Yn = Xn, and so, by the definition, TL(xn) = L(8)Yn· But IIL(8)Ynll ::; IIL(8)1I and IITdxn) II --+ 00, a contradiction. Thus TL E B(E). Clearly ()(TL) = L, and so Me(Qt) is a closed subalgebra of B(Ql) and () : B(E) --+ Mt(Qt) is a surjection. Again take L E Mt(Qt). For x E E with IIxII = 1, take>. E E' with (x, >.) = 1 and 11>'11 = 1. Then Ax = x, where A = x ® >., and so IITLxl1 = IIL(x ® >,)xll ::; IILII. This shows that () is an isometry. (ii) By 2.5.12(i), every multiplier on Qt is continuous. By 1.4.26, the specified map is an isomorphism. By (i), it is an isometry. 0
At a later stage, we shall study derivations from Banach algebras. As a harbinger of these later results, we prove by elementary means that all derivations on the Banach algebra B(E) are inner, and hence continuous. Theorem 2.5.14 (Chernoff) Let Qt be an operator algebra on a Banach space E, and let D : Qt --+ B(E) be a derivation. Then there exists T E B(E) such that
D(A) = AT - T A
(A
E Qt) .
Proof Fix Xo E E- and >'0 E E' with {xo, >'o} = 1, and set Po = Xo ® >'0, so that Po E J(Qt). By 1.8.2(i), PoD(Po)Po = O. Set Yo = PoD(Po) - D(Po)Po· Then D(Po) = PoD(Po) + D(Po)Po = PoYo - YoPo · Now set Dl (A) =
Then Dl : 21
--+
D(A) - (AYo - YoA)
B(E) is a derivation, Dl (Po)
(A E Qt) .
= 0, and
DI (APo) = Dl (A)Po (A
E Qt) .
The map 'f/l : APo t--+ Axo, QtPo --+ E, is a well-defined linear isomorphism. If AI, A2 E Qt with AIPO= A 2PO, then
DI(Al)xo
= DI(AdPoxo = DI(AIPO)xo = D1(A2PO)xo = DI(A2)xo,
and so 'f/2 : APo t--+ D1(A)xo, QtPo --+ E, is well-defined; clearly 'f/2 is linear. Set S = 'f/2 0 'f/ll : E --+ E, so that 8 is a linear map. For each x E E, define Ax = x ® >'0, so that Ax(xo) = x (x E E). Take x E E and A E Qt. Then 171(APoA x Po) = APox, and hence
S(APox) = 172(APoA x Po) = Dl(APoAx)xo. Also PoAx = (x, >'0) Po, whence Dl (APoAx) = (x, Ao)Dl (APo) = (x, Ao)Dl (A)Po = Dl (A)PoAx.
Banach and topological algebras
236
Since PoAxxo = (x, A)XO = Pox, we have SAPo = D1(A)Po (A E 2t). Apply this equation with A replaced by AAx: we obtain
SAAxPo = D 1(A)A x Po + AD1(A x )Po (x
E
E, A
E
2t).
Evaluate both sides of this equation at Xo: we have (SAAxPo)(xo) = (SA)(x), (Dl(A)AxPo)(xo) = D 1(A)x, and (AD1(Ax)Po)(xo) = A712(AxPo) = (AS)(x) because 711 (AxPo) = x. Thus SA = D 1(A) + AS (A E 2t). Finally, set T = Yo - S. Then T E .c(E) and D(A) = AT - T A (A E 2t). It remains to prove that T is continuous. Take A E E', and set A = Xo ® A. Let (xn) E co(I~, E). Then D(A)x n -+ 0 and T AXn = (x n . A)TxO -+ 0, and so (Tx n , A)XO = ATxn -+ 0 as n -+ 00; this shows that A 0 T is continuous. Hence T is continuous. 0
Corollary 2.5.15 Let E be a Banach space. Then each derivatwn on B(E) is mner. 0 The algebra B(E) is also defined when E is an arbitrary topological linear space. However, the following result shows that B(E) is usually not a topological algebra, and so the theory of non-commutative LMC algebras, for example, is rather disappointing.
Theorem 2.5.16 (i) Let (E, r) be a locally convex space, and suppose that the algebra E ® E' zs topologizable. Then there is a norm II . lion E such that the embeddmg (E, II·ID -+ (E, r) is continuous. (ii) Let (E, r) be a F'rechet space whzch is not a Banach space. Then the algebra E ® E' is not topologizable. In partzcular, B(E) zs not a topological algebra. Proof (i) For each Xo E E e , there exists AO E E' with (xo, Ao) i- O. Define K and PK as in 2.2.50. The argument of that proof now shows that PK is a norm, denoted by 11·11, say, on E. Set B = {x E E : IIxll ::; I}; by the analogue of (2.2.11), B is weakly bounded in (E, r), and so, by A.3.38, B is bounded in (E, r). This shows that the embedding (E, 11·11) -+ (E, r) is continuous.
(iii) It now follows from A.3.41 that E®E' is not topologizable. Since E®E' is a sub algebra of B(E), B(E) is not topologizable. 0 Notes 2.5.17 There is an extensive theory of Banach operator ideals. Let E be an infinite-dimensional Banach space. Then {T E B(E) : T
+ A(E)
E rad(B(E)jA(E))}
is the closed ideal of inessential opemtors; this ideal is contained in the strong radical of B(E). The set of operators in B(E) which map weakly convergent sequences into norm-convergent sequences on E forms the Banach operator ideal of the completely continuous opemtors on E. Further Banach operator ideals on E are the ideals of p-summing operators. For discussions of these ideals and of the other ones we have mentioned, see (Caradus et al. 1974), (Defant and Floret 1993), (Diestel and UhI1977), (Diestel et al. 1995), (Jameson 1987), (Palmer 1994), and (Pietsch 1980), for example. The Calkin algebra B(H)jlC(H) for a Hilbert space H was first studied in (Calkin 1941). It is a famous open problem whether or not there exists a Banach space E
Banach algebras of operators
237
such that B(E) = CIe ffi qE); a Banach space E such that B(E) = CIe ffi See) is constructed in (Gowers and Maurey 1993). Our 'nuclear algebra' of 2.5.4 is usually called thf' tensor algebra of thf' Banach space E (Helemskii 1989b, 11.2.20), but we have used that term for a different algebra. Eidelheit's theorem 2.5.7 is from (1940); this was probably the first paper that included 'automatic continuity' results for a Banach algebra Here are some examples, mainly taken from (Banach 1932), of Banach spaces E such that E ~ EffiE: (i) fP(S), where S is an infinite set and p E [1,00]; (ii) any infinitedimensional Hilbert space; (iii) C([O, aj), where a is an ordinal with W :-:; a :-:; WI or a is a singular cardinal; (iv) C(Q), where Q is an infinite, metrizable, compact space; (v) c(n)(H), for n EN; (vi) LP(H), for p E [1,00]. For a Banach space with a unconditional basis, B(E) has a continued bisection of the identity. On the other hand, for the spaces E = C([O, ,..j), where,.. is a regular cardinal with ,.. ~ WI, and E =:1, the James space (see 4.1.44), B(E) does not have a continued bisection of the identity; see (Kislyakov 1975) and (Loy and Willis 1989), respectively. There is a reflexive Banach space E with E 1- E constant C and a sequence (Xn) with the following
properties: (i) for each n E N, the map Xn : A -+ IC is an approximate character on B(E) of type (C,C/n2)j (ii) for each T E B(E) and n E N, we have IXn(T) - Xn+1(T)1 ~ C /n 2j (iii) for each T E K.(E), we have (Xn(T» E co; (iv) for each (Q,,) E Co, there exists T E A(E) with Xn(T) = 'L~n Qk/k2.
"TIl
We claim that A(E)nK.(E? has infinite codimension in A(E). In particular, K.(E) does not factor, and there are discontinuous point derivations on K.(E). To see this, take T E A(E) such that Xn(T) = l/nlogn (n EN); such an operator T exists by (iv). Assume towards a contradiction that T = 'L~=1 PjQj for some PI, ... , Pk, Ql ... ,Qk E K.(E). Then, since Xn satisfies properties (iii) and (iv) of an approximate character with c: = C/n2, there exists a constant C' such that I 12: Xn(Pj)Xn(Qj) k
1
nlogn
I ~ c'n 2
(n E N),
13=1
and so there exists P E {Pt, .... Pk,Ql ... ,Qk} with IXn(P)I2: 1/2k(nlogn)I/2 for infinitely many n E N. Since (X" (P» E Co by (iii), it follows that, for each such n E N, there t>xists m 2: n with 1
2k IXm(P) - Xm+l(P)1
1
2: m1/2(logm)1/2
>
(m
+ 1)1/2(log(m + 1))1/2
1
- (m + 1)3/2(log(m + 1»)1/2 '
and hence, by (ii), such that 4kC lIP"
2: m2IX>n(P)
- Xm+l(P)I2: m 1/ 2(logm)-1/2
2: nl/2(logn)-1/2.
But it cannot be that this holds for infinitely many n E N, and so T ¢ K.(E)2. A small variation of this argument establishes the claim. We note that the Banach space E does not have AP. 2.6
0
BANACH MODULES
The theme of the present section is that of Banach modules over a Banach algebra, and their role in the elucidation of the structure of Banach algebras. Banach modules and the related weak Banach modules will be defined in 2.6.1, and a large number of examples will be given in 2.6.2; these examples will be studied in greater detail later. The Arens products, which make the second dual A" of a Banach algebra A into a Banach algebra, will be defined in 2.6.16. The algebraic theory of §1.4 will be sharpened in the case of Banach algebras by the use of the theory of modules that has been developed to obtain various results which show restrictions on the structure of Banach algebras. Definition 2.6.1 Let A be an algebra, and let E be a Banach [an (F)-] space which is a left A-module. Then E is a weak Banach [weak (F)-] left A-module if the map pea) : x 1-+ a . x, E --+ E, is continuous for each a E A. A weak (F)-left A-module E is essential if AE = E. Suppose that A is a topological algebra. Then E is a Banach [an (F)-lleft A-module if the map (a,x) 1-+ a . x, A x E --+ E, is continuous.
239
Banach modules
Similar definitions apply to Banach right A-modules, etc. A [weak] Banach A-bzmodule is an A-bimodule which is both a [weak) Banach left A-module and a [weak] Banach right A-module. Let E be a weak (F)-A-bimodule. Then E is essential if AEA = E. Clearly a neo-unital module is essential. If A is unital and E is essential, then E is unital. Later we shall be particularly concerned with essential modules. The difference between Banach and weak Banach modules is very important. A Banach space E which is a left A-module is a weak Banach left A-module if, for each a E A, there is a constant C a with Iia . xii::; C a IIxll (x E E); in the case where A is a normed algebra, E is a Banach left A-module if there is a constant C such that I/a . xII ::; C I/a\l \lx\l (a E A, x E E). Thus a weak Banach left A-module E is a Banach left A-module if and only if the corresponding representation p : A --t B(E) is continuous. Let A be a normed algebra, and let (E, II· \I) be a Banach left A-module. For x E E, set Illxlll = sup{\lxll, Iia . xII : a E A[l]}. Then 111·111 is a norm on E which is equivalent to 11·11, and Ilia· xiii::; Ilalllllxlll (a E A, x E E). A similar result applies in the case where (E, II· I\) is a Banach A-bimodule, and so we shall suppose throughout that Iia . xii::; Ilallllxll ,
Ilx· all ::; II all \Ix \I
(a E A, x E E) .
(2.6.1)
Suppose that a, bE A, x E E, and kEN. Then \Ix -
ak • xii::; \Ix - a. xII (1 + lIall + ... + Ilak-IID .
(2.6.2)
Let A be a topological algebra, and let E be a [weak) Banach left A-module. Then E is a unital [weak) Banach left Ab-module with respect to the map (ael>
+ a, x)
f-+
ax
+a
.
x,
Ai>
X
E
--t
E.
Let A be a commutative algebra, and let E be a symmetric weak Banach A-bimodule (so that E is an A-module in the terminology of 1.4.2). Then E is a weak Banach A-module. Similarly, we refer to Banach A-modules, etc., in the case where A is also a topological algebra. Let A be an algebra, and let E and F be weak Banach left A-modules. The set of continuous left A-module homomorphisms from E into F is denoted by AB(E, F), so that AB(E, F) = AC(E, F) n B(E, F). Similarly, we define BA(E, F) and ABA(E, F), and we write AB(E) for AB(E, E), etc. Clearly AB(E, F) is a closed linear subspace of B(E, F), and AB(E) and ABA (E) are unital subalgebras of B(E). Two Banach left A-modules lA-bimodules) E and F are isomorphic, again written E ~ F, if there exists a bijection T E AB(E, F) [T E ABA(E, F»). We first give an elementary, but useful, construction involving modules. Let A be a Banach algebra, and let E be a Banach A-bimodule. As in 1.8.14, !2l = A 0 E is an algebra for the product (a,x)(b,y) = (ab, a· y +x· b)
(a,b E A, X,y E E).
Also, 21 = A EEloo E is a Banach space, and it is immediately checked that 21 is a Banach algebra; it is denoted by A EEl E.
Banach and topological algebras
240
There are many naturally occurring examples of Banach A-bimodules: indeed, the notion is ubiquitom; in our subject. Examples 2.6.2 (i) Let A be an (F)-algebra. Each left ideal I in A which is an (F)-space continuously embedded in A is an (F)-left A-module with respect to the multiplication in A. To see this, denote the topologies of A and I by T A and T[, respectively. Then the bilinear map
(a,x)
t-+
a . x,
(A,TA) x (I,TA)
---+
(I,T[)
is separately continuous. By A.3.39, it is continuous, as required. Similarly, an ideal in A which is an (F)-space continuously embedded in A is an (F)-Abimodule. For example, a Banach operator ideal on a Banach space E, such as (N(E) , II·IIJ. is a Banach B(E)-bimodule. In the case where I is a closed left ideal in A, A/lis an (F)-left A-module with respect to the left regular representation on A/I. For each closed ideal I in A, we obtain (F)-A-bimodulcs in a similar way. In particular, an (F)-algebra A is an (F)-A-bimodule over itself, and our terminology for essential bimodules coincides with that of 2.2.13. (ii) Let A be a Banach algebra, and let cp, 'ljJ E 4> AU {o}. The A-bimodule Ccp,'I/J (with the operations defined in (1.4.2» is a Banach A-bimodule. Clearly Ccp ~ A/M.., for cp E 4>A.
(iii) Let A be a commutative Banach algebra with 4> A :f:. 0, and let (E, 11·11) be a Banach space continuously embedded in (fOO(4>A), 1·IJ such that
lIafll ::; lalA IIfll
(a E A,
fEE).
(2.6.3)
Then E is a Banach A-module for the map (a, f) t-+ af. In this case, E is a Banach A-module of functions. In particular, X, f OO (4)A), C b(4)A), and CO(4)A) are Banach A-modules of functions. (iv) Let A be a topological algebra, and let E be a Banach left A-module. Set x . a = 0 (a E A, x E E). Then E is a Banach A-bimodule; it is a Banach nght-annihilator A-bimodule. A Banach space E such that A . E = E . A = 0 is an Banach annihilator A-bimodule. Suppose that J is a closed ideal in A and that E is a Banach A-bimodule. Then E is a Banach J-bimodule; in the case where J . E = E . J = 0, E is also a Banach (A/ J)-bimodule for the natural operations. (v) Let A be an algebra, and let E be a weak Banach left or right A-module. Define A . a and a . A, respectively, in E' for a E A and A E E' by (x, A . a) = (a . x, A),
(x, a . A) = (x . a, A)
(x E E) ,
(2.6.4)
essentially as in (1.4.8). Then E' is, respectively, a weak Banach right or left A-module. In the case where E is an A-bimodule, E' is also an A-bimodule. The space E' with these maps is the dual module of E. Note that A . a = p(a)'(A) in E', where p : A ---+ B(E) is the representation corresponding to the left module action on E; for each a E A, the map A t-+ A . a is continuous when E' has the (T(E', E)-topology.
Banach modules
241
Suppose that A is a Banach algebra and that E is a Banach left A-module, etc. Then E' is also a Banach right A-module, etc. In particular, the Banach space A' is a Banach A-bimodule when the module maps are given by the formulae
(a . A)(b)
= A(ba),
(A' a)(b)
= A(ab)
(b
E
A);
(2.6.5)
A' is the dual module of A. If A is unital, then A' is a unital A-bimodule. In the ca.',e where A is nonunital with adjoined identity e, take e' E A#' with (e, e') = 1 and e' I A = 0, and extend A E A' to an element of A#' by setting (e, A) = O. Then A#' = <ee' EBooA' as a Banach space, and (ae
+ a)
. be' + A) = (a')' + (a, A) )e' + aA + a . A.
(2.6.6)
Note that A' is not a submodule of A#'. Here is a specific example of the above situation. Let n be a non-empty, locally compact space, and let A = Co(n), so that A' = M(n), the space of measures on n, as in Appendix 4. Then the dual module actions on M(n) coincide with the natural action (j, f.L) ~ f f.L, where
(g, ff.L) =
l
fgdf.L
(g
E
Co(n».
Let E be a Banach left A-module. Then E" is a Banach left A-module, and we may regard E as a closed submodule of E". Indeed, we clearly have
a . A = p(a)"(A)
(a E A, A E E"),
where p is the corresponding representation; by A.3.56(ii), a . E" c E if and only if pea) is weakly compact. For each a E A, the map A ~ a . A, E" -+ E", is continuous when E" has the a(E", E')-topology. Let E be a Banach A-bimodule. Then the Banach spaces E', E", E"', ... are Banach A-bimodules, and the map
p;
r ~ r I teE),
EIII
-+
E' ,
is a projection with range E' and with IIPII = 1 which is an A-bimodule homomorphism; we have ker P = t(E)O, and so, as a Banach A-bimodule, EIII
= E' EB t(E)O .
(2.6.7)
The map P is termed the natural projection. The nth dual E{n} is a Banach A-bimodule for each n E N, and there is a similar projection from E{n+2} onto the canonical image of E{n} in E{n+2}. We now give a second example of the above situation. Let 8 be a semigroup, and let A = (£ 1(8), *,11,11 1) be the semigroup algebra, as in 2.1.13(v). The dual module of A is A' = £00(8), with the duality (j, A) = LSES f(8)A(8) for f E A and A E A', the module operations being specified by the requirements that
(t5t
.
A)(U) = A(ut),
(A' t5t )(u) = A(tU)
(t, u E 8, A E A').
(2.6.8)
The space co(8) is a closed linear subspace of A'. Now write As for the point mass at s E 8, regarded as an element of co(8). Then Dt . As = X[st-1j (s, t E 8) Where [srI] = {u E 8 ; ut = s}, and similarly for >'s . Dt. It follows easily that co(8) is a submodule of the bimodule A' if and only if, for each s, t E 8, the
242
Banach and topologzeal algebras
equations sx = t and xs = t each have only finitely many solutions for x. Of course this is the case for each cancellative semigroup S; in this latter case it is immediate to check that the dual module of co(S) is the algebra A itself, and so A is a dual module. In particular, in the case where G is a group, eo(G) is a Banach £ 1 (G)-bimodule for operations specified by the conditions that 8t . As = Ast-1,
As' 8t = 8t -l..
(8, t E S).
(2.6.9)
A generalization of these remarks will be given in 3.3.15. (vi) Let A be a Banach algebra, and let E be a [weak] Banach left A-module. The Banach spaces £P(N, E) (for 1 :S p:S 00) and eo(N, E) are defined in A.3.74. Define a . (xn) = (a . x n ) (a E A, (x n ) E £P(N, E)). (2.6.10) Then each £P(N, E) is a [weak] Banach left A-module, and eo(N, E) is a [weak] Banach left A-module which is a submodule of £OO(N,E). Similarly, E(n) is a [weak] Banach left A-module for each n E N. In the case where E is a Banach A-bimodule, lP(N, E), eo(N, E), and E(n) are also Banach A-bimodules. Let n E N and b1 , .••• bn E A. Then clearly { (al>'" ,an) E A(n) : tajbj =
o}
(2.6.11)
J=l
is Banach left A-module; it is a closed submodule of A(n). Let Ap be an Arens-Hoffman extension of a commutative, unital Banach algebra A by a monic polynomial p, as in 2.1.18(vi). Then Ap is a Banach A-module for the map given by a . (L~=o biXi) = L~=o abiXi. Let n be a non-empty, compact space, and set F = C(n. E). For a E A and f E F, define (a . f)(x) = a . f(x) (x En). Then F is also a Banach left A-module. (vii) Let A and B be (F)-algebras, and let () : A - t B be a homomorphism. As in 1.4.8(ii), B is an A-bimodule. In fact, B is a weak (F)-A-bimodule; if () is continuous, then B is an (F)-A-bimodule. In particular, we may consider the case where A is a subalgebra of Band () is the embedding of A into B. More generally, suppose that () is continuous and that E is a Banach Bbimodule. Then E is a Banach A-bimodule for the maps (a, x) t--+ ()(a) . x and (a, x) t--+ x . ()(a). (viii) Let A be a Banach algebra, and let E and F be Banach left A-modules. Recall from (1.4.12) that, for a E A and T E B(E, F), we set (a . T)(x) = a . Tx,
(T x a)(x) = T(a . x)
(x E E) .
(2.6.12)
Then B(E, F) is a Banach A-bimodule with respect to the maps (a, T) t--+ a . T and (a, T) t--+ T x a. Now suppose that E is a Banach A-bimodule. Then B(A, E) is a Banach A-bimodule in this way: here, (a . T)(b) = a . Tb and (T x a)(b) = T(ab) for a, bE A and T E B(A, E). For x E E, define j(x) : at--+ x . a,
A-tE,
(2.6.13)
Banach modules
243
so that j E ABA(E, BA(A, E)) and IIJII :::; 1. In the case where A is unital and E is a unital A-bimodule, B(A, E) is unital and j is an isometry, and so j(E) is a closed A-submodule of B(A. E); we have a short exact sequence
L
E :
0
---+
E
---L B(A,E) ~ B(A,E)jj(E) ---+ 0
(2.G.14)
of Banach A-bimodllies and continuous A-bimodule maps. Suppose that E is an essential Banach A-bimodule. Then we cla1,m that the map j : E' -+ BA(A,E'), where j(A)(a) = A . a (a E A, A E E'), is injective. For suppose that A E E' and j(A) = 0, so that A . a = 0 (a E A). Take (J E A and '!J E E. Then (a . y, A) = O. Since E is essential, (x, A) = 0 (x E E), and so A = 0, fl...') required. We identify B(A, A') with B2(A; C); the module operations hecome
(a· A)(b,c) = A(b,ca)
(a,b,c E A, A E B 2 (A;C)).
(A' a)(b,c) = A(ab, c)
(2.G.15)
Finally, suppose that A is a commutative Banach algehra and that E and F are Banach A-modules. Then, as in 1.4.13, AB(E, F) is a Banach A-module for the map (a, T) f-> a . T, and j(E) C AB(A, E). Ox) Let E be a Banach space, and let T E B(E). Then E is a unital weak qXl-module with respect to the map (p,x) f--+ p(T)J;, qXl x E --7 E (c/. Example 1.6.14). Let ~ he a Banach operator algebra on E. Then E is a Banach left ~-module for the map (T, x)
f--+
Tx ,
~
xE
-+
E.
In this case, E iH termed the natural module over~. If we then regard B( E) as a Banach ~-bimodule formed from the Banach left ~-module E as in (viii), then the module operations on B(E) are given by the usual product in B(E). The nuclear algebra E 0 E' is a Banach B(E)-bimodule for operations' that agree with the product in the ideal F(E). It followH from 2.5.5(ii) and A.3.69 that there is a unique continuous linear map Tr : E 0 E' -+ C such that
Tr (J; ® A) = (x, A)
(.1:
E E,
A E E') .
The map Tr is again called the tmce map; each continuous trace on E scalar multiple of Tr. Clearly
Tr(8 . T) = Tr(T . 8),
ITr(S· T)I :::;
1181111" IITII (8 E
E
®E', T
E
0 E'
is a
B(E)). (2.6.16)
In the case where E has AP, we obtain a trace on N(E). (x) Let A be a unital Banach algebra, let E be a unital Banach left A-module, and let n E N. Set ~ = Mn(A), an in 2.1.18(ii). The Banach space E(n) is a unital Banach left ~-module with reHpeet to the map n
. Xj, ..• ,
Lan) j=l
given by matrix multiplication.
244
Banach and topological algebms
(xi) Let U be an open set in en, and let E be a Banach space. Then the space C(U, E) of continuous E-valued functions on U is a Fn3chet C(U)-module for the product specified by
(J . F)(z) = f(z)F(z)
(J
C(U), FE C(U, E), z E U).
E
o
Similarly, O(U, E) is a Frechet O(U)-module.
Further examples of Banach modules over specific Banach algebras will be given in §3.3. The generically most important of the above examples are the dual modules considered in Example 2.6.2(v). \Ve shall now show that certain naturally occurring modules are dual modules. Proposition 2.6.3 Let A be a Banach algebra, let E be a Banach left A-module, and let F be a closed submodule of E. Then F and E j F are Banach left Amodules, and thezr dual modules are zsometrically 'Isomorphic as Banach right A-modules to E' j FO and FO, respectively. Proof Certainly F and E j F are Banach left A-modules, and FO is a right Asubmodule of E'. The isometric linear bijections F' ---t E' j FO and (E j F)' ---t FO of A.3.47 are right A-module homomorphisms. 0
In particular, in the case where I is a closed ideal in a Banach algebra A, we have the identifications I'=A'jIO
and
(AjI)'=Io.
In the following theorem, B(E, F') is a Banach A-bimodule with the products defined in (2.6.12); the theorem shows that B(E, F') is a dual module. Theorem 2.6.4 Let A be a Banach algebra, let E be a Banach left A-module, and let F be a Banach rzght A -module. Then:
(i) E@F zs a Banach A-bimodule for the products defined by a . (x 0 y)
=
(a . x) 0 y,
(ii) the map
T ---t
(x 0 y) . a
=x
0 (y . a)
(a E A, x E E, y E F);
TT' where
(y, Trx)
=
(x 0 y, r)
(x E E, y E F) ,
is an isometric A-bzmodule isomorphism from the dual A-bimodule (E@ F)' onto B(E,F'). Proof (i) Let a E A. By A.3.69, the maps pc(a) and Pr(a) of 1.4.9(ii) define elements of B(E@F), and the maps a I--> pc(a) and a I--> Pr(a) are continuous. As before, the result follows.
(ii) By A.3.70, the map
TT' (E@F)' ---t B(E, F'), is an isometric linear bijection; as in 1.4.14(i), it is immediately checked to be an A-bimodule homomorphism. 0
Similarly, we obtain a Banach left A-module E@F in the case where E is a Banach left A-module and F is a Banach space.
Banach modules
245
Corollary 2.6.5 Let A be a Banach algebm. Then A ®A is a Banach Abimodule, the pmjective induced product map 1fA : A ®A -+ A is a contmuous A-bimodule homomorphism, and 111: zs a closed submodule of A ®A. The dual module to A ®A is isomet1"tcally isomorphzc to B2 (A; q. 0 The following result is an easy verification. Proposition 2.6.6 (i) Let A be a commutatwe Banach algebm, let E be a Banach A-module. and let A E E'. Then there exists R>.. E AB(E, A') such that (a, R>..x)
=
(a . x, A)
(a E A, x E E) .
(ii) Let A be a Banach algebm, let E be a Banach A-bzmodule. and let A E E'. Then there exzsts R>.. E ABA(E, (A®A)') such that (a ® b, R>..x)
=
(b . x . a, A)
(a, bE A. x E E) .
o
Examples 2.6.7 (i) Let A be a Banach algebra which is a closed left ideal in a Banach algebra B, and let E be a Banach space. Then B ®E is a Banach left A-module for the product defined by a . (b ® x) = ab ® x
(a E A, bE B, x E E).
In particular, a module of the form A# ®E is a free Banach left A-module. Similarly, the modules A# 0 E ®A# are the free Banach A-bimodules. (ii) Let E be a Banach space, and set Qt = E ®E', the nuclear algebra. Then E is a Banach left Qt-module for the module operation given by
(x ® A) . y = (y. A)X
(x, y
E
E, A E E') ,
and its dual E' is a Banach right Qt-module. The bimodule operations described in 2.6.4(i) coincide with the product in Qt, and so, by 2.6.4(ii), the dual module of Qt is B(E, E") with the module operations a .T
=
(Ra)"
0
T,
T· a
=T
0
Ra
(a E Qt, T E B(E, E")),
(2.6.17)
where R is the map specified in 2.5.3(ii). In particular, in the case where E has AP, the dual module to N(E) is B(E, E") with the operations (S, T) ~ T 0 S and (S, T) ~ S" 0 Tj in general, the dual module to N(E) is a closed submodule of B(E, E"). Set K = ker R, a closed ideal in Qt, and take a E Qt, b E K, and T E B(E. E"). Then (ab, T) = (a, b . T) = 0 and (ba, T) = (a, T . b) = 0 by (2.6.17). Since B(E, E") is identified with the dual space of Qt, it follows that ab = ba = 0, and so K is an annihilator ideal in 2(. Now suppose that the Banach space E is reflexive and has AP. Then we have shown that the dual module of N(E) is B(E)j we identify the duality. Indeed, take T E B(E)j the element of N(E)' corresponding to T is AT, where
(x ® A, AT)
=
(Tx, A)
= '!'r(Tx ® A) = '!'r(T
0
(x ® A))
(x
E
E, A E E'),
Banach and topologzcal algebras
246 and so we have the following trace duality: (S, AT) = 'Ir(TS)
(2.6.18)
(S E N(E), T E 8(E».
(iii) Let E be a Banach space. Then we identified the dual of the Banach space A(E) = E®E' with I(E') in A.3.63: for r E A(E)'. we define Sr E I(F, E') hy setting (x. 8.,(>'») = (x 0 >., r) (x E E, >. E E'). It is easily checked that T(E') is the dual module of A(E) with respect to the module products defined h~' A . S = So A' and S . A = A' 0 S for each A E A(E) and S E I(E'). Now suppose that E' ha.'l AP and the Radon -Nikodym property. Then, as in A.3.71(iii), we have the following identifications of Banach spaces:
K(E)
= A(E) =
E®E' i
K(E)' = N(E') = E"@E' i
K(E)"
= 8(E") .
'Ih€' module products on 8(E") as the second dual of K(E) are given by the formulae A· T = A" 0 T and T· A = T 0 A" for each A E A(E) and T E B(E"). III the case where E is reflexive and has AP, the second dual of K(E) is B(E),
with the natural products.
0
Let A be a Banach algebra, let E be a Banach right A-module, and let F be a Banach left A-module. Modifying 1.4.9(iii), we set
N = lin {(x. a) 0 y - x 0 (a . y) : a
E
A, x
E
E, y
E
F},
where we take the closure in E @F. We now define the Banach space (2.6.19)
E@AF = (E@F)/N.
The coset in E ®A F containing the element x 0 y is denoted by x 0 A y. Now ~uppose that E is a Banach A-bimodule, so that E @F is a Banach left Amodule. Then N is clearly a submodule of E@F, and so E ®A F is a Banach left A-module. In particular, in the case where I is a closed ideal in A, we obtain the Banach left A-module I @A Fi clearly the map (Y E AB(I @F. F) with (Y(a 0 y) = a . y (a E A, y E F) is such that (Y I N = 0, and so there is an induced map
u E AB(J@A F, F)
with u(a 0A y) = a . y
(a E A, YEP).
(2.6.20)
Now suppose that El and E2 are Banach A-bimodules, that FI and F2 are Banach left A-modules, and that TI E ABA (E1 .E2) and T2 E AB(FI,F2)' Then, as in Appendix 3, there exists a map T I @T2 E A8(EI @FI ,E2 @F2), and T 1 @T2 induces a map T I @AT2 E AB(E1 @AFI.E2 @AF2) with (T1 @AT2)(Xl 0A X2) = T1Xl 0A T 2X2
(Xl EEl, X2 E E2)'
(2.6.21)
The following result follows as in 1.4.14(ii) and 2.6.4(ii). Proposition 2.6.8 Let A be a Banach algebra, and let E and F be rzght and 0 left Banach A-modules, respectively. Then (E@AF), ~ BA(E,F').
In 1.4.9(iv), we showed how an A-bimodule could be regarded as a left module over the algebraic enveloping algebra A#0A#oP. We require the analogous result for the enveloping algebra Ae = A# @A#op in the case where A is a Banach algebra; the result follows as in the algebraic case by using A.3.69.
247
Banach modules
Theorem 2.6.9 Let A be a Banach algebra, and let E be a Banach A-bzmodule. Then E is a Banach left Ae-module for the product defined by
(a I8l b) . x = a . x . b (a, bE A#, x E E).
o
Consider the special case in which E = A®AoP. Then A®Aop is a Banach left A ®AOP-module in the above sense; the module product is just the product in the algebra A ®AOP. Let E and F be Banach A-bimodules, and denote A ®AOP by B. Take T E ABA(E, F). Then T E BB(E. F). The converse holds in the case where E is essential. We maintain the notation B for A ®AOP in the next proposition. Proposition 2.6.10 Let A be a Banach algebra, let E and F be Banach Abimodules, with F essential. and let T E BB(E'. F'). Then T E ABA(E', F'). Proof Take a, b, c E A, A E E', and Y E F. First we see immediately that (a . A) . (b I8l c) = A . (b I8l cal in E'. Now we have (b . Y . c, T(a . A») = (b I8l c) . y, T(a . A») = (y, (T(a . A» . (bl8l c») = (y, T(a . A) . (b I8l c») = (y, T(A . (b I8l Co.))) = (y, T(A) . (b I8l ca»)
= (b . Y
. ca, T(A») = (b . y . c, a . T(A») ,
and so T(a . A) = a . TA because F is essential. Similarly T(A . a) and so T E ABA(E',F').
=
TA . a, 0
The following definition uses the notation of 2.6.2(vi). Definition 2.6.11 Let A be a Banach algebra, and let E be a weak Banach left A -module. Then null sequences in E factor if Co eN, E) = A . Co (1'\:f, E), and null sequences in E factor weakly if eo(N, E) = Aco(N, E). Thus null sequences in E factor if, for each (xn) in E with Xn -+ 0 as n -+ 00, there exist a E A and (Yn) in E with Yn -+ 0 as n -+ 00 such that Xn = a . Yn (n E N), and null sequences in E factor weakly if, for each such sequence (xn) in E, there exist kEN. 0.1,"" ak E A, and (Yl.n), ... , (Yk,n) in E with Yj.n -+ 0 as n -+ 00 for j E Nk such that Xn = E;=1 aJ • Yj.n (n EN). Proposition 2.6.12 Let A be a Banach algebra, and let E be a weak Banach left A-module in which null sequences factor weakly. Then there exists kEN such that, for each (xn) E co(N, E), there exist al,"" ak E A[k] and (Yl,n) , ... ,(Yk,n) E co(N, E) with 10
sup n
L: IIYj,nll $ sup Ilxnll j=l
n
10
and
Xn
=
L:aj . Yj,n
(n E N).
j=1
Proof Assume that the result fails. Then, for each kEN, there is a sequence (Xk,n) E co(N, E) such that sUPn IIXk,nll = k- 1 / 2 (k E N) and the conclusion fails for this sequence. Form a sequence (Xj) in E that contains every sequence (Xk,n : n E N) as a subsequence. We may factor (Xj) in the form
248
2:;=1 bj
Banach and topological algebras . Yj.n for some b1.... ,bk E A[k] and some (:Yl,n)' .... (Yk,n) E co(N, E)
with sUPn 2:~=1 IIYj,n II ~ sup" holds.
IIx n II·
This is a contradiction. and so the result 0
Proposition 2.6.13 Let A be a Banach algebra. Suppose that null sequences in A factor. Then every maximal nght ideal zn A zs closed. Proof Let lvI be a maximal right ideal in A, and take a E A and (an) in M with an -+ a as n -+ x. By hypothesis, there exist b, bo E A and (b n ) E co(N, A) such that a - an = bbn (n E N) and a = bbo. Set J = {x E A : bx EM}. By 1.4.30 (applied to AOP), either J = A or J is a maximal modular right ideal in A. In either case, J is closed in A. For each n E N, we have b(bo - bn ) = an E AI, and so bo = limn-+oo(bo - bn) E J. Thus a E lvI, and so M is closed. 0 Proposition 2.6.14 Let A be a Banach algebra, and let E be a Banach right A-mod'ule. SUT)pOSe that null sequences in A factor weakly. Then each module homomorphzsm from A to E zs continuous. In partzcular. each left multzplier on A zs continuous. Proof Let T E CA(A, E), and take (an) E co(N. A). There exist b1 , .•.• bl; E A and (Cl,n).' ... (Ck.n) E co(N. A) such that an = 2:~=1 bj . Cj.n (1£ EN). Now Tan = 2:;=1 Tb j
. Cj.n -+
0 as n
-+
:)0.
and so T is continuous.
0
Let A be a Banach algebra. We shall now define two products on the Banach space A". First, the product map rnA : A x A -+ A is a continuous bilinear map. and so, as in A.3.51. there is an extension of rnA to a continuous bilinear map m:A : A" x A" -+ A"; we define 0 W = m:4(, w) (. WE A"), For a E A and E A", we have (2.6.22) where . denotes the module operation in A". It follows from A.3.53 (taking El = ... = E6 = A and P = Q = R = S = rnA) that
( 0 w)
0
i = 0 (w
0
i)
(, Ill, i
E
A").
Since llmAil = 1, we have 11 0 wll ::; 1111 IIwil (, w E A"), and hence (A", 0) is a Banach algebra containing A as a closed subalgebra. It follows from (2.6.22) that A is an ideal in (A",O) if and only if both La and Ra are weakly compact operators on A for each a E A. Now let E be a Banach left A-module, and consider the continuous bilinear map B1 : (a,x) 1--+ a . x, A x E -+ E. Then Bl : A" x E" -+ E" is a continuolls bilinear map; we set . A = B 1 (,A) ( E A", A E E"). It follows from A.3.53 (with E1 = E2 = E4 = A, E3 = E5 = E6 = E. P = rnA, and Q = R = S = B 1) that ( 0 w) . A = • (w . A) ( , W E A" , A E E") , (2.6.23) and so E" is a Banach left (A", D)-module. Clearlya·A agrees with its previous definition in th.e case where a E A and A E E".
Banach modules
249
Again, let E be a Banach right A-module, and consider the continuous bilinear map B2 : (x. a) I-> x . a, E x A ---+ E, and similarly extend B2 to a map B2 : E" x A" ---+ E". It follows from A.3.53 (now with E2 = E3 = E5 = A, El = E4 = E6 = E, Q = mA, and P = R = S = B 2) that
A . ( D \[J) = (A . . M+A· '11)
(4),'11 E A". A,MEE"),
where the products 4> . 1\1 and A . '11 are those with respect to which E" is an (A",O)-bimodule. Theorem 2.6.17 Let A be a Banach algebm. Then the Jollowmg are equivalent:
(a) A is Arens regular; (b) Jor each '11 E A", the-map 4> J---+ '1104> is contmuous on (A", a(A", A')); (c) Jor each '11 E A", the map 4> J---+ 4>'11 zs continuous on (A",a(A", A')); (d) Jor each A E A'. the map a J---+ a . A, A --4 A', is weakly compact; (e) Jor each A E A', the map a J---+ A . a, A ~ A', zs weakly compact; (f) Jor each pair « am), (b n ) ) oJ bounded sequences in A and each A E A'. limm limn (amb n , A) = limn limm (ambn , A) whenever both the itemted limits exist; (g) there is a subset S oj A' with lin S = A' such that, Jor each pair oj bounded sequences m A and each A E S, limm limn {amb n , A) = limn limm {amb n , A) whenever both the itemted limits exist. Proof (a){:}(b){:}(c) These follow from the above remarks. (a)=>(d) Let (a o ) be a bounded net in A. Then (ao,) ha.'> a convergent subnet in (A",a(A",A')); in fact, suppose that aa ~ 4> E A". For each '11 E A", we have (W.A . a o )
= (aa, '11
. A)
--4
(if>, '11 . A) = (if>OW,A)
= (if>W,A) = ('11, A .
if»,
and so A . aa ~ A . if> in (AI,a(A',A"». This is the condition for the map in (d) to be weakly compact. (d)=>(b) Let A E A', and define p). : a J---+ A . a, A ~ A'. By A.3.56(ii), the map p~ : (A",a(A",A'» ~ (A',a(A',A"» is continuous. Let if> E A", and let Wa ~ '11 in (A",a(A",A'». Then (b) follows because (if> 0 '11 a, A) = (if>, p~ ('11 a)}
--4
(if>, p~ ('11» = (if> 0 '11, A) .
(d}¢:}(e) This follows because A is Arens regular if and only if regular. (a)=>(f)=>(g) These are immediate from (2.6.28).
AOP
is Arens
Banach modules
251
(g)=>(a) Assume towards a contradiction that (a) fails. Then there exist ' E A' with (.) =f. (IJI, >. . . . .) = limn(bn ,>. . am) = (IJI,>. . am) = (am, IJI . >.), and so limmlimn(ambn ,>.) = (.). But limn limm(ambn , >.) = (IJI, >.. ., A ---+ A', is weakly compact; the set of weakly almost periodic /unctionals in A' is denoted by WAP(A).
Clearly 3t(A") is the set of elements IJI E A" such that the map
, IIi E t(co)O) .
This shows that «£1)", 0) is a commutative Banach algebra, that £1 is Arens regular, that rad«£l)") = L(CO)O, and that (£1)"
= £1 EB rad ((£1)")
as a Banach algebra. In the terminology of §4.2, below, (£ 1)' = £00 = C({3N), and so (£ 1)" = M({3N) and rad «£ 1)") = M(BN \ N). 0 Theorem 2.6.23 (Young) Let E be a non-zero Banach space. (i) Let Qt be a Banach opemtor algebm on E such that Ql zs Arens regular. Then E is refiexwe.
(ii) Suppose that E zs refiexzve. Then E®E', N(E), K(E), and A(E) are each Arens regular. Proof (i) Suppose that E is not reflexive. By A.3.31, there are bounded sequences (xm) and (An) in E and E', respectively. such that limmlimn(xm, An} and limn lilllm(Xm. An} both exist, but are unequal. Choose y E E e and f..l E E' such that (y, f..l) = 1. and set Sm
= Xm 0
f..l
(m E N),
Tn
=y0
An
(n E N) .
Then (8m ) and (Tn) are both bounded sequences in Qt (by (2.5.3)). Define reT) = (Ty, f..l) (T E Qt). Then r E Qt', and, for each m, n E N, we have (Tn 0 8 m , r) = (xm' An)(Y 0 f..l, r) = (Xm' An). Thus clause (f) of 2.6.17 fails, and so, by 2.6.17, Ql is not Arens regular. Thus E is reflexive whenever Qt is Arens regular. (ii) Set Qt = E®E'. Since E is reflexive, it follows from 2.6.4(ii) that the dual module of Qt is B(E), where the left module operation is determined by the map T . (x 0 A) = Tx 0 A for T E B(E), x E E, and A E E'. We shall verify clause (e) of 2.6.17 for the algebra Qt. Fix T E B(E). Set K = {Tx0A : x E E[l], A E E[l]}' so that K c K(E). We first clazm that K is weakly sequentially compact in K(E). For let (Txn 0 An) be a sequence in K, where (x n ) is a sequence in E[l] and (An) is a sequence in E[l]. Since E and E' are reflexive, E[l] and E[l] are weakly sequentially compact by A.3.31, and so we may suppose that Xn --t x weakly in E and An --t A weakly in E'. To show that TX n 0 An --t Tx 0 A weakly in K(E), it suffices, by Rainwater's theorem A.3.29(v), to show that (Txn 0 An, A) --t (Tx ® A, A) for each A E exK(E)[l]. By A.3.64, each such A has the form It ® t(y), where y E E[l] and /L E E[l]. But now (Txn 0An, /L®t(y)
= (Txn'
/L)(Y, An)
--t
(Tx, /L)(y, A)
= (TX0A,
/L®t(Y) ,
and so the claim follows. Let L be the closure of the convex hull (K) in the weak or norm topologies on B(E); by Mazur's theorem A.3.29(ii), these two closures are the same set. By the Eberlein-Smulian theorem A.3.29(iv), K is weakly compact, and so, by
Banach and topological algebras
254
the KreTn-Smulian theorem A.3.29(iii), L is weakly compact. Clearly the set ({x®>.: x E E[ll' >. E E[ll}) is dense in ~[lJ' and so the set {T· a: a E ~[lJ} is contained in L. Thus this latter set is relatively weakly compact, 2.6.17(e) does hold, and ~ is Arens regular. It follows from 2.6.18 and 2.5.3(ii) that N(E) is also Arens regular. To show that K(E) is Arens regular, we verify the iterated limit criterion of 2.6.17(g), taking S = {>. ® ~(x) : x E E, >. E E'}; we note that linS = qE)' by A.3.64. Thus, let x E E and>' E E', and take (Sm) and (Tn) to be two bounded sequences in K(E). Then (SmTn, >. ® ~(x)) = (SmTnx, >.) by (A.3.12), and so (SmTn, >'®~(x)} = (Tnx, S:n(>.)}. The equality of the iterated limits now follows from A.3.31, which applies because E is reflexive. By 2.6.18, A(E) is also Arens regular. 0 Suppose that E is reflexive and has AP; we identify K(E)" with I3(E), as above. Then it is easily checked that both Arens products on K(E)" coincide with the natural product in I3(E). Proposition 2.6.24 Let E be a reflexive Banach space wzth AP. Then N(E) is a closed ideal in N(E)". Proof As in 2.6.7(ii), we identify N(E)' with I3(E). We have remarked that we must show that LT and RT are weakly compact operators on N(E) for each T E N(E); it clearly suffices to do this in the case where T is a rank-one operator, say T = Xo ® >'0· Let (Sa) be a bounded net in N(E). We may suppose that SaXO -+ Xl in (E,u(E,E'» and that S~(>.o) -+ >'1 in (E',u(E', E». For each U E I3(E), we have
(Sa 0 (xo ® >'0), U} = (U 0 SaXO, AO) (XO ® >'0) 0 Sa, U} = (Uxo, S~(>.O)}
-+ -+
(UXll >.o) = (Xl ® >'0, U), (UXO' >'l) = (Xo ® >'ll U),
and so Sa 0 (xo ® >'0) -+ Xl ® >'0 and (xo ® >'0) 0 Sa Lxo®>'o and Rxo®>'o are weakly compact. as required.
-+
Xo ® >'1 weakly. Thus 0
Further examples involving Arens regularity will be given in 3.2.37, 3.3.28, 4.1.45, and 4.4.34. Proposition 2.6.25 Let A be a Banach algebra with radical R, and write B for (A", 0). Then:
(i) an element a E A is left quasi-invertible if and only if a is left quaszinvertible in B; (ii) for each a E A, uB(a) = UA(a); (iii) (rad B) n A c R; (iv) (radB) n A = R in the case where A is commutative. Suppose, jurther, that (AIR)" is semisimple and Rn = 0, where n E N. Then: (v) (radB)n
= 0 and radB = R" = R(u), where u
= u(A", A').
Banach modules
Proof (i) Suppose that a is left quasi-invertible in B, so that there exists with ~ + a = ~ 0 a. For each \II E B, we have
255 ~ E
B
(fA - Ra)"(\11 - \II O~) = \II - \II 0 (~+ a - ~ Da) = \II, and so (fA - Ra)" : B ---+ B is a surjection. By A.3.48(vi), fA - Ra : A ---+ A is a surjection, and so there exists b E A with b+a = ba, Le., a is left quasi-invertible in A. The converse is trivial. (ii) and (iii) These follow easily from (i). (iv) Let a E R = .Q(A) and ~ E B. Then a . ~ = cI> • a because A is commutative, and so a . cI> E £l(B). Hence a E radB. (v) We have R(17) n = 0 by (2.6.28), and so R (17) C rad B by 1.5.6(ii). Let n : A ---+ AIR be the quotient map. Then n" : B ---+ (AIR)" is an epimorphism with R" = ker nil = R (17). Since (AIR)" is semisimple, it follows from 1.5.3(ii) that radB C ker nil. Hence radB = R(17). 0 In §1.4, we discussed simple modules over an algebra A, and in particular we gave Jacobson's density theorem as Theorem 1.4.32. We now give a sharper form of this result in the case where A is a Banach algebra.
Theorem 2.6.26 (Rickart) Let A be a Banach algebra, and let E be a simple left A -module. (i) There is a unique norm 11·11 on E such that (E, 11·11) is a Banach left A-module. (ii) A£(E) = Cle· (iii) Let {Xl, ... , xn} be linearly independent in E, and let Yl. ... , Yn E E. Then there exists a E A with a . Xj = Yj (j E Nn ). (iv) Let {Xl. ... , xn} and {Yl. ... , Yn} be lznearly independent sets zn E. Then there exists a E expA with a . Xj = Yj (j E N n ). Proof (i) Take Xo E E-, and set M = x6-, a maximal modular left ideal in A; by 2.2.28(i), M is closed. Since E ~ AIM, E is a Banach left A-module for the quotient norm 11·11. Suppose that (E, 111·111) is a Banach left A-module. Then Illxlll ~ Ilxlllllxolll (x E E), and so 111·111 is equivalent to 11·11. In this sense, 11·11 is uniquely specified. (ii) Let T E A£(E), and take Xo E E-. Since A . Xo = E, there exists ao E A with ao . Xo = Txo. For each X E E and a E A with a . Xo = x, we have Tx = a . Txo = aao . xo, and so IITxlI ~ Ilaollllxll. Thus T E B(E), and hence A£(E) C B(E). In particular, A£(E) is a normed algebra. By 1.4.31, A£(E) is a division algebra; by the Gel'fand-Mazur theorem 2.2.42(ii), A£(E) = Cle. (iii) This is now immediate from Jacobson's density theorem 1.4.32. (iv) Set F = lin {Xl. ... , x n , Yl, ... , Yn}. There exists R E Inv £(F) with .Rxj = Yj (j E N n ). By (2.4.10), Inv £(F) = exp£(F), and so R = exp T for some T E £(F). By (iii), there exists b E A with b . Xj = RXj (j E N n ), and now (expb) . Xj = (expT)(xj) = RXj = Yj (j E N n ), o so that a = exp b is the required element of exp A.
256
Banach and topological algebras
We give some applications of Theorem 2.6.26; the first will be required in Chapter 5.
Proposition 2.6.27 Let A be a Banach algebra, let E be a simple left A-module, and let {xn : n E N} be a linearly independent set zn E. Then: (i) there eX'tSts a E A with a . Xl = 0 and such that {a . Xn : n ? 2} is linearly zndependent in E; (ii) there ~s a sequence (an) zn A such that an'" al . Xm = 0 (m < n) and an ... al . Xn =I- 0 (n E N). Proof (i) By 2.6.26(iiiJ, there exists al E A with al . Xl = 0 and al . X2 =I- O. Now 2.6.26(iii) may be used to construct inductively a sequence (an) in A such that, for each n ? 2, we have Ilanll < 2- n , an . Xl = ... = an . Xn = 0, and (2.6.32) Set a = 2:::1 ai E A. Then a . Xl = 0 and a . Xn = 2:::::/ ai . Xn (n? 2), so tha.t a . X2 = al . X2 =I- O. Assume that a . Xm +1 E lin {a . Xl, ... , a . xm} for some m? 2. This contradicts the case n = m of (2.6.32), and so {a . Xn : n ? 2} is a linearly independent set in E. (ii) By induction, there exists (an) in A such that an'" al . Xm = 0 (m < n) and {an' .. al . Xm : Tn ? n} is linearly independent. 0
Theorem 2.6.28 (Aupetit) Let A be a semis~mple Banach algebra. Suppose that there ~s a real-linear subspace H of A such that A = H + iH and u( a) is fimte for each a E H. Then A is finite-dimensional. Proof Let a E A, and take b, e E H with a = b + ie. Set (nEN).
Fn={tElR:lu(b+te)l:s:n}
Suppose that to E lR\Fn . Then u(b+toe) is a finite set with at least n+I points, and so, by 2.4.7. there exists 8 > 0 such that lu(b + te)1 ? n + 1 for t E lR with It - tol < 8. Thus each Fn is closed in R By hypothesis, U{Fn : n E N} = JR, and so, by the category theorem. there exists kEN with int]RFk =I- 0. Define f: z 1-+ b + ze, C -+ A, so that f E O(C, A), and set
u(z) = logdiamk(u(f(z)))
(z E C),
where diamk denotes the kth diameter, as in A.l.IB. By 2.3.35, u is subharmonic on Co But diamk(u(f(t))) = 0 (t E Fk)' and so U == -00 on an open interval of R By A.2.20(ii), U == -00 on C, and so lu(f(z))1 :s: k (z E C). In particular. we have lu(a)1 = lu(f(i))1 :s: k, and hence A is spectrally finite. Now set An = {a E A : lu(a)1 :s: n} for n E N. Then, again by 2.4.7, each An is closed in A, and also U{ An : n E N} = A, and so there exists mEN such that int Am =I- 0, say ao E int Am. Take a E A, and set v(z) = logdiammu(ao
+ z(a -
ao))
(z E C).
Then v is subharmonic on C, and v == -00 on a neighbourhood of O. By A.2.20(i), v == -00 on C, and so diammu(a) = 0, i.e., lu(a)1 ::; m. We have shown that lu(a)1 ::; m (a E A).
Banach modules
257
Suppose that E is a simple left A-module, and that {Xl, ... ,xn } is a linearly independent set in E. By 2.6.26(iii), there exists a E A with a . x J = jXj for j E Nn • By 1.5.29(v), Nn c a(a), and so n S m. Thus dimE S m. Suppose that E 1 , ••• ,En are simple left A-modules such that Ef =f Ef for i =f j. Since each E j is finite-dimensional, 1.4.39 implies that there exists a E A with a . X = jx (x E E j ), and so again Nn C a(a). Thus n S m. Since A is semisimple, it now follows from 1.5.2(i) that the algebra A is finite-dimensional. 0
Corollary 2.6.29 Let A be a semisimple Banach algebra. (i) Suppose that A is spectrally jinite. Then A is jinzte-dimenswnal. (ii) Suppose that A has a linear mvolution and that a(a) zs jinzte for each self-adJoint element of A. Then A is jinite-dimenszonal.
Proof Apply the theorem with H respectively.
= A
and H = Asa in cases (i) and (ii), 0
Theorem 2.6.26 also leads to a characterization of the radical in Banach algebras.
Lemma 2.6.30 Let A be a Banach algebra, let E be a szmple left A-module, and let a E A. Suppose that there exists C > 0 such that v(a - b) S C for each bE A with a(b) = a(a). Then there exists 0: E C such that a . x = o:x (x E E). Proof Take b E A, and define
F(z) = exp(-zb)aexp(zb)
(z E IIbll/r, we have lI(za + b) ::; C IIbli. By 2.3.32(ii), the function z 1--+ lI(za + b) is subharmonic on C, and so, by A.2.21, this function is constant. Thus (b) holds. 0
Corollary 2.6.32 The following conditzons on a Banach algebra A are equivalent: (a) rad A = ,Q(A); (b) D(A) + D(A) c D(A); (c) D(A) . D(A) c D(A). Proof We may suppose that A is unital, with identity e. By 1.5.32(ii), radB c ,Q(B) for each algebra B. Certainly (a)*(b) and (a)*(c), and (b)*(a) by the theorem. Suppose that (c) holds, and take a, bE D(A). Then (e - a)-la and bee - b)-l belong to ,Q(A), and so (e - a)-lab(e - b)-l E D(A) by (c). Since e - (a + b) = (e - alee - (e - a)-lab(e - b)-l)(e - b), we have e - (a + b) E Inv A. In fact, e - qa + b) Thus ,Q(A) + D(A) c ,Q(A), and (b) holds.
c
Inv A, and so a + bE D(A). 0
Clearly each commutative Banach algebra A satisfies the conditions given in the above corollary, but the conditions do not characterize commutativity: 2.3.13(i) and 2.3.14(iii) (with S = §2) give non-commutative Banach algebras A with radA = ,Q(A) = {O}.
Theorem 2.6.33 (Aupetit) Let A be a Banach algebra. Then the following condztzons on A are equivalent: (a) A/radA is commutative; (b) II is uniformly continuous on A; (c) there exists M > 0 such that lI(a + b) ::; M(II(a) + lI(b» (a, bE A). Proof Set B = A/rad A, and write a for a + rad A. Suppose that (a) holds, and take a, bE A. Then VB (a + b) ::; IIB(a) + IIB(b) by (2.3.8), and so lI(a + b) ::; v(a) + lI(b) and III(a) - lI(b)1 ::; lI(a - b) ::; lIa - bll· Hence (b) and (c) hold. Suppose that (b) holds. Then there exists 8 > 0 such that III(c) - lI(d)1 < 1 whenever IIc - dll < 8, and so v(a - b) ::; lI(b) + Olall + 1)/8 (a, bE A). Suppose that (c) holds. Then lI(a - b) ~ M(II(a) + lI(b)) (a, bE A). Let a E A. In both cases it follows that there exists C > 0 such that v(a - b) ~ C whenever b E A with a(b) = u(a), for in this case v(b) = lI(a).
Banach modules
259
Let E be a simple left A-module. By 2.6.30, there exists a E 0 such that {b E A : dCa, b) < 8} c 8 nll where d is the metric on A. Take c E A·, and set p(z) = (a + zcr (z E q. Then p(z) = 0 whenever JzJ d(c, 0) < 8, and so p = O. Thus em = 0, and hence 8 m = A. By the Nagata-Higman theorem 1.3.33, A is nilpotent. 0 Theorem 2.6.35 (Dixon) Let A be an (F)-algebra.
0/ nilpotent ideals. nilpotent ideals in A.
(i) Each nil left ideal in A is contained in a unwn
(ii) The prime radicall.l3(A) is the union
0/ the
Proof (i) Let I be a nil left ideal in A, and let a E I. Then A#a C I, and so A#a is a nil algebra. Since A#a is an (F)-algebra for the quotient topology, A#a is nilpotent by 2.6.34, say (A#a)[ml = O. Set J = A#aA#. Then a E J, J is an ideal in A, and J[m] C (A#a)[m1A# = 0, so that J is nilpotent. (ii) By 1.5.26(i), I.l3(A) is a nil ideal, and so, by (i), I.l3(A) is contained in a union of nilpotent ideals. However, by 1.5.26(ii), each nilpotent ideal of A is contained in I.l3(A). 0 Corollary 2.6.36 Let A be a semiprime (F)-algebra. Then A has no non-zero nil left ideals. Proof By 1.5.25, A has no non-zero nilpotent ideals, and so A has no non-zero nil left ideals. 0
We can apply the two structure theorems 2.6.28 and 2.6.34 to show that a left Noetherian Banach algebra is necessarily finite-dimensional. Proposition 2.6.37 Let A be an (F)-algebra which is a Q-algebra, and let I be a left ideal zn A such that 1 zs fimtely generated. Then I zs closed m A. Proof We may suppose that A is unital. Let 21 be the algebra Mn(A), and let t be the identity of 21. Choose c > 0 such that det (t - x) E Inv A whenever d(xij, 0) < c (i,j E N n ), where x = (Xij) and d is the metric on A. There exist al,.'" an E 1 such that 1 = Aal + ... + Aa n . The linear map (Xl, ... ,xn ) t-+ Xlal + ... + xna n , A -+ 1, is a continuous surjection, and hence it is open. Thus, for each i E N n , there exist bi E I and Xib"" Xin E A with
and Set a A(n)
= (ab ... , an), b = (bI , ... , bn ), and x = (Xij). Then b and t - x E Inv21, and so a E 21 . be I(n). Thus 1 = I.
= (t -
x)(a) in 0
260
Banach and topological algebras
Lemma 2.6.38 Let A be a Banach algebra for' wh1ch the famzly of closed left ideals satzsfies the ascendzng chain condztion. Suppose that all left ideals of the form A(ze - a), where Z E C and a E A, are closed in A. Then A/radA is finzte-dimensional.
Proof Assume towards a contrarliction that A/rad A is infinite-dimensional. By 1.5.29(i) and 2.6.29(i), there exists a E A such that a(a) is infinite. Let {Zn : n E N} be a set of distinct points in oa(a) \ {OJ. It follows from 1.5.29(iv) that {zn : n EN} C oa(Ra). Take n EN. We have (ZnIA - Ra)(A) = A(ZneA -a), a closed subspace of A, and so, by 2.3.23(ii), Zn is an eigenvalue of Ra: take bn E A- with bna = znbn. Also, set I n = {b E A : b(zleA - a)··· (ZneA - a) = O}, so that (In) is an ascending chain of clos{''. Then U E AB(E,E) with IlUil = 1. Now take DE Zl(A,E), and define SD = U 0 D. Then SDa = (Da)(e~) (a E A) and SD E .c(A, E); further, (SlSD)(a, b)
= (Da)(e b) . b - (Da)(b) (a, bE A),
and so the map b f-+ (8 1S D )(a, b), A ~ E, is continuous for each a E A. This shows that SD E LI(A, E). If DE Zl(A, E), then SD E B(A, E).
Theorem 2.7.6 (Dales and Villena) Let A be a Banach algebm, and let E be a Banach A-bimodule. Then the map
£ : D ~ SD,
Zl(A, E) ~ LI(A, E),
is a linear z8omorphzsm, and the restrictwn map £ : Zl(A, E) ~ B(A, E) is a linear' homeomorphzsm.
Proof Certainly £ : ZI(A, E) ~ LI(A, E) is a linear map. We first prove that £ is a bijection. Let S E LI(A, E), and define D : A ~ .c(Ab, E) by the formula (Da)({3e~
+ b) =
S({3a + ab) - a . Sb
({3 E C, a, bE A) .
For each {3 E C and a, bE A, we have (Da)(l3e b +b) = {3Sa+ Sa· b- (SlS)(a, b), and so Da E B(AP, E) and D(A) C E. Clearly D E .c(A, E). Also, for each {3 E C and aI, a2, bE A, we have
(al . Da2 + Dal x a2)({3e P + b) = al . S({3a2 + a2b) - ala2 . Sb + S({3ala2 + ala2b) - al . S({3a2 = S({3ala2 + ala2b) - ala2 . Sb = D(ala2)({3el> + b) ,
+ a 2b)
Intertwming maps and denvations
265
and so al . Da2 + Dal x a2 = D(alaZ). Thus D E Zl(A, ii;). Finally, we see that SDa = (Da)(e b ) = Sa (a E A), and so SD = S. This shows that £ is a surjection. We now claim that
(Da)«(3e b + b) = (3Soa
+ SJ)(ab) - a . Sob «(3 E C, a, bE A).
Indeed the right-hand side of
(3Da(e b )
+ D(ab)(e b)
(2.7.2)
(2.7.2)
is equal to
a . (Db(e D )) = (3Da(e b ) + (a . Db + Da x b)(eb ) -
=
(3Da(e b)
+ Da(b) =
(Da)«(3e b
-
a . Db(e b )
+ b),
as required. It follows from (2.7.2) that £ is an injection. Suppose that D E Zl(A,E). Then IISDII :::; IIDII. Conversely, suppose that SD E SeA, E). Then it follows from (2.7.2) that
II(Da)«(3e b + b)11
for
(3 E C
and a, bE A, The result follows.
:::; IBllISDllllall + IISnllllabll + IISDlillalillbll :::; IISDII (1(31 + 211bll) Iiall :::; 21lSDllllallll(3eb + bll ' and so D is continuous with IIDII :::; 2I1SDII. 0
Corollary 2.1.1 Let A be a Banach algebm. Suppose that each denvatwn from A into a Banach A-bimodule is continuous. Then each lejt-intertwznzng map from A mto a Banach A-bzmodule zs continuous. 0 In fact, for maps with finite-dimensional range, there is a simple condition for the automatic continuity of homomorphisms, derivations, and intertwining maps, at least for unital, separable Banach algebras. We first give an easy result.
Proposition 2.1.8 Let A be a Banach algebm. Suppose that each homomorphism from A into a jinzte-dzmensional Banach algebm is continuous. Then each derivation from A into a finite-dimensional Banach A-bzmodule is continuous. Proof Let D be a derivation into a finite-dimensional Banach A-bimodule E, and set J={aEA:a· E=E· a=O}, that J is a closed ideal of finite co dimension in A. The algebra B = (AI J) ED E is a finite-dimensional algebra, and the map () : a ~ (a + J, Da), A -+ B, is a homomorphism. Clearly () is continuous if and only if D is continuous. 0 SO
'l'heorem 2.1.9 (Dales and Willis) Let A be a unital, sepamble Banach algebm. Then the following condztions on A are equivalent: (a) M2 is of finite codimension in A for each maximal ideal M of finite codimension in A;
266
Banach and topological algebms
(b) each homomorphism from A into a finzte-dimensional Banach algebm is continuous;
(c) each derivatzon from A into a finite-dimensional Banach A-bimodule zs continuous; (d) each intertwining map from A znto a finite-dimensional Banach A-bimodule is contmuous.
Proof By 2.2.30. (a){:}(b), and, trivially, (d)::::} (c). (a)::::}(d) Let S : A - E be an intertwining map, where E is a finitedimensional Banach A-bimodule. and set J = {a E A: a . E = E . a = o}. so that J is a closed ideal of finite codimension in A. By 2.2.30, J2 is closed and ha.'l finite codimension, and so, by 2.2.16(i). there exist mEN and M > 0 such that each a E J2 can be written as a = 2:7=1 ajbj , with alo ... , am, b1, .... bm E J and 2: =l lIajll lib) II :::; M lIali. But now, for such an element a E J2, we have
J
m
118all ::;
L
m
IIS(aj bj )1I =
)=1
L
II(olS)(a), b))11
j=l
m
:::; IWSII
L
lIajllllb j ll :::; M IWSllliall ,
)=1
and so S
I J2
is continuous. Hence S is continuous, and so (d) is proved.
(c)::::}(a) Assume towards a contradiction that M2 is of infinite codimension for a maximal ideal Af of finite codimension. By 2.2.29, there is an ideal I of finite codimension in A with M2 c I c M such that I is not closed. Set E = rad (AI 1), so that. E C Mil and E2 = 0; the quotient map is denoted by n: A - All. The space E is an A-bimodule for the operations given by a . x
= n(a)x,
X· a
= xn(a)
(a E A, x E E) .
For each x E E, the kernel of the map a 1---+ a . x contains A[, and hence is closed in A. This shows that the map is continuous, and so E is a Banach left A-module. Similarly E b a Banach right A-module, and so E is a Banach A-bimodule. By 1.5.19, AI I is decomposable: there is a subalgebra B of AI I such that AI I = B EB E, with the product (b1, xt}(b2, X2) = (bl b2, b1X2
+ X1b2)
(b 1 , b2 E B,
Xl,
X2 E E).
Let P: (b,x) - x, All - E, be the projection. Then D = Pon: A _ E is a derivation. We have (ker D)nM = I, so that D is discontinuous, a contradiction of (c). Thus (c)::::}(a). 0 We now give a condition for a continuous linear map to be a derivation; we adopt the notation j for the map of (2.6.13) and L:E for the short exact sequence of (2.6.14).
Intertwinmg maps and derivations
267
Proposition 2.7.10 Let A be a unital Banach algebra, let E be a umtal Banach A-bzmodule, and let T E R(A, E). Then the following are equivalent: (a) T is a denvation; (b) joT = -8T ; (c) q 0 8T = 0 and TeA
= O.
Proof Condition (b) states that j(Ta)(b) definition, j(Ta)(b) = Ta . band 8T (a)(b)
= (a
= -8T (a)(b) (a,b
E
A). But, by
. T - T x a)(b) = a . Tb - T(ab) ,
and so (a) and (b) are equivalent. Since LE is exact, (b) implies that q 0 8T = 0, and TeA = 0 by 1.8.2(i). Hence (c) holds. Conversely, suppose that (c) holds. For each a E A, 8T (a) E ker q = j(E), and so there exists Xa E E with j(xa) = -8T (a). We have
Xa
= j(xa)(eA) = -8T(a)(eA) = Ta -
a . TeA
= Ta
(a E A),
and so j(Ta) = j(xa) = -8T (a) (a E A), giving (b).
0
We defined point derivations in 1.8.7, and characterized them in 1.8.8. The analogous result to the latter for continuous point derivations is the following.
Proposition 2.7.11 Let A be a unital Banach algebra, and let cp E = lim", a", and IJI = lim,a b,a in (A",lT(A",A'», where (a",) and (b,a) are nets in A. Then D" (II> D IJI)
= D" (lim lim a",b,a) = lim lim D(a",b,a) '"
f3
= lim lim (a", . D(bf3)
'"
,a
+ D(a",) . b,a)
'" f3 by (2.6.26), and so D" is a derivation.
= II> . D"(IJI)
+ D"(II»
. IJI,
Banach and topological algebras
270
(ii) and (iii) Let q> E A" and A E
E{4},
and choose bounded nets (an,) in
A and (A/3) in E" with q> = limo aa and A = lim/3 A/3 in the weak* topologies. Then, using (2.6.26), we have
P(q> . A) = limlimP(aa . A/3) = lim lim aa . P(A{3) = q> . P(A) , a
/3
a
/3
giving (ii). Clause (iii) follows similarly by using the given assumption. (iv) By (i), D" : A" -- A{4} is a continuous derivation. Since A is Arens regular, it follows from (iii) that P : A{4} -- A" is an A"-bimodule homomorphism. SetD=PoD". 0
Proposition 2.7.18 Let A be a Banach algebra such that each continuous derivation on A is inner, and let T E B(A) be an automorphism with aCT) c S2r./3. Then there exists b E A such that (a E A).
T(a) = exp(-b)aexp(b)
Proof By 2.4.19(ii), there is a derivation D E B(A) such that T = exp D. By hypothesis, there exists bE A with D = Ob. We have T(a) = (expob)(a) =
L 00
n=O
=
1 n!
L( -1)1 (n). fi'ab n -) n
)=0
bn-j ) LLn ( TJb)J) a ( (n.)! J 00
by (1.8.4)
J = exp(-b)aexp(b) ,
n=OJ=O
o
as required.
Set Tt(a) = exp(-tb)aexp(tb) (t E JR., a E A), in the above notation. Then (Tt : t E JR) is a group of continuous automorphisms in B(A). It seems that the range of a derivation on a Banach algebra is necessarily 'small' in some sense; the next theorems explore this idea.
Theorem 2.7.19 (Kleinecke-Sirokov) Let D be a continuous derivation on a Banach algebra A, and let a E A. Suppose that either a . Da = Da . a or D 2a = O. Then Da E .Q(A). Proof In the case where D 2a = 0, we have Dn(a n ) = n! (Da)n (n 1.8.5(iii), and so Da E .Q(A) because II(Da)nlll/n ::;
(~!
r/
n
IIDlillall -- 0
as n --
E N)
by
00.
Now take a E A. Then La E B(A), and V : T 1-4 TLa - LaT is an inner derivation on B(A) with Vela) = o. Set S = V2(D) E B(A); we calculate that S = V(DLa - LaD) = DL~ - 2LaDLa + L~D, and so, in the case where a . Da = Da . a, we have D(a 2) = 2a . Da in A and Sb = D(a 2b) - 2a . D(ab) + a 2 • Db = 2a . Da . b + a 2 • Db - 2a(a . Db + Da . b)
+ a 2 • Db =
0
(b E A),
Intertwinzng maps and derivations
271
so that S = o. We have shown that 1J(D) E O(B(A)). However, we have 1J(D) = LDa because 1J(D)(b) = D(ab) - a . Db = Da . b = LDa(b) (b E A), and so II(Da)nlll/n -+ 0 as n -+ 00. This proves that Da E O(A). 0 We also obtain a third proof of 2.1.14(iii). For let A be a unital Banach algebra, and suppose that a,b E A and a E C with 8b(a) = ab-ba = aeA. Then a . 8b(a) = 8b(a) . a, and so aeA E O(A) by the theorem. Hence a = O.
Corollary 2.7.20 (Singer and Werrner) Let D be a contznuous derivation on a Banach algebm A wzth D(A) c 3(A). Then D(A) c rad A. Proof By the theorem, D(A) c O(A), and so D(A) c O(A) n 3(A) c rad A.
o Thus the only derivation on a Banach function algebra is o. In fact, it is also true that D(A) c rad A for an arbitrary, perhaps discontinuous, derivation D on a commutative Banach algebra A; this much deeper result will be proved as Theorem 5.2.48. However, the general version of part of 2.7.19 is an interesting open question.
Question 2.7.A Let D be a de'T"'tvation on a Banach algebm A. Suppose that a E A and a . Da = Da . a. Does zt follow that Da E O(A)? The conjecture that the answer to this question is positive has been called the unbounded Kleinecke- Szrokov conjecture. There is a slightly different proof of the theorem of Singer and Wermer in the case where A is a commutative Banach algebra. For take D E Zl(A, A) and cp E 4>A. By 2.4.19(i), exp(zD) E B(A) is an automorphism for each z E C. Define 00 (Dn) cpz(a) = cp((exp(zD))(a)) = cp(a) + ~ cp ,a zn (z E C, a E A).
n.
n=l
Then cpz E 4>A, and so Icpz(a)1 ~ Iiall. Also z cpz(a) is constant. In particular, Ji 2(J, E) . Suppose, jurther, that .J2 = J. Then there is an exact sequence 0---+ Ji2(AI J, E)
---+
Ji2(A, E)
---+
Ji 2(J, E).
Proof We shall construct linear maps 0'1, ... ,0'4 such that the sequence
0---+ ABA(J, E) ~ ZI(J, E) ~ KIKo ~ Ji2(A, E) ~ Ji2(J. E)
of linear spaces and linear maps is exact. Since E is an annihilator J-bimodule, we have Z1(J, E) = Ji 1(J, E) = {S E B(J, E) : S I J2 = O},
and so ABA(J, E) C Z1(J, E)j we choose 0'1 to be the inclusion map. Now take S E Z1(J, E). Since the space E is finite-dimensional, there exists R E B(A, E) with R I J = S. We define a2(S)=8 1R+Ko .
Clearly a2(S) E KIKo. We show that a2(S) is well-defined. For let RI E B(A, E) with R1 I J = S, and set R = R - R 1. Then R I J = and, in particular. R I J E ABA(J, E), so that 81 R E Ko. Thus a2(S) is indeed well-defined. It is immediate that 0'2 is a linear map and that ABA(J,E) C kera2' Now take S E ker 0'2, and choose R E B(A, E) with R I J = S. There exists T E B(A, E) such that T I J E ABA(J, E) and 81 R = 81T. This implies that
°
a . Sb - S(ab) = a . Tb - T(ab) (a E A. bE J), } -S(ab) + Sa . b = -T(ab) + Ta . b (a E J, bE A),
and so S E ABA(J, E), the image of 0'1. Thus the sequence is exact at ZI(J, E). For each T E K, define a3(T + Ko) = T+N2(A,E). Certainly a:i(T+Ko) is well-defined. and 0'3 : KIKo --+Ji 2(A, E) is a linear map. Let S E ZI(J, E), and take R E B(A, E) with R I J = S. Then (0'3 0 a2)(S) = 8 1 R+.N2(A, E) = 0, and SO ima2 C kera3' Conversely, suppose that T E K with T + Ko E kera3. Then there exists S E B(A, E) with T = 81S. But T I (J x J) = 0, and so S I J2 = 0, Whence S E Z1(J, E). Clearly T+Ko = rT2(S I J), and so kera3 C ima2. Thus the sequence is exact at KlKo. Finally, take T E Z2(A, E), and define 0'4
(T + ,N2(A, E»
= T
I (J
x J)
+ ,N2(J, E) .
Banach and topological algebras
276
Then again 0'4 is a well-defined linear map from 1{2(A, E) into 1{2(J, E). Suppose that T E K. Then (0'4 0 0'3)(T + Ko) = T I (.I x .I) + .N2(J. E) = 0, and so imO'a C ker0"4. Conversely, suppose that T E Z2(A,E) is such that T+.N2(A, E) E ker0'4. Then there exists S E 8(.1, E) with 01S = T in 8 2(.1, E). Let R E 8(A, E) with R 1.1 = S, and set U = T - 01 R. Then U I (.I x .I) = 0, so that U E K and 0'3(U + Ko) = U + .N2(A, E) = T + .N2(A. E), and so ker0'4 C im0"3' Thus the sequence is exact at 1{2(A,E). Now suppose that .12 = J. Then Zl(J, E) = {o}. Take T E Z2(AI J. E). and define p(T)(a. b) = T(a + .I, b + .I) (a, b E A). Then p(T) E K, and p induces a well-defined linear map
p: T + .N2(AI .I, E) ~ p(T) + K o,
1{2(AIJ, E)
-+
KIKo.
Suppose that T + .N2(AI.7, E) E ker p. Then there exists R E 8(A, E) such that R 1.1 E A8 A (J,E) and olR = T in 8 2(A,E). We have R 1.]2 = O. and so R I .I = 0 because .12 =.7. Thus there exists R E 8(AIJ, E) with R(a + .I) = Ra (a E A); T = 0 1 R. and so T E .N2(AIJ, E). Thus Pis injective. Now suppose that T E K. Then T I (A X .12 ) = T I (.]2 x A) = 0, and so T I (A x .I) = T I (.I x A) = O. This implies that the map
T; (a + J,b+ .I) ~ T(a,b),
AIJ x AIJ
-+
E.
is well-defined; clearly T E 8 2 (AI .I, E). We see that p(T) = T, and so it follows that p (T +.N2 (AI .I, E» = T + Ko. Thus Pis surjective. We have proved that KIKo ~ 1{2(AIJ, E), and so it follows from the first result that the second sequence is also exact. 0 The algebraic notions of a decomposable algebra and of split and singular extensions of an algebra were given in §1.5 and §1.9, respectively. There are analogous definitions for Banach algebras. Definition 2.8.5 Let A be a Banach algebra with radical R. Then A = B E!7 R is a strong Wedderburn decomposition of A if B is a closed .'Iubalgebra of A; A is strongly decomposable if it has a strong Wedderburn decomposition. Proposition 2.8.6 Let A be a non-radical Banach algebra such that rad A has finzte codimension in A. Then A is strongly decomposable, and there is a nonzero idempotent in A. Proof By 2.4.2. orthogonal idempotents can be lifted in A. and so this follows 0 from 1.5.9 and 1.5.18. Proposition 2.8.7 Let A be an algebrazc Banach algebra. Then there is a finitedimensional subalgebra B of A such that A = B E!7 rad A, and rad A is nilpotent. Proof Set R = radA. By 1.6.1l(ii), O'(a) is finite for each a E A, and so, by 1.5.29(i), AIR is a semisimple, spectrally finite Banach algebra. By 2.6.29(i), AI R is finite-dimensional, and so A is strongly decomposable. Let a E R. Then there exist n E N, a E C·, and q E Ca[X] such that an(aeA + q(a» = 0, and now an = 0 because aeA + q(a) E Inv A#. Thus R is a. nil ideal. By 2.6.34, R is nilpotent. 0
Cohomology
277
We shall be concerned with the questions whether a given (non-semisimple) Banach algebra is decomposable and whether it is strongly decomposable. Some preliminary information is given in the first theorem.
Theorem 2.8.8 Let A be a commutatzve Banach algebm such that A = B EEl f, where B is a closed subalgebm of A and f zs a closed zdeal of A wzth f C rad A.
(i) Q:(A) c B. (ii) Suppose that A zs unital and that a Ilanlllla-nll = o(n)
E Inv A zs such that
as n ~
00.
Then a E B.
Proof (i) It follows from 1.5.17 that Q:(A) C B. and so Q:(A) C B.
(ii) Let P be the projection of A onto B with kernel f, so that P is a continuous homomorphism. Set b = Pa and x = a-b. By 1.5.17, eA E B; we have b- 1 = P(a- 1), and so eA + b-1x = b-1a. Since b-1x E rad A, we have a(b-1a) = {I}. Moreover,
II(b-1atll ::; IIP(a-n)llllanll ::; Ilpllllanlllla-nll = o(lnl) By the Gel'fand-Hille theorem 2.4.22(ii). b-1a
as Inl ~
= eA. and so a = b E B.
00.
0
Definition 2.8.9 A Banach algebm A is spanned by its idempotents if Q:(A) = A. It follows immediately from the above theorem that a non-semisimple, commutative Banach algebra which is spanned by its idempotents is not strongly decomposable. We shall relate the theory of decomposable Banach algebras to the theory of extensions.
Definition 2.8.10 Let A be a Banach algebm. A Banach extension of A by f is a short exact sequence
L
= L(21; 1): 0 - d ~ 21 ~ A
-+
0,
Where f is a closed zdeal zn the Banach algebra 21 and t and q = q'lJ. are continuous homomorphisms. The extension is admissible zf q is admzsszble, and uniformly radical if f zs a uniformly radical ideal in 21. Two Banach extensions E(21; 1) and E(23; 1) are strongly equivalent zf there zs a continuo?)'s isomorphzsm 'I/J: 21 ~ 23 such that 'I/J(x) = x (x E f) and q'lJ. o'I/J = q'13. The extension E splits strongly if there is a continuous homomorphzsm () : A ~ 21 such that q 0 () = fA; () is a continuous splitting homomorphism.
We also refer to the Banach algebra 2l as a Banach extension of A. A Banach extension E is an extension of the algebra A in the sense of 1.9.4; the terms 'singular', 'nilpotent', 'radical', 'finite-dimensional', 'annihilator', and 'commutative' are still applicable. A nilpotent extension is clearly uniformly radical.
278
Banach and topological algebras
Note that E is admissible if and only if the dosed ideal I is a complemented l:mbspace of 2t; this is certainly the case when I is finite-dimensional. The seClIlPnee of Banach algebra..,;
o ~ Co --> £00 ~ goo/co
~
0
is not. admissible because Co is not complemented in ex; an example of a comIIllltativc, semisimple Banach algebra with a singular extension which is not cHlmissible will be given in 4.6.10. \~,re shall be interested in the following questions for a given Banach algebra A. \Yhen does every Banach extension in a particular class split or split strongly? \\'hen does every Banach extension in a particular class which splits also split strongly? \Ve first give an easy example of when strong splitting occurs. Proposition 2.8.11 Let G be an abelian group. Then each commutative, uniformly radlcal extension of £ 1 (G) splits strongly. Proof Let 2t be a commutative Banach algebra with radical R such that R is uniformly radical and 2t/ R = £ 1 (G). We may suppose that 2t is unital. Clearly {8. : f; E G} is a bounded subgroup of Inv£l(G), and~, by 2.4.24(ii), there is a group homomorphism rJ : G ~ (2t, .) such that 'fJ(s) = 88 (s E G) and SUP.sEG II I1(S) II < 00. Define () : ESEG Ct s 8s ~ E.~E:G Cts'T/(.'I), £l(G) ~ 2t. Then () is a continuous splitting homomorphism. 0 Let E = E(2t; 1) be a Banach extension of a Banach algebra A. Then splits strongly if and only if there is a dosed :,mbalgebra ~ of 2t such that 2t -= ~\f I; if () : A ~ 2t is a continuous splitting homomorphism, then ~ = ()(A). Clearly E(2t; rad 2t) splits strongly if and only if 2t is strongly decomposable. In the case where I contains a non-zero idempotent p such that J = pI + I p, E(2t; 1) splits strongly, for (e2( - p)2t(eQ{ - p) is then a closed subalgebra of 2t and 2t = (eQ{ - p)2t(eQ{ - p) G:J I. Suppose that E(2t; 1) is a singular Banach extension of a Banach algebra A. Then J is a Banach A-bimodule for the operations of (1.9.7) because there is a constant C such that, for each a E A, there exists b E 21 with q(b) = a and Ilbll S C lIall· Conversely, let E be a Banach A-bimodule, and set 2t = A EEl E. Then E(2t; E) is a singular Banach extension of A by E. Let A be a Banach algebra, kt E be a Banach A-bimodule. and take T E Z2(A, E). Define
E
2tT = A EElT E = {(a, x) : a E A, x E: E} , } lI(a,x)1I = Iiall + Ilxll . (a,x)(b,y) = (ab, a· y+x· b+T(a,b».
(2.8.6)
At:. in §1.9, 2tT is an algebra; we have
lI(a,x)(b,y)1I S Ilallllbll + lIallllvll + IIxlillbll + IITlillalillbll ~ (1 + IITII) lI(a,x)lIlI(b,V)1I , and so 2tT is a Banach algebra (with respect to an equivalent norm). Further, L(21T ; E) is a singular, admissible Banach extension of A.
Cohomology
279
Theorem 2.8.12 Let A be a Banach algebra, and let E be a Banach A-bimodule. Then the above map T 1-+ L:(2tT; E) from Z2(A, E) induces a bzjectzon between 1i2(A, E) [j[2(A, E)) and the family of equivalence classes of singular, admissible Banach extensions of A by E with respect to strong equwalence [equwalence). Proof Let T E Z2(A, E). Then 2tT is a singular, admis~;ible Banach extension of A by E. If T 1, T2 E Z2(A, E) with T1 - T2 = 81 8 for some 8 E 8(A, E), then the connecting map (a, x) 1-+ (a, x + Sa), 2tT, ~ 2tT2' is continuous, and so L:(2tT,; E) and L:(2tT2 ; E) are strongly equivalent. Conversely, let 2t be a singular, admissible Banach extension of A by E, and take Q E 8(A,2t) with q 0 Q = lA. Then the bilinear map T : A x A -. 2t given by (1.9.9) is continuous, and so T E 8 2 (A, 2t). The remainder is essentially the same as in 1.9.5. 0 Corollary 2.8.13 Let A be a Banach algebra, and let E be a Banach A-bimodule.
(i) The following condztions are equivalent: (a) 1i2 (A, E) = {O}; (b) each szngular, admisszble Banach extenszon of A by E splzts strongly.
(ii) The followmg condztions are equivalent:
(a) j[2(A, E)
=
{O};
(b) each singular, admisszble Banach extenszon of A by E splits.
0
We have analogous definitions of1i~(A, E) and j[;(A, E) to those of1i2 (A, E) and j[2(A, E), respectively, and also analogous results. For example, let A be a commutative Banach algebra, and let E be a Banach A-module. Then commutative, singular, admissible Banach extensions of A by E split [split strongly] if and only if H;(A, E) = {O} [1i~(A. E) = {O}]. We use the above corollary to exhibit an easy example of a commutative Banach algebra which is not decomposable; further examples will be given in 5.4.5. Indeed, set
B = {f
E
A(ii}) : f(O)
a closed subalgebra of the disc algebra B-module. The map
J.L: (f,g)
1-+
= 1'(0) =
O}.
A(ii}). and regard C as the annihilator
flll(O)glll(O),
B x B ~ C.
is a continuous. symmetric bilinear functional; it satisfie. H2(A, E) is injectzve. In the case where each mtertwining map from A into E is continuous, the above conditions are satisfied. 0
Cohomology
281
Theorem 2.8.17 (Bade, Dales, and Lykova) Let A be a unztal, separable Banach algebra such that lvJ2 zs of jinzte codzmensioT! in A for each ma.n-mal ideal At of jinzte codimension m A.
Then every jinzte-dzmensional extenswn of A
which splits also splits strongly. Proof We first prove that every radical, finite-dimensional extension E(!2t; I) of A which splits also splits strongly. Such a ra.dical extension is nilpotent; we proceed by induction on the index n E N such that In = o. By 2.7.9. each intertwining map from A into a finite-dimensional Banach Abimodule is continllous, and so the case where n = 2 follows immediately from 2.8.16. Suppose that E(~; 1) : 0 ~ I -.!.... ~ !!... A ~ 0 is a finite-dimensional extension such that I has index n ~ 3. and assume that the result holds for all nilpotent, finite-dimensional extensions of index at most n - 1. Set.J = I2. Then E(~I J; II J) is a singular, finite-dimensional extension of A, as in 1.9.8; the quotient map is 7r : ~ ~ ~/J. Let () be a splitting homomorphism for E(~;I). Then 7r 0 () is a splitting homomorphism for E(~/J;IIJ), and so there is a continuous splitting homomorphism, say 8, for E(~I J; IIJ). The finite-dimensional algebra I/.J is such that (II J)2 = 0, and so I jJ is a Banach A-bimodule for the operations analogous to those given in (1.9.7); we may take the module operations to be
a·x=8(a)x, Define D =
7r
0
() -
x·a=x(7r o ())(a)
(aEAxEIIJ).
8. Then, for a, bE A, we have
D(ab) = 8(a)«7r 0 ())(b) - 8(b)) + «7f 0 ())(a) - 8(a))(7r = 8(a)D(b) + D(a)(7r 0 ())(b) = a . Db+ Da . b,
0
())(b)
and so D : A ~ I1 J is a derivation. By 2.7.9, D is continuous, and so 7r 0 () is continuous. Set 1B1 = (7r 0 ())(A), so that IB 1 is a closed subalgebra of ~I I2 such that ~/I2 = 1B1 EB (III2). Now set It = 7r- 1 (lBt), so that It is a closed subalgebra of ~ containing ()(A). Since (q lit) 0 () = lA, the extension E(It; J) splits, and so, by the inductive hypothesis, E(It; J) splits strongly. As in 2.8.14(i), E(~; I) splits strongly, and so the induction continues. We have established the result for radical, finite-dimensional extensions. Now let E = E(~; l) be an arbitrary finite-dimensional extension which splits algebraically, say the splitting homomorphism is (). Set IB = ()(A), so that IB is a closed sub algebra of ~, and set J = I n lB. Then E(IB; J) is a finite-dimensional extension of A. Let x E J, and take (an) in A with ()(a n ) ~ x. Then an = (7r 0 ())(a n ) -> 7r(x), and so vA(a n ) -> 0, whence V21«()(a n )) -> 0 and, by 2.4.8, O"21(x) is a connected subset of C containing O. Since J is finite-dimensional, O"21(x) is finite, and so O"21(x) = {O}. This shows that each x E .1 is nilpotent, and so E(IB; .1) is a radical extension of A. We have shown that there is a continuous splitting homomorphism for E(IB; J), and this map is a continuous splitting homomorphism for E. Thus E splits strongly, as required. 0
282
Banach and topological algebras
Corollary 2.8.18 Let A be a cornrnutatzve. unital, separable Banach algebra such that all point derivations on A are continuous. Then every finzte-dirnensional extension oj A which splits also splits strongly. D
The condition in the above corollary that all point derivations on A be continuous is not necessary for the conclusion: we shall see in 4.1.42(v) that every finite-dimensional extension of e1 splits strongly, although there are discontinuous point derivations on £ 1. Proposition 2.8.19 Let A be a Banach algebra. Then the Jollowzng conditions are equivalent: (a) every finzte-dimensional. annzhilator Banach extension oj A which splzt,s also splits strongly; (b) every one-dimensional, annihzlator Banach extension oj A whzch splits also splits strongly; (c) N2(A, E) = N2(A, E) Jor each finzte-dimensional. annihilator Banach A -bimodule E; (d) A has the S-property. Proof (a)*(b) This is trivial. (a){::>(c) This follows from 2.8.16 because each finite-dimensional Banach extension of A is admissible.
(d)*(c) There exists M > 0 with II all 11' :::; M lIall (a E A2). Essentially as in the proof of 2.7.9, (a)*(d), we see that IISall :::; (M + 1) IWSllliall (a E A2), and so the map S : (A2, 11·11) --> E is continuous. Since E is finite-dimension~ S has a continuous extension to A, and so T E N2(A. E). (b)*(d) Assume that (d) fails. Since {a E A2 : lIall :::; I} is not bounded in (A 2 .11·111r)' it follows from A.3.38 that there is a linear functional>' on A such that>. I A2 is continuous with respect to 11·1111" but not with respect to 11·11. Define J.L = 81>.. Then J.L E N2 (A, Co) because>. I A2 is 1I·lIlr-continuous. By 2.8.16, it follows from (b) that J.L = 81 >'1 for some >'1 E A'. But >'1 A2 = >'1 I A 2, a contradiction. Hence (d) holds. D Corollary 2.8.20 Let A be a Banach algebra such that A is a finitely generated left ideal in A#. Then the comparison map t2 : 'Jt2(A, E) --> H 2 (A, E) zs injective for each finite-dtmensional, annihilator Banach A-bimodule E. Proof By 2.1.27, A has the S-property, and so this follows immediately from the theorem. D Proposition 2.8.21 Let A be a Banach algebra such that 'Jt2(A, Co) = {O}. Then A has the 1r-property. Proof Assume that A does not have the 1r-property. Then 1rA(A0A) is not closed in A, and so there is a linear functional>' on A such that>. I A2 is continuous with respect to 111·11111" but not with respect to 11·11. By hypothesis, 'Jt2(A, Co) = {O}, and so there exists >'1 E A' with 81>. = 81>'1, a contradiction, D as above.
Cohomology
283
Let A be a Banach algebra, and let E be a Banach A-bimodule. We explained in §1. 9 that .en (A, E) is an A -bimod ule for the prod uet * defined in (1. 9.12). Clearly 13n (A, E) is a submodule of .en(A, E), and 13n (A, E) is a Banach Abimodule. The connecting maps Ak,p defined in (1.9.11) identify 13k+ P (A, E) with 13k (A, 13P (A, E)), and so we have the following 'reduction of dimension' formulae analogous to 1.9.10. Proposition 2.8.22 Let A be a Banach algebra, let E be a Banach A-bimodule, and let k,p EN. Then fik+P(A, E) andfik(A, (13 P(A, E), *)) are linearly homeomorphzc as seminormed spaces. 0 We also have the following analogue of 1.9.12. Proposition 2.8.23 Let A be a unital Banach algebra, let E be a Banach Abimodule, and let n EN. (i) Suppose that A . E = 0 or E . A = O. Then fin(A, E) = {OJ.
(ii) fin (A, E)
~
fin(A, eA . E . eA).
(iii) Suppose that E is unital and M zs a maximal ideal of codzmension one in A. Then fin (A, E) ~ fin(M, E) and jjn(A, E) ~ jjn(M, E). 0
Proposition 2.8.24 Let A be a commutatzve, unital Banach algebra, and take cp,'l/J E A with cp -I- 'IjJ. Then fil(A,C"",,p) = fi 2 (A,C"",,p) = {OJ. Proof This follows as in 1.9.16(i) because the linear functional>' defined in 1.9.16(i) is continuous in the ease where J.L E Z2(A, C"",..p)' 0 The following version of the long exact sequence of cohomology theorem is proved in essentially the same way as 1.9.13. Theorem 2.8.25 Let A be a Banach algebra and let
O-E~F~G-O be an admissible short exact sequence of Banach A-bimodules. Then there are connecting continuous linear maps such that the sequence
o_
fio(A, E) ~ fio(A, F) ~ fio(A, G) ~ fil(A, E)- ...
_fin(A,E) ~ fin(A,F)
S
fin (A, G)
is an exact sequence of seminormed spaces.
S
fin+l(A,E) _
...
o
Corollary 2.8.26 Let A be a Banach algebra, and let n E Z+. Suppose that fin (A, E) = fin(A, G) = {OJ. Then fin (A, F) = {OJ. 0 The following reduction is useful when one is determining when a finitedimensional Banach extension splits. We omit the proof because we shall only Use the result in cases where the form of the product is obvious.
Banach and topological algebras
284
Proposition 2.8.27 Let A be a commutative Banach algebra. To establish that each finite-dimensional Banach extenswn of A splits, or splits strongly, zt suffices to establish the corresponding result for each extension I:(m; 1) for which the representatwns of m on I are gwen, for a E m, by the matrices cp(a) (
o ·· ·
o
Q12(a) cp(a)
·.. ·..
Q1k(a») Q2k(a)
. ..
·. .
..
0
· ..
cp(a)
cp(a)
and
(
o
,B12(a) cp(a)
·
..
.
·. .
0
.. .
··
o
·.. ·..
,B1k(a») ,B2k(a)
...
,
cp(a)
respectively, where cp E E (Ab), by setting Ab(el» = 0, and apply J.t = Ab®A to both sides of (2.8.7): we see that J.t(a 0 a) = A(a)2 and that all other terms are O. Thus A(a) = O. It follows that A2 = A. 0
Proposition 2.8.41 Let A be a Banach algebra.
(i) A is biprojective if and only ifJrA E ABA(A®A,A) is a retraction. (ii) Suppose that A zs biproJective. Then JrA(A®A) = A and A has the 7r-property.
(iii) Suppose that A is biproJectwe, and that I is a closed zdeal in A such that 1= AI. Then the Banach left A-module AI I is projectwe. Proof (i) Suppose that 7rA is a retraction. Then A is essential, and, by 2.8.37(i), A is left projective. Similarly A is right projective, and so, by 2.8.37(iii), A ®A is a projective A-bimodule. Since A is a retract of A ®A, A is itself a projective A-bimodule by 2.8.36. Conversely, suppose that A is biprojective. By 2.8.40, A is essential, and so, by 2.8.37(ii), there exists p E ABA (A, A®A®A) such that 'if 0 P = IA. By A.3.69, there exists 1f E ABA(A®A®A,A®A) with 1f(a®b®c) = a®bc (a,b,c E A).
Banach and
290
topolog~cal
algebras
Then if 0 p E ABA(A,A®A) and 1TA 0 7r 0 p = 7i' 0 P = lA, and so 1TA is a retraction. (ii) By the open mapping theorem A.3.23, this follows immediately from (i). (iii) Set E = AI I, an essential Banach left A-module, and let q : A --t E be the quotient map. Let 1TE E AB(A ®E, E) be the appropriate product map. By (i), there exists PA E ABA(A,A®A) such that 1TA 0 PA = IA. Now define P = (IA®q) 0 PA, so that P E ABA(A,A®E). For a E A and x E I, we have p(ax) = p(a) . x = 0, and so p I I = 0 because I = AI. Thus p induces a map PE : E --t A®E. It is immediately checked that PE E AB(A,A®E) and that 1TE 0 PE = IE' By 2.8.37(i), E is projective. 0 Corollary 2.8.42 Let A be a commutative Banach algebra which is biproJective. Then tIl A is discrete. Proof Set ~ = A ®A; by 2.3.7, tIl2l = tIl A X tIl A. By 2.8.41(i), there exists P E ABA(A,~) with 1TA 0 P = IA· For a,b E A and 'P,'IjJ E tPA. we have
'P(a)('P ® 'IjJ)(p(b)) = ('P ® 'IjJ)(p(ab)) = 'IjJ(a)('P ® 'IjJ)(p(b)).
(2.8.8)
Take
'0) ® (xo ® >.) (x E E, >. E E'), and in fact p E ABA(2t, 2t®2l). Thus 1T2l : 2t®~ --t ~ is a retraction. 0 Proposition 2.8.44 Let H be a separable Hilbert space. Then K(H) is a left projective Banach algebra. Proof The inner product on H is [" .J. Let (en) be an orthonormal basis for H, and, for n E N, define Pnx = [x, enJe n (x E H) and Qn = P 1 + ... + PnTake T E K(H), and set n
Un = LTPi ®Pi E K(H) ®K(H)
(n E N);
3=1
we shall show that the sequence (Un) is 1I·1I7r-Cauchy. Indeed, take c > O. Since TQn --t T as n --t 00, there exists no E N with IITQn - Til < c (n;::: no). For (E 1I' and n > m;::: no, we have IIE;=m+1 ~ 1 and
(jp,11
n
L
j=m+l
n
(iTPj
(T - TQno)
L j=m+l
(ipj
< c;
Cohomology
291
it follows from A.3.68 that IlUn - Um lLlI" < to (m, n ~ no), and so (Un) is indeed 11·ILlI"-Cauchy. Thus E;1 TPj®Pj converges in (K(H) 0K(H).II·lln)' say to peT). Clearly p: K(H) ---+ K(H) 0K(H) is a bounded linear map which is a left K(H)-module homomorphism. and
(71'
0
p)(T) =
7rKCH)
~TPj ®Pj )
(lim n~oo ~ j=1
for T E K(H), and so 71'
0
P=
h:'(H)'
= lim
~TP} =
n-+oo ~
j=1
lim TQn = T
n-+oo
o
Thus K(H) is left projective.
The following definition should be compared with Definition 1.9.19. Definition 2.8.45 Let A be a unital Banach algebm. A projective diagonal for A zs an element U E A0A such that 7rA(U) = eA and a . u = u . a (a E A).
Thus a projective diagonal for A is an element u such that
t.
llaj 1IIIbJ II < 00,
00
t.
ajb j =
lOA,
00
l:aaj®bj=l:aj®bja j=1 j=1
I
= E;:1 aj
® bj in A 0 A
(2.8.9)
(aEA).
is a diagonal Let E N. We noted in 1.9.20 that d = (E~j=1 eiJ ® eJi) for the full matrix algebra Mn. Let p E [1,00]. As in 2.1.18(ii), the algebra Mn is a Banach algebra with respect to the norm II· lip' formed by regarding Mn as an algebra of operators on (en, II· lip)'
n
In
Proposition 2.8.46 Let n E Nand p E [1,00]. Then Ildlin = 1 as a proJective diagonal for (Mn.II·ll p ) Proof Fix k E N n , and set Xj = ej.j+k and YJ = ej+k,J for j E N n , where j is calculated modulo n in N n . It is easy to see that = 1 p
J=l
and so, by A.3.68, IIE7=1 Xj ® Yj 1111"
+k
(( E 11'),
p
= 1. The result follows.
o
Definition 2.8.47 Let A be a Banach algebm. Then the homological bidimension of A zs
dbA = min{n
E
Z+ : Hn+1(A, E)
= {OJ for each Banach A-bimodule E}.
The algebm A is contractible if db A = O. Thus A is contractible if and only if H1(A, E) = {OJ for each Banach Abimodule E. It follows from 2.8.22 that Hn+1(A, E) = {OJ for each Banach A-bimodule E and each n ~ db A. We note that dbA = dbA#.
Banach and topological algebras
292
It is of interest to characterize Banach algebras within special classes that have a particular bidimension, that are left projective, and that, are biprojective, and we shall at least make some remaIks on this at some later points. (See 3.3.32, 4.1.42(vii), and 4.2.31, for example.) The first result, characterizing contractible Banach algebras. is related to 1.9.21. Theorem 2.8.48 Let A be a non-zero Banach algebra. Then the following condztwns on A are equwalent: (a) A is contractible; (b) A is umtal and has a projective dwgonal m A 0 A; (c) A is umtat and biprojectwe; (d) A# is bzprojectwe.
Proof (a){::}(b) This is the same as the proof of the equivalence of (a) and (b) in 1.9.21, save that we work with Banach A-bimodules and continuous derivations. (b)~(c) Suppose that u is a projective diagonal in A0A. Then the map defined by p: a ~ a . 1.£, A ~ A ®A, shows that 7rA is a retraction, and so A is biprojective. (c)~(b) Let p E ABA (A. A 0 A) be such that 7rA 0 P = lA, and set 'U = p(eA)' Then u is a projective diagonal for A in A ®A. (c)~(d) This is trivial. (d)~(a) Let E be a Banach A-bimodule. Then E is a Banach A#-bimodule. and so 1{1(A#.E) = {O} by the implication (c)~(a) for A#. By 2.8.23(iii). 1{l(A, E) = {O}. 0
Corollary 2.8.49 Let A be a contractible Banach algebra. Suppose ezther that A is commutative or that A has CAP as a Banach space. Then A zs semisimple and finite-dimensional. Proof By 2.8.48, A is unital, biprojective, and has a projective diagonal 1.£ in A ®A with the form in (2.8.9), say. Suppose that A is commutative. By 2.8.42, A is discrete, and hence finite. Thus there exists n E N such that A = en Efj rad A as a Banach space. Suppose that A has CAP as a Banach space. Then there is a net (Ta,) in K:(A) such that Toa -+ a (a E A). Define Soa = 2::;':1 ajTa(bja) (a E A). Then (So) C K(A), and limo Soa = 2::;':1 aJbja = a (a E A). There exists Uo : A0A ~ A such that Uo(a i8l b) = aTob (a, bE A). We have
Soa=Uo(u· a)=Uo(a ·1.£)=a· Uo (taii8lbj) =a· SoeA
(aEA),
)=1
and so So ~ IA in B(A). Thus fA E K(A), and so A is finite-dimensional. In both cases, there is a continuous projection P : A ~ rad A. Now define Q : a ~ E';laiP(b)a), A ~ radA. Then Q is also a continuous projection. As in the proof of 1.9.21, Q = 0, and so A is semisimple. 0
293
Cohomology
Definition 2.8.50 Let A be a Banach algebm, and let E be a Banach left Amodule. A complex
(0
+-
E
+-
P)
(2.8.10)
is a projective resolution of E if zt zs admissible and zf earh of the modules Pn for n E z+ zs a pr'oJective Banach left A-module. The length of the resolution lS the smallest n ;:::: -1 such that Pk = 0 (k > 11) or 00 zf there is no such n. Similar definitions apply to Banach right A-modules and A-bimodulcs. ~ ~ For example, consider Pn = AD ® ®j=IA (n EN), Po = A I> ® E. and P- I = E. Then each Banach left A-module Pn for n E Z+ is projective Set a-I = 7r, and define an : Pn + 1 -+ Pn for n E Z+ by
(-n )
an(a®al®'" ®anH®x)=aal®a2®"'®an+l®x n
+ ~) -l)j a ® al 0 ... ® ajaJ+I ~ ... ~ an+1 ® x J=l
+( _l)nHa ® al ® ... ® an+l . x. We clazm that the corresponding resolution (2.8.10) is admiHsible. To see this, we use the bounded linear operators Q-l : x ~ el> ® x, E -+ Pu, and Qn : (ae!>
+ a) ® al ® ... ® an ® x
~ el> ® a ® al ® ... ® an ® x,
Pn
-+
Pn+1 ,
for n E Z+. Then Qn-l 0 an-I + an 0 Qn = I p " (n E Z+), as required. Thus we obtain a projective resolution of a Banach left A-module E, called the bar resolutzon. Let A be a Banach algebra, let E and F be Banach left A-modules, and let o + - E + - P be a projective resolution of E as in (2.8.10). Then we may form the complex of Banach spaces
o ~ AB(Po, F)
~ AB(PI , F) ~ AB(P2 , F)
.!!:4 ... ,
where Dn(T)(x) = T(dn(x» (x E P n+1 ) for T E AB(Pn , F). The corresponding cohomology groups are defined to be ExtA(E,F)
= ker Dn/imDn -
1
(n E N),
with Ext~ (E, F) = ker Do = AB(E, F). We state without proof the following fundamental theorem and corollary. Theorem 2.8.51 (Helemskii) Let A be a Banach algebm, and let E and F be Banach left A-modules. Then Ex(.t(E, F) for n E Z+ depends only on E and F, and not on the projective resolution occurrtng in the rlefimtion. 0 Corollary 2.8.52 Let E be a Banach left A-module. Then the following conditions are equivalent: (a) E is a projective Banach module; (b) ExtA(E, F) = {O} for each n E N and each Banach left A-module F; (c) Ext~(E, F) = {O} for each Banach left A-module F. 0
Banach and topological algebras
294
Let A be a Banach algebra with enveloping algebra Ae = A#0A#oP. As in 2.6.9, a Banach A-himodule E is regarded as a Banach left Ae-module. Clearly E is a projcctive A-bimodule if and only if E is a projective left Ae-module. The following theorem gives the conncction between the Ext groups and the continuous cohomology groups; in the next two results, B(E, F) is a Banach A-bimodule for the operations given in (2.6.12).
Theorem 2.8.53 Let A be a Banach algebra, and let n E N.
(i) For all Banach left A-mod1tles E and F, 'Hn(A.B(E,F))
~
Ext'A(E,F).
(ii) For each Banach A-bimodule E such that E is unital
~f
A is unital,
'Hn(A,E)~Ext'Ap(A#,E).
0
Corollary 2.8.54 (Kaliman and Selivanov) Let 2l be a Banach opemtor algebm on a Banach space E. Then 'Hn(21, B(E)) = {O} (n EN). Proof By 2.8.38, E is a projective Banach left 2l-module. and so, by 2.8.52, Ext!1(E. E) = {O} (71 EN). By 2.8.53(i), 'Hn(21, B(E)) = {O} (71 EN). 0 In particular, all continuous derivations from 2l into B(E) are inner, a result already obtained in 2.5.14. Let A be a Banach algebra, and let E be a Banach left A-module. Then the homological dzmension of E, denoted by dhAE, is the minimum of the lengths of the projective resolutions of E, so that dhAE = 0 if and only if E is projective. and dhAE = n if a.nd only if Ext1+1(E, F) = {O} for each Banach left A-module F and Ext'A (E, G) =I- {O} for some Banach left A-module G.
Definition 2.8.55 The global homological dimension of A is dg A = sup{ dhAE : E is a Banach left A-module}. Thus dgA is the minimum Tl E Z+ such that 'Hn+1(A,B(E,F)) = {O} for all Banach left A-modules E and F. We clearly have dg A ~ db A. It follows from 2.8.53(ii) that, for a Banach algebra A, db A is the minimum value of n E Z+ such that A# has a projective resolution (using A-bimodules) of length n. The following striking theorem suggests a reason why biprojective algebras are considered. The resolution constructed in the following proof is called the entwining resolution.
Theorem 2.8.56 (Helemskii) Let A be a biprojective Banach algebra. dbA ~ 2, and 'H3(A, E) = {O} for each Banach A-bimodule E.
Then
Proof We first define some A-bimodules. We set: Po = Po 1 ffi Po 2, where PO,l = A~0A# and P~2 = A; P l = Pl,l ffi P l ,2, where Pl,l' = A#'0A and P I ,2 = A0A#; P 2 = A0A. Since A is biprojective, it follows from 2.8.37, (iv) and (v), that each of Po, P l l and P2 is a projective A-bimodule.
Cohomology
295
'Ve now define maps 8, do, and d 1 such that
L :0 ~ A# ~ Po ~ PI ~ P2~O is a projective resolution of A#. The maps 8, do, and d 1 are continuous linear operators which satisfy the following conditions:
= ab+c (a,b E A#, c E A), do(a0b,c0d) = (a0b- c0d. -ab + cd) (a,d 8(a0b,c)
d 1 (a0b)=(a0b,a0b)
E A#. b,c E A).
(a,bEA).
It is easily checked that these maps are each A-bimodule homomorphisms and that :E is a resolution of A#. For example, for each a, dE A# and b, c E A, we have (8 0 do)(a 0 b, c 0 d) = (ab - cd) + (-ab + cd) = O. To show that:E is admissible, we define continuous linear maps Q : A# ~ Po, Qo : Po ~ PI, and Q I : PI ~ P2 such that I A # = 8 0 Q, I Po = do 0 Qo + Q 0 8, I pt = d 1 0 Q1 + Qo 0 do, and Ip2 = Ql 0 d 1. It suffices to suppose that A is non-unital. We define: Qa=(e,4.0a,O) QO«aeA
+ a) 0
b, c)
Ql«aeA
= (-eA 0 ab - eA 0
+ a) 0
(aEA#)j
c, -a cg b)
b, c 0 d) = a 0 b
(a E C, a, c E A, bE A#) j
(a E C, a, b, c E A. dE A#).
We check that the required equations are satisfied, and so :E is admissible. We have constructd the required projective resolution of A#.
0
We have discussed contractible, or 'cohomologically trivial', Banach algebras in 2.8.48. A much more important class of Banach algebras consists of the amenable ones, those that are 'cohomologically trivial with respect to dual module....,·. Indeed, the class of amenable Banach algebras is of grea.t significance in many diverse areas of the theory.
Definition 2.8.57 Let A be a Banach algebra. The weak bidimension of A is dbwA = min{n E Z+ : 1-£n+1(A,E') = {OJ
for each Banach A-bimodule E}.
Then A is a.menable if dbwA = 0, weakly amenable if 1-£1 (A, A') simplicially trivial if 1-£n(A, A') = {OJ (n EN).
= {OJ, and
Clearly dbwA ::::; dbA. The Banach algebra A is amenable if and only if
?i 1(A, E') = {OJ for each Banach A-bimodule E. By 2.8.34, 1-£n+1(A, E') = {OJ
for each Banach A-bimodule E and each n ~ dbl.l.A. It follows from 2.8.33(iv) that A is amenable if and only if ?i 1 (A, E) = {O} and im do is closed in E for each E. Clearly a contractible Banach algebra is amenable and an amenable Banach algebra is simplicially trivial. The justification for the term 'amenable' will become apparent at 5.6.42. There are commutative, amenable algebras which are not biprojective (see 5.6.3), and commutative, biprojective algebras which are not amenable (see 4.1.42). A commutative Banach algebra A is weakly amenable if and only if ZI (A, A') = {a}.
Banach and
296
topolog~cal
algebras
Proposition 2.8.58 Let A be a Banach algebra. Then:
(i) A is amenable iJ and only iJ A# is amenable; (ii) zn the case where A zs umtal, A zs amenable zJ and only zJ1iI(A, E') = {O} Jor every unital Banach A-bimodule E.
Proof Since (e . E . e)' = c . E' . e, these follow from 2.8.23
o
Proposition 2.8.59 (Gourdcau) Let A be a Banach algebra.
(i) The Jollowzng condztwns are equzvalent: (a) A is amenable; (b) 1i1(A. E") = {O} Jor each Banach A-bimodule E;
(c) JOT each Banach A-bimodule E and each D E Zl(A, E), there exists a bounded net (xn) in E such that Da = lima(a . Xc> - Xo; . a) (a E A).
(ii) Suppose that (A", 0) is amenable. Then A is amenable. (iii) Suppose that SeA. A') = W(A, A') and that (A". 0) is weakly amenable. Then A is Arens regulaT and weakly amenable.
Proof (i) (a)=:;.(b) Thil:; is immediate. (b)=:;'(c) We have t 0 D E ZI(A, E"), where t : E ~ E" is the canonical embedding. By (b), there exists A E E" with (t 0 D)(a) = a . A-A· a (a E A). Set m. = IIAII, U = E[m]' and a = a(E", E'). By A.3.29(i), A E t(U) "". Now fix aI, ... ,an E A. Then the set V = 117=1 (aj . U - U . aj) is a convex subset of E(n). and (Dal ..... Dan) belongs to the weak closure of V. By Mazur's theorem A.3.29(ii), (Dal, .... Dan) belongs to the II· II-closure of V. Thus, for each finite subset F of A and c > 0, there exists xP,c E U such that
IIDa - (a . XP.c - xP,c .
a)11 < c
(a E F).
The family of such pairs (F, e) is a directed set for the partial order :::: given by (Fl.el):::: (F2,e2) if FI C F2 and el;::: e2' Clearly (xp,c) is the required net. (c)=:;.(a) Let D E Zl(A, E') for a Banach A-bimodule E, and let (Ac» be a bounded net in E' such that Da = liillo;(a . Ao; - Ao; . a) (a E A). By passing to a subnet, we may suppose that Ao; ~ A in (E',a(E', E», and then D = 8>. E NI(A, E'). Thus A is amenable. (ii) Let E be a Banach A-bimodule, and take DE Zl(A, E'). As in 2.6.15(iii), E'" is a Banach (A",D)-bimodule; by 2.7.17(i), D" : (A", D) ~ E'" is a continuous derivation, and so, by (i), (a)=:;.(c), there is a bounded net (Ao) in E'(/ such that Da = limo(a . Ao - Ao . a) (a E A). By passing to a subnet, we may suppose that Ao -+ A in (EIIf,a(EIIf,E"». For each a E A, we have a· Ao-Ao; . a ---+ a· A-A· a in (EIIf, a(EIIf, E"». and so Da = a . A-A· a. Let P : E'" ---+ E' be the natural projection. Then Da = a· peA) -peA) . a (a E A), and so D E Nl(A,E'). Thus A is amenable. (iii) By hypothesis, condition (d) of 2.6.17 is satisfied, and so, by that result, A is Arens regular. The dual module of A" is AIIf; set 0'= O'(A", A').
Cohomology
297
Let D E Z1(A, A'). Take q), \.II, Y E A", and then take bounded nets (a o,) and (bf3) in A with ao --> q) and b{3 --> \.II in (A",O"). For each a, we have lim (Y, Da o {3
.
b(3) = lim (b{3 0 Y. DaOl.) = (\.II 0 Y, DaOl.) (3
because the map Ry is continuous on (A",O"). Thus lim lim (Y, DaOi. . b{3) O!
(3
= (lJI 0
Y, D"(q»)
= (Y,
D"(q» . lJI) .
Similarly, we have lim{3 (Y, aa . Dba) = lim{3 (Y 0 aOl.' Db(3) = (Y 0 a D"(\.II)) for each a, and so limOl.lim{3 (Y, ao 0 q), D"(lJI)) = (Y. q) . D"(lJI)) because the map Ly is continuous on (A", 0") by 2.6.17 and because D"(lJI) E A' by A.3.56(ii). It now follows that Q ,
(Y, D" (q) 0 lJI»)
= lim lim (Y, a a = (Y,
q) .
D(aOl.b{3))
D"(lJI)
= lim lim (Y, aa a a
+ D"(q»)
. Db{3
+ Dao
.
b(3)
. lJI) ,
and so D" E Z1(A", A"/). Since A" is weakly amenable, D" E NI(A", Alii), and so D E NI(A, A'), where we are using the projection P E ABA(A''', A'). 0 Theorem 2.8.60 Let A be an amenable Banach algebra, let E be a Banach right A-module. and let F and G be Banach left A-modules. Then each admissible
short exact sequence of Banach left A-modules
L :0
---+
E' ~ F
-I-.. G ---+ 0
splits strongly. Proof Since L is admissible, there exists QI E B(G, F) with T 0 QI = Ie. As in Example 2.6.2(viii), B(G, F) is a Banach A-bimodule with respect to the maps (a, R) ~ a . R and (a, R) ~ R x a of (2.6.12). Define D:a~a·
QI-Q1 xa,
A-->B(G,F),
so that D is a continuous derivation. For a E A and z E G, we have
(To Da)(z) =T(a· QIz-QI(a· z)) = a . (T 0 Q1)(Z) - (T 0 Qd(a . z) =a' z-a· z=O,
because T E AB(F, G)
and so (Da)(G) C kerT = SeE'). We are making the identification S(E') 9:! E/, and so we may say that D : A --> B(G, E') is a continuous derivation. By 2.6.4(ii), B(G, E') is a dual module, and so, since A is amenable, there exists Q2 E B(G, F) with Q2(G) C kerT and Da = a . Q2 - Q2 x a (a E A). Set Q = Q1 - Q2 E B(G,F). Then To Q = To Q1 = Ie. For a E A, we have a . Q = Q x a, and so Q E AB(G, F). Thus L splits strongly. 0 Definition 2.8.61 A Banach algebra A is biflat if the dual map 71"~ is a coretraction in ABA(A',(A®A)').
298
Banach and topological algebms
It follows from 2.8.41(i) that a biprojective Banach algebra is biflat. Proposition 2.8.62 A biftat Banach algebm zs simplicially trivial. Proof By hypothesis. there is a continuous A-bimodule homomorphism R: (A 0 A)' -+ A' such that R 0 7rA = IA" Let n E N, and take T E zn(A, A'). In the case where n ~ 2, define S: ACn-1) -+ (A0A)' by requiring that (b®c, S(a1, ... ,an -d) = (c, T(a1, ... ,an- b b» (a1, ... ,an-1, b,cE A). We compute bn - 1S by using Definition 1.9.1: for a1,"" an, b, c E A, we have (b®c, (b n - 1 S)(a1, ... ,an ») n-1 = (b 0 cab 8(0,2, . .. , an»
+L
(-I)j (b ® c, SCab ... , ajaJ +1, ... ,an)) j=1 + (_1)n (anb®c, S(al, ... ,an-d) n-1 = (cal. T(a2, ... ,an .b» + L(-I)j (c, T(a1, ... ,aJ aj+1, ... ,an ,b» j=l + (-It (c, T(ab.·., an-b anb» = (c, (b n T)(a1"'" an, b) - (_l)n+1 (bc, T(a1, .... an» = (_I)n (be, T(ab ... , an)) = (_1)n (b 0 c, (7rA'
We conclude that (_I)n(b n - 1S) = T = Ro «_I)n(b n- 1
s»
7rA'
0
0
T)(ab.·., an» .
T, and so
= bn - 1«_I)nRo S) E Nn(A,A').
In the case where n = 1, define Ji-o(b, c) = (c, Tb) (b, c E A). Then clearly /-Lo E 8 2 (A, q, identified with (A0A)', and T = bRCI'o) E Nl(A. A'). In each case, 'Jtn(A, A') = {O}, and so A is simplicially trivial. 0 We shall see in 4.1.42 that there are commutative. weakly amenable Banach algebras for which 'Jt2(A, A') i= {a}. Theorem 2.8.63 Let A be a weakly amenable Banach algebm. Then: (i) A is essential; (ii) there are no non-zero, continuous pomt derivatwns on A; (iii) zn the case where A is commutative, Zl(A,E) = {a} for each Banach A-module E. Proof (i) Assume towards a contradiction that A2 i= A. Take ao E A \ A2, choose Ao E A' with Ao I A2 = 0 and (ao, Ao) = 1, and define D = Ao 0 AO : a
1-+
(a, Ao}AO,
A
-+
A' .
Certainly D is a continuous linear map. For a, b E A, we have D( ab) = 0 and (c, a . Db)
+ (c,
Da . b) = (ca, Db)
+ (be,
= (ca, Ao)(b, Ao)
Da)
+ (be,
Ao}{a, Ao) = 0
(c E A)
because Ao I A2 = 0, and so a . Db + Da . b = 0. We have proved that D E Zl(A, A'). Now (ao, Dao) = 1, but (0.0, b,,(o.o)} = (A E A'), and so
°
299
Cohomology D
tt ,Nl(A,A'), a contradiction of the fact that Jil(A, A') =
{a}. Thus A2 = A. (ii) By (i), there are no non-zero, continuous point derivations on A at O. Let d be a continuous point derivation on A at
A. Then the map D = d0 . E K' is such that a . >. = A . a (a E 21.). However, the space of continuous traces on 21. has dimension 1. Hence, if dimK 2: 2 (which implies that E does not have AP), K does not have the trace extension property, and so, by (iii), above, N(E) = 21./ K is not weakly amenable.
Cohomology
301
Proposition 2.8.67 Let A be a Banach algebra such that A# is weakly amenable and 1{2(A, Co) = {o}. Then A is weakly amenable. Proof We may suppose that A is non-unital. Consider the short exact sequence
L :0 ~ A ~ A# ~ Co ~ 0 of Banach A-bimodules, where t is the natural embedding. The sequence E is admissible, and hence so is its dual E/. By 2.8.25, we have a long exact sequence which contains the subsequence ... ~ 1{1(A,A#/)~1{1(A,A')~1{2(A,Co)~'" .
By 2.8.23(iii), 1{l(A, A#/) = 1{l(A#, A#/), and so 1{l(A, A#/) = {a}. 1{2(A, Co) = {O}, and so 1{l(A, A') = {o}. Thus A is weakly amenable.
But 0
Lemma 2.8.68 Let A be a weakly amenable, commutative Banach algebra, let I be a closed ideal in A, and let E be a Banach I -module. Then D I 14 = 0 for each D E Zl(I,E). Proof Set F = AB(I, E), so that F is a Banach A-module for the product specified by (a . T)(b) = T(ab) (a E A, bEl). The map j : E --+ F such that j(x)(a) = a . x (a E I, x E E) belongs to AB(E, F) (cj. 2.6.2(ii), and so j 0 DE Zl(I, F). Define D: (a, b) ~ (j
0
D)(ab) - b . (j
0
D)(a),
I x A
--+
F;
clearly D E B2(I, A; F). By 1.8.4(i), D(a, b) = a . (j 0 D)(b) (a, bEl). Take a E 12. By 1.8.4(ii), the map b ~ D(a,b), A --+ F, is a derivation. Since A is commutative and weakly amenable, it follows from 2.8.63(iii) that this map is zero, and so D(I2 x A) = O. Thus, for a E 12 and b, eEl, we have ac . Db
and so 1 3
.
= (j
0
D)(b)(ac)
D(I) = 0, whence D
= (a . (j I 14 = O.
0
D)(b»(c)
= D(a, b)(c) = 0, o
Theorem 2.8.69 (Gf!zmbrek) Let A be a weakly amenable, commutatzve Banach algebra, and let I be a closed zdeal m A. Then: (i) I is weakly amenable if and only if 12 = I;
(ii) I is weakly amenable in the case where I has finite codzmension in A. Proof (i) Suppose that I is weakly amenable. Then 12 = I by 2.8.63(i). Conversely, suppose that 12 = I, so that 14 = I. Let D E Zl (1, I'). Then D I 14 = 0 by 2.8.68, and so D = O. Thus I is weakly amenable. (ii) Suppose that I has co dimension one in A. By 2.8.63(i), A2 rt I, and so, by 1.3.37(iii), I = M
and 1> . al ... an belong to A whenever al,'" ,an E A and 1> E A{2n}. Let A.l be the space of functionals in A{2n+l} which annihilate the copy of A in A{2n}. Then A{2n+l} = A' EB A.l as Banach A-bimodules. and so 'H I (A,A{2n+I}) = HI(A,A') EBHl(A,A.l). By hypothesis, H1(A,A') = {a}, and so it suffices to show that HI (A, A.l) = {o}. Let DE ZI(A,A.l), and let a,b E A[n]. Then
+ (1) . a, Db) = 0, and so D = O. The result
(1), D(ab)) = (b . 1>, Da) and so D(ab) = O. Thus D I A2n
= 0,
follows.
0
Notes 2.8.78 The ~tudy of the continuous cohomology groups rC'(A, E) for a Banach algebra A and a Banach A-bimodule E was initiated by Kamowitz (1962) (for commutative algebras); the continuous homology groups H,,(A, E) appear first in (Guichardet 1966). An important and very influential early memoir in this area is that of Johnson (1972a); this memoir used 'direct methods' to calculate cohomology groups, and, in general, we have followed this approach. Somewhat earlier than 1970, Helemskii and his Moscow school began to develop the alternative approach through relative homological algebra: this approach is expounded in (Helemskii 1993, Chapter VII) and, in fuller detail, in (Helemskii 1989b), where the history and many references are given. The corresponding theory for more general topological algebras originates with J. L. Taylor (1970, 1972). A synthesis of these complementary approaches is now being achieved, forming a subject that is called topologzcal homology; this subject is already too large to permit a comprehensive survey. An introductory account of cohomology theory is given in (Bonsall and Duncan 1973, Chapter VI); for important surveys and lists of open problems, see (Helemskii 1984, 1989a, 2000). The cohomology spaces H" (A, A') are closely related to certain cyclic cohomology groups HC"(A) (Helemskii 1992), this notion is related to important work of Connes (see (1994, IILl.a)) and Tsygan (1983). Some methods for computing cyclic cohomology groups of Banach algebras are given in (Lykova 1998). Let 8 be the free semigroup on the generators ao, bo, aI, bl , a2, b2, ... , subject to the relations bl O. Then there exist u E A[n] and bEl such that lIa - ua + bll < c/2(1 + m), and there exists v E A[m] with lib - vbll < c/2. Now v u E A[m+n+mn] and Q )
Iia -
(v u)all
s II (a -
ua + b) - v(a - ua
< (l+m)c
- 2(1 + m) 'rhe result follows from 2.9.14(ii).
+ b) II + lib - vbll
c_
+2-
c.
o
Proposition 2.9.21 Let A and B be norrned algebras. Suppose that A and B have left approximate identities of bounds m and n, respectively. Then A®B has a left approximate identity of bound mn.
312
Banach and topological algebras
Proof Take 2:7=1 aj ® bj E A ® B, and set r = max{lIaJII, IIbjll : j ENd· There exist u E A[m) and 17 E B[n) such that Ilaj - uaJIl < e/(3r + l)k and IIbj - vbjll < e/(3r + l)k for each j E N k . We now have k
k
2:aj ®bj - (u®v) 2:aj ®bj j=l j=l k
::; L (1laj -
uaj 1IIIbj II
+ lIaj IIllbj
- vbj II
+ IlaJ
-
uaJ 1IIIbj - vbj
II) < e.
j=l
o
Since u ® v E (A®B)[mn), the result follows from 2.9.14(ii).
Our first factorization theorem requires two easy lemmas, giving estimates that we shall use. Lemma 2.9.22 Let A be a non-umtal Banach algebra. and let E be a Banach left A-module. Take m ~ 1 and "I E (0, l/(m + 1)). and let u E A[m)' Then: (i) (1 - "I)eA
11«(1 - "I)eA + "I1.l)-1 II ::; (1 - "I - m"l)-l; exists TJ > 0 such that IIx - «(1- "I)eA + "Iu)-l . xII < e
+ "IU E Inv A#,
(ii) for e > 0, there whenever x E E with Ilx - u . xii
and
< TJ.
Proof (i) We have 11"1(1- "I)-lull ::; "1(1- "I)-1 m from (2.1.18).
(ii) Set v = (1 - "I)eA IIx - v- J
•
xii::; (1 -
< 1 and so the result follow!'!
+ "IU E Inv A#, and take x
"I - Tn"l)-lliv . x - xII
E E. Then
= "1(1- "I -
m"l)-llix - u . xii,
o
and so the result follows by taking rl = e(l - "I - m"l)/'Y.
Lemma 2.9.23 Let A be a non-unital normed algebra wzth a left approX'lmatf identzty of bound m, and let E be a Banach left A-module. Take al,"" a p E A and Xl, ... , Xq E AE. Then, for each e > 0 and ko E N, there eX'lsts u E Arm] wzth
max {liai - ukaill. IIXj - uk .
Xjll: i E N
p,
j E N q , k E Nko}
< e.
Further, Xj E A . x J (j E N q ).
Proof There is a net (u o ) in A[m) such that lim(~ uaa = a for each a E A. By (2.6.2), limo u~a = a for each a E A and kEN. Further, limo u~ . X = x for each x E AE and kEN. Thus we may take u = U o for a suitable a to achieve the required inequality. 0 Theorem 2.9.24 (Cohen, Hewitt, Allan and Sinclair) Let A be a non-umtal Banach algebra with a left approximate identity of bound m, let E be a Banach left A-module, and let (an) be an unbounded, increasing sequence in (1,00). Then, for each x E AE, no E N, and e > 0, there exist a E A[m] and (Yn) in A . x such that:
BoundeAi approxzmate idelltzties and factorizatwn (i) x
= an . Yn (n
313
EN);
Ilx - Ynll ::; c (n E Nno ); IIYnll ::; a~ Ilxll (n EN). Proof We may suppo~e that Ilxll = (ii)
(iii)
1 and that e < min{l,a~ - 1 : n EN}. Choose, = 1/4m, so that, < l/(m + 1). For each u E A[m], define
feu) = «1 -,)eA
+ ,u)-1
E Inv A#;
by 2.9.22(i), III(u) II ::; 4m/(3m - 1) ::; 2. We first inductively choose a strictly increasing sequence (jk : kEN) in N such that jl ~ no and an > 3k + 1 (n ~ jk) for each k- E N; such a choice is possible because On ----> 00 as n ----> 00. We shall next inductively define a sequence (Uk) in A[m] such that, for each kEN, the element o,k defined byak = (l-,)k eA + 2:;=1 ,(1-,)j- l Uj belongs to Inv A# and satisfic~ the conditions that
Ila;;111 ::; 3k
and
lIa k':' 1· x- ak xii < ;k j .
(j E
Nj~);
(2.9.1)
here we are setting ao = CA. It is immediate from 2.9.22(ii) and 2.9.23 that there exists Ul E A[m] such that (2.9.1) holds for k = 1, where we use (2.6.2); in fact, 1l = IIf(ul)ll::; 2. Now take kEN, and assume that Ul •... , Uk have been defined in A[m] so that al •... ,ak E Inv A# and (2.9.1) holds. Set
IIa li
k
g(1.1) = (1 _,)k eA
+ feu) L ,(1
_,)j-1Uj
(u E
A[m])'
j=1 Then lIak - g(u)11 ::; 2:;=1,(1 - ,)J-1IIUj - f(u)Ujll (U E A[m])' and so, by 2.9.22(ii) (applied with E = A), 2.2.36, and (2.1.2), there exists 171 > 0 such that g(u) E Inv A# and Ila kj - g(u)-jll ::; e/2k+2 (j E NjH ,) whenever we have
lIuj - UUj II < "II (j E Nk). Also by 2.9.22(ii) and (2.6.2), there exists "12 > 0 such that Ilx-f(u)j.
xii <e/2 k +2 (lla;;jll+1)
(jENjk+,)
whenever Ilx - U . xII < "12. By 2.9.23, there exists U E A[m] ~uch that both lIuj - uUjll < "II (j E Nk) and IIx - u . xII < "12: this element U is our choice of Uk+1' We have «(1 -,)CA + ,Uk+r)g(Uk+1) = ak+1, and so, for each j E N iH1 ,
xii = Ila;;j . x - g(Uk+1)-j f(uk+r)j . xii . x- g(Uk+1)-i . xii + Ilg(uk+1)-jllllx - f(uk+1)j . xII
Ila;;j . x - a;!1 . ::; lIa;;j
< c/2k+ 2 + c/2k+2 = c/2 k + 1 . Also Ila;;~111
::; IIf(Uk+1) IllIg(Uk+d- 111
::; 2
(lIa;;111 + 1) ::; 2(3k + 1) ::; 3k+1 .
lienee the inductive construction continues.
314
Banach and topological algebms
Define a = l:j:l 'Y(1 - 'Y)i- 1UJ = limk-+x. ak· Then a E A[m]' It follows from (2.9.1) that, for each n E N, the sequence (a;n . x : kEN) is a Cauchy sequence in E, and so it converges to an element, say Yn. of A . x. By (2.9.1). Ilx - Ynll ::; c (n E Nil)' Suppose that n E Nil' Then II Yn II ::; 1 + c < n~ by thE:' restriction on the value of c. Now suppose that n E Uk, ... ,jk+l-1}. Then it follows from (2.9.1) and the choice of Jk that
IIYnll ::; Ila;n . xii + c::; IIYnll ::; Q~ IIxll (n EN).
Ila;l r + 1 ::; 3nk + 1 ::; Q~.
Hence This completes the proof of the theorem.
o
The following corollaries are trivial in the case where A is unital, so the proofs can assume that A is non-unital. Corollary 2.9.25 Let A be a Banach algebm with a left approximate identity of bound m. For each a E A and c > 0, there exist b E A[m] and (c n ) in Aa with a = bnc-n (n E N) and lIa - clil < c. 0 Corollary 2.9.26 Let A be a Banach algebm with a left approximate identity of bound m, and let E be a Banach left A-module. Then AE = A . E is a closed, weakly complemented submodule of E, and, for each J: E E with IIxll < 1 and each c > 0, there exist a E A and Y E such that x = a . y, IIx - yll < c. and Ilallllyll < m.
rx
Proof We prove that A . E is weakly complemented in E; the remainder is immediate from the theorem. Let (eo:) be a left approximate identity in A[m], and define a net (To:) in B(E')[m] by setting 1-;"(>') = >. . eo: (>. E E'). By A.3.35, the ball B(E')[m] is wo*-compact, and so there exists T E B(E')[m] such that we may suppose that wo*-limo: To: = T. For each a E A, x E E, and A E E', we have
(a . x, TA)
= lim(a '"
. x, A . eo:)
= lim(er>a
. x, A)
= (a
. x, A),
0:
and so TA - >. E (A . E)o. The map T - IE' is a projection of E' onto (A . E)O, and so A . E is weakly complemented in E. 0 Corollary 2.9.27 (Johnson) Let A be a Banach algebm with a bounded approximate identity, and let E be a Banach A-bimodule. Then A . E . A is a closed, weakly complemented submodule of E, and
'lin (A, E') = 'lineA, (A . E . A)')
(n E N) .
Proof Set F = A . E, a closed submodule of E, and set G = ElF. By 2.9.26, G' = FO is complemented in E', and so 0 ---+ G' ---+ E' ---+ F' ---+ 0 is an admissible short exact sequence of Banach A-bimodules. Clearly A . G = 0, and so, by 2.9.9, 'lin (A, G') = {O} (n EN). By 2.8.25, we have 'lin (A, E') = 'lin (A, F') (n EN). The result follows by using the same argument 'on the other side'. 0
Bounded approximate identitzes and factorization
315
Corollary 2.9.28 Let A be a Banach algebm with a bounded approximate identzty, and suppose that 1-£1 (A, E') = {O} for each essential Banach A-bimodule E. Then A is amenable. 0 Corollary 2.9.29 Let A be a Banach algebm with a bounded left approximate zdentity, and let E be an essentwl Banach left A -module. (i) E is neo-unital, and null sequences in E factor.
(ii) Suppose that K is a non-empty, compact subset of E. Then there exzst a E A and a compact subset L of A . K such that the map x I-> a . x, L ---+ K, is a homeomorphism. Proof (i) Certainly E is neo-unital. Set F = co(N, E), as in 2.6.2(vi). It follows from 2.9.23 that F is essential, and hence neo-unital.
(ii) Set F = C(K, E), so that, as in 2.6.2(vi), F is a Banach left A-module. Since K is totally bounded, it is easy to see that F is essential. Let f : K ---+ E be the identity map. By 2.9.26, there exist a E A and 9 E A . F such that a . 9 = f, so that a . g(x) = x (x E K). Set L = f(K). Then L has the required properties. 0 The projective induced product map
I'/r
1f A
= ker1fA of A®A were defined in 2.1.25.
and the projective diagonal ideal
Corollary 2.9.30 Let A be a Banach algebm with a bounded left approximate zdentity. Then the following results hold. (i) A factors, A has the S-property, the complex LA : 0
---+
I'/r = ker1fA ~ A®A ~ A -~ 0
(2.9.2)
is an exact complex of Banach A-bzmodules, and L~ zs admzssible.
(ii) A has factorization of pairs, and null sequences in A factor. (iii) Let S be a countable subset of A. Then there exists a E A with S c aA. (iv) Suppose that A is sepamble and that a E A. Then there exists bE O(A) with a E bA. (v) Let E be a Banach space. Then the Banach left A-module A ®E is neounital, and null sequences in A ®E factor. (vi) Let E be a Banach left A-module. Then the map 1f E AB (A®E, AE) with 1f( a ® x) = a . x (a E A, x E E) is a contmuous, open surjection onto A . E. (vii) Set L = lin{ab®c - a® be: a,b,c E A}. Then I'/r n (A ® A) = Land I'/r = L. (viii) Every maximal nght ideal in A is closed. (ix) Let E be a Banach right A-module. Then each module homomorphism !rom A to E is continuous. In particular, each left multiplier on A is continuous. Proof (i) We show that E~ is admissible; the remainder is immediate from 2.9.26.
316
Banach and topological algebras
Let (eo,) be a bounded left approximate identity for A. Then (eo ® eo) is a bounded net in A 0 A, and so we may suppose that (eo: ® eo) converges in the weak* topology on (A0A)", say to u. Define p; (A0A)' -+ A' by setting (a, peA)) = (u, a . A)
(a E A, A E (A0A)').
Clearly p E B«A0A)',A' ). For a E A and A E A', we have (a, (p07r~)(A)) = (u. 7r~(a· A)) = lim(7rA(eo:®eo:), a·A) = lim(e~a, A) = (a, A), 0
0:
and so p 0 7r~ is the identity on A'. Thus L:~ is admissible. (ii) Since A is essential. this follows from 2.9.29(i). (iii) Set S = {an; n EN}. Take (en) E (lR.+.y'" with (enan) E co(N, A). By 2.9.29(i), there exists a E A such that (enan) Ea' co(N, A), and now S c aA. (iv) Let S be a countable, dense subset of A. By (iii), there exists b E A with S U {a} C bA. Clearly b E !1(A). (v) Take Z = L:;=l aj ® x J E A ® E. There exist a, bl , ... , bn E A such that a J = abj (j E N n ). Thus Z EA· (A ® E). and so A0E is essential. By 2.9.26, A 0 E is neo-unital. (vi) Clearly the map 7r is a continuous surjection; by the open mapping theorem A.3.23, 7r is open. (vii) Clearly L C I'/T n (A ® A). Let L:7=1 aj ® bj E I'/T n (A ® A), and set E = {(Xl,"" Xn) E A (n) ; L:;=l xjbj = O}, so that E is a Banach left Amodule, as in 2.6.2(vi). and (al."" an) E E. Clearly E is essential, and so there exist a E A and (CI,""Cn ) E E with (a!. ... ,an ) = a· (Cll."'Cn ); we have L:j=l aj ® bj = L:j=l (acj ® bj - a ® cjbj ) E L, and so /'/T n (A ® A) c L. Thus I'/T n (A ® A) = L. We have L C I'/T' Now take Z E /'/T' Then there exists (zn) in A ® A such that /lzn - z/I'/T -+ 0 as n -+ 00. We have (7rA(Zn)) E co(N, A), and so, by 2.9.29(i). there exist a E A and (b n ) E co(N, A) with 7rA(Zn) = a . bn (n EN). But now (zn - a ® bn ) C /'/T n (A ® A) = Land Zn - a ® bn -+ Z, and so Z E L. (viii) and (ix) By (ii), null sequences in A factor, and so these follow from 2.6.13 and 2.6.14, respectively. 0 Corollary 2.9.31 Let A be a Banach algebra with a bounded approximate zdentity, and let E be an essential Banch A-bimodule. Then E zs neo-unital, A 0 A"P is neo-unital, and multipliers on A are continuous. 0 Corollary 2.9.32 Let R be a commutative, radical Banach algebra, let a E R, and let I be a non-zero, closed ideal in the quotzent Banach algebra aR. Then / does not have a bounded approximate identity. Proof Assume towards a contradiction that I has a bounded approximate idenBy 2.9.24, applied with tity. Then I factors, and so a ¢ ~(R). Take x E (an) = (lIanll-I/n), there exist bE R, C > 0, and (cn ) C R such that
r.
x = anbnacn ,
1Ic..11
~C
lIanll- 1
(n E N) .
Bounded approximate
But now
~dentitzes
Ilxll ::; c Ilbnllliall
--t
and factorization
0 as n
--t 00,
317
the required contradiction.
0
Theorem 2.9.33 (Esterle) Let A be a commutative Banar;h algebra contaming a continuous sernzgroup (a(, : ( E II) such that {a(, : ( E II n ]))} zs bounded and a E O(A). Then M(A) is a closed subalgebra of S(A), and there zs a semigroup (T(, : (E IT) in M(A) such that TO = lA, T(, = a(, « E II), and ( f-4 T O. Clearly there exists M > 0 such that lIa 0, there exists T E 2l[mJ with liT' A - All < e. We have
I(TxJ, A) - (Xj, A)I = I(xj, T'A - A)I < e
(j E Nn ),
and so (Xl, ... , Xn) belongs to the a(F, F')-closure in F = E(n) of the convex set ~[mJ . (Xl> ... , Xn). By Mazur's theorem A.3.29(ii), (Xl, ... , xn) belongs to the
I· \I-closure of this set, and so there exists (To) in 2l[mJ with Taxj
~ Xj (j E Nn ).
As in A.3.60(i), we see that, for each compact set K c E and each e > 0, there exists T E 2l[mJ with IITx - xII < e (x E K). Since 2l C K(E), it follows as in (i) that 2l has a bounded left approximate identity. By 2.9.3, 2l has a bounded approximate identity. 0
Banach and topological algebras
320
Theorem 2.9.38 Let A be an Arens regular Banach algebra.
(i) Suppose that A has a bounded approximate identity. Then the Banach A-bimodule A' is neo-unital. (ii) Suppose that A has a sequential approX'tmate zdentzty (en) bounded by m and that A is weakly sequentially complete. Then A has an identztyeA with IleA II S m and such that en -+ eA weakly.
Proof (i) Let (e be a bounded approximate identity in A. By 2.9.26, A . A' is a closed linear subspace of A'. By Mazur's theorem A.3.29(ii), A . A' is a(A',A")-closed. Let A E A'. By 2.6.17, (a)*(d), the map a 1-+ a . A, A -+ A', is weakly compact, and so we may suppose that the net (e", . A) is a(A', A")convergent. For each a E A, we have (a, e A) = (ae Q , A) -+ (a, A) because (e is a bounded right approximate identity, and so e", . A -+ A in a(A',A). Hence eo: . A -+ A in a(A', A"), and A EA· A'. Thus A . A' = A'; similarly A' = A' . A, and so A' is neo-unital. Q )
Q
•
Q )
(U) Take A E A'. By (i), there exist a E A and J.L E A' with A = a . J.L, and then (en, A) = (ena, J.L) -+ (a, J.L). Thus (en) is weakly Cauchy in A. Since A is weakly sequentially complete, (en) is weakly convergent, say en -+ e in (A,a(A,A ' ). Certainly lIell S liminfn-->oo lien II Sm. Let a E A. For each A E A', we have (ae n , A) = (en, A . a) -+ (e, A . a) and (ae n , A) -+ (a, A), and so ae = a. Similarly ea = a, and so e = eA. 0 Theorem 2.9.39 (,Ulger) Let A be an Arens regular, weakly sequentially complete Banach algebra with a bounded approxzmate identity. Then A zs unital. Proof Suppose that A has an approximate identity bounded by m, and assume towards a contradiction that A does not have an identity. By 2.9.17, there is a non-zero, closed subalgebra B of A with a sequential approximate identity bounded by m. By 2.6.18, B is Arens regular, and B is weakly sequentially complete. By 2.9.38(ii), B has an identity. say Pl E B[m]' We now construct inductively a sequence (Pi) in (J(A) \ {O}) n A[m] with Pi i= Pj and PiPj = PiVj for i,j EN with z i= j. Indeed, assume that Pb'" ,Pk have been defined, and set S = {a E A : apk = Pka = a}, a closed subalgebra of A. Then certainly Pk is an identity for S, and so, by assumption, S i= A; choose ao E A \ S. By 2.9.17, there is a closed subalgebra B of A such that {Pb ... , Pk, ao} C Band B has a sequential approximate identity bounded by m. As before, B has an identity, say PH1, with IIPk+111 S m. For j E N k , PjPk+l = Pk+1Pj = Pj and PJ i= Pk+l' This continues the inductive construction. Define P = lin{Pi : i EN}. Then P is a closed, commutative subalgebn· of A, with bounded approximate identity (Pi). By 2.9.38(ii), P has an identit) ep and Pi -+ ep weakly in P. For i E N, set cJ>i = {ep E cJ> p : ep(Pi) = I}, so that each cJ>i is an open and closed subspace of the compact space cJ> p. Clearly cJ>i C cJ>i+l (i EN). For each ep E cJ>p, ep(Pi) -+ ep(ep) = 1 as i -+ 00, and so ep(Pi) = 1 eventually. Thus U{cJ>i : i E N} = cJ>p. There exists kEN with cJ>k = cJ>p, and Pk+1 = Pk. By 1.5.7(ii), Pk+1 = Pk, a contradiction. Thus A is unital. 0
Bounded approximate identities and factorization
321
'Ve now give our becond factorization theorem; it is again convenient to present some elementary estimates in a lemma.
Lemma 2.9.40 Let A be a normed algebra, and let a, c E A, Tn, n E N, Z E C, r E lR.+., and E > O. (i) There exists TJ > 0 such that + b)k - ak - bkll < E whenever k E Nn and b + eA E Arm] with Ilabll + llball < TJ· (ii) There exists TJ > 0 such that
II(a
II (exp a - exp(a + b»)cll
+ IIc(expa - exp(a + b))11 < E whenever b + eA E Arm] with Ilabll + Ilball + IIcbll + Ilbcll < TJ· (iii) There eX'tsts TJ > 0 such that lIexp«(a + zeA)) - exp«(a + zeA
+ b») II
::; (€ + el(l(m+l) - 1) exp(~«(z»)
whenever b + eA E Arm] with lIabll + IIball < TJ and ( E ][)leO; r). Now suppose, further, that E is a Banach left A-module and that x E E. (iv) There e.rists TJ > 0 such that II(a+zeA)k. x-(a+zeA+b)k. whenever k E Nn and b + eA E A[m] with IIball ( v ) There exists TJ > 0 such that
+ lib
. xII < TJ.
II (exp«((a + zeA)) . x - exp«((a + zeA
whenever b + eA E A[m] with IIball
+ lib
xII .Bn) for each n E Nj the choice of such a sequence (.Bn) is possible because at ~ 00 as t ~ 00. We shall next inductively define a sequence (Uk) in A[m] such that, for each kEN, the elements ak defined in A# by ak = e- k expVk, where Vk = 2:;=1 Uj. satisfy the conditions that: lIaLI - ail!
~ ;k + (e/(/(m+l) -
Ila;~1
.
Ila;2 1 ej - a;lejll
x - a;( .
1)
e(1-k)lR(
xii < ;k
+ Iha;21 -
E Jl)I(Oj .Bk)) j
{(
{( E Jl)I{Oj .Bk)) j
eja;lll ~ ;k
(j E N k ).
Here we are setting ao = eA, and, of course, we define so that (ai : C; E C) is an analytic semigroup.
(2.9.3) (2.9.4) (2.9.5)
ai to be e-(k exp«(Vk) ,
Bounded approximate identities and factorization
323
We first utilize (iii) and (v) of 2.9.40 with a = 0, z = 0, r = fil, and with c replaced by c/2, and choose b to be UI - eA; by (iii), (2.9.3) holds (for k = 1) for each choice of Ul E Arm], and, by (v), (2.9.4) holds provided that, further, Ilx - Ul . xII is sufficiently small. By 2.9.40(ii) with c = el, (2.9.5) holds provided that lIel - uIeIi! + lIel - el ulil is 5ufficiently small. We can thus make an appropriate choice of Ul. Now take kEN, and assume that UI,"" Uk have been defined in Arm] flO that (2.9.3), (2.9.4), and (2.9.5) hold. We utilize (iii) and (v) of 2.9.40 with a = 11k, Z = -k, r = fJk, and with c replaced by €/2 A'+ 1 , and choose b to be Uk+1 - eAi (2.9.3) and (2.9.4) hold (for k + 1) provided that
+ IIx - Uk+1 • xII is sufficiently small. Further, by 2.9.40(ii) with a = -Vk and c = ej, where j E N k +1, (2.9.5) holds for k+ 1 provided that lIeJ -uk+lejll + Ilej -ejUk+11l IIVk - Uk+1 Vk
II + IIvk -
Vk U k+111
is sufficiently small. We can make a choice of Uk+l E A[m] to satisfy this finite number of constraints simultaneously. The inductive construction continues. By (2.9.3), the sequence (a~ : n E N) is a Cauchy sequence in A#, uniformly on compact subsets of II. Thus (a~ : n E N) is convergent in A# for each ( E II, say to a M/ cos'1/;. Then (a~ : (E II) is the required semigroup. (ii) We apply 2.9.41 in the case where B = A. By 2.1.9, there is an equivalent algebra norm on A such that (a( : ( E S,p) is bounded by 1 in the new norm. By 2.9.41(v), (a l / n ) is a sequential approximate identity for A; this is not affected by the rescaling. 0 Corollary 2.9.44 Let A be a commutative Banach algebra. Then the following are equivalent: (a) there is a non-zero, closed ideal I in A such that I contains a bounded approximate identity; (b) there is a non-zero analytic semigroup (a( : ( E II) in A such that (at: t E JR+e) is bounded.
Bounded approximate zdentities and jactorizatwn
325
Proof (a)=?(b) This is immediate from 2.9.43. (b)=?(a) Set I = U{a(A: C; E II}. Then I is a non-zero, closed ideal in A. For each C; E II and bE A, we have a 1/ n Q,(b = Q,(+1/nb ~ a(b as n ~ 00, and so (a 1 / n ) is a bounded approximate identity for I. 0 We have seen that a Banach algebra that has a bounded left approximate identity factors. Let A be a Banach algebra, and consider the following statements about A: (a) A has a bounded left approximate identity; (b) there exists Tn ~ 1 such that, for each a E A and e > 0, there exist b E A[m] and c E A with a = be and Iia - ell < e: (c) null sequences in A factor: (d) null sequences in A factor weakly; (e) A has factorization of pairs; (f) A factors: (g) A factors weakly. It follows from 2.9.24 that (a)=?(b). Conversely, suppose that (b) holds. Then, for each Q, E A and c > 0, there exists b E A[m] with Iia - ball ::; me, and so A has left approximate units of bound m; by 2.9.14(H), A has a left approximate identity of bound m. Thus (b)=?(a). It is now easy to see that we have the following implications: (a)
¢=:}
(b)
==?
(c)
==?
(d) ~
(e)
==?
(f)
n
(g)
~
Further, (d)+(e)=?(c). There is a counter-example to every converse implication. However, as indicated by 2.2.16. there are some special implications which hold for separable Banach algebras. and so we should particularly like to find counter-examples to these implications which are either commutative or separable, or. preferably, both; it is not known whether the latter is always possible. "Ve shall sec later that the natural weakest hypothesis for several automatic continuity results h,; that (d) hold; it would be particularly interesting to know if (d) is implied by any of (e), (f), or, especially, (g) for some classes of Banach algebras. Examples 2.9.45 Let G be a totally ordered 171-grouP (as in §1.2), and let 8 be the semigroup G+·. The semigroup algebra £ 1 (8) is a commutative Banach algebra; we shall note in 4.5.5 that £1(8) is semsimple. We have remarked that ll(8) does not have approximate units. However, we claim that null sequences in £ 1 (8) factor. For let {Jn : n EN} be a countable set in £ 1 (8). Then T = U{supp j n : n E N} is a countable subset of 8, and so there exists s E 8 with s « T . For each n E N, there exists gn in £1(8) with fn = I5 s * gn and IIgnl11 = IIfn111' and this implies the claim. This commutative, non-separable example satisfies (c), but not (b).
Banach and topological algebras
326
Now take T to be the semigroup (G+ x G+)·. Then (l(T) is a commutative Banach algebra, and, essentially as above, we St.,'e that null sequences factor weakly in ll(T). However, el(T) does not factor. This commutative, nonseparable example satisfies (d), but not (f). The commutative. separable, semisimple Banach algebra e1 (Q+.), to be discussed in §4.7. does not satisfy (a); it is not known wht>ther or not this algebra factors, or even factors weakly. 0 Examples 2.9.46 An example of a (non-commutative) four-dimensional Banach algebra which factors, but which does not have factorization of pairs, was noted in 1.3.7(vii); this example has neither left nor right approximate units. Thus (f)=fo(e). An example of a (non-separable, non-commutative) Banach algebra which factors weakly. but which does not factor, was noted in 2.2.51; by 2.6.12, null sequences do not factor weakly in this example. Thus (g)=fo(f) and (g)=fo(d). It will be shown in 3.3.30 that the augmentation ideal LACG), where G is a locally compact group, always factors weakly, and in 5.6.42 that this algebra has a bounded left (or right) approximate identity if and only if G is amenable. and so we obtain separable (non-commutative) counter-examples to the implication (g)*(a); for each non-amenable group G, it is not known whether or not these algebras always factor. An example of a (non-separable) uniform algebra which factors, but which docs not have factorization of pairs, and for which null sequences do not factor weakly, will be noted in 4.3.23. Thus (f)=fo(e) and (f)=fo(d) for commutative Banach algebras. 0
We now seek commutative. separable counter-examples to the various implications. Before giving the next example. we make a preliminary remark. Suppose that, for each n E N, (An, II· lin) is a Banach algebra with an identity en such that lIenli n = n. Set 2t = co(N, An), so that, as in 2.1.18(iii), 2t is a Banach algebra with respect to the norm specified by lI(a n )1I = sUPn lIa n li n . Then 2t does not have bounded approximate units. For assume that 2t has approximate units of bound m. Then each An has approximate units of bound m. and so. by 2.9.11, lIe n li n :::; m for each n E N, a contradiction. Example 2.9.47 (Willzs) First fix n E N, and define a weight Wn on the group (Q, +) by setting wn(t) = n (t E Q+) and wn(t) = 1 (t E Q-.). As in 2.1.13(v). An = e1 (eQ, w n ) is a commutative Banach algebra, and An is separable; the Horm on An is denoted by 1I·ll n . The identity of An is 15o, and IIc50 l n = n. Let Uk) be a null sequence in An with sUPk Ilfklln = 11n, say. Then there exists k n EN with kn 2: n and IIfklin < 11n/n (k 2: k n ). Also, there exists tn E Q+. such that
IiI5t ...
* !kiln < 1I!kll n + 11n (k E Nk,,). * fk, so that IIgkll n :::; n IIfkll n . Clearly (gk)
For kEN, set gk = c5tn is a null sequence in An with sUPk IIgkli n < 217n. Now let 2t be formed as above. Then 2t is a commutative, separable Banach algebra which does not have bounded approximate units. We now claim that null sequences in 2t factor. Indeed, let U(k» be a null sequence in 21, say f(k) = (/k,n : n E N) for kEN. For each n E N, set
Bounded approx't1nate zdentzties and factorizatwn
327
= sUPk 1I/k,nll1/' SO that limn -+ ov Tin = O. Then there exist tn E Q+e and a null sequence (gk.n : kEN) in .e 1 (Q) with IILt..Iln = 1. with sUP/r Ilgk,n lin < 2.,."" and with fk.n = Lin * gk.n (If EN). For kEN, set g(k) = (.,.,;;1/2 g/r,n : n EN); clearly Ilg~k) < 2T1~/2 -+ 0 as
Tin
lin
n -+ 00, and so g(k) E 21. Take E > O. Then there exists ko E N such that Ilgk,nll" < E (k ~ ko, n EN), and so (g(k)) is a null sequence in 21. Also, set h = (.,.,~/2Ltn)' so that hE 21. Clearly U(k)) = h . (g(k)) in 21, establishing the claim. This commutative, separable example satisfiC's (c), but not (a). 0 Example 2.9.48 (Willis) For each v E [1. oc), define weights w" and T" on the group (Q, +) by setting w,,(t) = v (t E Q+) and w,,(t) = 1 (t E Q-e) and by set.ting T,,(t) = v (t E Q n [I/v, (0)) and T,,(t) = 1 (t E Q n [-00, I/v)). Take A to be the set of elements E:'=o fnX" in the algebra f I (Q) [[X]] such that IIUn)ll" =
z::
Ifo(t)1 w"
+
tEIQ!
f (z:: k=I
1/k(t)1 T,,(t/k))
., in (E',a). Take x = a . y E E, where a E A and y E E. Then (x, D/l-eo.) = (y, D/l-eOl. . a) = (y, D/l-(eo.a) -+ (y, Dp.a)
Banurh and topological algebms
:330
and HO (x, A) = (y, Df£a). Thus DILen -+ A in (E'.a), and so A iH independent of the choice of the bounded approximate identity. Set DJ-L = A. Then we have (y. D!J.a) = (a . y, DJ-L) = (y, DJ-L . a) (a E A. y E E), and hence
DJ-L . Certainly
J-L2
= (L 2• R 2 )
0.=
Df£a = D(La) - J-L . Da.
(a E A).
(2.9.9)
D : M(A)
-+ E' is a linear map. Now take J-Ll = (L].R]) and in M(A). Then, using (2.9.9). we sec that, for each a E A.
D(PIJ-L2) .
0.=
Df£I!J.2a = D!J.l (L2a) + J1.1 . D!J.2a
= (DJ-Ll . Jl2
+ J-Ll
• DJ-L2) . a
by 1.8.3
by (2.9.8) .
and so D(J1.1P,2) = i\L1 . It2 + J1.1 . DJ-L2 because E is essential. Thus D is a derivation. Let a. E A. and set It = (La, Ra). Then
Dp, . b = Df£b = D(ab) - a . Db = Do. . b (b
E
A) ,
and so Dp, = Da. Thus D I A = D. Suppose that D : M(A) -+ E' is a derivation such that D I A = O. Then. for each J-L E M(A). we have
Dp,=a-limDp,· eO! =a-lim(D(J1.. eo) -Jl,' DeO!) =0, O!
'"
and so D = O. Therefore D is the unique derivation extending D. Suppose that D is inner, so that D(a) = a· A-A' a (a E A) for some A E E'. Then the inner derivation J1. I--t J1. . A- A . Jl. M(A) -+ E', is a derivation which extends D, and so iH equal to D, which is thus inner. Suppose now that D is continuous. Let Jl."I = (L"I' R"{) -+ J-L = (L, R) in (M(A),so), and take x = a . y E E. Then (x. DP,"I) = (y, D(L"Ia) - Jl"{ . Da)
-+
(y, D(La) -I)' . Do.) = (x. DJ-L),
and so D: (M(A), so) -+ (E', a) is continuous. It follows from the closed graph theorem A.3.25 that D : M(A) -+ E' is continuous. That = IIDII when A has an approximate identity of bound 1 follows 0 from 2.9.49(ii).
IIDII
The above result implies that 1-{l(A, E') ~ 1-{l(M(A), E'). The following theorem extends this result. Theorem 2.9.54 (Johnson) Let A be a Banach algebra with a bounded approximate identity, let E be an essential Banach A-bimodule, and let n E N. Then 1-{n(A. E') ~ 1-{n(M(A), E').
We may suppose that n ;::: 2. In the proof, we shall write M for M(A), regard A as a subalgebra of M, and write 6 for 8n - 1 : Bn-1 (A, E') -+ Bn (A, E'). By 2.8.33, Bn-1(A, E') is the dual space of B n- 1(A, E), and a net (8"1) converges to 8 E .8"1-1 (A, E') with respect to the weak* topology if and only if (x, 81'(a1,"" an» -+ (x, 8(a1, ... , an» for each x E E and 0.1,.'" an E M. We take (eO!) to be a bounded approximate identity in A.
Bounded appro:nmate identities and factorization
331
Lemma 2.9.55 Let T E zn(M,E') and BE Bn-I(A,E') with 6B = T I A(nJ. Then ther-e exists S E Bn-l(M, E') such that 68 = T and S I A(n-l) = B. Proof We claim that, for each j E Z;;_l' there exists Bj E Bn-l(M,E') with: (a)j
6Bj
I Xj = T I Xj;
(b)1
Bj
I A(n-I) = B I A(n-l) .
Here X 1 = A(n-j) x M(j), with Xo = A(n). First define Ua(al, ... ,an-d = B(aleo:. ... ,an-Iea ) (al, ... ,an-1 EM). Then (Ua) is a bounded net in Bn-l (M, E'). and so we may suppose that (U"') is wcak*-convergent, say to Bo E sn-I(M,E'). For al, ... ,an-l E A, we have lima B(ale n , ... , an-leo,) = B(aI, ... , an-I) in (E', 11·11), and hence
(x, Bo(al, ... , an-I»
= lim(x, a
Bo(ale(., ... , an-len)
= (x,
B(al,"" an-t})
for each x E E. Thus Bo I A(n-l) = B. Also 6Bo I A(n) = 6B = T I A(n), and so Bo satisfies (a)o and (b)o. Take j E Z;;_2' and assume that Bj satisfies (a)j and (bk Set V = T - 6B1 , and, for al, ... , an-j-2 E A and an- 1 -1,"" an-I EM, define Ua(al, ... , an-I) = V(ale a ,··., an-j-2ea, ea , an-j-l,··· ,an-I). Again, we may suppose that (Ua ) is weak*-convcrgent, say to R E Bn-I(M, E'). Note that R I A(n-l) = o. Now take bI, ... ,bn - J- 2 E A and bn-j_l, ... , bn - l E M. Then lim(Ua(bl, ... , bn - l ) - V(b l , ... , bn - j - 2, ea , bn-J-l. ... , bn - l )) a
=0
in (E', 11·11). Thus, for each x E E, we have
(x, R(bI,"" bn - 1 - 2, bn- j - l .... , bn- I )
= lim(x, a
V(b l , ... , bn - j - 2, ea , bn - J- I , ... , bn - l )}.
Next take al, ... , an-j-l E A, an-j, ... , an EM, and x E E. We have
(x, (6R)(al"'" an-j_I, an- J , ... , an) =
li~ \ x. al +
.
V(a2,"" an-j-l, ea , an-j,"" an)
n-j-2
L
) (-l)iV(al, ... , ai-I, aiai+I,···, an-J-I. ea , an-j,···, an)
i=l
(2.9.10) the remaining terms that arise from the expansion of 6R are zero by (a))' We have 6V = 8T = 0 because T E zn(M, E'), and so, in particular, (8V)(aI, ... , an-J-I, ea , an-j, ... , an) = 0
for each
Q.
By comparing (2.9.10) and (2.9.11), we see that (x, (6R)(ar, ... , an-j-I, an-j, ... , an) = lim(x, (-l)n-iV(al,' .. , an -j-2, an-i-Ieet:) an_j, ... , an» a
(2.9.11)
Banach and topological algebms
332
Set Sj+1 = Sj + (-l)n-jR. Then Sj+l satisfies (a)J+l and (b)j+1' The claim follows by induction. Now define S = Sn-l, so that 8 E 8 n - l (M,E') and 8 I A(n-l) = S. Also (T-8S) I X n- l = O. For a, al,"" an E M, we have 8(T-8S)(a, al,··. ,an) = O. Take a E A and al, ... ,an E M. Then a . (T - 8S)(al,'" ,an) = 0, and so 88 = T. Thus S has the required properties. 0 Proof of Theorem 2.9.54 For kEN, let nk : 8 k (M, E') ---7 8 k (A, E') be the restriction map. For T E 8 k (M, E'), we have 8k (nk(T» = nk+l (8 kT), and so nk maps cocycyles into cocycles and coboundaries into coboundaries. Thus these maps induce a continuous linear operator n : 1{n(M, E') ---7 1{n(A, E'). It is immediate from the lemma that the map n is injective. Now take S E zn(A, E'). Then 8n S is the restriction to A(n+1) of the zero element of 8 n+1 (M, E'), and so, by the lemma, there exists 8 E zn(M, E') with nn(S) = S. Thus the map n is surjective. We have constructed the required linear homeomorphism. 0 Theorem 2.9.56 (Gr!/lnbrek) Let A be a Banach algebm with a bounded right approximate zdentity, and let E be a Banach right A-module. Then
1{n(A. A®E) = 1{n(A, (A®E)') = {O}
(n E N).
Proof By 2.8.33(iv), it suffices to prove that 1{n(A, (A®E)') = {O} (n EN). Let (en) be a right approximate identity of bound m for A. Take T E zn(A, (A®E)'). For each a, define
Qn : (al,"" an, x)
I->
(en ® x, T(al,"" an»),
A(n) x E
---7
C.
Then Qn E (8 n (A, E»' = sn-1(A, (A®E)') and IIQnll :::; m IITII. Thus there exists Q E (8n (A, E»' such that limn Qn(al, ... , an, x) = Q(al,"" an, x) for each (al, ... ,an,x) E A(n) x E. Let a, aI, ... ,an E A and x E E. Essentially as in 2.8.62, we have
(a®x,(8 n - 1Qn)(al, ... ,an ») = (-l)n(aeQ®x, T(al, ... ,an ») ---7 (-l)n(a ® x, T(al, ... , an»), and so T = (_1)n8 n- l Q E Nn(A, (A®E)'). Thus 1{n(A, (A®E)') = {O}.
0
There are important relationships between the notions of bounded approximate identities and of amenability of a Banach algebra. The following theorem is a key result; a characterization of amenability in terms of bounded appproximate identities will be given in 2.9.65, below. Theorem 2.9.57 (Johnson) Let A be an amenable Banach algebm. has a bounded approximate identity, and A factors.
Then A
Proof Let E be the. Banach space A', with the module operations x, where a x >. = a . >. and >. x a = 0 for a E A and >. E A', so that E is a Banach right-annihilator A-bimodule, as in 2.6.2(iv). The module operations that make
Bounded appT'Oximate zdentitzes and factorization
333
E' into the dual module of E are also denote by x, and t denotes the canonical embedding of A into E' = A". For a, bE A and A E E, we have (t(ab), A}
=
(a, b . A)
=
(a, b x A)
=
(t(a) x b, A} and
(a x t(b), A}
=
(b, A x a) ,
and so t : A ---> E' is a derivation. Since A is amenable, there exists cI>o E E' such that t(a) = cI>o x a - a x cI>o (a E A), and then a = cI>o . a (a E A). As in 2.9.16(i), A has a bounded left approximate identity. Similarly, A has a bounded right approximate identity, and so, by 2.9.3, A has a bounded approximate identity. 0 Let I be a closed right ideal in a Banach algebra A. Then
L: 0 ~ I ~ A ~ A/I ~ 0 is a short exact sequence of Banach right A-modules (where t is the injection map and 7f is the quotient map). The dual sequence 'L-' is a !:ihort exact sequence of Banach left A-modules, and this sequence is admissible if and only if I is weakly complemented in A.
Theorem 2.9.58 Let A be a Banach algebra, and let I be a closed nght ideal in A. (i) Suppose that I has a bounded left appT'Oximate identity. Then I is weakly complemented in A, and the dual sequence 'L-' splzts strongly. (ii) Suppose that A is amenable and that I is weakly complemented in A. Then I has a bounded left approximate identity, and I factors.
Proof (i) By 2.9.16(i), (I", 0) has a left identity, say cI>o. Define Q : I' ---> A' by setting (a, QA) = (cI>o, a . A) (a E A, A E I'). Certainly Q E B(I', A'), and (a, t'(QA)} = (cI>o, a . A) = (cI>o . a, A) = (a, A) (a E I, A E I'), and so t' 0 Q is the identity on I'. Thus I is weakly complemented in A. Further, (b, Q(a . A)}
= (cI>o.
ba . A)
=
(ba, QA)
=
(b, a . QA)
(a, bE A, A E I'),
and so Q E AB(I', A'). Thus 'L-' splits strongly. (ii) Since I is weakly complemented in A, 'L-' is admissible, and so, !:iince A is amenable, it follows from 2.8.60 that 'L-' splits strongly; take Q E AB(I', A') such that t' 0 Q is the identity on I'. By 2.9.57 and 2.9.16(i), (A",O) has a left identity, say cI>o. For a E I and A E I', we have (a, A)
= (cI>o .
a, QA)
= (cI>o,
Q(a . A)}
=
(Q'(cI>o)' a . A}
=
(Q'(cI>o) . a, A},
and so Q'(cI>o) . a = a. Thus Q'(cI>o) is a left identity for (I", 0). By 2.9.16(i), I has a bounded left approximate identity. 0
Corollary 2.9.59 Let A be an amenable Banach algebra, and let I be a closed ideal in A. Then the followmg are equivalent:
(a) I has a bounded approximate identity; (b) I is weakly complemented in A; (c) I is amenable.
334
Banach and topological algebras
Proof (a){:}(b) This follows easily from the theorem by using 2.9.3. (c),*(a) This is 2.9.57. (a),*(c) Let E be an essential Banach I-bimodule; by 2.9.28, it suffices to show that 1i 1 (I,E')={O}. Take DEZ1(I,E'). By 2.9.51, E is a unital Banach M(I)-bimodule, and, by 2.9.53, there exists i5 E Zl (M(!), E') with D I 1= D. There is a continuous homomorphism () : A --> M(!); by 2.8.64(ii). ()(A) is 0 amenable, and so D I ()(A) is inner. Thus D E Nl(I, E'), as required. In particular, each closed ideal of finite codimension in an amenable Banach algebra has a bounded approximate identity. Thus, by 2.9.36, every finitedimensional Banach extension of an amenable Banach algebra splits strongly. Corollary 2.9.60 Let A be an amenable Banach algebra. Then: (i) either A is semiszmple or rad A is infinite-dimensional; (ii) AI I is semisimple for each closed ideal I of finite codimension in A. Proof (i) Set R = rad A. Suppose that R is finite-dimensional. By 2.9.59, R has a bounded approximate identity, and so R = R n (n E N). However, R n = 0 eventually, and so R = O. (ii) By 2.8.64(ii), AI I is amenable. The result now follows from (i). 0 Corollary 2.9.61 Let A be a unital Banach algebra which is a directed union of a family of pliable and amenable subalgebras. Then A 1S plwble. Proof This follows from 2.9.60(ii) and 2.2.24.
0
Corollary 2.9.62 (Johnson) Let A and B be amenable Banach algebras. Then A ®B is amenable. Proof By 2.9.57, A and B each have a bounded approximate identity, and so, by 2.9.21, A ®B has a bounded approximate identity. By 2.8.58(i). A# and B# are amenable, and A ®B is a closed ideal in A # ®B#. Thus, by 2.9.59, it suffices to suppose that A and B are unital. Let E be a Banach A®B-bimodule, and let D E Zl(A®B,E'). Then D I (A ® e B) E Zl (A ¢9 e B, E'), and so, since A is amenable, there exists A E E' such that D I (A ® eB) = 8.\. By replacing D by D - 8.\, we may suppose that D I (A ® eB) = O. Set F = lin {(a ® eB) . x - x . (a ® eB) : a E A, x E E}, a closed linear subspace of E. For a E A, bE B, and x E E, we have (eA ® b) . «a ® eB) . x - x . (a ® eB)) = (a ~ eB) . Y - Y . (a ® eB),
where y = (eA ® b) . x E F, and so F is a left (eA ® B)-module. Similarly, F is a right (eA ® B)-module, and so F is a Banach (eA ® B)-bimodule. Further, for a E A and bE B, we have «a®eB) . x -
X·
(a®eB), D(eA ®b»
= (x, D(eA®b)· (a®eB)-(a®eB)' D(eA®b)) =0
because D(a ® b) = D(eA ® b) . (a ® eB) = (a ® eB) . D(eA ® b), and SO D : eA ® B -+ FO = (ElF)' is a continuous derivation. Since B is amenable,
Bounded approximate identities and factorization
335
there exists J.L E FO with D I (eA ® B) = Ow By replacing D by D - 0J1, we may suppose that D I (eA ® B) = O. But now D = 0, and so thf:' original derivation is inner. 0 There is a remarkable number of different characterizations of amenahle Banach algebras that arise in different contextsj we now present some of these characterizations. Recall first from 2.8.48 that a Banach algebra A is such that 1{1 (A, E) = {OJ for every Banach A-himodule E if and only if A is unital and has a projective diagonal in A @Aj a related condition gives an intrinsic characterization of amenability. Recall that 11'A : A@A -+ A denotes the projective induced product map.
Definition 2.9.63 Let A be a Banach algebra. (i) An approximate diagonal for A is a bounded net (u a ) in (A ® A," '11,,) such that. for each a E A, we have lim(ua . a - a . u a a
a
E
)
=0
and
lim 7lA(ua )a n
= a.
(ii) A virtual diagonal for A is an element M of(A@A)" such that, for each A, we have a . M = 1\.1 • a and 11"~ (M) . a = a.
Lemma 2.9.64 Let A be a Banach algebra. Then A has an approximate diagonal if and only if A has a virtual diagonal. Proof Let (uo:) be an approximate diagonal for A, and regard (no:) as a bounded net in (A @A )". Then (uo:) has a weak* -accumulation point. and each such point is a virtual diagonal for A. Conversely, Huppose that A has a virtual diagonal M E (A @A )". Then, by A.3.29(i), there is a bounded net (u a ) in A ®A that converges to M in the weak* topology. Clearly, for each a E A, the net (u a . a - a . uo:) converges to 0 in the weak topology of A @A and the net (11" A(ua)a) converges to a in the weak topology of A. Now fix F = {(11, ... ,an} C A and € > O. The bounded net
(Uo: . a1 - a1 . U a ,11"A(ua )a1 - al,"" Uo: . an - an . Ua , 11"A(ua )a n
-
an)
convergf:'B to 0 in the space «A@A)xA)(n) with respect to the weak topology. By Mazur's theorem A.3.29(ii), there is a convex linear combination VF,c of elements in the set {u(~} such that IIvF,€ . a - a . vF,€1I < € and II11"A(vF,€)a - all < € for a E F. As in 2.8.59, we obtain a net (VF,c) in A ® A that is an approximate diagonal for A. 0 Let M he a virtual diagonal for A. Then there is an associated approximate diagonal (u a,) in A (9 A such that (11"A(U a » is a bounded approximate identity in A. Now suppose that E is an essential Banach A-himodllle. Then we see inunediately that lima 11"A(Uo:) . A = limo: A . 11"A(Uo:) = A (A E E'). Recall that 11< = ker 11"A is a cl08ed left ideal in the Banach algebra A @A op • SUppose that A has a bounded approximate identity, so that 11"A : A 0 A -+ A is a surjection and the short exact sequence L:A is exact. By A.3.48(vi), the map 11"'..1 : (A0A)" -+ A" is a surjection, and I:; = ker11"~; I:; is a Banach A-bimodule.
336
Banach and topological algebras
Theorem 2.9.65 (Helemskii, Johnson) Let A be a Banach algebra. Then the following conditions on A are equivalent:
(a) A is amenable; (b) A has an approximate diagonal; (c) A has a virtual diagonal;
(d) A has a bounded approximate identity, and Jil(A, I:{) = {O}; (e) A has a bounded approximate identity and A is biftat; (f) A has a bounded approX'tmate identity and 11{' has a bounded right approximate tdentity. Proof (b)¢:}(c) This is 2.9.64. (a)=>(d) By 2.9.57, an amenable Banach algebra has a bounded approximate identity; certainly, Jil(A, I:{) = {O}. (c)=>(a) By 2.9.28, it suffices to show that Jil(A, E') = {O} for each essential Banach A-bimodule E. Let E be such a bimodule, and let M be a virtual diagonal for A, with associated approximate diagonal (uoJ Take DE Zl(A, E'). For each x E E, define Ax(a, b) = (x, a ·Db) (a, bE A). Then Ax E B2 (A,C); we regard Ax as an element of (A~A)'. Now define (x, >.) = (M, Ax) (x E E). Then clearly>. E E'. For each a, b, c E A and x E E, we have (b ® c, A a .x - x.a ) = (a . x - x . a, b . Dc) = (x, b . Dc . a - ab . Dc)
and (b®c, a· Ax -Ax' a)
= (b®ca-ab®c,
Ax)
= (x,
bc· Da+b· Dc· a-ab· Dc),
and so (b ® c, A a . x- x.a ) = (b ® c, a . Ax - Ax . a) - (x, bc . Da). Now take a E A and x E E. Then (u o" Aa x-x a) = (u a , a . Ax - Ax . a) - (x, ll'A(U a ) . Da),
and so (a . x - a . x, >.) = (M, a . Ax - Ax . a) - (x, Da) = -(x, Da) .
Thus Da = a . >. - >. . a. We have shown that D E Nl(A, E'). (d)=>(c) Let (e a ) be a bounded approximate identity for A. As in 2.9.30(i), we may suppose that (e a ® ea ) converges in the weak* topology on (A~A)", say to u. Take a E A and >. E A'. Then (ll'~(u . a - a . u), >.) = (u . a - a . U,ll'A(>'))
= lim(ll'A(e a ® eaa - ae a ® ea ), >.) = lim(e~a - ae~, >.) = 0, a
a
and SO u . a - a . u E ker 1l'~ = I:{. The map a 1-+ u . a - a . u, A continuous derivation, and so, by (d), there exists v E I:{ such that u .a- a . u= v . a- a . v
(a
E
A).
-t
I:{, is a
Bounded approX'tmate identities and factorization Set M = 7r~ (M) =
U -
337
v E (A0A)". Certainly a . M = M . a (a E A). Also we have and so, for each a E A and >. E A', we have
7r~ (u),
(7r~(u)
. a, >.) = (u, 7r~(a . >.») = lim(ea ®e a • 7r~(a . >.)) a
= lim(e~a, >.) = (a, >.) . a
Thus
7r~ (M)
. a = a (a E A). We have shown that M is a virtual diagonal.
(e)=>(c) Let (e a ) be a bounded approximate identity for A. Since A is bifiat, 7r~ has a left inverse, say p E ABA«A0A)',A). We may suppose that (p'(e a )) converges in the weak* topology on (A 0 A)", say to M. For each a E A and A E (A0A)', we have
= lim(p'(e a ), a
(M, A· a)
A· a}
= lim(e a . p(A) . a} = lim(ae a , p(A)) = (a, a a
p(A)} ,
and similarly (M, a . A) = (a, p(A)). Thus a . M = M . a (a E A). Essentially as in the implication (d)=>(c), 7r~(M) . a = a (a E A), and so M is a virtual diagonal. (a)=>(f) Certainly A has a bounded approximate identity. The Banach algebra AOP is amenable, and so, by 2.9.62, A 0 AOP is amenable. By 2.9.30(i), 17r is weakly complemented in A 0 A, and so, by the 'opposite' form of 2.9.58(ii), 17r has a bounded right approximate identity. (f)=>(e) Let (u a ) be a bounded right approximate identity for 17r . Then we may suppose that (u a ) converges in the weak* topology on 1~, say to u. Define a : 1~ ~ (A0A)' by setting (v, a(A)) = (u, A . v) (v E A0A). Then a is a continuous linear map. For v E 17r and A E 1~, we have
(v, (t.' and so t.' we have
0
0
a)(A))
= (u,
A . v)
= lim(vu a , A} = (v, a
A),
a is the identity map on 1~. Further, for v, w E A 0 A and A E 1~,
= (u, A . wv) = (wv, a(A)} = (v, a(A) . w}, and so a is a right A 0 A-module homomorphism. Thus 2:~ splits strongly as a short exact sequence of Banach right A 0 A-modules. This implies that there is a right A 0 A-module homomorphism p : (A 0 A)' ---+ A' such that (v, a(A . w)}
p 0 7r~ is the identity on A'. Since A has a bounded approximate identity, A is an essential Banach A-bimodule, and so, by 2.6.10, P is a Banach A-bimodule homomorphism. Thus 7r~ is a coretraction in ABA(A', (A A),), and so A is biflat. 0
o
Corollary 2.9.66 Let A be an amenable Banach algebra, and let I be a closed, left ideal with a bounded approximate zdentity. Then I is amenable. Proof By 2.9.65, A has an approximate diagonal, say (u a : 0: E S), in A ® A, lVith lIu a 117r ::; C. Let (e,B : (3 E T) be an approximate identity for I of bound m. Define Va,,B,,,( = ua(e,B ® €"() (0: E S, (3, "I E T), taking the product in the algebra A ® A. Then (va,,B,"( : S x TxT) is a bounded net in (I ® I, 11'11",)·
Banach and topological algebras
338
We write e for eA E A#. For each a E I, we have lIa . va ./3,"'( -
Va
./3,') .
all7r = II (a ® e)ua(e,a ® e"'() ~ 1I«a ® e)ua
~
m2
-
ua (e/3 ® e"{)(e ® a)ll7r
ua(e 0 a))(e/3 ® e"'()II 7r
+ II ua «e/3 ® ae"'() - e/3 ® e"{a)lI7r 11a . U U a . all7r + Cm Ilae",( - e-yall . all7r = O. Q
-
,
and so lima ,/3,"'( lIa . va ,{3,"'( - v a ,(3,,,,( Also, for each a E I, we have 7r[(va ,/3,"'() = 7rA(ua,)e,ae"'(, and so 11 7r I(vQ
,/3."{)a - all ~ Crn lIe"'(a - all
+ C lIe(3a -
all
+ II7rA(u n )a -
all·
Thus lim ,/3,"'( 7r[ (v Q .(3."'()a = a. We have shown that (v a .(3,"'( : S x TxT) is an approximate diagonal for I. By 2.9.65, I is amenable. 0 Q
Notes 2.9.61 The concept of a bounded approximate identity first arose in harmonic analysis; see §3.3. For a discussion of (bounded) approximate identities and their historical significance, see (Bonsall and Duncan 1973, §11), (Doran and Wichman 1979), (Hewitt and Ross 1970, §32), (Palmer 1994, Chapter 5), and (Reiter and Stegeman 2000). The notions of approximate units and their connection with approximate identities arose in particular in (Reiter 1971); see (Doran and Wichman 1979, §9). I do not know an example of a Banach algebra which has approximate units, but no approximate identity. Theorem 2.9.9 is (Johnson 1972a, 1.5). and 2.9.12 is taken from (Doran and Wichman 1979, §32); for variations 011 2.9.12, sec (Dixon 2001). Proposition 2.9.16 extends (Civin and Yood 1961, Lemma 3.8). Proposition 2.9.3 is from (Dixon 1973b), where the following result is also proved. Theorem (Dixon) (i) There is a Banach algebra with a sequential left approxtmate identity and a sequential right approximate identity, but no approximate identity. (ii) A normed algebra with a bounded left approximate identity and a right approximate identity has an approxzmate identity. 0 The papers (1973 b, 1978) of Dixon also investigate the norms of various approximate identities. The following theorem on the impossibility of renorming Banach algebras to achieve a certain bound on an approximate identity is proved in (Dixon 1978). Theorem (Dixon) For each m > 1, there is a separable Banach algebra (A, ""D with a sequential bounded left approximate identity of bound m such that there is no norm III . III which is equivalent to " ." and !s such that (A, ",, III) has a bounded left appronmate identity of bound strictly less than m. 0 Dixon also shows in (1978) that, for each ml > m2 > 1, there is a (non-separable) commutative Banach algebra with an approximate identity of bound m 1, but such that there is no approximate identity of bound m2 in any equivalent norm on A. There is a detailed account of different proofs of a large variety of factorization theorems, with careful historical references, in (Doran and Wichman 1979). The ancestor of all these proofs is that of Cohen in (1959); the extension to the module form is due to Hewitt (see (Hewitt and Ross 1970, (32.22»). Our version of the factorization theorem 2.9.24, which involves 'power-factorization', is from (Allan and Sinclair 1976); a somewhat shorter proof is given there in the case where the algebra A is commutative. A variant of 2.9.24 is proved by Grl1lnbrek in (1982); here, the condition on A is strengthened by requiring that A be a commutative, radical Banach algebra, but the condition
Bounded approxLrnate identitif's and fad or'ization on the left approximate identity i::; w('akened to th 0 as 1(1 -+ <X! with ( E Sv' 0 Analyticity of the semigroup is important in this result: Esterle shows in (1981 c) that, given any continuous function t ....... at, JR!+ -+ JR!+, there is It eommutative, radkal Banach algebra R containing an infinitely differentiable sero.igroup (at: t E JR!+.) such t that < at (t 2: 1). The relations between properties (a) to (g), given immediately aft.er 2.9.44, have been explored by many authors. Examples 2.9.47 and 2.9.48 are from (Willis 1992a); other examples with various additional properties are also constructed in this interftlting Paper. There are no known counter-examples in the da.{d); (g),*(d). However, it is likely that such counter-examples will be constructed.
lIa l!
340
Banach and topological algebras
It is particularly important to decide whether null sequences factor weakly in a com-
mutative, separable Banach algebra which factors weakly. For some related results, see (Dixon 1990). Theorem 2.9.49(iii) is from (McKilligan 1973), and 2.9.51 is from (Johnson 1964). Theorem 2.9.53 in the ca..'!e where D is continuous is (Johnson 1972a, 1.11); the general case is an abstract version of (Willis 1986, Lemma 3.4). Theorem 2.9.54 is (Johnson 1972a, 1.9), and 2.9.56 is essentially (Grlllnbrek 1995, Lemma 6.4); 2.9.57 and 2.9.62 are from (Johnson 1972 a, 1.9, 1.6, and 5.4). Theorem 2.9.58 is related to results in (Helemskii 1989b, Chapter VII, §1.4) on flat modules: see also (ibid., VII.2.31) and (Curtis and Loy 1989). The following theorem on simplicial triviality is proved by Lykova and White in (1998); in fact, more general results are proved. Theorem (Lykova and White) Let A be a Banach algebra. and let I be a closed 'Ideal w'ith a bounded approximate ~dentzty. Then: (i) A is szmplicially trivwl whenever both I and AI I are simphczally trivial; (ii) 1-["+1 (AI I, (AI I)') = 11." (I. I') (n E N) whenever A is simplicially trivial; (iii) there is a long exact sequence
... 11. n - 1 (I,1')
--->
11.fl(AII, (All)')
--->
(iv) dbwI ~ dbuoA and dbu·(AI 1) ~ dbwA.
11.1t(A, A')
--->
11. n (I,1')
---> ... ;
o
Definition 2,9.63 is from (Johnson 1972b). The history of 2.9.65 is rather complicated. The equivalence of (a) and (b) (and 2.9.64) were established in (Johnson 1972b) Most of the equivalences in 2.9,65. and other equivalences. are given in (Helemskii 1989b, §VII.2.3), with references to the original papers; note that Helemskii in (1989b. VIL2.16) and (1993, V] I. 1.59) defines a Banach algebra A to be amenable if A# is biflat. The work of Helemskii, approaching amenability through the notion of flatness, was contemporaneous with that of Johnson. A key earlier paper establishing the equivalence of (a) and (e) of 2.9.65 is (Helemskii and Shelnberg 1979); see also (Racher 1981). The proofs in (Helemskii 1989 b) are different from ours, being more' homological'; our proofs are related to those in (Curtis and Loy 1989). An extension of part of 2.9.65 for biflat algebras is given in (Selivanov 1996, Theorem 6). Indeed, let A be a biflat Banach algebra. Then: dbwA = 0 if A has a hounded approximate identity; dbwA = 1 if A has a bounded left approximate identity or a bounded right approximate identity, but no bounded approximate identity; dbwA = 2 if A has neither a bounded left approximate identity nor a bounded right approximate identity. Corollary 2.9.66 is from (Gr(IJnbrek et al. 1994). Johnson (1996) has introduced the notion of symmetric amenability: a Banach algebra is symmetrzcally amenable if it has an approximate diagonal consisting of symmetric tensors. Most, but not all, amenable Banach algebras are symmetrically amenable; symmetrically amenable Banach algebras have hereditary properties similar to those of amenable Banach algebras.
3
Banach algebras with an involution
There is an additional algebraic operation on an algebra that we have not so far considered for Banach algebras; thh, is the operation of taking an involution, as defined in 1.10.2. Indeed, we briefly discussed *-algebras in §1.lO. We shall now consider Banach and other topological algebras which have an involution. A surprising amount of the theory applies in exactly this setting, without any requirement that the involution be related to the topology of the algebra. Nevertheless. we prefer to define a Banach *-algebra in such a way that the involution is an isometry, and thus sometimes refer to 'Banach algebras with an involution' for the more general class. A general theory, leading to the 'GNS representation' of a Banach *-algebra. is given in §3.1. In §3.2 and §3.3, we shall discuss the two most important classes of Banach *-algebras. the class of C* -algebras, in §3.2, and the cla.'is of group algebras Ll(G), where G is a locally compact group. in §3.3. There is a very substantial and rapidly growing literature concerning C* -algebra.'i. and this corpus contains several major texts; there are also some comprehensive texts on group algebras. For this reason, we shall not attempt any geupral account of these classes here, but we shall merely, in this chapter, give their most basic properties and establish some quite specific results that will be needed later. Let A be a *-algebra. We recall that the subsets Asa, A+, and Apos were defined in §1.lO: see 1.10.1, (1.10.5), and (1.10.6), respectively. 3.1
GENERAL THEORY
In this section, we shall define Banach and topological *-algebras, prove the Shirali-Ford theorem that a Banach algebra with an involution is hermitian if and only if it is symmetric, and develop the GNS representation of a Banach *-algebra as an algebra of operators on a Hilbert space.
Definition 3.1.1 A Banach *-algebra zs a Banach algebm which has an isometric involutwn. Let (A, 11·11) be a Banach algebra with a continuous involution. Then there is a norm 111·111 on A equivalent to 11·11 such that (A, III· liD is a Banach *-algebra: indeed, set IIlalll = max{llall, Ila*lI} (a E A). However, a Banach algebra may have a discontinuous involution. For let A be an infinite-dimensional Banach
Banach algebras with an involution
342
algebra Huch that A2 = O. and take {:en: n E N} U {Yl' : ~( E f} to be a basis for A cOllsiHting of elements of norm ouc. Definc
Y; = y"(
bE f),
X2n = nX2n-1.
X~n-l = x2n/n
(n EN).
Then * extends by conjugate-linearity to a discontinuous involution on A. An pxample of a Banach algebra with diseontinuous involution in which the multiplication iH not trivial will be given in 5.6.83(iii).
Proposition 3.1.2 Let A be a Banach algebm with a unique complete norm such that A has an involution *. Then there is an equzvalent norm on A such that (A. *) is a Banar;h *-algebm. Proof Let 11·11 be the given norm on A. The function 111·111 : a ~ Ila¥11 is a complete algebra norm on A. By hypothesis, III· III is equivalent to II . II, and so the involution is continuous. 0 Definition 3.1.3 A topological *-algebra is a topologzcal algebm whzch has a continuous znvolution. An LMC *-algebra is an LMC algebm which is also a topological *-algebm. We define (F)-*-algebms, locally convex *-algebms, and Frechet *-algcbras similarly. Let A be a topological *-algebra. Then the subspace Asa is closed in A, and the closures of *-ideals and *-subalgebras are *-ideals and *-subalgebras, respectively. Suppose further that A is a Q-algebra. Then it follows from 1.1O.5(i) and 2.2.28(i) that each maximal modular *-ideal in A is closed. Let A and B be Banach *-algebras. As in 1.10.4, A ® B is a *-algebra with respect to an involution such that (a ® b)* = a* ® b* (a E A, b E B). Clearly this involution is isometric on (A ® B, 11·1I 7r ), and so it extends to an isometric involution on (A®B. II· 117r)'
7
Examples 3.1.4 (i) Let X be a topological space. Then the map f ~ is a continuous involution on C(X), and so C(X) is an LMC *-algebra. Clearly C(X)sa = C(X.JR). Since 1 + 1* f is invertible for each f E C(X). the *-algebra (C(X). *) is symmetric. The involution is isometric on (Cb(X), I· Ix ), Cb(X) and Co(X) are symmetric Banach *-algebras, and Cb(X)+ = Cb(X)pos = Cb(X, JR+) .
en.
The algebra C(U) is a Frechet *-algebra for each non-empty, open set U in (ii) Let S be a subset of e such that S is symmetric with respect to the real axis. For a function fEes, define
f*(z) = I(z)
(z
E S).
Now take A to be A(jj)) or A+(jj)). Then the map I ~ f* is an isometric involution on A, and so A is a Banach *-algebra; the involution is proper, and IE Asa if and only if f([-I, 1]) c JR. Clearly Z E A sa , but u(Z) = jj), and so A is not hermitian. The only elements of Apos are the constant functions with values in jR+ .
General theory
343
Let A be a Banach *-algebra, and let a E A. Suppose that J E O(U), where U is a non-empty, open set in IC, symmetric with respect to the real axis, with U J a(a). Then J(a*) = (J*(a))*. In particular,
exp(a*) = (expa)*,
sin (a*) = (sina)*,
cos (a*) = (cosa)*.
(3.1.1 )
(iii) Let H be a Hilbert space with inner product [., .]. For y E H, define = [x, y] (x E H). Then J : H ....... H' is an isometric, conjugate-linear bijection. It follows from the Riesz theorem A.3.14 that, for each T E B(H), there is a unique clement T* E B(H) such that
(Jy)(x)
[Tx, y] = [x, T*y]
(x, Y E H):
indeed, T* = J- 1T' J. The element T* is the (Hilbert space) adjoint of T, and the map T 1--4 T* is an isometric involution on B(H). In the case where H has finite dimension n, B(H) is *-isomorphic to Mn (so defining a norm 11·112 on Mn, as in 2.1.13(ii)), and the adjoint coincides with the object defined after 1.10.2. We clazm that the following 'automatic continuity' result holds. Suppose that 8, T E £(H) are such that [Tx, y] = [x, 8y] (x, y E H). Then 8, T E B(H) and 8 = T*. For suppose that Xn ....... 0 and TX n ....... Z in H. Then, for each y E H, we have [Txn, y] = [xn' 8y] ....... 0 and [Txn' y] ....... [z, y] as n ....... 00, and so z = O. By the closed graph theorem A.3.25, T is continuous. Similarly 8 is continuous, and the claim follows. It is for this reason that we only define T* for a bounded linear operator T on H. The Banach *-algebra (B(H), *) is the prototype of a C*-algebra; it will be shown (in 3.2.3(v) and 3.2.8, respectively) that B(H) is symmetric and that B(H)+ = B(H)pos. Note that, for each x E H with Ilxll = 1, the map T 1--4 [Tx, xl is a state on B(H). (iv) Let w be a weight on a semigroup 8. The weighted semigroup algebra A = (£1(8, w), *, II·IL) has a natural involution
t:
LQ 0 Lasos, S
5 1--4
A ....... A.
The algebra (£ 1 (Z+), *, t) is *-isomorphic to (A + (IDi) , . , * ), and so the algebra A is not necessarily hermitian. Now let §2 be the free semigroup on two generators u and v. Then the map # : §2 ....... §2, defined by the requirements that u# = v and v# = u and that the formula (st)# = t#s# (s,t E §2) hold, induces an isometric involution, also denoted by #, on £1(§2). For each word W E §2, let n,..(w) and nv(w) be the total number of times that the letters u and v, respectively, occur in w. For each (1, (2) E IDi 2, the function "{
~
1:
Qwuw:
! : ! } " {,..nu('w),..nv(W) } wEI ~2 1--4 ~ QU''>1 '>2 : w E I!:! ~2 ,
is a character (and each character arises in this way). Thus a( o,..v) J IDi by 1.5.28. Since otfv = ouv, the algebra £1(§2, #) is not hermitian. (v) Let G be a group. We noted in §1.1O that the algebraic group algebra Coo(G) is a *-algebra. Now suppose that w is a symmetric weight on G, so that, by definition, W = w. Then the involution extends in such a way that the Weighted group algebra i1(G,W) is a Banach *-algebra. 0
Banach a;lgebrus with an involution
344
Theorem 3.1.5 (Ford) (i) Let A be a unital Banach algebm wzth an involution, and let a E Asa with a(a) C C\IR-. Then there exists a unique b E A with b2 = a and a(b) en. Further, bE Aba n {ay. (ii) Let A be a Banach algebm with an mvolution, and let a E Asa wzth v( a) < 1. Then there exists b E Asa wzth bob = a. Proof (i) By 2.4.18, there exists a unique b E A such that b2 = a and a(b) en. We also have (b*)2 = a* = a and a(b*) C n. and so, by the uniqueness of b, b = b* E Asa. As in 2.4.18. bE {ale. (ii) By (i), there exists c E A!. with c2 = eA -a and a(c) en. Set b = e A -c. Then b E Asa and bob = a. 0 Corollary 3.1.6 Let A be a Banach algebm with an mvolution, and let>. EPA' Then:
(i) 1(a*xa, >')1:::; (a*a, >.)v(a:) (a E A. J: E Asa); (ii) 1(a*ba, >')1:::; (a*a, >.)v(b*b)1/2 (a, bE A); (iii) >. 1 A(rad A)A = O. Proof (i) Let a E A and x E Aba; we may suppose that veX) < 1. By 3.1.5(ii), there exist y. Z E Asa with 2y - y2 = x and 2z - Z2 = -x. Set u = a - ya and 'V = a - za. Then we see that u*u = (0* - a*y)(a - ya) = a*a - a*xa and v*v = (a* - a*z)(a - za) = a*a + a*xa. Thus (a*a. >.) ± (a*xa, >.) : : : O. and fiO
>')1:::; (a*a. >.). (ii) Let a,b EA. It follows from the Cauchy-Schwarz inequality 1.10.13(iii) that I(a*ba, >')1 2 :::; (a*a. >.)(a*b*bo, >'), and, by (i), (a*b*ba, >.) :::; (a*a, >.)v(b*b). The result follows. (iii) By (ii), (a*ba, >.) = 0 (a E A, bE radA), and so the result follows from (1.10.4). 0 I(a*xa,
Let A be a unital Banach algebra with an involution, and let a E A.sa with v(a) < 1. By (2.3.8), v(a 2 ) < 1, and so, by 3.1.5(i), there exists b E Asa with b2 = eA - a 2. Set u = a + ib and v = a - ib. Then '11, t' E U(A) because ab = ba. and a = (u + v) /2. Since A = Asa + iAs... , it follows that every element is a linear combination of four uni taries; in particular, A = lin U(A) .
(3.1.2)
Vie now turn to the study of Banach algebras with a hermitian involution.
Proposition 3.1. 7 Let A be a Banach algebm wzth a hermitzan invol1Liion, and let B be a closed *-subalgebm of A. Then aRea) C aA(a) U {O} C aB(a) U {O}
(a E B).
Proof First suppose that A is unital and that e A E B, and take b E B n Inv A. Then bb* E Inv A, and hence aA(bb*) C IR·. It follows from 2.3.21(iii) that bb* E Inv B, and hence that b- 1 = b*(bb*)-l E B and b E Inv B. This implies that aB(a) = aA(a) (a E B) in this case. The non-unital case follows. 0
General theory
345
Proposition 3.1.8 Let A be a Banarh algebra with an mvolution. Then the mvolutzon 1,8 herm1,tian 1,f and only if
v(a)2 ~ v(a*a)
(a E A).
(3.1.3)
Proof We may suppose that A is unital, with identity e. Suppose that A is hermitian, and asf>ume towards a contradiction that (3.1.3) does not hold: there exists a E A with 1 E a(a) and v(a*a) < 1. By 3.1.5(i), there exists h E Asa n Inv A such that h 2 = e - a*a. Set b = (e + a*)(e - a) and c = (e - a)(e + a*). Then we have b = h 2 + a* -
(l
= -ih(k
+ ie)h.
where k = ih-l(a* - a)h- I . Clearly k = k*, and so -i E p(k) because A is hermitian. It follows that b E Inv A. Since v(aa*) < 1 (by 1.5.29(ii)), it follows similarly that c E Inv A. But now e - a has a left and right inverse, and so e - a E lnv A, a contradiction of the fact that 1 E a(a). Thus (3.1.3) holds. Conversely, suppose that (3.1.3) holds, and assume that A is not hermitian. Then there exists a E Asa with a(a) ct. }R; we may suppose that i E a(a). Choose t E}R with 1 + 2t > v(a)2. and set b = a + ite. Then i(l + t) E a(b), and so
(1 + t)2 ~ V(b)2 ~ v(b*b) = v(a 2 + t 2e) ~ v(a)2 < 1 + 2t + t 2 ,
+ t2
by (3.1.3) by (2.3.8)
o
a contradiction. Thus A is hermitian.
Proposition 3.1.9 Let A be a Banach algebra with a hermztian involutzon. Then: (i) v(ab)
~
(ii)
+ Apos
Apos
v(a)v(b) (a, bE Aa); cApos.
Proof We may suppose that A is unital, with identity e. (i) Take a, bE Asa. By 3.1.8 and 1.5.29(ii), we have
v(aTtbn ) ~ v(bn a2Tt bn )1/2 = v(a 2n b2Tt )I/2
(1£
E
N),
and so, by induction on mEN, v(ab) ~ v(a 2"'b 2m )1/2m (rn EN). Thus
I "' Il
v(ab) ~ a 2
l 2m /
I b2'" Il
l 2m /
~ v(a)v(b)
as rn ~
00,
giving (i).
(ii) Take a, b E Apos. Then a + b E Asa, and so a(a + b) c }R because A is hermitian. For each ~ > 0, we have ~(~e + a + b) = (~e + a)(e - hk)(8e + b), Where now h = (8e + a)-Ia and k = b(8e + b)-I. By 2.4.4(iv), a(h)
C
{tj(8 + t) : t
E }R+},
and so a(h) C [0,1) and v(h) < 1. Similarly v(k) < 1. Since h, k E Asa, it fallows from (i) that v(hk) < 1, and so e - hk E Inv A. Thus 6e + a + bE Inv A, and so -~ ¢ a(a + b). Thus a(a + b) C lR+, and a + bE Apos. 0
346
Banach algebras 'Unth an involution
It follows from (ii), above, that A+ c APOH for a Banach algebra with a hermitian involution. We showed in 1.l0.22(iii) that a symmetric *-algebra is hermitian. We shall now prove the converse result for Banach algebras. Note that we do not require that the involution be continuous. Theorem 3.1.10 (Shirali-Ford) Let A be a Banach algebra with a heT7nitian involutwn. Then A u; symmetric. Proof We may suppose that A is unital. with ide-ntity e. Let a E A, so that a(a*a) C JR, and set m = sup a(a*a). We clatm that m 2': 0 and v(a*a) = m. For set a = h + ik. where h, k E Asa. Since a(h) C R we have a(h 2 ) C JR+ by 1.6.11(i), and similarly a(k 2 ) C JR+. By the definition of m. aCme - a*a) C JR+. We have me + aa* = 2h2 + 2k2 + (me - a*a): by 3. 1.9(ii) , aCme + aa*) C JR+, i.e., a(aa*) C [-m. Xl). But a(a*a) = a(o.a*), and so m 2': 0 and v(a*a) = m. as required. Take a E A with v(a*a) < 1 and set b = 2a(e + a*a)-1. We claim that v(b*b) < 1. Since a*a and (e+a*a)-1 commute, we have b*b = 4a*a(e+0.*a)-2, and so a(b*b) = {4t/(1 + t)2 : t E a(a*a)} C (-00, 1). Thus sup a(b*b) < 1, and so v(b*b) < 1 by the first claim. Assume towards a contradiction that A is not symmetric. Then there exist a1 E A and t1 < 0 with v(aia1) < 1 and t1 E a(aiad. Inductively define a n+1
= 2an (e + o.~an)-l ,
tn+1
= 4tn/(1 + t n )2
(n E N).
For each n E N, tn E a(a~an) and, by the second claim, v(a~o.n) < 1. Clearly (t n ) is decrea.'ling and there exists N E N with t /Ii > -1 and t N +1 ~ -1, a contradiction of the fact that t N + J E a ( a +1 aN +1 ) . Thus A is symmetric, and the theorem is proved. 0
tv
Corollary 3.1.11 Let A be a Banach algebra with an involution. following are equivalent:
(a) A is symmetric; (b) A zs hermitian; (c) v(a)2 ~ v(a*a) (a E A).
Then the
0
In the remaining re!:lults of this section, we do suppose that the involution is continuou!:l. Let A be a Banach *-algebra. It follows from the Cauchy-Schwarz inequality 1.1O.13(iii) that each positive functional on A is a separable map relative to the product on A. Now suppose, further, that A is unital, and take A EPA. Since v(b*b) ~ IIbll, it follows from 3.1.6(ii) that I(b, A}I ::; (eA' A) IIbll (b E A), and so A is continuous and II All = (e A, ).). Thus the state space (cf. 1.lO .12) of A is SA = {A
E
A' : 11).11 = (eA' A) = 1, A(A+)
C
JR+}
C
KA,
where KA was defined in (2.3.1); SA is a convex, closed subset of (Al1l' a(A', A», and so SA is compact.
347
General theory
The *-radical, *-rad A, of a unital *-algebra A was defined in 1.10.15 as n{IA : A E SA}, where IA = {a E A : (a*a, A) = OJ. Suppose that A is a Banach *-algebra. Then each h is closed, and so *-rad A is a closed ideal in A. Let B be a *-subalgebra of A, and take b E radB. Then b*b E O(A), and so bElA (A E SA). Thus radB c *-radA, and so B is a semisimple algebra in the case where A is *-semisimple. Theorem 3.1.12 (Kelley and Vaught) Let A be a umtal Banach *-algebra. Then *-radA = {a E A : -a*a E A+} . Proof Suppose that ao E A with -a(jao E A+. For each A E SA, we have A(A+) C ]R+, and so (a(jao, A) = o. Thus ao E *-rad A. Conversely, suppose that ao E *-radA. Define f-l(a) = d(a.A+) (a E Asa). Then f-l is a sublinear functional on A sa , and so, by A.3.16(i), there is a reallinear functional A on Asa with (aoao, A) = f-l(a(jao) and (a, A) S f-l(a) (a E Asa); extend A to a linear functional, also called A, on A. For each a E A, we have - (a*a, A) S -f-l(a*a) = 0, and so (a*a, A) ::::: O. This shows that A EPA, and so (aoao, A) = o. Thus -a(jao E A+. 0
Corollary 3.1.13 Let A be a unital Banach *-algebra which that A+ is closed. Then A ~s *-semisimple.
~s
ordered and such 0
Lemma 3.1.14 Let A be a umtal, hermitzan Banach *-algebra, and let I be a proper left zdeal zn A. Then there exists A E SA such that I C h. Proof Define AO : (aeA,a) f---> a, ]ReA 0lsa --+ lR.. Then AO is a real-linear functional on ]ReA o Isa. For each a E ]ReA 0lsa , we have a - (a, AO) E I, and so (a, AO) E (T(a); if also a E Apos, then (a, AO) E ]R+. By 3.1.9(ii), Apos is a convex subset of Asa> and ApOb - eA is absorbing in Asa because A is hermitian. By the Hahn-Banach theorem A.3.16(ii), there is a real-linear functional A : Asa --+]R extending AO with A(Apos) c ]R+; extend A to a linear functional on A = Asa 0 iAsa . Then A EPA, and so A is continuous. Since (eA' A) = (eA' AO) = 1, certainly A E SA. For a E I, we have a*a Elsa because I is a left ideal, and (a*a, A) = (a*a, AO) = o. Thus lelA. 0 Theorem 3.1.15 Let A be a hermitian Banach *-algebra. Then *-radA = radA.
Proof We may suppose that A is unital. We have noted that rad A C *-rOO A. Let a E *-rad A, and let M be a maximal left ideal of A. By 3.1.14, there exists A E SA with M c h; since M is maximal and eA rt. lA, necessarily M = h, and so a E M. By 1.5.2(ii), a E roo A, and so *-radA C radA. 0
We now give the GNS representation of a Banach *-algebra; it is analogous to the Gel'fand representation for commutative Banach algebras of 2.3.25. (The letters G, N, S honour Gel'fand, Naimark, and Segal, who developed the theory.)
348
Banach algebras with an involutzon
Definition 3.1.16 Let A be a *-algebra. A *-representation of A on a Hilbert space H zs a *-homomorphism 7r : A -+ B(H)j the *-representation Z8 faithful if 7r is a monomorphism. Let A be a unital *-algebra. Then a *-representation 7r of A on a Hzlbert space H is universal if 7r zs unital and each state on A has the form a t--+ [7r(a)x, x] for some x E H with IIxil = 1. A *-representation of A is, in particular, a representation of A in the sense of 1.4.7. Let 7r : A -+ B(H) be a unital *-representation of a unital Banach *-algebra A, and take x E H with Ilxll = 1. Then A : a t--+ [7r(a)x, x] is a state on A and 117r(a)xI12 = (a*a, A) ~ lIal1 2 (a E A). Thus 117r1l = 1. Suppose that the *-representation 7r is faithful, and that a E *-rad A. Then 7r( a) = 0 and a = 0, so that A is *-semisimple. Let A be a unital Banach *-algebra, and let A E SA. It follows from (1.10.7) that the formula
[a+h.,b+IA]=(b*a,A)
(a,bEA)
(3.1.4)
gives a well-defined value to [a + h, b + I A]; clearly [" .] is an inner product on the space A/IA. The corresponding norm on A/h is denoted by 11·112' and the completion of A/ h with respect to this norm and inner product is defined to be the Hilbert space (HA' 11·112)' Next, set 7r A(a)(b + I A) = ab + IA (a, bE A). (3.1.5) Let a E A. Then 7r A (a) is a well-defined linear operator on A/h. For a, bE A. we have II7rA(a)(b + IA)II~ = lab + h, ab + I A] = (b*a*ab, A) ~ (b*b, A)v(a*a) by 3.1.6(i) ~ IIal1 2 11b + IAII~ , and so 7rA(a) is continuous on (A/lA, 11.11 2) with II7rA(a)11 ~ lIall. Hence 7rA(a) extends to an element of B(HA ). Clearly 7rA : a t--+ 7rA(a), A -+ B(HA ), is a unital homomorphism such that II7rAII = 1, and so HA is a Banach left A-module. For a, b, c E A, we have
[7r A(a)(b + h), c + h] = (c*ab, A) = (a*c)*b, A) = [b + lA, 7rA (a*)(c + h)]. and so 7rA(a)* = 7rA(a*). Thus 7rA is a *-representation of A on H A . Set XA = eA + h E H A. Then 7r A(a)xA = a + IA (a E A), and so we have 7rA(A)XA = A/h and 7rA(A)XA = H)... Moreover,
[7r)..(a)xA' x A] = [a + h,
eA
+ I A] =
(a, A)
(a E A) ;
(3.1.6)
in particular, IIx)..lI~ = (eA,A) = 1. Suppose, further, that A is a tracial state on A. Then h is a closed ideal in A, and A/his an A-bimodule. As above, HA is also a Banach A-bimodule for maps satisfying a· (IT+ h.) = ab + f).. and (b + h) . a = ba + h for a, bE A. This remark will be used in 5.6.76.
349
General theory Let {H y : 'Y E r} be a family of Hilbert spaces. Define H = f
[(x,),), (y')')] = ~)x')', y')'j
2
(r, H'))
and
«x')'). (y')') E H) .
')'Er
Then H i~ a Hilbert ~pace for the inner product [ " .j: H i~ called the direct sum of the Hilbert ~paces H')': Now ~uppose that 7r')' i~ a unital *-representation of the unital Banach *-algebra A on H')' with /l7r')'11 = 1 for each 'Y E r. For each a E A, define
7r(a)«x')')) = (7r')'(a)(xl'))
«x')')
E
H).
Then 7r(a) E 8(H) with 117r(a)/I :::; /lall, and the map 7r : A ...... 8(H) is a unital *-representation of A on H with /l7r/l = 1; it is called the direct sum of the *-representations 7r1" and it is denoted by E& 7rT For)' E SA, let 7r,\ and H,\ be as above, and then set H = f2(SA, H,\). Then 7r = E&7r,\ is a continuous, unital *-repre~entation of A on H, and it follows from (3.1.6) that 7r is universal, where we are now regarding x). as an element of H. Let a E A. Clearly (a*a,).) = 0 (). E SA)-if and only if 7r(a)x = 0 (x E H), and so (3.1.7) ker7r = n{I,\ : ). E SA} = n{ker>. : ). E SA}. We have proved the following result. Theorem 3.1.17 Let A be a unital Banach *-algebra. Then there is a universal
*-representation of A on a Hilbert space. Furthermore, the followmg conditions on A are equzvalent: (a) there exzsts a fazthful, unzversal *-representation of A; (b) A has a fazthful, unital *-representatzon on a Hilbert space; o (c) A zs *-semzszmple. Corollary 3.1.18 Let A be a unital, semiszmple, hermztzan Banach *-algebra.
Then A has a faithful, universal *-1'epresentation on a Hzlbert space.
0
Let A be a Banach *-algebra. We have defined (in §1.lO) a linear involution E A", define If>* E A" by ~etting (If>*, ).) = (If>, ).-+ v(a*a)1/2 was recognized by Pt8.k (1970, 1972) and Palmer (1972). For a history and generalizations 1D. (Palmer 2001).
350
Banach algebms with an involution
of this result, see (Doran and Belfi 1986) and (Palmer 2000). It seems to be unknown whether or not the Shirali-Ford theorem holds for complete LMC *-algebras. Palmer (2001, 9.7.1) defines the reducing ideal of an arbitrary *-algebra A as the intersection of the kernels of the *-representations of A on a Hilbert space, and studies this concept. This ideal coincides with our *-rad A in the case where A is a unital Banach *-algebra. The characterization of *-rad A in 3.1.12 is from (Kelley and Vaught 1953). The GNS construction of 3.1.17 is extensively discussed in the literature, but it is sometimes given just for C*-algebras. The Hilbert space of 3.1.17 was formed by taking the direct sum of the spaces H).. for .A E SA. The pure states of A are the extreme points of SA; a state .A is pure if and only if the unital *-representation 'Ir).. on H).. has no non-trivial, closed invariant subspace for each 'Ir)..(a). One also obtains a faithful, unital *-representation by taking the sum over the pure states (Helemskii 1993). Clearly a *-semisimple Banach *-algebra A ha..'l an auxiliary norm 111·111 such that Illa*alll = IIIall1 2 (a E A); Banach *-algebras with such a norm are called A*algebras. Let A be a unital Banach algebra. In 2.4.40, we defined H(A), the closed, reallinear subspace of A consisting of the hermitian elements. Let J(A) = H(A) + iH(A), a closed linear subspace of A. Then each element of J(A) has a unique representation in the form h + ik, where h, k E H(A), and the map * : h + ik 1---+ h - ik is a continuous involution on J(A). In general, J(A) is not a subalgebra of A. However, in the case where h2 E H(A) whenever h E H(A), then (J(A), *) is a Banach algebra with a continuous involution, and J(A)sa = H(A). The algebra A is a V-algebra if A = J(A).
See (Bonsall and Duncan 1973, §38) and (Palmer 1994, §2.6). 3.2
C* -ALGEBRAS
The present section is devoted to C* -algebras, with some brief remarks on the subclass of von Neumann algebras. Definition 3.2.1 A C* -algebra is a Banach *-algebm A such that
Ila*all
=
lIall 2 (a E A).
(3.2.1)
An algebra norm on a *-algebra satisfying (3.2.1) is a C*-norm. Let A be a *-algebra, and let 11·11 be an algebra norm on A such that lIall 2 ~ lIa*all (a E A). Then lIall 2 ~ lIa*lIlIall, and so Ila*1I = Iiall and the involution is isometric; also lIa*all ~ lIall 2 , and so (3.2.1) holds. The involution on a C*-algebra is certainly proper in the sense of 1.1O.7(ii), and so, by 1.10.9, finite-dimensional C* -algebras have the form Mnl ... , nk EN. Let A be a non-unital algebra which is a C* -algebra, and let A : a 1-+
La,
A - B(A) ,
be the left regular representation. Since IIal1 2 ~ IILalllla*1I (a E A), A is an isometry. We identify A# with {adA + La : a E C, a E A}, where IA is the identity of B(A). Then A# is a Banach *-algebra with respect to the involution alA + La 1-+ lilA + La * and the operator norm from B(A); it is easily checked that this norm is a C*-norm on A#. Thus we may regard A as a closed ideal in the unital CoO -algebra A # .
C*-algebms
351
Examples 3.2.2 (i) The algebras Cb(X) and Co(X) for X a completely regular space are commutative C* -algebras with respect to the involution f ~ 1 of 3.1.4(i).
(ii) The standard example of a non-commutative C*-algebras is 8(H), where H is a Hilbert space, as in 3.1.4(iii); for T E 8(H), T* is the adjoint of T, and we have IITI12 = sup [Tx, Tx] = sup [T*Tx, x] :::; IIT*TII .
"x"Sl "x"Sl Thus 8(H) is a unital C*-algebra with respect to this involution. Let T E 8(H). The following standard facts are easily seen: T is self-adjoint if and only if [Tx, x] E lR. (x E H); T is normal if and only if IITxll = IIT*xll (x E H); T is unitary if and only if IITxl1 = IIT*xll = Ilxll (x E H). Further, T E J sa (8(H)) if and only if T is an orthogonal projection onto a closed linear subspace F of H, and then F = {x E H: Tx = x} = {x E H: IITxl1 = Ilxll}. For a closed subspace F of H, let PF be the orthogonal projection onto F. Then F E Lat T if and only if T P F = PFT. Let 2( be a *-closed subalgebra of 8(H), let x E H, and define F = 2( . x, where 2l . x = {Ax: A E 2l}. Then FE LatT for each T E 2l, and so PF E 2l". (iii) Let H be a Hilbert space. A continuous rank-one operator on H is determined by xo, Yo E H. Indeed, we have Xo 18) Yo : x ~ [x, Jyo]xo, H ---> H, where .I was defined in 3.1.4(iii). The adjoint of Xo 18) Yo is Yo 18) Xo. Thus F(H) is a *-ideal in 8(H). Since H has AP, the nuclear algebra of H is N(H) and A(H) = K(H); (N(H), II· IIv) is a Banach *-algebra and (K(H), 11·11) is a C*algebra. As in Example 2.6.7(ii), we can identify 8(H) as the dual space of N(H); the duality is given in (2.6.18) as the trace duality (S, T) = Tr(TS)
(S E N(H), T E 8(H)).
Let (e v ) be an orthonormal basis of H, and let T = Xo
Tr(xo
18) Yo)
= [XO,
./yo]
18) Yo.
Then
= L[(XQ 18) yo)e v , ev ] .
Thus, for each T E N(H), we have Tr(T)
= L[Tell,ev ] and Tr(TT*) = Tr(T*T) = L IITe l1 2 ~ O. ll
Hence Tr is a positive trace on (N(H) , *) in the sense of 1.10.12.
o
Proposition 3.2.3 Let A be a C* -algebm.
(i) Let a E A be normal. Then IIa 2 11
= Ila*all and v(a) = Iiali.
Ila11 2 •
(ii) For each a E A, v(a*a) = (iii) Let p E Jsa(A) \ {O}. Then Ilpll = 1. (iv) Suppose that A is unital, and let a E U(A). Then (v) The *-algebm A is hermitian and symmetric.
II all
= 1 and a(a) cT.
Banach algebras with an involutwn
352
(vi) Let a E A and
A) zs an zsometnc *-isomorphzsm. Proof By 2.3.25,9 is a homomorphism and lI(a) = lal1>A (a E A). By 3.2.3(vi), 9 is a *-homomorphism and A is self-adjoint. Since each a E A is normal, lI(a) = lIall (a E A) by 3.2.3(i), and so 9 is an isometry and A is closed in CO(cJ>A). By the Stone-Weierstrass theorem A.l.1O(i), A is dense in CO(cJ>A), and hence A = CO(cJ>A). 0 A subalgebra of a C* -algebra which is both II· II-closed and *-closed is a C* -subalgebra. For example, pAp is a C* -subalgebra of A for each projection p E Jsa(A).
Let A be l'1 C*-algebra, and let a E A. The C*-subalgebra Co[a,a*j polynomially generated by a and a * (cf. 2.2.7 (i» is denoted by Co (a ); Co (a) is the smallest C*-subalgebra of A containing a. Note that it follows from 3.2.5 that {aY = Co(a)C whenever a is normal. In the case where the C*-algebra A is unital, C*(a) = qa,a*j is the smallest unital C*-subalgebra of A containing a. Now suppose that a is normal. Then B = CoCa) is a commutative C*-algebra, and cJ> B is homeomorphic to a B (a) \ {O} because the map
aB(a) U {O},
is injective by 3.2.3(vi), and hence a homeomorphism; aB(a) \ {O} = aA(a) \ {O} by 3.l.7, and so 9: CoCa) ----> Co(aA(a) \ {O}) is an isometric *-isomorphism.
Theorem 3.2.7 Let a be a normal element of a C*-algebra A. Then there is a unique *-homomorphism e a : Co(a(a) \ {O}) ----> A such that ea(Z) = a. Further, Sa is an isometry wzth range Co (a). Proof Set e a = 9- 1 in the above notation; e a has the required properties. Suppose that (} : Co(a(a) \ {O}) ----> A is also a *-homomorphism such that 6(Z) = a. By 3.2.4, (} is continuous, and (} and Sa agree on the dense subalgebra C[Z, Z] of Co(a(a) \ {O}), and so (} = ea. 0 The map Sa is the continuous functwnal calculus for the normal element a. We usually write f(a) for ea(f). The map f I--t (ea' (7»* is also a continuous functional calculus for a, and so, by the uniqueness assertion in 3.2.7, we have f(a*) = (f*(a»*, where f* was defined in 3.l.4(ii). In the case where the algebra A is unital, the map e a : C(a(a» ----> C*(a) is a unital, isometric *-isomorphism such that ea(Z) = a, and so e a extends the analytic functional calculus of 2.4.4 for a. In this case, a(f(a» = f(a(a»
(f E C(a(a») ,
and g(f(a» = (g 0 J)(a) whenever f E C(a(a» and g E C(f(a(a))). For eJcample, let 0 be a non-empty, compact space and take f E C(O). Then 81(9) = 9
0
f
(g E C(a(f))).
Banach algebras with an involution
354
Let A and B be C*-algebras, and let B : A ~ B be a *-homomorphism. For each normal element a E A, B(a) is normal in B, and aB(B(a) \ {OJ C a A(a) \ {OJ. Set C = Co(aA(a) \ {OJ). Then B 0 e a and ee(a) I Care *-homomorphisms from C into B which agree on Z, and so B 0 e a = eO(a) I C. Thus
f(B(a)) = B(f(a»
(f
E
Co(aA(a) \ {O})).
(3.2.3)
In particular, since each
O. Take b E B ha with b = eg. Then Illal (e ± b) S; 1 and lal b -/ O. where e is the identity of A#. By (3.2.6), lIa(e ± b) II S; 1 and ab -/ 0. But 2a = (a + ab) + (a - ab). a contradiction of the fact that a E eXA[l]' Thus f(cf>B) C {O, I}, and so p = f. whence p2 = p. Thus p E Jba(A). Similarly q E Jsa(A) . We have established the claim. We now have (a - ap)*(a - ap) = O. and so ap = a; similarly qa = a. Now take c E (e - q)A(e - p) n A[l]' Then r*a = a*c = 0, and so Iia ± cl1 2 = II(a
± c)*(a ± c) II
= lIa*a + c*cll = 1
by 3.2.12(v). Hence c = 0 because a E eXA[l]' Thus (e - q)A(e - p) = O. Let (eO') be a bounded approximate identity for A. Then (e -q)ea(e-p) = 0 for each n, and so e a = eaP + qe a - qeaP -+ P + q - ql). Hence p + q - qp is the identity of A. 0 We now obtain the second main representation theorem for C* -algebras, the non-commutative Gel'fand-Naimark theorem.
Theorem 3.2.29 Let A be a unital C* -algebra. Then there is a Hilbert space H and an isometrtc, umversal *-zsomorphism 7r fmm A onto a C* -subalgebm of B(H). Proof By 3.2.13, A is Hlemisimple. By 3.1.17, there is a faithful, universal *-representation 7r of A on a Hilbert space H. By 3.2.23, 7r is isometric and 7r(A) is a C*-subalgebra of B(H). 0 We present one explicit example of a C* -algebra.
Example 3.2.30 Set A = e=CN, MIn) and I = co(N, MIn). As in 2.1.18(iii), A is a Banach algebra, and I is a closed ideal in A; as in §1.lO, A is a *-algebra. Clearly A is a C* -algebra. The algebra A is the C* -direct product and the algebra I is the C*-direct sum of the algebras MIn. We regard each MIn as a subalgebra of I and of A. For each n E N, set In = {a = (am) E A : an = O}. Then In is a closed ideal of finite codimension in A with A = In El1 MIn. Since MIn is a simple algebra, In is a maximal ideal in A. We shall prove that each maximal ideal of finite codimension in A is of the form I ko for some ko EN. Indeed, let M be a maximal ideal of finite co dimension in A, say AIM ~ MIko' so that dim (AIM) = k~. For each n E N, either MIn n M = 0 or MIn n M = MIn'
C* -algebras
363
and Mn n M = 0 only for n ~ ko. Set L = h n··· n h o ' Then the identity of the closed ideal L is eL = (0, ... ,0,1,1, ... ). Write 7r : A -+ AIM for the quotient map; (Eij : i,j E Nko+d is the standard system of matrix units for Mko+l. For each n ~ ko + 1, choose any ordered (k o + I)-tuple of distinct elements from Nn , and let tn : Mko +l -+ Mn be the embedding onto the corresponding rows and columns. For each z,j E Nko+1' set p~;) = 0 E Mn (n E Nko) and
p~7) = tn(Eij) (n ~ ko + 1), and then set Pij = (p~;) : n EN), so that Pi) EA. The elements Pij generate a subalgebra, say B, of A, and B ~ M ko +1 ' It cannot be that ker (7r I B) = 0, for this would imply that dim (AIM) ~ (ko + 1)2. Thus 7r I B = O. A speczal pro]ectzon in A is an element P = (Pn) E L such that each Pn is a diagonal matrix in Mn with diagonal terms 0 or 1. The special projection p has bounded rank if sup rn < 00, where rn is the rank of Pn. It is clear from the above paragraph that 7r(p) = 0 for each special projection P of bounded rank. For n ~ ko + 1, we write n = (ko + I)u + v, where u E N and v E Z~_l' For j E Nko+1' we define qJ n ) to be the matrix in Mn taking the value 1 at diagonal positions (j - I)u + 1, ... ,ju and the value 0 elsewhere, whilst pen) is defined to be the matrix in Mn taking the value 1 at diagonal positions (k o + I)u + 1, ... ,n (in the case where v ~ 1). Then q~n) + ... + qk:~l + pen) = En. the identity matrix in Mn, and, for each i,j E Nko+l, there is a unitary matrix u~;) with u~':')*q(n)u~n) = q~n) For n < k0 define q~n) = pen) = u(,:,) = 0 for i , ]' E Nn' 13 ) I) I' ,) 0) Now set qi = (qi n) : n EN), P = (p(n) : n EN), and Uoj = (u~;) : n EN), so that ql + ... + qko+1 + p = eL and u.ijqjUij = qi. Since p is a special projection of bounded rank, we have 7r(p) = 0, and so {7r(ql), ... ,7r(qko+1)} is an orthogonal set of idempotents in Mko with kO+l
i=l By 1.3.19, necessarily 7r(q,) = 0 for somej E N ko +1' But now, for each i E Nko+l, We have 7r(qi) = 7r(uij )7r(qj)7r(Uij) = 0, and so 7r(qi) = 0 for i E Nko+1' Thus 7r(eL) = 0, as required. It follows that eL E M, and hence that M = Mk o' It is now clear that each maximal ideal of finite codimension in A has the specified form, and, in particular, each such ideal has an identity. The algebra .A has exactly one simple *-representation of dimension n for each n E N. 0 Let (H, [ ., .]) be a Hilbert space. As in Appendix 3, the strong operator topology on B(H) is defined by the seminorms T ....... IITxll for x E H, and the Weak operator topology on B(H) is defined by the seminorms T ....... J[Tx, y]1 for :t, Y E H. These topologies are denoted by so- and wo-, respectively; clearly the 8()..topology is stronger than the wo-topology. It is easily seen that B(H)[lJ is \IVa-compact. Let So, To E B(H). Then the inequality II(ST - SoTo)xll ~
IISIIII(T -
To)xll
+ II(S -
So)Toxll
(S, T E B(H), x E H)
(3.2.8)
364
Banach algebras with an involutzon
shows that the map (8, T) 1-7 8T, 8(H)[1] x 8(H) --+ 8(H), is so-continuous. The involution T 1-7 T* on 8( H) is wo-continuous, but it is only so-continuous in the case where H is finite-dimensional. For each n E N, the space H(n) = £2(N n , H) is also a Hilbert space. Theorem 3.2.31 Let H be a Hilbert space. (i) A linear functional on 8(H) is so-contmuous if and only if it is wocontinuous; each such functional has the form T 1-7 Ej=l[Txj, YJ] for some Xl.··· ,Xn,Yl,'" ,Yn E H. (ii) Let K be a convex set in 8(H). Then K (so) = K (WO) Proof (i) The functionals described are clearly wo-continuous, and hence socontinuous. Take). E (8(H), so)'. Then there exist Xl>"" Xn E H with
(T E 8(H». Define'IjJ : T 1-7 (TXl,"" Tx n ), 8(H) --+ H(n), so that 'IjJ is a continuous linear operator, and define A on 'IjJ(8(H)) by setting A('IjJ(T» = (T, ).). Then A is well-defined and linear; clearly, IIAII ~ 1. By the Hahn-Banach theorem A.3.19 and the Riesz theorem A.3.32(ii), there exists Y = (Yll"" Yn) E H(n) with A('IjJ(T» = ['IjJ(T), y]. Thus (T, ).) = Ej=l[TXj, Yj) (T E 8(H)). (ii) By A.3.19, this follows immediately from (i). 0 The following result is von Neumann's double commutant theorem. Theorem 3.2.32 Let H be a Hzlbert space, and let 2t be a unital C* -subalgebra of 8(H). Then meso) = m(wo) = 2t cc • O/(WO) Of CC P roof Clear1y O/(so) ~ C ~ C ~ . Let R E 2t CC and x E H, and set F = 2t . x; X E F because IH E 2t. Since PF E 2tc , we have RPF = PFR, and so Rx = RPFx = PFRx E F. Thus there exists 8 E 2t with II(T - S)xlI < 1. Now let T E 2t ('c and Xl,"" Xn E H, and set X = (Xl> ... , Xn) E H(n). The map 'IjJ : S 1-7 (8ij 8), 8(H) --+ 8 (H(n») ~ Mn(8(H» ,
is a unital *-monomorphism, and so, by 3.2.23(ii), 'IjJ(2t) is a unital C*-subalgebra of 8(H(n». An element (8ij ) of 8(H(n» belongs to ('IjJ(2t»C if and only if each 8ij belongs to 2tc , and so 'IjJ(T) E 'IjJ(2t) CC. By the result already established, but noW applied to the algebra 'IjJ(2t)CC, there exists 8 E 2t with II ('IjJ(T) - 'lj;(8»xlI < 1 in H(n). It follows that II (T - S)Xj II < 1 in H for each j E N n . This proves that T E meso). 0 A function f E C(lR) is strongly continuous if, for each Hilbert space H, we have J(T-y) ~ J(T) whenever T-y ~ T in 8(H)sa.
C* -algebras
365
Lemma 3.2.33 Each function in C o(1R.) is strongly continuous.
Proof Let A denote the set of strongly continuom; functions on JR.. Clearly A is a uniformly closed subspace of C(JR.) with 1, Z E A, and it follows from (3.2.8) that fg E A whenever f E An Cb(JR.) and g E A. Set fo(t) = t/C1 + t 2) (t E JR.), and let 8, T E B(H)sa. Then fo(8) - fo(T) = (IH + 8 2)-18 - T(IH + T2)-1 = (IH + 8 2)-1(8 - T + 8(T - 8)T)(IH + T2)-1. Let x E H, and set y = (IH + T2)-1X and z and II(IH + 8 2)-1811 ~ 1, we have
= Ty.
IIUo(8) - fo(T))xll ~ 11(8 - T)yll
Since II(IH
+ 8 2 )-111
~ 1
+ II(T - 8)zll ,
and so fo(Ty) ~ fo(T) whenever T-y ~ Tin B(H)sa. It follows that fo EA. Thus go = 1-Zfo E AnCo(JR.). Since the set {fo, go} ofreal-valued functions sep~rates the points of JR. and go(t) > 0 (t E JR.), it follows from the StoneWeierstrass theorem A.l.1O(i) that Co(JR.) cA. 0 We use the above lemma to establish Kaplansky's density theorem. Theorem 3.2.34 Let H be a Hzlberl space, and let 2l be a unital C* -subalgebra of B(H). Set s.B = ~(so). Then 2l[1J is so-dense in s.B[1J.
Proof First take T E s.B sa . Then there exists (T-y) c !.2l with T-y ~ T. We have * wo T ,and so T E""Sa o;-(wo) . B y 3231(") o;--(so) . T-y----t .. 11, T E~sa Next take T E (s.B",a)[lJ' Then there exists (T-y) C !.2lsa with T-y ~ T. Let / E Co(JR.) with f(t) = t (t E lI) and IflR = 1. By 3.2.33, f(T-y) ~ f(T). But -:-=--:-- (so) /(T) = T and U(T-y)) C (2lsa)[1J' and so T E (!.2lsa)[1J . The algebra M2(!.2l) is a C*-subalgebra of M2(B(H)) ~ B(H(2»), and clearly -(so) M2(2l) = M2(s.B). Let T E s.B[1J' and set
8 =
We have shown that 8
E
(~* ~) E (M2(s.B)sa)[1J'
(M2(2l)sa)[1J (so). By considering the (1,2)th positions
in the appropriate matrices, we see that T
E
!.2l[1J (so), as required.
0
We next define the most important subclass of the class of C* -algebras, the class of von Neumann algebras. nefinition 3.2.35 A C* -algebra A is a von Neumann algebra if there is a 8anach space E such that E' ~ A (as a Banach space). Let A be a von Neumann algebra. By the Krein-Mil'man theorem A.3.30(i), tlJcA[1J =1= 0, and so, by 3.2.28, A is unital. For example, Co is not a von Neumann algebra.
Banach algebra.'! with an involution
366
Let H be a Hilbert space. Then l3(H) ~ N(H)' is a von Neumann algebra. The weak* topology a(l3(H),N(H» specified by this duality is the a-weak topology on l3(H); temporarily it is denoted bya. Suppose that T"( ~ T in l3(H). Then, for each x, y E H, (x ® y, T"() -+ (x ® y, T), and so [T"(x, yJ -+ [Tx, yJ. Thus the identity map (l3(H), a) -+ (l3(H), wo) is continuous, and wo is weaker than a. The unit balll3(H)[lj is compact with respect to a, and so, by A.1.7, the topologies a and wo from l3(H) agree on l3(H)[lj' Let A be a unital C* -algebra, and let 1f' be a universal *-representation of A on a Hilbert space H. Then it is clear from 3.2.17 that the a-weak and wo topologies coincide on 1f'(A). Let H be a Hilbert space, let 2{ be a wo-closed, and hence a-closed. C*subalgebra of l3( H), and define mand (0 s~t) as in A.3.45. Then mis a closed linear subspace of N(H), and (Om)O = 2{a = m. Set E = N(H)/o2{, and let q : N(H) - ? E be the quotient map. By A.3.47(ii), q' : E' -+ (0 m) is an isometric linear bijection, and so m= E' is a von Neumann algebra.
°
°
°
°
Theorem 3.2.36 Let A be a unital C* -algebra with universal *-representatiort 1f' : A -+ l3(H). Then A is Arens regular, A" is a von Neumann algebra, and there is an isometric, unital *-representation 7f : A" -+ l3(H) such that 7f I A = 1f' and 7f(A") = 1f'(A)Cc.
Proof By 2.6.15, there is a continuous homomorphism 7f : (A",O) - ? l3(H) with !l7fH = 111f'1I and such that 7f extends 1f'; we recall the formula. Take x, y E H. Then Jy E H', and x . Jy E A' is defined by
(a, x . Jy) = (a . x, Jy) = [7r(a)x, y] and then [7f(4.»x,
yJ =
(a
E
A),
(4.>, x . Jy) for 4.> E A". For each a E A, we have
(a, (x . Jy), ,\) = 0 (,\ E SA), and so 4.> = O. Thus 7f : A" -+ l3( H) is an injection, and 7f : (A", a(A", A'» -+ (7f (A"), wo) is a homeomorphism. By 3.2.32, we have 7f(A") = 1f'(A)cC' = 1f'(A) (so). Take T E (1f'(A)CC)[lJ. By Kaplansky's density theorem 3.2.34, there exists 4.> E (A"hl] with 7f(4.» = T, and so 7f: (A", II'I!) -+ (l3(H), II'I!) is an isometry. Let CP, IJI E A". Suppose that aa -+ 4.> and b{3 - ? IJI in (A", a(A", A'». For each ,\ E SA, we have
(4.> 0 IJI,'\) = lim lim lim[1f' a {3 (aa b,8 , ,\) = lim a {3 (aab{3)x>. , x>.J = limlim[1f'(b{3)x>., 1f'(aa)*X>.J = [7f(IJI)x>., 7f(~)*x>.J a {3 = lim lim[1f'(b{3)x>., 1f'(aa)*X>.J {3 a
= (4.> 0 IJI,'\) ,
C* -algebras
367
and so .p D III = .p 0 Ill. Thus A is Arens regular. It follows that (A", D) is a Banach *-algebra, and that ::;r is a *-homomorphism, and so A" is a von Neumann algebra. D Corollary 3.2.37 Let A be a C· -algebra. Then A zs Arens regular, and A" zs a von Neumann algebra. Proof This follows easily by Ilsing 2.6.18 and 2.9.16(iii).
D
Definition 3.2.38 Let A be a C· -algebra. Then A" is the enveloping von Neumann algebra of A.
In the case where A = C(O) is a commutative C* -algebra, A" is also a commutative C* -algebra, and so, by 3.2.6, A" = C(n) for a compact space this algebra will be discussed further in §4.2. For example, the enveloping von Neumann algebra of Co is £00. Similarly, let A = £OO(N,Mn ) and f = co(N.Mn ), as in 3.2.30. Then f is a C* -algebra, and A is the enveloping von Neumann algebra of f. It is a theorem of Sakai that each von Neumann algebra can be represented as a wo-closed unital C*-subalgebra of I3(H) for some Hilbert space H. Thus a unital C* -algebra A is a von Neumann algebra if and only if there exist a Hilbert space H and a unital *-monomorphism 7r : A --+ I3(H) such that 7r(A) has any of the following equivalent properties: (a) 7f(A)CC = 7r(A) ; (b) 7f(A) is wo-closecl in I3(H); (c) 7f(A) is so-closed in I3(H). We shall not use this result.
n;
Proposition 3.2.39 Let A be a C*-algebra.
Then M(A) is a unital, closed
C· -sub algebra of A". Proof It follows from 3.2.21(ii), 3.2.37, and 2.9.50 that M(A) is a unital, closed subalgebra of A", and it is easily checked that M(A) is a *-subalgebra. D
Theorem 3.2.41, below, will be required in §5.6. In the proof, we shall appeal to some results from the standard theory of projections in a von Neumann algebra; these results are collected without proof in Theorem 3.2.40. Let R be a von Neumann algebra, and let a E Rand>. E R'. Then we write: Wa(R) = ({vav· : v E R, v·v = eR}); W>.(R) = ({v* .
>. . v: v
E R,
v·v = eR}).
Finite and properly infinite algebras were defined in 1.3.21. Theorem 3.2.40 Let R be a von Neumann algebra. (i) There are central projections p, q E R such that p
+q
= eR, such that
pR is a finite von Neumann algebra, and such that qR is a properly infinite von Neumann algebra. (ii) Suppose that R is a finite algebra. Then there is a bounded linear projection S of R onto 3(R) such that S(ab) = S(ba) (a, b E R) and such that S(a) E Wa(R) (a E R). D
Banach algebras with an involutwn
368
Theorem 3.2.41 Let R be a von Neumann algebra. (i) Suppose that R is properly infinite. Then 0 E Wa(R) for each a E R. (ii) There is a bounded linear map T : R -+ 3(R) with T(a) E Wa(R) (a E R) and T(ab) = T(ba) (a, bE R). (iii) Let .x E SR. Then there is a positive trace T in Rhl n W",(R) (0-), where a = a(R',R).
Proof (i) Take a E R; we may suppose that Iiall = 1. By 3.2.11, there is an orthogonal sequence (Pn) in Jsa(R) and a sequence (v n ) in R such that vnv~ = Pn and v~vn = eR for each n E N. Define 1
n
an = - L vjav; n
(n
E N),
J=1
so that (an) C Wa(R). It is sufficient to prove that lIanll -+ 0 as n -+ 00. Since a*a :::; eR, we have vJa*av; :::; VjV; = Pj (j E N) by 1.10. 11 (i). Let n E N. Then n
n
0:::; L vja*avj :::; LPj.
j=1
j=1
jll
By 3.2.3(iii), 111:}=I P = 1, and so, by 3.2.12(ii), 111:}=1 vja*av;1! :::; 1. For each j, k E Nn with j t- k, we have V;Vk = V;VjV;VkVZVk = V;PjPkVk = 0, and so n2a~an = 1:}=1 vja*av;' Thus lIa~anll :::; 1/n 2 , and lIanll :::; lin. Hence lIanll -+ 0 as n -+ 00, as required. We have shown that 0 E Wa(R). (ii) By 3.2.40(i), there are central projections p, q E R such that P + q = eR, pR is a finite von Neumann algebra, and qR is a properly infinite von Neumann algebra. By 3.2.40(ii), there is a bounded linear map S : pR -+ 3(pR) such that SCab) = S(ba) (a, b E pR), such that Sea) = a (a E 3 (pR» , and such that Sea) E Wa(pR) (a E pR). Set Ta = S(pa) (a E R). Then
T(a)b = S(pa)pb = pbS(pa) = bT(a)
(a, bE R) .
Clearly T(R) C 3(R), T(ab) = T(ba) (a, bE R), and Ta E Wa(pR) (a E R). It remains to see that Ta E Wa(R) (a E R). Take a E Rand € > O. Since Ta E =w:-a-'-(p-=R:-.'"), there exist OI, . . . ,am E (O,lJ with 1:7=1 aj = 1 and WI, ... ,Wm E pR with wjWj = p (j E Nm ) such that m
Ta -
L
ajwjpaw;
0, let V be a symmetric, compact neighbourhood of ee such that 11 - A(8)1 < c (s E V), and choose 9 E Coo with suppg c V, with g(S-I) = g(s) ~ 0 (8 E G), and with l(g) = 1. Then 11 - 0'1 = 11(g) - .l(g) I < c, and so 0: = 1. By A.4.9, Coo is dense in L1, and so t he result follows.
(iii) For a
E G, we have
1
1
f(sa) dm(s) =
e
G
f(s-la)A(8- 1 ) dm(s)
= [J(8- 1 )A(8- 1 a- 1 ) dm(8) = A(a- 1 ) [f(S) dm(s).
o
(iv) This is the same as (iii).
Recall from §1.2 that, for a function f on a group G, ](8) = f(8- 1 ) (8 E G). In general, the condition that f E LP(G) (for p E [1, does not imply that ] E peG), but this is true in the case where G is unimodular. Let G be a locally compact group with product rr : (s, t) 1-+ st, G x G ---T G. We now recognize M(G) as a Banach algebra. The technical lemma which underlies the definition ofthe product in M(G) is the following. Let J1" v E M(G), and let J1, x v be the Borel product of J1, and v on G x G, a..)} * IL, >.)
(f
E
L1(G), >.
E
LOO(G), IL
E
M(G).
(3.3.8)
Let G be a group. As in (2.6.8), set (os· f)(t) = J(ts) and (f . os)(t) = J(st) for s,t E G and f E CP. Thus f . 08 = 8 8 -,f = OS-1 * J for s E G and (f . os) * 9 = (f * g) . 08 whenever the convolution products are defined.
Group algebras
377
Let G be a locally compact group. First, suppose that A E U>O(G). Then Os . A, A . 8.. E L'X> (G), and these clements of UX) (G) are the same as the corresponding elem('llts defined by (3.3.8). Second, suppose that f E V(G), where p E [1,00). Then 08 • f, f . Os E £P(G) with 1/8., . flip = ~(S-l)l/p IIfllp and IIf . osllp = IIfli p. Further, we have the following important continuity result.
Proposition 3.3.11 For each p E [1,00) and f E LP(G), the maps S I--> 0.. . f, f . 88 , S I--> Os * f, and 81--> f * lis from G mto V(G) ar'e contmu()us.
S I-->
Proof First take g E Coo, say with suppg = K, and let U be a compact, symmetric neighbourhood of ea. For'f/ > 0, set
w = {s E G : Ig(ts) -
g(t)1 < 'f/
(t
E
K . Un.
Since K . U is a compact set, W ENe. Suppose that sEW n U. Then gets) = g(t) = 0 for t E G \ (K . U), and so 1I0s . 9 - gllp :::; (m(K . U»l/p'f]. Now take f E LP, So E G, and c > O. By A.4.9, there exists g E Coo with IIf - gllp < c. There is a symmetric neighbourhood V of ee such t.hat 118u . 9 - gllp < c for U E V, and ~(t-l) < ~(SOl) + 1 for t E V' . So. Then, for each t E V . So, we have
1I0t . f - 880 • flip:::; 118t . f - 8t . gllp
O. For each f E U(G), there exists V ENG such that lu . f - fiG for each u
E
Ll(G)+ wzth suppu
C
+ If
. u - fiG < c
(3.3.20)
V and fGu = l.
Proof (i) Since the maps a ...... Saf and a ...... sa f are continuous, there exists V E NG such that IISaf - flip < c/2 and lisa f - flip < c/2 whenever a E V. For each u E (L 1)+ with fGu = 1 and each 'If; E Lq, we have
I
Il(u *p f - f)(t)'If;(t) dm(t) = I i i u(t)(f(C 1s) - f(s))'If;(s) dm(t) dm(S)/
~l
u(t) ( l I(Sd - f)(s)'If;(s)1 dm(S)) dm(t)
~l
u(t) IISd - flip II¢II q dm(t)
by HOlder's inequality A.4.2(i), and so we have IIu *p f - flip < c/2 in the case where suppu C V. Similarly, IIf *p u - flip < c/2, and (3.3.19) follows.
(ii) For each f E U(G), there exists V ENe with 168 • f - fiG < c/2 and If· 68 - f/ < c/2 for s E V. For u E (L 1)+ with fGu = 1 and suppu C V, we have lu . f - fiG ~ sup If(st) - f(s)1 u(t) dm(t) < c/2,
r
8EGiG
and, similarly,
If . u - fiG < c/2,
giving (3.3.20).
o
'l'heorem 3.3.23 Let G be a locally compact group. Then there is a net (e",,) in Coo(G)+ such that e:'" = e"" in L1(G) and fG e"" = lIe",,1I1 = 1 for each a and such that (e",,) is a bounded approximate identity in Ll(G) for the Banach
Banach algebras wzth an involution
384
Ll(G)-bimodules (LP(G), *p) for p E [1,00), (Co(G), .), and (U(G), .). Each of these bimodules is a neo-unztal Ll (G) -bzmodule, and Ll (G) factors. Let G be a metrizable locally compact group. Then L1 (G) has a sequential bounded approximate identity.
Proof Let F be the family of compact, symmetric neighbourhoods of eG in G, and set F1 ~ F2 in F if F2 c Fl' Then (F,~) is a directed set. For each FE F, choose UF E with suppUp c F and fG Up = 1, and set VF = (up + u'F)/2, so that Vp E v} = VF, and fG VF = Ilvplll = 1. Take f E LP and c > 0, and let V ENG be as specified in 3.3.22(i). Choose Fo E F with Fo c V. Then, for each F E F such that F ~ Fo, we have IIvp *p f - flip + Ilf *p VF - flip < E, so that (vp) is an approximate identity for (LP, *p), being in particular an approximate identity for Ll. Similarly, (vp) is an approximate identity for (Co, . ) and (U, . ). That the specified bimodules are neo-unital follows from 2.9.31. In the case where G is metrizable, eG has a countable base of neighbourhoods, and so we can choose the net (eo,) to be a sequence. 0
ccto
ccto,
The approximate identity (e Q
)
of the above theorem was chosen to be in
Coo(G)+; equally, (eo,) could have been chosen in Ll(G)+ to have the form
XKo./m(KQ)' where (KQ) is a net of compact, symmetric neighbourhoods of ec;. It follows from 3.3.13(i) that RU(G) is a Banach left L1(G)-module for the module operation *. By a similar argument to the above, Ll(G) contains a bounded left approximate identity for RU(G), and so, by 2.9.29(i), RU(G) is a neo-unitalleft L1 (G)-module. Corollary 3.3.24 Let J.L E M(G). Suppose that J.L J.L=O.
*h=
0 (h E Coo(G)). Then
Proof Let (e Q ) be a bounded approximate identity for the bimodule (Co, . ), with (e Q) contained in Coo. For each f E Co, (e Q . f, J.L) = (f, J.L * eQ) = 0 by hypothesis, and so (f, J.L) = limo (eo' f, J.L) = O. Thus J.L = O. 0
In particular, M(G) is left (and right) faithful over Ll(G) in the terminology of 1.4.5. A linear subspace F of L1(G) is leJt-invarzant [right-invariant] if Sa(F) c F [sa (F) c F] for all a E G, and F is translation-invariant if it is both left- and right-invariant. Corollary 3.3.25 A closed subspace of Ll(G) is translation-invariant if and only if it is an ideal. Proof Suppose that F is a closed ideal in L1, and take (eo,) to be an approximate identity in L1. Then 8a * f = limo (8a * eo p(s)(g + I), G - All, is continuous, that each pes) is an isometry, and that the left regular representation of A on AI I is given by p(f)(g + 1)
=i
f(s)p(s)(g
+ I) dm(s)
(f, 9 E A).
There is an inner product on the finite-dimensional space AII such that each element of p( G) is a unitary operator; the space AI I with this inner product is a Hilbert space H. For f, 9 E A, we have p(f*)(g + 1) = i
f(S-I)6.(S-I)p(S)(g
+ 1)dm(s)
= if(s)p(s-1)(9+I)dm(s)
= i
f(s)p(s)*(g
+ I) dm(s)
= i (f(s)p(s»*(g + 1) dm(s)
by3.3.6(ii) because pes) is unitary = p(f)*(g
+ 1),
and so p is a *-homomorphism from A onto a C*-subalgebra of B(H). Since A has an approximate identity, 1= ker p and AI I ~ peA) by 2.9.2(ii). 0 Thus I is a *-ideal and All is *-isomorphic to peA). By 3.2.37, all C*-algebras are Arens regular; however, we shallllow show that the group algebras Ll (G) are not Arens regular unless G is finite. It will then follow from 2.6.18 that an infinite-dimensional group algebra is not the quotient of any C* -algebra. Theorem 3.3.28 (Young) Let G be an inJinzte locally compact group. neither Ll(G) nor M(G) is Arens regular.
Then
Proof Assume towards a contradiction that Ll (G) is Arens regular. By 3.3.23, LI(G) has a bounded approximate identity. By A.4.4, Ll(G) is weakly sequentially complete as a Banach space. Thus, by 2.9.39, Ll (G) has an identity, and 80 G is discrete (and infinite). It is immediate that there are subsets {xn : n E N} and {yn : n E N} of G fUch that Xl = YI = eG and Xi,Yj, =1= Xi2Yh whenever (il,jl) =1= (i2,h) in N2.
Banach algebras with an involution
386
The sequences (b xn ) and (b yn ) are II· II-bounded in .e 1 (G). Define h( x) = 1 if x = XrnYn for m, n E fit with m < n, and h(x) = 0 otherwise, so that h E .eOO(G). Then lim lim(bxm 8y", h) = lim lim h(xrnYn) = 1, n
m
m
n
lim lim(8xm 8yn , h) = lim lim h(xmYn) = 0, n
n
m
m
and so condition (f) of 2.6.17 fails. It follows that .e 1 (G) is not Arens regular, a contradiction. Thus L1 (G) is not Arens regular. By 2.6.18, the algebra M(G) also fails to be Arens regular. 0 Let 1 denote the function constantly equal to 1 on G, regarded as an element of LOO(G).
Definition 3.3.29 Let G be a locally compact group. Then the map
i{)o : f
t-+
fa
f(s) dm(s) = (1, 1),
L1(G) --de,
is the augmentation character on L 1 (G) and its kernel L5 (G) is the augmentation ideal of Ll(G). The augmentation character is indeed a character on Ll(G), and the augmentation ideal is a maximal modular ideal in Ll(G). It is immediately checked that
f . 1 = 1 . f = i{)o(f)l (f
E
Ll(G)),
. 1
=
1 .
= (, 1)1
( E L1(G)/I).
(3.3.21 )
Theorem 3.3.30 (Willis) Let G be a locally compact group. Then LA (G) factor'S weakly. Proof Define J = {/l EM: /l(G) = O}; clearly, J is a closed ideal in M and L1 n J = LA. Let f E LA. Since L1 has a bounded approximate identity, by 2.9.31 there exist u, v E L1 and 9 E LA with f = u * 9 * v. We have 9
= gl -- g2 + i(g3 -- g4)
for some gl,g2,93,g4 E (L1)+. Since fdg1 -- 92) = fR(JG9) = 0, there exists (\' E 1R such that fG gl = fG g2 = a. Similarly fG g3 = fG g4 = /3, say. Since be -- a- 1 g1 E J and iia-1g1ii1 = 1, by 2.4.1 with f = (1 -- Z)1/2 there exists /l1 E J with /ll * /ll = be -- a- 1 gl. Similarly there exist /l2,/l3,/l4 E J with /l2 * /l2 = be -- (\'-l g2 , with /l3 * /l3 = be -- /3-1 93 , and with /l4 * /l4 = be -- (J-lg4· We now have
9 = a(8e -- a- l g2) -- a(be -- a-lgl) + i/3(8e -- /3-lg4) -- i/3(be = a/l2 * /l2 -- a/l1 * /l1 + i/3/l4 * /l4 -- i/3/l3 * /l3 , and so 9
E
J2 and f
E
(u
* J) * (J * v)
C
(LA)2, as required.
--
/3-lg3)
o
It will be shown in 5.6.42 that LA( G) has a bounded (left or right) approXimate identity if and only if G is amenable (see 3.3.48, below); the following
387
G1'OUP algebras
significant extension of 3.3.30 applies for each locally compact group G. The proof of (i) is omitted; part (ii) follows from A.3.43 and 2.9.29(i). Theorem 3.3.31 (Willis) Let G be a locally compact group, and let I be a closed ideal of finite codimension in Ll(G).
(i) Suppose that G is a-compact. Then there is a closed left ideal J with bounded right approximate zdentzty and a closed right ideal K with bounded left approxzmate identity such that I = J + K; (ii) co(N, I) = co(N, I) . I
+I
. eo(N, I) and 12 = I.
0
Theorem 3.3.32 (Helemskii) Let G be a locally compact group. (i) The Banach algebra L1 (G) is left projective. (ii) The Banach algebra L1 (G) is biprojectwe if and only if G zs compact. Proof Set A = Ll(G). By 3.3.23, A is essential.
(i) By 2.8.37(i), we must show that 7l'A is a retraction in AB(A ®A, A). Take a compact set K in G with m(K) = 1, and, for f E A, define p(f)(s, t) = XK(C 1)f(st)
(s, t E G).
Then p(f) E Ll(G x G) with IIp(f)lll S Ilfll l , and p E B(A, A and s, t E G, we have
* g)(s, t) =
p(f
fa
®A).
For f, g E A
f(u)g(u-lst)XK(C l ) dm(u) = (f . p(g»(s, t)
by (3.3.16), and so p E AB(A, A®A). For f E A and s E G, we have f(s)
and so 7l'A
0
=
fa
f(S)XK(S-lt) dm(t)
= 7l'A(P(f»(S)
by (3.3.18).
P = lA. Thus 7l'A is a retraction in AB(A®A, A).
(ii) Suppose that G is compact. By 2.8.41(i), we must show that 7l'A is a retraction in ABA(A®A, A). For f E A, define (p(f»(s,t) = fest) (s,t E G). Thenp(f) E Ll(GxG) with IIp(f)111 S Ilflll' and p E B(A, A®A). Essentially as above (choosing K = G), we check that 71' A is indeed a retraction. Now suppose that A is biprojective, and let 'Po be the augmentation character of A. By 2.9.5, C<po is a projective Banach A-module, and so there exists a map p: C -+ A such that 'Po(fo) = 1 and 'Po(f)fo = f * fo (f E A), where fo = p(l). For each a E G and f E A, we have 'Po(f)fo
= 'Po(Saf)fo
and so SafO = fo (a 8()
E G).
= Saf
* 9 = Sa(f * g) = 'Po (f)Safo ,
Thus fo is a non-zero, constant function in A. and
G is compact.
0
Corollary 3.3.33 Let G be a compact group. Then db£1(G) S 2. Proof This is immediate from 2.8.56 and 3.3.32(ii).
o
388
Banach algebras with an involution
We now place one toe in the ocean which is representation theory for groups and group algebras. Let G be a locally compact group, and set H = L2(G), so that H is a Hilbert space with respect to the inner product [., .], where
[r.p, 1P] =
l
a}. Then E(JL) is a countable set, and IJLI (xK) = 0 whenever x E G \ E(JL). Let L E.c. We now define a linear functional AL on I by the formula (JL, A L ) = JL(K Certainly AL E I' with
(JL * v, A L)
= fa
F :n E
n L) =
IIALII S
faxKndx) dJL(x)
(JL E 1).
1. Let JL, v E I. Then
faXKnL(XY) dJL(x) dv(y)
= fa v(x-1(K n L»
For each x E G with X-I ¢ E(JL), we have Ivl (x- 1 K) countable set, and so IJLI (E(v)-I) = O. Thus
I(JL*v,AL)1 S I/ll (E(v)-I)
dJL(x).
= O. Also, E(V)-1 is a
IIvll =0.
We have shown that AL I 12 = O. Again take L = Km(ib"" i m ) E.c. We define a sequence (JLn : n ? m) of (positive) measures in I. Let n ? m and i E N~. We define JLn.i whenever Kn(i) c L to be the restriction to Kn(i) of mG divided by 4n - mmG(Kn (i», so that JLn,i(Kn(i» = 4m- n ; we know that mc{Kn(i» > 0 because intKn(i) "/0. We now define
JLn = L)JLn,i : i E N~, Kn(i) c L}
(n E N),
that each JLn is a positive measure with IIJLnll = Iln(G) = 1. Since G is nondiscrete, each JLn belongs to I. The sequence (I1n) ha.') a weak*-accumulation Point, say JLL, in M(G). We show that JLL E I. First let x E K n L. For each n ? m, there exists a llnique i E N~ with x E Kn(i). Take In E Co(G) with InCx) = 1, InCG) c 1I, and In(Kn(j» = {O} for each j E N~ with j "/ i. Then, for each r ~ n, we have fGlndJLr $ JLr(Kn(i)) $ 4m- n , and so JLLC{x}) S JGlnrlJLL $ 4m-n. This
80
392
Banach algebm.s with an involution
for each n 2: m, and so 11 d {x}) = O. Certainly 11 d {x}) = 0 for each Thus ilL E [. We finally prove that AL -I- O. Fix n E N. For each f E Co(G) ~uch that f(K.,) = {I} and f(G) c IT, we have (f,llr) = 1 (r 2: n), and so (f,IlL} = 1. ThlI~ IldU) 2: 1 for each open neighbourhood U of K m , and so l1,dK.,) = 1. Since (Kn : n E N) is a decreasing ~equence with n~=l Kn = K, it follow~ that IldK) = 1. Hence (ilL, A L} = 1, and so AL -I- O. Now let {Ln : n E N} be a pairwise di~joint ~ubfamily of .c. Then it is clear that (/1,L m, ADJ = 8m ,., for m. n E N, and HO {ADn + [2 : n E N} is a linearly independent ~et in [/[2. The result follow~. 0 hold~
x E
G\ (KnL).
We note that each measure I1,L that arises in the above construction belong~ to Mcs(G). We next identify the multiplier algebra M = M(Ll(G)) of Ll(G). Since Ll (G) is left and right faithful, M is a Banach algebra by 2.5.12(i), and, since Ll(G) has an approximate identity of bound 1, the natural embedding of Ll(G) into M is an i~ometry by 2.9.43(i). Theorem 3.3.40 (Wendel) Let G be a locally compact group. Then M(Ll(G)) ~s zsomet'f"lcally *-isomorphic to M(G). Proof Let (eo) be an approximate identity of bound 1 in L1. Let T E MR(Ll). By 2.9.30(ix), T E 8(L1). Then (Teo) is a net in M, and IITeoll ~ IITII for each a. By A.3.20, we may ~upppose that Teo --+ 11 in (M, (J) for some 11 E At with 111111 ~ IITII, where (J = a(Af, Co). Take 9 E Co and h E L1. We have Teo. * h = T(eD! * h) --+ Th in L1, and so
(g, Th} = lim (g, Tea a
* h} = lim (h
. g, TeIY.} = (h . g, /l}
01
= (g,
11
* h}
.
Thus Th = 11 * h (h E Ll) and IITII ~ 1IJ.l11. It now follows from 3.3.24 that the map 11 ~ L I " AI --+ M.e(Ll), is an isomorphism. By 1.4.26, the regular homomorphism 11 ~ (Lp., Rp.). 111 --+ M(L 1), is also an isomorphi~m; clearly this map is an isometry. Let (£. R) E M(LI), say (L, R)(f) = (11 * f, f * 11) (J E L1), where J.l. E M. Then (L, R)*(J) = (R*, L*)(J) = «(J*
* 11)*, (J.l * j*)*) =
and so the identification is a *-isomorphism.
(J.l*
* f, f * 11*)
(J
E [}),
o
From now on, we shall identify the multiplier algebra of L1(G) with M(G). The (two-sided) strong operator topology was defined in §2.9; we have 110 ~ Pif and only if 1101. * f --+ 11 * f and f * 110 --+ f * 11 for each f E LI(G). It follows from 3.3.11 that the map s ~ 8s , G --+ (M(G), so), is continuous. Clearly a so-closed subspace of M(G) is closed in M(G). By 2.9.16(ii), L1(G)" has a mixed identity, say q,o. As in 2.9.49(iii), there is an isometric embedding J.l ~ J.L • q,o = J.L ... , 8 m E G I with BI = lin {ql (OSI)' ... ,qI (o",J}. Similarly. there exist tI, .... tn E G 2 with B2 = lin {q2(OtJ ..... Q2(OtJ}. Take rEG = G I . G 2 . Then Or E lin {c5 8Jtk ,J : j E N m . k E N n }. and so £l(G) = lin {OSjtk,I : j E NIT" k E Nn }, giving the result. 0 {gl
There are certain special results which hold in the case where G is a compact group. Of course. there is an enormous literature on the very important topic of representation theory for compact groupSj we shall merely establish a form of the Peter -Weyl theorem that we shall require. Note that, for a compact group G, we have G(G)
c
LX!(G)
c
L2(G)
c
Ll(G) ,
11/111 ~ 11/11z (f E L2(G», and 11/11z ~ 1/1e; (f E G(G». By 3.3.4, the group G is unimodular, and. by 3.3.14, I * 9 E G(G) with II * 910 ~ 11/11z 11911 2 for 1,9 E L2 (G). In particular, (L2 (G), * ) is a Banach algebra. Definition 3.3.42 Let G be a compact 9rouP. Then I E G( G) is a representative function if lin {os * I : 8 E G} is a finite-dimensional space. The set of repr'Csentative functwns is R( G). It is clear that R(G) is a HeIf-adjoint, unital ~mbalgebra of G(G) (with respect to the pointwise product). We shall often write R for R(G). Lemma 3.3.43 Let G be a compact group, and take IE G(G). Then I E R(G) il and only if there exist n E Nand 91, ... ,9", hI, ... ,hn E G(G) with
fest)
" = "L,9j(8)h j (t) (8,t
E G).
(3.3.22)
j=I Proof Suppose that I satisfies the given condition. Then certainly I E R. Conversely, take fER·. Then there exist 81,.'" 8 n E G such that the set {681 * j, ... , OSn * f} iH a basis for lin{ Os * f : S E G}. For each 8 E G, there exist unique numbers 91(8) .... , gn(8) with OS-1 * I = I:,]=I 9j(S)Osj * f. Since the map S I-> Os * I, G -> L1, is continuous. the functions 91, ... ,9n are continuous. The function I satisfies (3.3.22) with h j = os) * I (j E N n ). 0
For IE Ll(G), define W f
= lin{6s
* f * Ot : s, t E G}, so that IE Wf'
Lemma 3.3.44 Let G be a compact group. (i) The following conditions on I E C(G) are equivalent: (a) W f is finitedimensional; (b) the space lin{J * 8t : t E G} is finite-dimensional; (c) I E R(G). (ii) For each I E R(G), Wf is a finite-dimensional ideal in M(G). (iii) R(G) is a *-ideal in M(G).
Group algebras
395
Proof (i) By 3.3.43 applied to G and to GOp, (b)-(c), and clearly (a)::::}(c). To show that (c)::::} (a), take a basis {hl, ... ,hn } of lin{ds * f : s E G}. For each j E Nn , we have hJ E R, and so the space Fj = lin{hJ * dt : t E G} is finite-dimensional. Since Wf C EJ=l F j , (a) follows. (ii) For fER, Wf is a finite-dimensional linear subspace of Ll, and so Wf is closed; Wf is translation-invariant, and so, by 3.3.25, Wf is an ideal in Ll, and hence, by 3.3.26, in M. (iii) Certainly R is a right ideal in M. Take fER as in (3.3.22). Then j*(st) = EJ=l hj(s)g;(t) (s,t E G), and so j* E R. Thus R is *-closed, and hence a left ideal. 0 Theorem 3.3.45 Let G be a compact group. For each f E Ll(G), the operator Tf : g t--t f * 9 on L2(G) is compact. Proof Set H = L2. First suppose that f E C, and take c > O. There exists U E NG such that If(s) - f(t)1 < c whenever s, t E G with srI E U. For IP E H[l) and such s, t E G, we have ITf(IP)(s) - Tf(IP)(t) I S
fa If(su- l ) - f(tu-l)lllP(u)1 dm(u) S c 111P111 S c.
Also ITf(IP)IG S IflG !llPill S IfI G· Thus the set Tf(H[l)) is pointwise bounded and equicontinuous in C, and so, by Ascoli's theorem A.1.10(iii), it is relatively compact in C. Hence Tf E K(H). Now suppose that fELl. There is a sequence Un) in C with fn --+ f in Ll. By 3.3.34, Tfn --+ Tf in 8(H), and so Tf E K(H). 0 Theorem 3.3.46 (Peter-Weyl) Let G be a compact group. Then R(G) is dense in Ll(G). Proof We shall first show that R is and 9 E C. Then (IP
II· 112-dense in
H = L2(G). Take IP E H
* g)(s) = fa lP(t)g(C l s) dm(t) = fa lP(t)(d.. * g*)(t) dm(t) = [IP, ds * g*].
Now suppose that 1P0 E H with [1P0, h] = 0 (h E R). Then 'Po * R = 0 by 3.3.44(iii). Set fo = lPo * 1P0, so that fo = fo E CeLl, and fo * R = O. By 3.3.34 and 3.3.45, T,o is a compact, self-adjoint operator in 8(H). Assume towards a contradiction that there exists ( E a (Tfo) \ {O}. Then F = {1/J E H : fo * 1/J = (1/J} is a non-zero, finite-dimensional space. We have Fe fo * L2 C C and F * ds c F (s E G), and so FeR by 3.3.44(i). For each 1/J E F, (1/J = fo * 1/J = 0, a contradiction. Thus a(Tfo) = {O} and v(Tfo) = O. By 3.2.3(i), Tfo = o. By 3.3.34, fo = 0, and hence 1P0 = O. It follows that R is dense in H. Now let fELl, and take c > O. There exists 9 E L2 with Ilf - gill < c. We have shown that there exists hER with IIh - gll2 < c, and then IIf - hill < 2c. -Thus R is dense in Ll. 0
Banach algebras with an involution
396
Theorem 3.3.47 Let G be a compact group. Then there is a family F of continuous, finite-dimensional representations of M(G) such that n{ker p: P E F} = {O}. Proof Let fER. By 3.3.44(ii), Wi is a finite-dimensional ideal in M. We denote by Pi the left regular representation of M on Wi. Take J.L E n{ker Pi : fER}. Then J.L * R = 0, J.L * L1 = 0 by 3.3.46, and so J.L = 0 from 3.3.24. Thus n{ker p: P E F} = {O}. 0 We shall also require a few basic results from another substantial area, the theory of amenable groups. Let X be a closed, left-invariant linear subspace of V>O(G), and take A EX'. Then 88 • A is defined on X for s E G by the formula
(A, 88
•
A) = (A . 88 , A)
(A
E
X) .
Definition 3.3.48 Let G be a locally compact group, let X be a closed, selfadJoint, left-invanant subspace of Loo(G) with 1 E X, and let A E (X,II·lloo)'. Then A is a mean on X if (1, A) = IIAII = 1, and A is left-invariant if 88 • A = A (s E G); a mean whzch is left-invanant is a left-invariant mean. The group G is amenable if there zs a left-invariant mean on Loo(G). Thus A is left-invariant if (A, A) = (A . 88 , A) (s E G, A E X), where we recall that (A . 88 )(t) = A(st) (s, t E G). The set of left-invariant means 011 X is denoted by £(X); we write £(G) for £(Loo(G». A mean on Loo(G) is right-invariant if (A, A) = (88 • A, A) (s E G, A E X), and mva1iant if it is both left- and right-invariant. Historically, the motivation for considering left-invariant means was the following. Let v be a finitely-additive function defined on the measurable subsets of G and taking values in IT, with v(G) = 1 and with v(N) = 0 for each locally null set N. Define A on the linear space 8 of measurable functions with finite range by setting
Then A is continuous on (8,11.11 00 ), and 8 is dense in Loo(G), and so A extends to a continuous linear functional on Loo(G). Clearly A is a mean on Loo(G), and each mean arises in this way. The function v is left-invariant (Le., v(s· V) = v(V) for s E G and V E B( G» if and only if A is left-invariant. A continuous linear functional A on Loo(G) is a mean if and only if (1, A) = 1 and (A, A) 2: 0 whenever A E Loo(G)+. ' Let X be as in 3.3.48. Then the set of means on X and the set £(X) are compact, convex subsets of (X', O"(X', X». Let G be a locally compact group. Then G is amenable as a discrete group if the discrete group Gd is amenable, so that there is a left-invariant mean all foo(G).
Group algebras
397
Proposition 3.3.49 Let G be an amenable locally compact group. Then there is an invarzant mean on G. Proof Set A = L I , so that A' = LOO and A" = Loci; we shall work in the Banach algebra (A", 0), where 0 denotes the first Arens product. First, we note that M . (A' of differentiable and Lipschitz functions in §4.4. 'Ve shall also develop the theory of the Stone-Cech compactification of a completely regular space in §4.2. In §§4.5 -4.7, we shall discuss algebras in which the product is originally defined by convolution, considering the case of the group algebra L1 (G) of a locally compact abelian group G in §4.5, the case where the underlying semigroup is Z or Z+ in §4.6, and the case where it is lR. or lR.+ in §4.7. In particular, we shall consider various radical convolution algebras in §4.6 and §4.7. In §4.8, we shall develop the basic theory of prime ideals in general commutative Banach algebras and, especially, in the algebras Co(H). In §4.9, we shall give a classification theory for commutative, radical Banach algebras developed by J. R. Esterle; this cla.. 0 and an approximate identity (eo,) for M", in J", with sup lie", II < m",. For each compact subset K of
0
lIallp = lI allp/2::; L IIfj(3),,(j) II j=l
and so
lIallp::; IIlallln,p.
DC
p/2
::; L 11/3(;)11 1I,(j)11 ' P
j=l
P
Suppose that p E [1,2). Then DC
n=1
Ilall l
::;
f:: 11f3(J),(i) III ::; f:: Ile(j) 1121I,(j) 112 ~ f:: 11j3(j) lip 11,,(j) lip' 3=1
J=1
1=1
and lIalll ::; Illallln.p. The result follows from (2.1.15). (ii) The followmg are equivalent: p = 1; fP has the S-property; t P has the 7r-property. This is immediate from (i): II· lip is equivalent to (iii) fP is weakly amenable, but not amenable.
II· lip if and only if p =
1.
That tv is weakly amenable follows from 2.8.72(i) because CP is spanned by its idempotents; fP is not amenable because it does not factor. (iv) Each finite-dzmensional f'xtenszon of fP splits. Clearly each maximal ideal in (fP)# has an approximate identity, and so this follows from 4.1.41.
(v) For p E (1. Xl), there eX'lst one-dimensional, annihilator extenswns of fP which do not spbt strongly; each finite-dimensional extension of f 1 splzts strongly. If p > 1, then £P does not have the S-property, and so, by (iv) and 2.8.19, there is a one-dimensional, annihilator extension which docs not split strongly. The algebra £1 does have the S-property, and so, by (iv), 2.8.19, 2.8.13(i), and 2.8.28(iii), every finite-dimensional extension splits strongly. (vi) For p E [1,00], M(£P) is isometrically isomorphzc to £00.
For each a E eoo , we have Let E M(£P) with IIL",II ::; laiN" Conversely, take 'I' E M(£P). For each kEN, we have T(ok) = akok for some O'k E C with IQkl ::; IITII. Set a = (ak). Then a E £00 and laiN::; IITII. Also, T = Lo.. (vii) £v is biprojective if and only if p = 1. If p > 1, then £v does not have the 7r-property, and so £v is not biprojective.
424
Commutative Banach algebra8
In 2.1.24(i), we identified e1 ®e1 with fl (N x N). In this case, the continuous linear map p : (nn) f--> (nn8(n.n»' (1 ----4 (I(N x N), is a right inverse for the projective induced product map 1f : fIg, e 1 ----4 e 1, and so, by 2.8.41 (i), e 1 is biprojective. (viii) Let C be the annzhzlator e2 -module. Then N 2 (C 2 ,q is not closed zn Z2 (t 2. q, and 1{2 (e 2 ,q -::j:. {O}. For each kEN, define ).(k): (nn) and 81 ).(k) E N2(e 2, q. Define
f-->
~ nnf3n JL(n,f3 ) = ~ n 1/ 2 71=1 Then J-l E Z2(£2, q, and 81).(k)
1I(J-l-81 ).(k»(n,{j)ll:::;
f
n=k+l
----4
-L~=18n/nl/2, so that
(a./1
).(k)
E (f2)'
2
E e ).
J-l in 8 2(£2, q because
':~1;':::; (k+ 11)1/2 IInll 2 11f3'b
(n,/3Ee 2 ).
Assume that J-l = ( 1). for some). E «(2)'. Then )'(8k) = _1/k 1 / 2 (k EN). a contradiction of the fact that ()'(8",» E e 2 • Thus J-l ~ N2(e 2 ,q. The result follows. In fact, by 2.8.21 and (ii), 1{2(fP, q -::j:. {O} for p > 1. (ix) Let p, q E [1, (0) wzth p :::; q. Then the quotient algebra t q /£P zs normable if and only if q :::; 2p. If q :::; 2p, there i8 an algebra norm
III· III
on e q such that
II,all' = IIalip (a E ep). ' SUPPOHC that q :::; 2p. Then (e q )[2 j = t q / 2 c i p • and HO f q /e p has zero multiplication, and hence is normable. By (i) and 2.1.28. there is an algebra norm ",, "' on f2p with the required restriction, and eq c e2p when q :::; 2p, HO that III . III is the required norm OIl f q. Suppose that q > 2p, and take c > 0 such that 2p(c + l/q) < 1. Let (Sn) be a sequence of disjoint subsets of N such that L{l/k : k E Sn} is divergent for each n E N. and set (k E Sn), (k E N\ Sn). Then (an) is an orthogonal sequence in eq, and (an) C ane p. To obtain a contradiction. assume that e q /e p is normable. By 2.1.3(ii), a;' E e p eventually. But, for each n E N, a~ ~ ep by the choice of Sn. (x) For' each p > 1, 1{2(fP,e q ) -::j:. {O}, where q is the conjugate zndex to p, and so e p zs not simplicially tnvzal. Since £1 C p, there is a natural embedding of 8-(CP,Cq) into 8-(Cl,COO),
e
in the notation of (2.8.2); the connecting maps in the complex 8-(CP,Cq) are denoted by 80,). Set Ep
= 8 2 (fP, fP; q n 8 2 (£P, t P ; q.
Then Ep is a Banach space, and there are natural, injective, norm-decreasing embeddings 8 2 (CV, q --+ Ep --+ 8 2 (lP, q.
AIgebms of functwns
425
Take J.l E Ep- Then, by (2.8.3), 8tl)J.l E B3(f P, q, and 8tl)J.l E ker8l1)' and so 8t1)J.l E ker8lp)' Suppose that 8t1)J.l = 8tp)v for some v E B 2 (f P , q. Then v = J.l because ker8t1) = {O} by (iii). Thus, to prove the result, it suffices to find J.l E Ep such that J.l is not the restriction of a continuous bilinear functional on fP
x
fP.
First, fix p E (3, (0), choose 8 E ((p - 3)/p, (p - 2)/p). and define
J.l(a, 8) =
t an~n
(a E
n
n=l
t P , 13 E
fP/2).
By the Cauchy-Schwarz inequality, 2::=1 IXnYnznl :::; IIxliru /lyl/rv I/z/l s whenever l/r+ lis = 1 and l/u+ l/v = 1. Take a E fP and!3 E f P/ 2 , and apply this with Xn = l/no, Yn = an, Zn = 13n, r = p/(p - 2), s = p/2, u = (p - 2)/(p - 3), and V = P - 2. Since p8/(p - 3) > 1, we have (l/nO) E t p /(p-3), and so
1J.l(a,p)l:::; 11(1/nO)ll p /(P_3) I/al/ p 11 8 11 p /2
.
Thus J.l E B2 (fP. [15; q. Similarly, J.l E B 2 (f P, fP; q, and so f.1 E Ep. Now choose E: E (l/p, (1 - 8)/2), and set an = 13" = line (11, EN). Then a,13 E f P , but the sum
diverges because 8 + 2E: < 1. and so J.l has no continuous extension to fP x Second, fix p E (2,3], and define
fP.
00
n=1
Then 1J.l(a, (3)1 :::; lIa/l p /l13/1 q :::; /la/l p 1113l1 p/2' where we note that p/2:::; q because J.l has the required properties is now as above. Third, fix p E (1,2] (so that q ~ 2), choose 8 E (l/q, 2/q). and define
p :::; 3. The argument that
J.l(a,fJ) = Set C =
L { Im-nl a m13n ° : m,n E N, m =f. n}
(22::=11/nO q )1/ q ; C
(a E f P , iJ E
f1).
is finite because 8q > 1. Then
00
1J.l(a, (3)1 :::; C lIolip
L
Ifin I = C lIali p 1113111
(a
E f P,
fi
E f
1),
n=l
and so J.l E B2(f P ,f P;q; a.. before, JL E Ep. Now choose E: E (l/p. 1- 8/2), so that 8 + E: > 1, and define a = 13 = (l/n e ). Then 0,13 E fP. However 00
L
n=m+l
1
1m _
nlo
1 ne
00
1
~ n=m+l L no+ e ~
1
1
(8 + E: - 1) (m + l)o+e-l
Commutative Banach algebras
426
for each mEN, and so we have '"
~
m#n
oman
1m - nl8 2:
~
(~
1
~ me
~
m=l
m=II+1
1 1) 1 1m _ nl8 n e 2: (8 + € - 1)
~
~
m=l
1
(m
+ 1)8+2e-1
The latter series diverges because (j + 2€ -] < 1. Thus It has no continuous ext(>nsion to f P x £ p • In each case, we have found a It E Ep with the required properties. 0 Example 4.1.43 (Dixon) Let U, X, and Z be Banach spaces. each a copy of f 1. and set A = U x X x Z as a Banach space, with lI(u. x, z)11 = IIuI1 1 + Ilxll l + Ilzll1; we regard U. X, and Z as closed subspaces of A. We defiue a product in A. We begin by specifying a sequence (Zr) in Z. For each r E N. Ilzrlll = 1 and Zr ~ £1/2; further, the supports of the elements Zr are pairwise disjoint. The product of two elements x, y E X is xy E Z, and the product of 1./" v E U is ~ u(k)v(k) ~ k Zk, k=l
so that Zr E A2 (r EN). All other products are O. so that XU = U X = 0 and AZ = ZA = O. Clearly A2 C Z, A3 = O. and A is a commutative, separable Banach algebra. Trivially, each element Z E Z can be expressed as 1::1 XiYi with Xi, Yi E X and 1::1 Ilx;III IIYilll = IlzlII' and so Illzllltr = IIzII 1 and A has the 7r-property. Now take r E N, and suppose that Zr = 1:~1 XiYi + 1:';=1 UiVj, where m. n. E N, Xi, Yi EX (i E N m ), and u)' 1Ij E U (j E Nn ). Set x~ = Xi I supp Zr and = Yi I SUppZr for i E Nm . Then
Y:
Zr
~
=
I
I
~xiYi i=l
r
and so Ilzrlltr 2: r the S-property.
=
~
uj(r)vJ(r)
J=l
r
+~
n
II
j=l
j=l
L lLj(r)Vj(r) ::; L
Zr·
IIujl1111 vjll1 '
= r'lIzrlll' This is tlUe for each r
E N, and so A does not have 0
Example 4.1.44 Consider the space bv of sequences a = (an) such that 00
lIall bv = sup lanl nElIi
+ 2: lak+! -
akl
0 and a sequence (a(n») E Coo such that (a:(n) , >.} > e and 113 = 1 for each n E N and such that distinct elements of this sequence have disjoint supports in N. Define {3 = L:~=l a(n) In; clearly the sum converges in [00. We calculate N({3). Let P = (PI, ... ,Pk) E P, and consider the sum for N({3,p). A term apjl may be non-zero either because {p"Pj} has nonempty intersection with the support of at most one element a(n), or because the two elements Pi and PJ belong to the supports of two distinct elements, say a(m) and a(n). In the first case, the sum of the contributions to 2N({3,p)2 is at most L:~11/k2. In the second case, the sum is at most
Ila(n)
lap; -
00
(1 1)2
(Bonsall and Duncan 1973, §23), (Gamelin 1969), (Hewitt and Ross 1970, §39), (Ric-kart 1960. §§3.2, 3.7), and (Stout 1971, §7), for example. Note that an algebra A may be a Banaeh function algebra on a compact space X such that A is a uniform algebra, without being a tmiform algebra On X. Most authors require that a uniform algebra on a compact space be unital. but this is not necessarily the case with our definition. For let M = {f E A(iD» : 1(0) = O}, and set A = M I '['. Then A is a non-unital uniform algebra on '['. The notion of an abstract Segal algebra in 4.1.8 is developed from (Burnham 1972); see also (Reiter 1971) and (Reiter and Stegeman 2000, §§6.2, A.3). Theorem 4.1.10 was shown to me by F. Ghabramani. In the ~iterature, a regular Banach algebra is sometimes termed completely reg~lar; the term Silov regular is also used because these algf'bras were first studied in (Silov 1947). The concept of complete regularity also applies to general, non-commutative algebras; see (Palmer 1994, §7.2). The tf'rm strongly r-egular was introduced (in the COntext of uniform algebras) in (Wilken 1969). Proposition 4.1.17 is due to Albrecht (1982); the present elementary proof is takpn !rom (Neumann 1992). Further discussion of the maximum regular subalgebra is given In (Laursen and Neumann 2000, §4.3). It is an interesting open question whether this 8Ubalgebra necessarily coincides with the closed subalgebra consisting of the functions in A which are continuous on 4> A with respect to the hull-kernel topology.
Commutative Banach algebms
432
Theorem 4.1.19 is from (Bade and Curt.is 1966), and Proposition 4.1.21 extends a result of (Bade and Curtis 1960a). Proposition 4.1.30 is the localization lemma of (Hewitt and Ross 1970, 39.21) and (Reiter and Stegeman 2000, 2.1.8); 4.1.38 is from (Bade and Dales 1992) and 4.1.39 is (Reiter and Stegeman 2000, 2.6.12), where it is attributed to Katznelson. The latter result implies that there are at least c closed ideals K of A with I eKe J, for there is clearly a closed ideal K t for each dyadic rational t E H such that Ko = I, Kl = J, and Ks C K t (s < t), and we set Kt
=
n
{K. : s dyadic rational with s ~ t}
for each t E H; {Kt : t E H} is the required family of closed ideals. It will be proved in 4.3.10 that a unital, normal uniform algebra on a compact space is natural. However, an example of a unital uniform algebra A on a compact space X which is regular on X, but which is not natural, is given in (Hoffman 1962). A normal Banach function algebra on a compact space is not necessarily natural; see 4.5.34. Proposition 4.1.41 is from (Bade et al. 1999, §4), where more general results are proved. Clauses (ii), (iv), and (v) of Example 4.1.42 are from (Helemskii 1964, 1970) and (Johnson 1968). clause (ix) is from (Esterle 1978d), and clause (x) is developed from (Aristov 1993). Example 4.1.45 is built on the famous James space .:J from (James 1950); see (Megginson 1998, §4.5). The results related to the structure of .:J as a Banach algebra are from (Andrew and Green 1980), where the automorphism group of .:J is identified. Examples 4.1.43 and 4.1.46 were shown to me by P. G. Dixon and J. F. Feinstein, respectively. 4.2
ALGEBRAS OF CONTINUOUS FUNCTIONS
Our first specific examples of Banach function algebras are the algebras CoCO) for a locally compact space OJ by the commutative Gel'fand-Naimark theorem 3.2.6, these algebras are exactly the commutative C* -algebras. We shall develop the theory of the Stone-Cech compactification j3X of a completely regular space X, and of the second dual C(O)" of C(O), where 0 is compact. We shall also make a few further remarks about general C* -algebras. Some topological terms that are used are defined in Appendix 1. Theorem 4.2.1 Let 0 be a non-empty, locally compact space. (i) Co(O) is a natuml, regular uniform algebm on o. (ii) For each closed subset S of 0, the ideal 1(8) ofCo(O) has an approX'tmate zdentzty of bound 1 contamed m J(8). (iii) Let 1 be an ideal m CoCO) with hull 8. Then J(8) C 1 c 1(8); if 1 is closed, then 1 = 1(8). (iv) Spectml syntheszs holds for CoCO), and CoCO) is strongly regular. (v) CoCO) is a strong Dztkin algebm and has bounded relative units. (vi) The multzplier algebm of CoCO) is isometrically isomorphic to Cb(O). (vii) C(lI) is unitally polynomially genemted by {Z}, and C(T) is polynomially genemted by {Z, Z-l}. Proof (i) Certainly CoCO) is a regular uniform algebra. Now suppose that 0 is compact. Then C(O) satisfies the conditions specified in 4.1.5(ii), and so C(O) is natural. The general case follows from 4.1.11(i).
Algebras of continuous functwns
433
(ii) The family IC of compact subsets of n which are disjoint from 8 is a directed set with respect to inclusion. For each K E 1C, take IK E J(8) with fK(K) C {I} and IfKlo ~ 1. Then II - IKlin --+ 0 for each I E I(8), and so (fK : K E 1C) is the required bounded approximate identity. (iii) Since Co(n) is regular, it follows from 4.1.20(iv) that J(8) C I C I(8). By (ii), J(8) = I(8), and so I = I(8) in the case where I is closed. (iv) and (v) These are immediate from (ii) and (iii). (vi) Set M = M(Co(f2». Clearly each I E Cb(n) defines Lf E M with IILfl1 ~ 1/10.· Conversely, let T E M. Then T = Lf for some I E Cb(n) with 1/10. ~ IITII· (vii) This is the Stone-Weierstrass theorem A.1.lO(i), where we note that the polynomials in Z and Z-1 on '][' are just the trigonometric polynomials. 0 Theorem 4.2.2 Let n 1 and n 2 be non-empty, compact spaces. Then C(n 1) and C(f22) are isomorphic ~f and only zf f21 and n 2 are homeomorphic. Proof Let 1] : f21 --+ n 2 be a homeomorphism. Then we see immediately that the map I f--+ I 0 1], C(n 2 ) --+ C(f2 1), is an isomorphism. Let () : C(f22) --+ C(n 1) be an isomorphism. Then (}X : 0 (x EO); (c) Co(H) has a seq'uentzal bounded approximate identity.
Then the
Proof Suppose that n is cy-compact, say (Kn) is a compact exhaustion of n. For each n E N, there exi!lts en E CoCO) such that e.. (n) ell, cn(Kn) c {I}. and cn(O \ Kn+r) = {OJ. Then (en) is a sequential bounded approximate identity for CoCO). Set I = ~:'=1 c nl2n. Then I E CoCO) and f(x) > 0 (x EO). ThuS (a)=?(b) and (a)=?(c). Suppose that I E Go(H) and I(x) > 0 (x EO), and set Kn = {x En: II(x)1 ~ lin}
Then each Kn is compact, and
n = U:'=l Kn
(n E N).
is cy-compact. Thus (b)=?(a).
Algebms of continuous functions
439
Suppose that (en) is a sequential bounded approximate identity for CoCO), and set Kn = {x EO: len(x)1 2: 1/2} (n EN). Then each Kn iH compact. Take x E 0 and f E CoCO) with f(x) = 1. Then cn(x)f(x) ---> f(x) as n ---> 00, and so en(x) ---> 1. It follows that x E Km for some mEN, and so 0 = U~l Kn. ThuH (c)=?(a). 0
Proposition 4.2.15 Let 0 be a a-compact. locally compact, non-compact space. Suppose that cp E PO\O, that f E C(PO), and that cp E Zn(f). Then there exzsts V E /V'" such that f 1 (V \ 0) = o. Proof The space 0 has a compact exhaustion, Hay it is (Kn). For each n E N, set Fn = Kn n Zn(f) and
Un = {x E intKn +1 : If(x)1 < lin}, so that Un is an open neighbourhood of F," and take gn E C(PO) such that 9n(PO) C [0. 2-n). gn(Fn) C {2- n }. and gn(PO \ Un) = {OJ. Set 9 = L:=l gn' Then 9 E C(,BO) , and Zn(g) n Zn(f) = 0. By 4.2.8, Zn(g) n Zn(J) = 0; set
ao \
V = Zn(g), so that V E /Vcp. Take '1/) E V \ O. For each mEN, there exists Xm E (0 n V) \ Km+l with If(x m ) - f(1/') 1 < 11m. For n E Nm , gn(X m) = 0 because Xm rf: K m+1 • but g(x m ) > 0, and so there exists n > Tn with 9n(X m) > 0: we have Xm E Un, and so If(xm)1 < lin < 11m. Thus If (1/1) 1 < 21m (m EN). It follows that f(1/I) = 0, and so f 1 (V \ 0) = 0, as required. 0
For the definition of an F-space, sec A.1.11 (vi).
Proposition 4.2.16 Let 0 be a a-compact, l()(,Ally compact, non-compact space. Then PO \ n is an F -space. Proof Let U and V be open F".-sets in PO\O with UnV = 0. Since ,a0\0 is compact, U and V are a-compact. Since un V = 0, it follows from 4.2.14 that there exists f E Co(U U V, JR) with f( 0 (cp E U) and f((c) Let I EM"'. Since JR. is a metric space, there exists 9 E C(JR.) with Z(g) = clJR(JR. \ Z(f». We have t1JR. = Z(f) U Z(g). Assume that c.p E Z(g), so that 9 E M.p, and set h = /1/2 + /9/ 2 • Then hE M'P, and
Z(h)
= Z(f) n Z(g) = aRZ(f) ,
a contradiction of (b) because intJRa]RZ(f) = 0. Thus I E J'" because we have established that r.p E t1JR. \ Z(g) C int,BRZ(f). (c)=>(a) This is immediate. (d)=>(b) Assume that there exists I E M'" with intJRZ(f) = 0. For each co~ponent interval (a, b) of JR. \ Z(f), take (xn) and (Yn) in (a, b) with Xn --+ a and Yn --+ b. The union of these sequences forms a discrete subset D of JR., and Z(f) C clJRD. Thus c.p E Z(f) c D, and so r.p is not a remote point of t1JR., a contradiction of (d). (b)=>(d) Assume that there exists a discrete subset D ofJR. with r.p E D. Since JR. has no isolated points, intRclRD = 0. Take 9 E C(JR.) with Z(g) = clJRD. Then r.p E Z(g), so that 9 EM"', but intJRZ(g) = 0, a contradiction of (b). Thus r.p is a remote point of t1lR.. (c)=>(e) Since M'P is a prime ideal in C(JR.), J'P = J'PnC(t1JR.) = M'PnC(t1JR.) is prime. (e)=>(d) Assume that r.p is not a remote point of t1JR., say r.p E D, where D is a discrete subset of JR., and take I E Cb(JR.) such that, for each tED, 1(8) = s - t on a neighbourhood of t. Then 1+1- E J'P' but 1+ (j. J", and 1- (j. J'P' and so J", is not a prime ideal, a contradiction of (e). 0 Corollary 4.2.28 (CH) The space t1JR. contains remote poznts. Proof For each f E C(JR.), set Uf = X \ ax (Z(f) n X). Then Uf is a dense, Open subset of X, and so. by 4.2.22, there exists r.p E n{Uf : I E C(JR.)}. Clearly
r} : f E F}; by hypothesis, each Ur is open in 0, and clearly U 1 = 0. Define
g(x) = sup{r ElI: x E Ur } ElI. If g(x) E (r, s), then x E Ur \ Us, and, if x E Ur \ Us, then g(x) E [r, s]. Let Xo E 0, and take a neighbourhood V of g(xo) in R. Then there exist r, s E R with g(xo) E (r, s) C [r, s] C V. Since Ur \ Us is open, Xo E Ur \ Us, and Ur \ Us C g-l([r, s]) C g-l(V), we see that 9 is continuous at Xo. Thus 9 E C(O,lI). Let h E C(O, 1I) with h ~ f (f E F). Assume that there exists Xo E n with h(xo) < g(xo). Then h(xo) < r for some r with Xo E Ur . Let W be a neighbourhood of Xo with h(x) < r (x E W). Then there exists x E W with f(x) > r for some f E F, and so r < h(x), a contradiction. Thus h ~ 9 and
9 = supF. (ii) As in §3.2, therE' is a faithful representation 7r of C(O) onto a wo-closed subalgebra, say 21, of B(H) for some Hilbert space H. By 3.2.9(i), the map 7r: C(O, R) - t B(H) is isotonic. Let F be a family in the lattice C(O)+ such that F is bounded above by 1; we may suppose that f V 9 E F whenever f, 9 E F. Let 9 be the directed set of all finite subsets of F (where 9 is ordered by inclusion); for each G E Q, set fe = sup G. Then (7r(fG) : G E Q) is a net in 21 n B(H)[l). Since B(H)[l) is wo-compact, this net has a wo-accumulation point of the form 7r(fo), where 10 E C(n)~r Fix x E H, and set r = [7r(fo)x, xl. Assume that there exists I E F such that s > r, where s = [7r(f)x, xl. Then there exists G E 9 such that lEG and [7r(fG)x, xl < s, a contradiction because I ::; IG. Thus 10 ~ I (f E F). Also, for each s < r, there exists I E F such that [7r(f)x, xl > s. Thus, if 9 E C(O)+ is such that 9 ~ I (f E F), then [7r(g)x,xl ~ r. It follows that 10 = supF in C(O)+. By (i), 0 is extremely disconnected.
445
Algebras of contmuous functzons
(iii) Let By 3.2.36, the C*-algebra C(O) is Arens regular, and its second dual
C(O)" is naturally identified with the von Neumann algebra C(O)CC C B(H), where C(O) has its universal representation on a Hilbert space H. The C*algebra C(O)" is commutative, and so, by (ii), it has the form C(O) for a certtin extremely disconnected compact space O. 0 We now identify an important C*-subalgebra of C(O); as in Appendix 3, we denote by Bb(n) the algebra of bounded Borel functions on 0, so that Bb(O) is a C*-subalgebra of «(00(0), I· In)· For f E Bb(O), define w(J) by
(w(J), /L) =
l
f d/L (/L E l\1(n)) .
Clearly w(J) E M(n)'.
Proposition 4.2.30 Let 0 be a non-empty, compact space. Then the map
W:f
1--+
w(J),
Bb(O)
-+
C(O)" ,
is an isometric *-isomorphzsm zdentifymg Bb(O) as a closed C* -subalgebra of C(O).
Proof Clearly the map W is linear and II wII ::; 1. For each x E 0, we have (w(J), 6x ) = f(x), and so W is an isometry. For f, 9 E C(O) and /L E M(O), we calculate that
(w(J)
0
w(g), /L) = (w(J), w(g) . /L) =
In fgd/L = (w(Jg), /L),
and so W is a homomorphism; W is a *-homomorphism with closed range.
0
We conclude our present study of the algebra. 0, choose a finite subset Fo of n such that IO:xl < c (x E 0 \ Fo). Take finite subsets FI, F2 of 0, each containing Fo, and set K = FI D. F2. Then. by A.3.68, we have
Ilx~, "x'x
0
'x - x~, "x'x 0
,1 ~ Ill;"xu ,1 C(02) is naturally identified with a proper dense subalgebra of C(Hl x H2) called the Varopoulos algebm (Helemskii 1989b, 11.2.50); there is an isometric isomorphism of C(Ol) ®C(02) onto C(H1 XH2) (Palmer 1994, 1.10.21).
Uniform algebras 4.3
447
UNIFORM ALGEBRAS
Let n be a locally compact space. Uniform algebras on n, as defined in 4.1.1, are closed algebras of (eb(n), I· Ix ) which separate strongly the points of n. In this section we shall define various standard uniform algebras, and we shall examine their character spaces, the existence of bounded approximate identities in maximal ideals, and the theory of point derivations on these algebras; the key notion is that of the Choquet boundary.
Definition 4.3.1 Let A be an algebra of functions on a topological space X. (i) A subset S of X is a peak set for A if there exists f E A such that f(x) = 1 (x E S) and If(y)1 < 1 (y E X \ S); the functwn f peaks on S. (ii) A poznt x E n is a peak point for A if {x} is a peak set, and x is a strong boundary point for A if, for each U E N"x, there exists f E A with f(x) = If Ix = 1 and Iflx\U < 1; w'Nte So(A) for the set of peak points for A. (iii) A subset S of X zs a boundary for A zf, fOT each f E A, there exists xES with If(x)1 = If I· (iv) The intersectwn of all the closed boundaries for A is the SHov boundary for A, denoted by r(A). Clearly each peak point for A is a strong boundary point for A. Suppose that x E X is a strong boundary point for a Banach function algebra A and that {x} is a Go-set. Then x is a peak point for A: if {x} = n:'l Un, where Un E N"x (n EN), choose Un) in A so that fn(x) = Ifnln = 1 and Ifnln\U" < 1, and set f = L~=l fn/2n Ilfnl!. so that a multiple of f peaks at x. A peak point belongs to each boundary for A, and a strong boundary point belongs to each closed boundary for A, and hence to the SHov boundary. It is easy to see that x E r(A) if and only if, for each U E N"x, there exists f E A with {y EX: If(y)1 = Iflx} c U. and it follows that rCA) = r (A), where A is the uniform closure of A in eb(X). We shall see in 4.3.7 that the set rCA) is non-empty and is itself a closed boundary in certain cases. Let A be a unital uniform algebra on a compact space n. Recall from (2.3.1) that KA = {A E A' : IIAII = (1, A) = I}; we have KA = (exKA), where the closure is taken in the weak* topology. By the Hahn-Banach theorem A.3.16(i) and the Riesz representation theorem AA.lO(ii), for each A E KA there exists JL E M(n) such that (I, A) = In f dJL U E A) and IIJLII = 1; as in 3.2.14, JL is a probability measure on n. Such a measure JL is a representzng measure for A on n. In particular, for each x E n, we have ex E KA, and ex has the representing measure 8x • the point mass at x; in general, ex has other representing measures. Lemma 4.3.2 Let A E exKA. Then there f'.xiBts x E unique representing measure for A on n is 8:r.
n
such that A = ex. The
Proof Let JL be a representing measure for A on n. For each Borel set E C with JL(E) f/. {a, I}, we have A = JL(E)AI + JLcn \ E)A2' where (f, AI)
=
JL(~) Ie f dJL
and
(f, A2) = JL(n \ E)
k'E f
dJL
n
Commutative Banach algebras
448
for f E A. Since A1,A2 E KA and A E exKA, we have A = Al
Ie f
dp, = /-teE)
10 f d/-t
= A2, and so
(J E A) .
This implies that each f E A is constant almost everywhere with respect to /-t, and so J.L = Ox for some x E O. Thus A = ex. The argument also shows that Ox is the unique representing measure for x on O. 0 In the case where A is the uniform algebra C(O), we have KA = SA by 3.3.7, and cI>A C exKA by 1.1O.22(ii), and so, by the lemma, cI>A' = exKA: the pure states on C(O) are exactly the characters ex: f I---> f(x) on C(O).
Definition 4.3.3 Let A be a umtal uniform algebra on a non-empty. compact space. Then the Choquet boundary, ro(A), of A is exKA . Proposition 4.3.4 Let A be a unital umform algebra on a non-empty, compact space O. Then ro(A) cO is a boundary for A. Proof Let f E A, and take yEO with If(y)1 =
K
= {A
E
KA :
(J,
A)
/fIn.
Set
= f(y)}.
Then ey E K, and so K is non-empty. Also K is a compact, convex subset of (A',CT(A',A)) and so, by the KreIn-Mil'man theorem A.3.30(i), K has au extreme point, say Ao. But then AO E exKA, for, if AO = tAl + (1- t)A2' where Al,A2 E KA and t E II, then
If In = 1(1, Ao)1 :::; t 1(1, A1)1 + (1 - t)
I(J,
A2)1 :::; If In
'
and so (J, AO) = (j, AI) = (J, A2), whence A1, A2 E K and AO = Al = A2' Thus, by 4.3.2, there exists x E ro(A) such that ex = AO and If(x)1 = /fIn. 0
Theorem 4.3.5 Let A be a unital uniform algebra on a non-empty, compact space O. Then the followmg conditwns on x E 0 are equzvai1;nt: (a) x E ro(A); (b) the unique representing measure for ex on 0 is ox; (c) there exist a, /3 wzth 0 < a < /3 < 1 such that, for each U E N'x, there exists f E A wzth /fIn:::; 1, f(x) > /3, and Iflow < Ct.; (d) x is a strong boundary point for A; (e) Mx has a bounded approximate identzty.
Proof (a)::::}(b) This is contained in 4.3.2. (b)::::}(a) Suppose that AI, A2 E KA with ex = (AI + A2)/2, and let /Jo1 and J.L2 be representing measurE'J> for Al and A2, respectively. Then (J.L1 + J.L2)/2 is a representing measure for ex, and so o:r = (J.L1 + J.L2)/2. Since J.L1 and J.L2 are positive measures, J.L1(E) = /Jo2(E) = 0 for each Borel subset E C 0 \ {x}, and so J.L1 = J.L2 = Ox and Al = A2 = ex. Thus ex E exKA . (c)::::}(b) Assume towards a contradiction that v is a representing measure for ex on 0 with v f= ox, say v( {x}) = c, where c < 1, and set J.L = (v - cox )/(I- c),
Unilorm algebras
449
so that J.l({x}) = O. For each U E N x , there exists I E A with I(x) > {3, and 1110\U < 0', and so (3 < I(x)
=
1110
= 1,
r I dJ.l = 1 I dJ.l+ r I dJ.l ::; J.l(U)+aJ.l(X\U) = a+(l-a)J.l(U). u lo\U
io
Thus J.l(U) > ({3 - a)/(l - a), and hence J.l({x}) ~ ({3 - a)/(l - a) > 0, a contradiction. This proves that 8x is the unique representing measure for ex. (b)=*(c) For h E C(O, R), set
p(h) = inf{lRI(x) : I
A, lRl
E
~
h}.
Then p is a sublinear functional on C(O, lR). Fix u E C(O, R). The linear functional .A : au f----t ap(u) is defined on lRu. By the Hahn-Banach theorem A.3.16(i), .A extends to a linear functional A on C(O,lR) with (h, A) ::; p(h) (h E C(O, lR». For each h E C(O, lR), we have
(h, A) = -(-h, A)
~
-pC-h) = sup{lRI(x): I E A, lRI::; h}.
In particular, (h, A) ~ 0 if h ~ 0, and so, by the Riesz representation theorem A.4.lO(i), there is a positive measure J.l on 0 such that
(h, A) =
In hdp,
(h
E
C(O, lR».
It follows that lRI(x) = Io(lRf)dJ.l (f E A), and this implies that J.l is a representing measure for ex. By (b), J.l = 8x . Since (u, A) = -p(-u), we have
u(x) = sup{lRI(x) : I
E
A, lRl ::; u}.
(4.3.1)
Take any a, {3 with 0 < a < (3 < 1. By Urysohn's lemma A.1.2(i), there exists u E C(n, lR) with u ::; 0, u(x) = 0, and u(y) < loga (y E n \ U). By (4.3.1), there exists 9 E A with lRg ::; u and lRg(x) > 10g{3. Set 1= expg. Then I has the required properties. (c)=*(d) Let a and {3 be as specified in (c). First choose r < 1 with
1 - {3 (3-a
r 1-r'
-- j), and so Ifj(y)1 -s; 1 (j E N) and If(y)1 -s; 1. Thus f(x) = If In = 1 and Iflnw -s; 0./13, and so x is a strong boundary point.
(d)=>(e) For each U E N x , there exists fu E A with fu(x) = Ifuln = 1 and Ifulnw < 1. Consider the set B = {I - flJ : U ENe, n EN}. Clearly B C Mx and Igln -s; 2 (g E B). Take h E Mx and E > O. Then there exists U E N x with Ihlu < E. Choose n E N with IflJluw < e/(Ihln + 1), so that If{}hln < e. We have shown that B is a bounded approximate unit for Mx. By 2.9.15, Mx has a bounded approximate identity. (e)=>(c) Let (j,,) be an approximate identity of bound m for Mx. and set a = 1/2(1 + m) and 13 = 1/(1 + m). For each U E Nx , there exist functions gl, ... , gk E Mx with the property that, for each yEn \ U, there exists j E Nk with Igj(y)1 > 1. Choose Vo so that Igj - f"ogjl < 1/2 (j E Nk). and set f = (1- f"o)/(l +m), so that f E A, If In -s; 1, and f(x) = 13. For each y E fl\U, we have I(giI)(y) I < 1/2(1 + m), and so If(y)1 < o.. Thus (c) holds. 0
Corollary 4.3.6 Let A be a umtal uniform algebra on a non-empty, compact space fl, and suppose that x E n is a strong boundary point. Then there are no non-zero point derivatwns at x. Proof By the theorem, Mx has a bounded approximate identity, and hence At; = Mx by 2.9.30(i). 0 Corollary 4.3.7 Let fl be a non-empty, compact space. (i) Let A be a unital uniform algebra on fl. Then ro(A) = rCA), and r(A) is a closed boundary for A. (ii) Let A be a unital Banach function algebra on fl. Buppose that fl zs metrizable. Then Bo(A) = rCA). Proof (i) By 4.3.4, fo(A) is a boundary for A, and so fo(A) is a closed boundary. Since rCA) contains every strong boundary point for A, we have f(A) = fo(A). (ii) Fitst suppose that A is a uniform algebra. Since fl is metrizable, each strong boundary point is a peak point for A, and so ro(A) = Bo(A). ClearlY each boundary for A contains Bo(A), and so Bo(A) is the minimum boundary for A.
451
Umform algebras
In the case where A is a general unital Banach function algebra on n, take Xo E r(A) and Uo E N xo ; we ::;hall show that Uo n So(A) I- 0. There exist::; go E A with 1901n = 1 and 19010Wo < 1/2. By the first remark, So (::4) is a boundary for A, and so there exists Xl E uonSo (A) with 190(Xl)1 = 1: we may suppose that 90(Xl) = l. Take U l E NeLl with U l C Uo. There exists !I E::4 with
1!Ilo =
and
!I(xd = 1.
Choose gl E A with 19110Wl < 1/2, with 91 (Xl) = 1. and with 19l1n < 2, and set G 1 = 90 + 1]191 E A, where '7111911\ E (0,1). We have IG 1 (x)1 efinition 4.3.8 Let A be a umtal uniform algebra on a non-empty, compact Bpace n, and let
A. A probabilzty measure J-t on n is a Jensen measure for Ip if log I 0 and u E P. and so P is a convex cone in E. Suppose that u E N and that IflO: -< exp u for some f E A. Then If I -< 1, and hence Icp(f) I < 1, so that 'U f/. P. Hence N n P = 0. By the Hahn -Banach theorem A.3.17(i), there exists A E E' with "All = 1 and (f, A) < 0 :::; (g, A) (f E N, g E P); A is a positive functional on E, and so, by A.4.lO(i), there is a probability measure j.t on n with (h, A) = In hdj.t (h E E). Take f E A with cp(f) = 1. For each c > 0, the function log(lfl + c) belongs to P, and so In logOfl + c) dj.t 2: O. Thus In log If I dj.t 2: O. By applying this inequality to f jcp(f) in the case where O. Since A is normal on n, there exists f E A with f(F) = {I} dnd f(G) = {a}. Then
In 1)1 :::; In
log Icp(f)1 :::; log Icp(1 and so both cp(f) =
a and cp(f) =
log If I dJl
=
-00,
log 11 - fl dj.t =
-00,
1, a contradiction. Thus A is natural.
0
We now consider some standard uniform algebras on compact subsets of en.
en.
Definition 4.3.11 Let K be a non-empty, compact subset of Then: (i) P(K) is the set of functions on K which are the uniform limits of the restrictions to K of polynomials; (ii) R(K) is the set of functions on K which are the uniform limits of the restrictions to K of rational functions of the form pjq, where p and q are polynomials and a ¢ q(K); (iii) A(K) is the set of continuous functions on K which are analytic on intK.
Uniform algebras
453
Clearly each of P(K), R(K), and A(K) is a uniform algebra on K, and P(K) c R(K) c A(K) c G(K). The algebra P(K) is unitally polynomially generated by {Zl •... , Zn}. We shall determine the character spaces of these algebras in certain cases. For a compact set K c cn. the polynomiaily convex hull of K is
R=
{z
E
c n : Ip{z)1 ::; IplK
(p E
qx 1 , ... , Xn])} ,
and K is polynomzally convex if R = K. It will follow from 4.3.12(ii), below, that the present usage of the notation R coincides with that in Appendix 1 in the case where K c C. Proposition 4.3.12 (i) Let K be a non-empty. compact set in C n . Then
R=
P(K) =
O"p(K) (Z1' ... , Zn).
(ii) Let K be a non-empty, compact set m Co Then R zs the union of K and the bounded components ofC\K, and r(P(K)) = av, where V 'tS the unbounded component ofC \ K.
(iii) Let K be a non-empty, compact set in Co R(K) zs natural, and r(R(K)) = DK.
Then the uniform algebra
Proof (i) By 2.3.30(iii), P(K) = O"p(K)(Z1, .. " Zn). By 1.6.17, each character on the polynomial algebra qx1 •... , Xn] has the form Cz for some z E cn, and ez extends continuously to P(K) if and only if Z E R.
(ii) By 2.3.21(iv), O"p(K)(Z) is the union of K and the bounded components of C \ K; by (i), O"P(K)(Z) is the polynomially convex hull of K. Clearly r(p(K)) c avo Take Zo E av and c > 0, and set U = ]]J)(zo; 2c). Choose (1 E ]]J)(Zo:c) n V, and then choose (2 E K with 1(2 - (11 = d«(],K), so that (2 E U. Define f = (Z - (J}-1. so that f E P(K) by 2.4.4. We see that IflK = If«(2)1 > Ifl Kw ' and so Zo E r(P(K)). (iii) Clearly O"R(K)(Z) = K. Let tp E R(K)' and set Z = 1. it is difficult to describe R for K C 1(:"; for example. polynomial convexity is no longer a topological invariant. It is a long-standing open question whether or not P(J} = C(J) for each polynomially convex arc J in I(:n; see (Stout 1971, §30) for some partial results. This is closely related to thE' famous question whether therE' is a natural uniform algebra on II other than C(ll). Proposition 4 3.14 is Arens's theorem (Arens 1958b). (GamE'linI969. 11.1.9), (Stout 1971, 24.6). In the case where K is a compact subset of 1(:" for somf' n > 1. q) A(K) need not be homE'omorphic with a subspace of 1(:" . For Swiss cheeses. see (Stout 1971, §24), in particular. The implication (b)~(c) in 4.3.17(i} is (Browder 1969, 3 a.11), and the other implications follow from 4.3.5, 4.3.17(ii} then follows from (Gamelin 1969, II.ll.4). Example 4.3.17(iii) is in (WermE'r 1967), and t.he basic example 4.3.17(iv) of a normal uniform algebra A with A oF C(q)A} is due to McKissick; for a simplification of the clas~ieal proof (Stout 1971, 2743), see (Korner 1986). The infinite-order highf'r point derivation of 4.3.17(iv) W8.'> constructed by O'Farrell (1979). Explicit conditions involving analytic capacity for there to be a non-zero, continuous point derivation at a point of K on R(K) are given in (Hallstrom 1969). For a study of HOC(llJ» as a Banach algebra, see (Hoffman 1962. Chapt.er J 0) Thpre is a short. proof, due to Wolff, of Carleson's corona theorf'm 4.3.19(i) in (Gamelin 1980). The analogous question when the disc llJ) is replacE'd by llJ)n for n 2: 2 is opf'n. Thf' point
(A.II·II) such that 17(0) = r.p and f 0 17 E O(lIJ)n) for ea('h f E A. Let n be a non-empty. compact space, and lE:'t A hC' a natural uniform algpbra on H snch that spectral synthesis holds for A. It is not known whether or not necc.."lsarily A = C(H). It is also unknown whether or not a normal uniform algf'bra must be strongly regular at each peak point.
4.4
ALGEBRAS OF DIFFERENTIABLE AND LIPSCHITZ FUNCTIONS
The next Banach function algebras that we shall consider arc algebras of differentiable functions on compact intervals I of R First we shall study the a.lgpbra 0(71) (1) of n-times continuously differentiable functions, and then we shall introduce some algebras D(1; (MIr )) of infinitely-differentiable functions; these latter examples have been used as counter-examples to a number of conjectures. We shall also consider related algebras consisting of Lipschitz functions on a compact metric space and of absolutely continuous functions and of continuous functions of boundE'd variation. Let I be a compact interval of JR, and let n E N. As in A.3.72, 0(71) (1) d(,Ilotm; the Banach space of all n-times continuously differentiable functions on I with respect to the norm II· lin' Also 0(:>0)(1) = n{c(n)(1) : n EN}. We shall often write o(n) and O(oc) for o(n) (II) and O(oc)(II), respectively.
Theorem 4.4.1 Let n E N. Then (o(n) (1), II· lin) is a natural, T'f'guiar, selfadJomt, 1tnztal Banach lunctwn algebra, unitally polynomzally generated by {Z}. Proof It is easily checked that o(n)(1) L" a Banaeh algebra with respect to the norm II· lin ' and certainly the algebra 0(71) (1) is self-adjoint and regular on [. Let IE o(n)(1), and take c > O. By the Stone-Weierstrass theorem A.1.W(i), there exists a polynomial q with I/(n) - qlI < c. Fix a E I, and let p be the polynomial with p(7I) = q and p(k) (a) = I(k)(a) (k E Z~_l)' Then III - plln < CE for a constant 0, and so the polynomials are dense in o(n)(1). Hence o(n)(1) is unit ally polynomially generated by {Z}. Since a(Z) = I, it follows from 0 2.3.30(iii) that 0(71)(1) is natural on I.
Algebms of different~able and Lipschitz functwns
459
Definition 4.4.2 Let n E N and to E I. Thf'n, for k E z;t ,
(j E
Mn,k(tO) = {J E e(n) (I) : J(j)(to) = 0
ztn.
Each Mn,k(tO) is a closed ideal in e(n)(I), and
Mn.n(to) c Mn.n- 1 (to) c ... C Mn,l(tO) C Mn,o(to) with dim(Mn.k(to)/Mn,k+l(tO)) = 1 (k E Z~_l); the ideal Mn,o(to) is the maximal ideal at to. Similarly, we now write In(to) for the ideal of functions vanishing on a neighbourhood of to, so that In(to) C Mn,n(to). In particular, e(n)(I) is not strongly regular. We write Mn,k for Mn,k(O) and I n for In(O). For each k E Z+ and any function J which has a kth derivative at 0, we write
8k (J) =
~!J(k)(O).
Clearly (15k : k E z;t) satisfies the condition on (d k : k E z;t) in equation (1.8.15), and so (15k : k E z;t) is a higher point derivation of order n on ern) at O. Further, 80 , ... , 8n are continuous on e(n). Each J E e(1I) has a Taylor expanswn: for each tElL there exists St E [0, tJ such that 11-1
J(t) = ' " 8j (J)t j ~
+ ~ J(n)(St)t n ,
(4.4.1 )
n!
)=0
where IJ(n)(St) - f(n)(o)l- 0 as t - 0+, and so, for each k E z;t. we have k
J
k
= I: 8j (J)Zj + RkJ = I: 8j (J)Zj + Zk . SkJ,
(4.4.2)
j=O say, whcre RkJ E Mn,k and tj(Sd)U)(t) f E Mn,k, where k E Z~_l' then
IfU)(t)1 = O(tk+1- j and, if
JE
)
as t
as t - 0+ for j E
--+
0+
(j E Zt+l) ,
--+
0+
(j
zt.
Thus, if
(4.4.3)
M 1I .n , then
IJU)(t)1 = o(t n- j ) as t Let k E z;t and
Jg =
--+ 0
J, 9
E
Z~).
E M 1I .o. Thcn
k+1 (i_1 ) 2k ( k t; ~ 8 (J)8i - (g) Zi + i~2 iEk 8 (J)8 j
(4.4.4)
J
j
i- j
) (g) Zi
k
(4.4.5)
+ "f)8;(J)(Rkg) + 8i (g)(RkJ))Zi + (R,J)(Rkg). i=l We shall use the fact that, in the case where a function h E e(ll) is differentiable on (0,1] and h'(t) --+ C as t --+ 0+, then h E e(l) and h'(O) = C. The analysis of the algebras ern) will be assisted by consideration of a certain auxiliary algebra An, which we now define.
Commutative Banach algebras
460
Definition 4.4.3 Let n E N. Then An is the set of n-times continuously differentzable functions 9 on (0, 1] such that
tkg(kl(t)
->
0
as t
->
0+
(k
E Z~).
For 9 E An, set
Again it is easily checked that (An, 111·llln) is a Banach algebra, and it follows immediately from 4.1.5(ii) that Aft is natural on][, and so An is natural on (0,1]. The map is a linear homeomorphism by the above consequences of Taylor's formula, and so there are constant.s k n and Kn such that Illf /znliin
:s kn 11/11n' II/l1 n :S Kn IIII/Znlii n
(f
(4.4.6)
E Mn,n)·
It follows that AIn,T! is an An-module: if I E An and 9 E Mn,n, then Ig E Mn,n, and (4.4.7)
za
> 0, and so ZOI is a
The real-valued function belongs to An for each a multiplier on Mn,n' Fix an increasing function 'I/J E C(ool(lR+) with
'I/J(t)
=1
:s 1/2),
(t 2 1) and 'I/J(t) = 0 (D:S t
and set Nk = I'I/J(kll n (k E Z+). For each c > 0, define 'l/Jg(t) Then Wg I [0, e/2] = 0, Wg I [e, 1] = 1, and we have
IW~kl(t)1 :s ~:
and
= 'I/J(t/c) (t
Itk7/J~kl(t)l:s Nk
E IT).
(4.4.8)
for k E Z+ and t E IT.
Lemma 4.4.4 The net (7/Jg : c > 0) zs a bounded approxzmate identity m (An, III'IIITJ and an approximate identity in (Mn,n, II· lin) contained in I n . Further, (zn1/Jg: e > 0) is bounded in (c(n l , II· lin)'
:s
Proof By (4.4.8), IIIWgili n "£;=0 Nk/k!, and so (Wg) and (zn7/Jg) are bounded in (AT" 111·lll n) and (c(n l , II· lin)' respectively. For 9 E An and k E Z;i, we have k
IZk (g(k l - (7/Jgg)(k») In
:s IZk (g(k l _7/Jgg(k l ) In + ~ k
::;IZkg(kll
+ Nk [O,EJ
L
j=l
(~) J
G) IZk7/J~jlg(k-jt
IZk-jg(k-jll
by (4.4.8), and so 1119 -7/Je911In -> 0 as e -> 0+. Take f E Mn,n' By (4.4.6), 7/Jd -> f in (Mn,n, I/·I/n)'
[O,gJ
0
Algebras of differentiable and Lipschitz functions
461
Theorem 4.4.5 Let n E N and to E II. (i) Suppose that k E Nn and f E M n,k-1(tO). Then
If(k-m)(t)1 ~ It - tolm If(k)
II
(t E II, mE
zt)·
(ii) In(to) = Mn,n(to). (iii) Each closed ideal J of c(n)(ll) with ~(J) = {to} has the forn~ Mn,k(t o) for some k E z;t . (iv) Mn,n(tO)2 = M".n(tO)[2] = (Z -to1)nMn.n(to); for each sequence (fk) zn eo(N, Mn,n(to)), there eX'lst 9 in Mn,n(to) and (hk) in eo(N, Mn,n(to» such that
(Z - to1)n /k = ghk
(k E N) .
(v) For k E Z;'_l' Mn.k(tO)2 = M n,k(tO)[2] = (Z - to1)k+l Mn,k(tO)' (vi) Mn,O(tO)2 = {f E Mn.l(tO): f(n+l)(to) exists}; if g,h E Mn,o(to)' then
(gh) (n+l) (to) =
t
(n; l)g(r) (to)h(n+l-r) (to) .
(4.4.9)
Proof We take to = 0 for convenience. (i) Apply (4.4.1) with f replaced by f(k-m) and n by m. (ii) This is immediate from 4.4.4. (iii) By (ii) and 4.1.20(i), J::) Mn,n, and so J = Mn,k for some k E
z;t.
(iv) By 4.4.4. An has a bounded approximate identity, and so these results follow easily from 2.9.29(i). (v) Since Zk+l E Mn,k, we have Zk+lMn .k C J\;[!;~. Now take Jr, 12 E M n.k , and set h = Jr12/Zk+1, ~ith h(O) = O. For i = 1,2, we have fi = E?=k+1 OJ (fi)Zi + Rnfi' in the notation of (4.4.2). By (iv) , RnJr . Rn12/Zk+ 1 E Mn,n C Mn,k, and so h E )\;[n,k. (vi) Define K = {f E M n,l : f(n+1) (0) exists}. Take g, hE M.".o, ann. set 1 = gh. Then, for t E (0,1]' we have
j(n)(t) =
tu
(;)g(r) (t)h(n-r)(t)
= get) (h(n)(o) + 0(1») + (g(n) (0) + 0(1») h(t)
+ as t
-+
~ (;) (g(r) (0) + g(r+1)(0)t + o(t») (h(n-T') (0) + h(n-r+l) (O)t + o(t»)
0+. Thus
L(n)(t)
~ l(n)(o) = g~t) +
(h(n) (0) + 0(1») + (g(n) (0) + 0(1»)
h~t)
~ (;)(g(r+l)(O)h(n-r)(o) + g(T) (O)h(n+l- T)(0») + 0(1),
462
Commutatwe Banach algebras
and so
10
L 8r (g)8 +!-r(h)
as t - ? 0 + . n r=l We have shown that f E K and (4.4.9) holds. Now take f E K, and set 9 = f/Z, with g(O) = O. We shall show that 9 E Mn,o: we may suppose that f E AIn,n' For k E z;t, we have
(f(n)(t) - f(n)(O»/t
9
-?
(k) _ f(k) Z
k (-l)jk! f(k- j ) Zj+1 .
+?= (k - J.)! 3=1
By (4.4.4), g(O) = '" = g(n-1)(0) = O. By successive applications of L'Hopital's rule, f(n-j)(t)/t j +! - ? f(n+1) (O)/(j + I)! as t - ? 0+ for.i E z;t, and so 9 E e(n) with g(n) (0) = o:f(n+1)(o) for some a E~. (By (4.4.9), a = l/(n + 1).) Hence 9 E Mn.o and f E ZMn,o = M~.o. 0 Corollary 4.4.6 (i) Each contznuous point derivation on c(n) at to has the form f 1--+ of'(to) for some a E Co (ii) There are discontinuous point derwations on ern) at to. Proof (i) It follows from 4.4.5(vi) that Afn ,o(to)2 = .l\1n,l(tO)'
(ii) Set M = Mn,o(to). For a E (0,1), define feAt) = It - toln+o d, then
Ie.
(t E I). TheJl E D, and, if d > 0
465
Algebras of differentiable and L-ipschitz functions
and r < d, then fe; fj; D. Thus O'D(Z) = {z E C : d(z, 1) ~ d}. By 2.3.21(iv) and 2.3.30(iii), if>Dp = O'D(Z). 0 Let D = D(I: Uvh» and d = d(M,,), as above. Suppose that d> O. For each fED and tEl, the series 1 L ,f(k)(t)(z - t)k 00
k=O
k.
converges uniformly on the closed disc D(t; r) for each r < d, and the sums agree with f(s) for s E I with Is - tl < r, and so together the sums define an extension of f which is analytic on int O'D(Z), In the case where f E D p , f has an extension in A (if> D p ) • The result about polynomial approximation follows from a certain 'dominated convergence principle', 4.4.14, which we shall first establish. Set D = D(I; (Mk». For k E Z+ and tEl, define
Llk(t) : f ....... f(k\t),
C(k)(I)
C.
---+
Regard Llk(t) as an element of D'; we claim that lim -hI (Llk(t + h) - Llk(t» = Ll k+1(t)
h--+O
Indeed. for each fED with
in (D',
II·ID.
(4.4.10)
Ilfll ~ 1,
I(f' Llk(t + h~ - Llk(t) _ Llk+ 1 (t») =
1 1~(I(k)(t + h) -
f{k)(t) - hf(k+l)
(t»1
~ ~ Ih llf(k+ 2 >ir ~ ~Mk+2Ihl , and so (4.4.10) follows. For k E Z+ and A E D", set Ak(t) = A(Llk(t». Since A is continuous, it follows from (4.4.10) that A~ = A k+1, and so Ak = A6k). Thus
P: A ---+ Ao,
D"
---+ c(OO) (1)
,
is a continuous linear map. (In fact, P is a projection of norm 1 onto D.)
Let (In) be a sequence in D. Then (In) is (Mk)-dominated if there exists ({3k) C ~+ with ""£';'=0 {3k/ M k < 00 and IfAk)II ~ {3k (k E Z+, n EN). Proposition 4.4.14 Let (In) be a sequence in D. Suppose that (In) ~s (lVh)dominated and that fn ---+ f pomtwise on I. Then fED and fn ---+ f in D. Proof There exists ({3k) with {3 = ""£~o{3k/Mk < 00 and IfAk)II ~ {3k' We have IIfnll ~ {3 (n EN), and so Un : n E N} may be regarded as a bounded set in D". Let S be the set of O'(D",D')-accumulation points of the sequence (In), and, for each A E S, let (Ak) be as described above. Since fn ---+ f pointwise on I, we have Ao = f, and so f E C(oo)(I) and Ak = f(k) (k E Z+). Thus peA) = f for each A E S. Take kEN and tEl. The set of accumulation points of the sequence (f~k\t) : n E N) is {A(Ak(t) : A E S}, and so f~k)(t) ---+ f(k)(t) as n ---+ 00. In
Commutative Banach algebms
466
fact, since the sequence (1/~k+l)IJ is bounded by (1k+l, we see that I~k) ~ I(k) uniformly on I as n ~ x, and I/(k) II ~ 13k, so that lED. Take 6 > 0, and choose ko E N so that 2:%"=ko fJk/Mk < no E N so that
I/~k) - I(/ 1,
Let lED with 11/11 = 1. For 0 > 1, set la(t) = I(t/o) (It I ~ 0) and define ga = 10. I I, so that I~k)(t) = o-kl(k)(t/o), la E Do., and go. E D with Ig~k)II ~ /!(k) Since go. ~ I pointwise on I as 0 ~ 1+, we have go. ~ I in D by the dominated convergence principle 4.4.14. Thus it suffices to show that each go. belongs to Dp. Fix Q > 1. Consider first the analytic case. By 4.4.13, go. has an extension to a function which is analytic on a neighbourhood of aDp (Z). By the functional calculus 2.4.4, go. E Dp, as required. Now consider the non-analytic case. The (truncated) Poisson integral for la is defined for y > 0 by
II"
lct
where
Py
(w) = ~ (w2 : y2) = 2~i (w ~ iy - w~ iY)
is the Poisson kernel (cf Appendix 2). The function P y is analytic on the strip {w E C : I;}wl < y}, and so, by A.2.14, Uy is also analytic on this strip. In particular, each Uy is analytic on a neighbourhood of I = aDp(Z), and sa Uy E D p .
Algebras of dzJjerentiable and LipHr:hitz functwns
467
The value of Uy on IR is given by the formula (4.4.11)
and so Uy
--->
gn pointwise on [
8.'>
Y
--->
0+ hy A.2.:17(i). We shall now estimate
IU~kl II for kEN. Indped, from (4.4.11), we have U~kl(x)
k'-l
=
L
(!~j)(-O:)P'~k-j-l)(:r + a) - Ifjl(a)p~k'-j-l)(x
j=O
+
i:
-0'))
IJ:)(s)Py(x - s) ds.
For each r EN, we have p(rl(t) _ (-1)r! ( y
and so
-
Ip~rl(x ± 0')1 ~ r!/n(a -
1
_
1 ) (t + iyy+l
(t - iyy+l
2ni
ly+1 (x E I). Thus
~ ~ ~ (k - j -1)! II(Jll IU(k)1 Y I n ~ aJ(O' -1)k-) I
1J,(kll
+
ct
=
fh,
[-a.a]
say.
Since (k!/kh)l/k ---> 0, there is a constant c > 0 such that cMr/r! ~ 1/e r for each r E N, where e = (a - 1)/0'. We have ~ M J because 11111 = Land
I!(jlll
Alj ~ Mk/Mk-j ~ d1ke k - j /(k - J)!
(j E Zk-l)'
Thus
t~k ~
2cMk n
~~ L
j=o
aJ
(_e_) a -
1
k-j
+ II(k) I ~
2cAh . n
I
and so ~~O i3k/Mk < 00. We have shown that each subsequence of (Uy and so Yct E D p, as required.
:
~k + II(kl I O'
1
.
y > 0) is (Afk)-dominated, 0
The following result is now immediate from 4.4.13 and 4.4.15. Theorem 4.4.16 (i) Let (1tfk ) be an analytic seA]uence. isomorphic to a natural Banach functwn algebra i5 on K
=
Then D(I; (Mk )) is
{z E C : d(z,I) ~ d(Mkl} '
and jj is contained zn A(K).
(ii) Let (Mk) be a non-analytic sequence. Banach function algebra on I.
Then D(I; (Mk)) is a natural
o
We now study further the natural Banach function algebras D(I; (Mk» in the case where (Mk) is non-analytic.
468
Commutative Banach algebras
Take kEN and to.·.·. tk E !R+ with 0 S to S tl S ... S tk, and throughout set dj = t J - t j - I ' We inductively define h(to, ... , tk) as follows:
= 1;
Io(to)
j
Ik(tO, .... td=
=
h(to, tl)
tl ds = tl -
ito
to:
tk
Ik- 1(t O,.··,tk-2,S)ds
(k~2).
tk-l
Note that h(to, ... , tk)
t
I->
~
0 and that the function
Ik(to, ... ,tk-l,t),
[tk-l,OO) ->!R+,
is increaHing. Also h(to,· .. , tk-l, tk-I) = 0,
Ik(tO,"" tk-2, tk-2, tk) ~ h(to ..... tk-2, tk-l, tk), and Ik(to, ... ,tk-3, tk-3, tk-3, tk) ~ h(to, ... , tk-:i. tk-2, tk-2, tk)' etc. The importance of these numbers lies in the following formula, which holds for j E C{k) (!R+) and k ~ 2:
(4.4.13)
The formula followH by induction and integration by parts. In the special ca. a decreasing sequence. Set Ok
= ('£;=1 J..Lj) -1,
so that Q:k
-+
0 as k _ 00.
Fix kEN, and define to, ... , tk so that to = 0 and tj = t j - 1 + Q:kJ..LJ; clearly tk = 1, so that fU)(tk) = 0 (j E Z+). By (4.4.13) and 4.4.17, we have
If(O)I::;
Lkit] j=l
k
!f(j)(t)!Ij - 1 (t 1 , ..• ,tj-l,t)dt::; LMjQ:kJ..Ljlj-l(tJ, ... ,tj) j=1
tJ-l
k
::; LJ..L1··· J..L j M j D.kf.lJ(eO'.kP-l ::; M 1 Q:k(1- eQ:k)-I. j=l
Since Q:k _ 0 as k _ 00, necesssarily f(O) =
o.
o
Proposition 4.4.19 Let (lvh : k E Z+) be a sequence zn lR+· wzth Mo = 1 and 2:.':=0 Mk/Mk+l < 00. Then there exists f E d~)(lR) such that f(O) = 1, I(t) = f(-t) (t E lR), and /f(k)/lR ::; 2-kMk (k EN). Proof Since
2:.':=0 lvh/Mk+l
0 and a > 0 such that Mk-2/aMk-l (k?: 2).
r:;:1 J.Lk < 1, where we define J..Lk =
470
Commutative Banach
Define 90 define
=
X[-l,l], so that 9o(t)
9k(t)
I
t +J'A
1
= -2-
p·1e
= go(-t)
9k-I(S)ds
aIgebra,,~
(t E ~), and then inductively
(k E N, t E~).
(4.4.14)
t-l1k
so that gk E C{k-l)(~) (k EN). Note that, for each k E Z+, we have gk(t) = ] fOl It I 1 - I::7=1 ILJ. and gk(t) = 0 for It I 2': 1 + I::~=1 ILj, and so gk(O) = 1 and supp 9k C [-2,2]. Also. by induction. gk(t) = gk( -t) (t E ~) for each k E Z+. Take k 2': 2 and t E R By (4.4.14). we have
s
g~(t)=-21 (gk-l(t+lLk)-gk-l(t-!1k)), ILk
(4.4.15)
and so g[k-J)(t) = (gk~-;.2\t + ILk) - gk~-;.2)(t - ILk)) /2/lk" whence
Igk-l (k- 2l l < 1 _ ... < _ IgAo(Ie-Ill IR . Then D(I; (Mk)) is a natural, regular Banach functwn algebra on I. Proof Set D = D(I; (M k )). By A.1.26(ii), the sequence (Mk) is non-analytic. Thus, by 4.4.16, D is a natural Banach function algebra on [. We show that D is regular. Take to E [ and c > O. By 4.4.19, there exists J E C(oc) (JR) with J(t) = 0 (t < to), J(to+c) > 0, and IJ(k) IIR :5 2- k Mk (k EN). Define get) = J(t + c)J(2to + c - t) (t E I). Then g E D, g(to) > O. and supp g C [to - c, to + c). It follows that D is regular. 0 For example, take Jl,h = (k!)a (k E Z+), where a > 1. Then D(I; (Mk )) is a natural, regular Banach function algebra. These algebras contain closed prime ideals of infinite codimension. By combining several of the abovE' results, we obtain the following classification.
Theorem 4.4.23 Let I be a compact interval of JR, and let (Mk ) be a logarithmically convex, differentwble sequence such that (Mk/k!)l/k ---+ cx> as k ---+ oc. Then D(I; (JI,lk)) tS a natural Banach function algebra; it is e1ther quasi-analytic (iJE~l Mk/Mk+l = (0) or regular (ifE';'=lMk/Mk+1 < (0). 0 The Banach spaces LiPaK and liPaK for a compact metric space (K, d) are defined in A.3.73; in particular, for f E Lipa:K, we have Ilflla: = IflK + Pa(f), Where If(x) - f(Y)1 } Pa(f) = sup { d(x, y)a: : x, y E K, x t= y .
Commutative Banach algebras
472
Theorem 4.4.24 (Sherbert) Let (K. d) be a non-empty, compact met1ic space. For each (} E (0.1], (LiPnK.II·ll n ) zs a natuml. regular, self-adjoint, umtal Banach function algebra on K. For each a E (0,1), (liPaK, 11.11 0 ) is a natural, T(~gular. self-adjomt, umtal Banach subalgebra of LiPoK. Proof Set A = LiPoK for 0 < (} ::::; 1 or A = liPoK for 0 < (} < 1. It is straightforward to check that A is a self-adjoint Banach function algebra on K. Suppose that f E A and Z(J) = 0. Then II f E A, and so, by 4.1.5(ii), A iH natural on K. By considering the functions x ~ min{d(x, xo)S 18.1} for suitable f3 and 8, we see that A is regular. 0 Lemma 4.4.25 Fm' each finite SlLiJSet F of K and each h E LiPnK. there eTlsts f E LiPIK 'Ulzth f I F = II I F and Ilfllo ::::; 211 h llu· Proof For y E F, define pace, and Co(W) is a closed linear subspace with dual space if:>ometrically isomorphic to (1\1 (W), III . III). where IIIILIII = max{IIJL I KII· IIIL I VII} Fix () with 0
contained in Co(W). By A.3.I9. each A E (lip",K)' ha.'i a norm-preserving extension to a continuous linear functional on (Co(W), 11·11), and so, by AA.lO(ii), there exists J1 E M(W) with (f, A)
=
/w
1 dJL
(f E liPn K )
and with IIJLII = IIAII. Such a IIleasure Jl is a correspondmg measure for A. Lemma 4.4.26 (i) Let JL E M(W), and take c > O. Then there exists v E M(K) such that (4.4.20)
(ii) Let v E M(K), and take c support such that
> O. Then there exists A E M(K) with finzte (4.4.21)
(iii) Let JL E M(W) be such that fw hdJL = 0 (h E Lip",K).
I\v 1 dJ.L
= 0 (f
E
lip",K).
Then also
Proof (i) Take L to be a compact subset of V such that 1J1,1 (V \ L) < c, and set (f, 111) = IKUL 1 dJL (f E C(K)) and 8 = inf{d(y. z)'" : (y. z) E L}. Then l(f, 111)1
:s 111KuL IIJLII :s
(1 + ~) IIJLlllflK
(f
E
C(K)).
and so 111 E C(K)'. By the Ricsz theorem A.4.lO(ii), there existf:> v E M(K) f:>uch that (t, 111) = iK J dv (f E C(K)); inequality (4.4.20) follows. (ii) Define B = {h E Lip",K : IIhll", :::; I}. Then B is a pointwise bounded, equicontinuous set in (C(K), I·IK)' and f:>0, by A.1.lO(iii), B is totally bounded in (C(K), I·I K ); take T to be a finite subset of S such that, for each h E B,
474
Commutative Banach algebras
there exists Uh E T with Ih - 'uhlK < £/4(IIvil + 1). By A.4.ll(ii), there pxists AE M(K) with finite support such that I/AII :::; IIvil and I[
U
dA - [
U
dvl
O. By (i) and (ii). there exists A E M(K) with finite'support
F such that ifw hdJL - fK hdAi < £ IIhil a (h E LiPnK). Take h E Lip", K. By 4.4.25, there exists , E LiP! K Clip", K with , I F = h I F and with l/fllr.. :::; 211hll",· We have
IJv JdJL fw J dJL = 0 by ifw hdJLi :::; 3£ IIhll""
But
['dAI
hypothesis, and
< £ "'"", :::; 2£ IIhi/r.. .
fK' dA = fK hdA,
and so it follows that
giving the result.
0
It follows from (i) and (ii), above, that lin{£x : x E K} is dense in (lip",K),. We shall require the following approximation theorem for Lipschitz functions. Theorem 4.4.27 Let K be a non-empty, compact metric space, and take E (0, 1). Suppose that P is a linear subspace lip",K and that there is a K and each, E lip",K, there constant C such that, 'or each finite subset F exists 9 E P wzth 9 I F I F and IIglia :::; C "'"",. Then P is. dense in lip",K.
Q:
=,
0'
0'
Proof Take
a
Theorem 4.4.34 (Bade, Curtis, and Dales) Let K be a non-empty, compact metric space, and take Q E (0,1). Then Jip",K is Arens regular, and the Bana,ch algebra (Jip",K)" is isometrically isomorphic to LiPoK. Proof Set A = liPoK. For E A", define
-r( 0, and choose ko E N with L:~ko 'f/k < C. By (ii), for each kEN, there exists nk E N with vark(f - fgn) < c/ko (1£ ~ nk). Set no = max{nJ .... , nko }' Then, for each 1£ ~ no, we have IIf - fhnll var :::; 4c, and so IIf - fh n IIvar -7 0 as 1£ -7 00. Thus S is a Ditkin set. (v) This is immediate from (iii). 0 Notes 4.4.36 The basic properties of the algebras c(n) (I) are given in (Rickart 1960. A.2.4); the results generally go back to SHov (1947). Most of 4.4.5 and 4.4.7 can be found in (Bade and Curtis 1978a) or (Dales and McClure 1977a). It is shown in (Ouzomgi 1984) that null sequences in the Frechet algebra Moo.",,(O) factor. The algebras D(I; (Mk » were introduced in (Dales and Davie 1973); in fact, analogous algebras D(X; (Mk» are defined on more general compact plane sets X. The polynomial approximation theorem 4.4.15 is due to O'Farrell (1983). Theorems 4.4.21 and 4.4.22 are close to the classical Denjoy-Carleman theorem; see (Rudin 1974,19.11)' where the proof uses the theory of Fourier and Laplace transforms. We have chosen to present more elementary proofs taken from (Mandelbrojt 1952, Chapitre IV): 4.4.18 follows the method of Bang (1946). An advantage of our proof of 4.4.18 is that it applies to the algebras DeX; (Mk» for more general spaces X (Dales and Davie 1973). The quasi-analytic Banach function algebras have been used to give counter-examples to a number of questions. For example, the following result is given in (ibid., §3). Let F be defined on lIJ), and let A be a Banach function algebra on ~ A. We say that F operates on A if F olE A whenever 1 E A and 1 (~ A) C lIJ). Then there is a natural quasi-analytic algebra D on II and a discontinuous function F on lIJ) such that F operate.s on D. Also, there is an example of a natural Banach function algebra D = D(X; (1\Ih» on an uncountable, compact space X such that ro(D) is countable. For a > 1, set A", = {J E D(ll; «k!)"'» : 1(/ 1'2
E
r) ,
= 1 (s E G) and that
(8 E G, l' E r).
The topology on r is that of uniform convergence on compact subsets of G. With this topology, r is also an LeA group; it is the dual group of G, often denoted by G. For each s E G, the map l' 1---+ (s,1'), r ~ T, is a continuous character on r. The Pontryagin duality theorem asserts that each continuous character on r has this form, and that the topology of uniform converg~ce on compact subsets of r coincides with the original topology on G. Thus G = f = G: if r is the dual of G, then G is the dual of r. It is standard that the dual of a compact group is discrete and that the dual of a discrete group is compact.
Definition 4.5.2 Let G be an LCA group. For' f E L1(G), the Fourier transform of f Z5 the function or:F f defined on r by
1
1('}) =
1 e
f(s)(-s, 1') ds
(l'
E
r).
(4.5.5)
For fL E .M (G), the Fourier transform (or Fourier-Stieltjes transform) of II, is [J, or :FfL defined on r by
[J,(l')
=
fa ~
(-s, 1') dfL(S) --
(l'
E
r).
For example, if 5 E G, then 6A'Y) = (5,1'). If fL = fLj, where f /J,f = We have -;;- = /l for fL E M(G).
1.
(4.5.6) E
1
L (G), then
482
Commutative Banach algebras We introduce the following standard notation:
A(r) = B(r)
{i: f E Ll(G)} = F(Ll(G)),
= {J1:
j.L E
M(G)}
= F(M(G».
Theorem 4.5.3 Let G be an LCA group. Then: (i) each
J1 E
B(r) zs bounded and unzformly contmuous, and F: (M(G), *,11·11)
-+
(Cb(r), ., 1·lr)
is a norm-decreasmg homomorphism; (ii) B(r) zs self-adjoint and translatwn-invanant on
r;
(iii) (Riemann-Lebesgue lemma) A(r) C Co(r); (iv) A(r) zs self-adJomt, translatwn-invanant, separates strongly the points of r, and zs dense in Co (r). 0 We next recall the standard proof that A(r) is a natural Banach function algebra on r. As in (1.2.2), we denote the shift operator on G by Sa: we have
(Saf)(s) = f(s - a)
(s
E
G).
Theorem 4.5.4 Let G be an LCA group. Then each character on Ll(G) has the form f t--+ f( 'Y) for some 'Y E r. Proof Set A = Ll(G), and let cp E CPA. Take fo E A with cp(fo) =f 0, and define 'Y(s) = cp(S-sfo)/ O. (4.5.13)
(t = 0).
27f Then T is the 'trapeziulll function' given by:
(Iyl
~
1),
Iyl ~ 2), (Iyl ;::: 2). (1
~
(4.5.14)
Proposition 4.5.11 (i) Let 9 E c5~) (JR.). where n ;::: 2. Then there exists a function f E Ll(JR.) n Co(JR.) 'Unth f = o(z-n) as It I --+ 00 s'llch that = g.
i
(ii) Let f E Ll(JR.), and set get) = -itf(t) (t Then ~s dzfferent'table, and = g.
i
l'
E
JR.). Suppose that 9
E
Ll(JR.).
(iii) Let h --+ f in Ll (JR.). and let (gk) be a sequence in LOO(JR.) and 9 E LOO(JR.) be such that gk --> g pomtwzse on JR. and also 119kll"" ~ I\gl\oo (k EN). Then Uk * gd(O) --. (f * g)(O). Proof (i) Set f(t) = g( -t)/27f. Then f E Co(JR.). and f by integration by parts. In particular. f E Ll(JR.).
(ii) For
-
11
(e- itu -l)/u
--+
as It I --+ x
E JR. with lJ i- Yo, we have
y. Yo
icy) - icyo) = y - Yo Since le- iu
= o(z-n)
11t\
~
-it as u
1
00
-00
f(t) (e-itCY-YO) Y - Yo
1)
e- iYOf dt.
(u E JR.), the integrand is dominated by Ig(t)l. Since --+ 0, the result follows from the dominated convcrgenC'e
theorem. (iii) We have
IU
* g)(O)
~ Let k
--+ 00.
* gk)(O)1 ~
i:
- Uk
i:
i:
If(8)g( -8) - 1k(8)gk( -8)1 d8
If(s)llg( -8) - 9k( -s)1 ds +
Igk( -b)llf(8) - fk(s)1 ds.
In the first integral on the right, the integrand is bounded by
211911 00 If(s)l, and it converges pointwise to 0, so the integral tends to 0 by the dominated convergence theorem. The second integral on the right is bounded by 11911 00 IIIk - fill' and this tends to O. 0
486
Commutative Banach algebms
Second, take G = ']['. Then Haar measure on G is d()/21r, and the formula for cOllvolution is (4.5.15) We shall usually identify [-1r,1r] with '][' and write f«() for f(e iO ); in this case, equation (4.5.15) becomes
(f
* g)«()
1 = 21r
111" -11" f«() - cp)g(cp) dcp
«() E (-IT, lTD·
For each k E il, the map z 1-+ zk, '][' ~ '][', is a cOl~inuom, character on ,][" and every continuous character has thil:l form. Thus '][' = il, and the Fourier transforms are given by
(Ff)(k) = f(k) =
~ 111" 2IT
-11"
f«()e- ikO d()
and
j1(k) = f'j[' e- ikO dJL«()
l1r
(4.5.16)
for k E il, f E Ll(']['), and JL E M(,][,). The constants f(k) and j1(k) are the Fourier coefficient8 of f and JL, respectively. Thus A(il) and B(il) are Banach sequence algebras on il in the bense of 4.1.34. Let E be a Banach space, and let T E B(E). The (analytic) functional calculus for T was defined in 2.4.3 as a unital homomorphism 8 : Vu(T} ~ B(E) with 8(Z) = T. In certain cases, there il:l an extension of this calculus. The spaces C(k}(K) and C(k}(U) for compact subsetl:l K and open subsetl:l U of]R2 are defined in Appendix 3. Definition 4.5.12 Let E be a Banach space, let T E B(E), and let k E il+. Then a C(k}-functional calculus for T is a continuous, unital homomorphzsm 8 : C(k}(]R2) ~ B(E) with e(Z) = T. The opemtor T is generalized scalar if there zs a C(kLju7lriional calculus for T for some k E il+.
The operator T has a C(kl-functional calculus if and only if there is a compact set K in ]R2 with aCT) C int K and a continuous, unital homomorphism 8 : C(k}(K) ~ B(E) with 8(Z) = T. For example, by 3.2.7, each normal operator on a Hilbert l:lpace il:l a generalized scalar operator. Proposition 4.5.13 Let E be a Banach space, and let T E Inv B(E) be such that IITnll+IIT-nli = O(nk) as n ~ 00. Then T has a C(k+ 2Lfunctional calculus. I
Proof Take f E C(k+2}(]R2), and set 9 = f I ']['. Then Inl k+2 Ig(n)1 :::; If(k+2}1'Il' for nEil. Define 8(f) = L~=_oog(n)Tn. Then 118(f)1I :::; Clf(k+2}I'!I' for some constant C, and so 8 : C(k+2}(]R2) ~ B(E) is continuous linear operator. Clearly e is a unital homomorphism. 0
Finally, take G = Z. Then of course convolution is given by
(f
* g)(n) =
L j+k=n
f(j)g(k)
(n
E
il) ;
(4.5.17)
Abelzan grvup algebras
Z=
we have
487
T, and the Fourier transform of I = (Ok) E f 1 (Z) takes the form ex;
(Ff)(O) = 1(0) =
I>l!ke-ikO
(0 E (-71",71"]).
-00
Theorem 4.5.14 (Bernstein) zs a constant CrY. such that
FOT
each n E (1/2,1]' LiPa T c A(T), and there
(4.5.18)
Proof Take
I
E
LiPa T. For each () E JR, we have
00
L
le ikO
11 2 11(k)1
-
2
=
IISol - III~ ::; ISol - II~ ::; 1()1 2 II/II~ . (t
k=-oo
Now fix m E Z+, and set 0 = 271"/(3· 2m). For each k E Z such that {2m, ... , 2rn+l}, we have le iko - 11 ~ J3, and so
Ikl E
L
l1(k)1
2
::;
3c1T2mrY.II/II!
2"'::;lkl Xp, is continuous. We shall estimate I = JK IIFy ® Gyll 1r dy. Indeed, I
= [IIFYllp IIGy ll q dy
~ ( [ IIFYII~ dY) lip (i IIGYII~ dY) l/q
by Holder's inequality. Now
[IIFyll~ dy = [
( [ Ih(xY)h(xW
= [ ([ Ih(xy)I P dY) and, similarly,
JK IIGy"~ dy =
Ih(xW dx
= IIhll~ 111211: '
IIg111~ IIg211~. Thus
I ~ IIhll p 1112 lip IIg1ll q IIg211 q =
It follows from AA.16 that the clement H = [Fy 0Gy dy = belongs to Xpo
dX) dy
IIh 091!l1r 1112 o g2 II
l
Fy 0Gy dy
71'
•
494
Commutative Banach algebras
We now show that 7rp(H) = 7rp (fI 0 gd7rp (J2 0 g2). Indeed, for each x E f, we have
7rp(Jl 0 gd(X)7rp(J2 0 g2)(X) = = =
=
II II l (t
fl(XY).Ql(y)!2(XZ)g2(Z) dydz
fl(xzy)gl(Zy)!2(XZ)g2(Z) d1Jdz
t
Fy(xy)Gy(z) dZ) dy
(7rp (Fy 0 G y» (x) d1J = 7rp(H)(x)
by A.4.17, as required. The result now follows easily by the use of A.4.9.
o
Thus the Fourier algebra A 2 (r) of f is an algebra. Let G be an LCA group, with dual group f. It follows immediately from 4.5.10 that A 2 (r) coincides with A(r); we now have a generalization of A(f) to the case where f is not nec(>ssarily abelian. Theorem 4.5.31 Let f be a locally compact group, and let p E (1,00). Then Ap(f) is a self-adjoint, natural, strongly regular Banach function algebra on f. and Ap(r) is dense m (Co(r).I·l r ). Proof The algebra Ap is certainly self-adjoint. and it is obvious that the algebra Ap,oo of elements in Ap of compact support is dense in Ap. Now take x E f; we shall show that J x = Mx. Indeed. take f E M; and c E (0, Iflr); we may suppose that f E Ap,oo. Define
Clearly W is a compact neighbourhood of er. Let V be a symmetric, open neighbourhood of er with V c W such that If I", v < c and mr(V) ::; 1. and set 9 = fXx v· Set n = Xv/mr(V) and set h = (J - g) * u, so that hEAp; clearly hE J x , IIgli v < cmr(V)l/p, IIullq = mr(V)l/p, and
IIlf - hili p
::;
Ilif -- f * uili p + IIglip lI ul q :s c
l
u
+ c = 2c.
It follows that J x = Mx, as required. By 4.1.32(i), Ap is natural on f and strongly regular. By the,Stone-Weierstrass theorem A.1.1O(i), Ap is dense in Co(r). 0
The following result on bounded approximate identities will not be proved. Theorem 4.5.32 (Leptin, Herz) Let f be a locally compact group. Then the following are equivalent: (a) f is amenable; (b) Ap(f) has an approximate identity of bound 1 for each p E (1,00); (c) Ap(r) has a bounded approximate identity for some p E (1,00). 0
Abehan group algebras
495
We conclude this section by examining a specific example originating with Mirkil.
Example 4.5.33 We start with the Banach space (£2(1['),11,11 2), again identifying 1[' with [-7T. 7T). Write S = [-7T /2, 7T /2), and set M
= {J
E £2('JI') : 1 I S E C(S)} .
For 1 E Al, set 11111 = 111112
+ Ills =
vk (L:
11(0)1 2 dO) 1/2 + Ills·
(i) (M, *, II·ID 'tS a commutatwe Banach algebra (for an equivalent norm). Certainly (M, 11·11) is a Banach space. Let 1, g E At, and set h = f * g. By 3.3.14, h E C('JI') c £2('JI') with IIhlb :-::; Ihl'Jr :-::; IIfll2l1glb, and so IIhll :-::; 211111211g112 :-::; 21111111g11 .
(ii) The trigonometric polynomials are a dense lmeaT' subspace of M. Indeed. given 1 EM and c > 0, there exists 9 E C('JI') with 9 I S = 1 I Sand 1\9 - fl12 < c, and there is a trigonometric polynomial h with Ih - gl'Jr < ej we have IIh - fll < 3£. We now identify M with its algebra of Fourier transforms on Z. Thus we have: (iii) M ~s a strongly regular Banach seq'uence algebra on Z. and M2 C ll(Z). Set A = M#, so that A is a strongly regular Banach function algebra on Zoo; the ideal J oo corresponds to the space of trigonometric polynomials. There is an obvious isometric linear isomorphism from M onto a closed linear subspace of the Banach space E = £2(1[') EBl C(S). Since E' = £2('JI') EBoo M(S), each element of lv/' can be represented by a measure of the form v=gdO+/-L,
(4.5.20)
where 9 E £2(1['). /-L E M(S), and Ilvll = max{lIglb, II/-LII}; the pairing is
(1. v) = 21
111" I(O)g(O) dO +
7T -11"
r1(0) d7i(O)
Js
(f E M, v EM').
In the following, we write rZ + s for the subset {rn + s : n E Z} of Z. Set + 2) U {x}, and F = 2Z U {oo} = F1 U F2. (iv) The sets F1 and F2 are of synthesis for A. First take v E M' with v I J(Fl) = O. If k E Z \ 4Z, then (Zk, v) = 0 because Zk E J(Ft} , and, if k E 4Z, then e ik «()-1I"/2) = eik () (101 ::; 7T). Thus (S1I"/21, v) = (1, v) for each 1 E J oo , and hencE' for each 1 E M. It follows that v(T + 7T /2) = v(T) (T E B'Jr), and so, in this case, v has the form 9 dO for some 9 E £2('JI'); using (4.5.7), we have Fl = 4Z U {oo}, F2 = (4Z
(1, v)
= :7T
Since g(k) = 0 (k
E
1:
f(O)g(O) dO
=
kf;oo j(k)g(k)
Z \ 4Z), we have (1, v) = 0 for each
(f
f
E E
M) . J(Ft); it follows
496
Commutative Banach algebras
that v I I(F!) = 0, and hence that J(F!) = I(Fi ). This provel-J that Fi is of synthesis for A. For each v EM', we have v I J(F2) = O[v I I(F2) = 0] if and only if Z2 v I J (F!) = 0 [Z2 V I I (F!) = 0], and so F2 is also of synthesis for A. (v) The set F is of non-synthesis for A, but each compact subset of F \ {oo} is of synthes%S. Define go on '][' by
go(fJ)
=1
(lfJI:::; rr/2),
go(fJ)
= -1
(rr/2 < IfJl :::; rr).
Thcn go E A!. We have
go(k) =
~ 17r g(fJ) cos kfJ dfJ =
k: sin ( k2rr)
(k
E
ze) ,
with go(O) = 0, and so "§o I F = 0 and go E I(F). Take /-Lo = 87r / 2 + L7r/2' so that /-Lo E M(S). We have
(Zk,/-LO) = e ik7r / 2 + e- ih / 2 = 2cos(krr/2) and so (Zk, /-Lo)
= 0 for
I
= O.
+ g( -rr /2) =
2.
k E Z \ 2Z, whencc /-Lo J(F) (gO, /-Lo) = g(rr /2)
(k E Z). However,
Thus go E I(F) \ J(F), and F is of non-synthesis. Clearly compact subsets of F \ {oo} are trivially of synthesis. Note that we have
J(Fi
U
F2) = J(H)
n J(F2) 0, and set k m = (2m + 1)2r. For m sufficiently large,
l[(km)1 < Elk
m
by (iii), and h(km ) = 0 because k m E 2Z. Also k m (1 + 2- r ) E 2Z + 1, and so I§;, (k m ) I = 1/km 7r, whereas. for s E Zr-1, we have km (1 + 2- 8 ) E 2Z. and so iis(k m ) = O. Thus larl < 7rE. Hence a r = 0, and then we see successively that a r -1 = ... = a1 = 0, giving the claim. (x) The algebra 2(
~s
decomposable, but zs not strongly decomposable.
Set 9Jt = M /.J(F) C 2(. Then 9Jt2 n!.R = 0 because go 1. liP, and so, by 1.5.16, 2( is decomposable. Since A is spanned by its idempotents, the same is true of 2(, and so, by 2.8.8(i), 2( is not strongly decomposable. In summary: A is a strongly regular Banach function algebra on cI> A = Zoo. but A is not a Ditkin algebra; there are closed subsets F1 and F2 of cI> A, both of synthesis, but such that F = F1 UF2 is of non-synthesis; the algebra 2( = A/.J(F) is decomposable. but not strongly decomposable. We shall note in 5.1.20 that 2( does not have a unique complete norm. 0 Notes 4.5.34 The early material on Ll(G) for G an LCA group is quite classical; see (Graham and McGehee 1979), (Hewitt and Ross 1979, 1970), (Reiter and Stegeman 2000), (Rudin 1962), and (Zygmund 1959), for example. For the algebra A('JI') specifically, see (Kahane 1970). There is a structure theorem for LCA groups. A topological group G is compactly generated if it contains a compact subset K such that the subgroup generated by K is G. For the following, see (Hewitt and Ross 1979, §§5, 9). Theorem (i) Each LCA group G has a compactly generated subgroup H such that G / H is discrete.
(ii) Each compactly generated LCA group is topologically isomorphic to a group of the form IRm x zn X K. where m, n E Z+ and K ts a compact group. 0 For a discussion of the algebra M(G), where G is a LCA group, see (Graham and McGehee 1979), (Laursen and Neumann 2000), and (Rudin 1962), for example: the remarks after 4.5.6 are proved in (Graham and McGehee 1979, Chapter 8). The fact that there is a non-zero, continuous point derivation on Al(G) whenever the group G is non-discrete is due to Brown and Moran (1976): see (Graham and McGehee 1979, 8.5.4). Results on M(G) are also given in (Rudin 1962, §5.3) and the memoir (J. L. Taylor 1973). For an extensive discussion of generalized scalar operators, see (Laursen and NeuIllann 2000, §1.5). Proposition 4.5.14 is due to Bernstein; the result is sharp because the classical example of Hardy and Littlewood, L ::'=1 ein log ne in9 In, belongs to LiPl/2'JI', but not to A('JI'). Indeed liPl/2'JI' ¢.. A('JI') (Kahane 1970, p. 15). See the treatise (Zygmund 1959, Chapter V) for this and many other special series. The proof of 4.5.18 follows that of (Bachelis et al. 1972); 4.5.21 is from (Hewitt and Ross 1970, (32.47». The literature on sets of synthesis in A(r) is enormous; see, in particular, (Graham and McGehee 1979, Chapter 3), (Reiter and Stegeman 2000), and (Hewitt and Ross 1970, Chapter 10), where detailed histories are given. Theorem 4.5.23(i) is the famous theorem of Schwartz (for n ~ 3)-the converse, for n = 2, is due to Herz. Malliavin's
498
Cornrnutatwe Banach algebras
theorem, 4.5.23(ii), is proved by Varopoulos's method in (Hewitt and Rosb 1970) and in several ways in (Graham and McGehee 1979); for 4.5.23(iii), seC' (Kahane 1970, p. 52); for 4.5.23(iv), see (Korner 1973, 1991), (Kaufman 1973), and (Graham and McGehee 1979, §4.6). Let G be an infinite, metrizable LCA group. Then it is shown in (Graham 2001) that G has an infinite, closed subset F such that: (i) F is not a Helson set; (ii) A(F) is Arens regular; (iii) (A(F)",O) is not Arens regular. There are relatively few closed sets S in r such that the ideal 1(8) in A(r) has an approximate identity in J(8), and certainly there are Ditkin sets without this property; for a summary of results on these sets, see (Graham and McGehee 1979, §3.1). There are two major open questions about A(r). The first is whether or not each set of synthesis is a Ditkin set; the seeond is whether or not the union of two sets of synthesis is also of synthesis. It was a major achievement of Drury and Varopoulos to prove that the union of two Helson sets for A(r) is also a Helson set; see (Graham and McGehee 1979, Chapter 2). Define BoOR) = B(IR) n Co (IR). Then Bo(IR)# is a normal Banach function algebra on IR'OO, but it is not a natural algebra (Graham and McGehee 1979, 8.2.3). Let G be an arbitrary locally compact group. Then the centre AG = 3(LI(G» of L I (G) is a commutative Banach algebra: it is non-zero if and only if G is an IN group. When it is non-trivial, AG has properties similar to those of the group algebra of an LCA group. It is shown in (Liukkonen and Mosak 1974) that AG is a semisimple Banach *-algebra with a bounded approximate identity; we regard AG as a Banach function algebra on Aa' Then AG is regular and spectral analysis holds for AG; for a large class of groups G, AG is strongly regular, but an example given in (ibid.) shows that this is not always the case. Segal algebras on an LCA group are defined in (Reiter and Stegeman 2000, §6.2); the notion originates with Reiter. The multiplier algebras of the Segal algebras Sp(G) are discussed in (Larsen 1971, Chapter 6); it is shown that, in the case where G is noneompact and non-discrete, the multiplier algebra M(8p (G» is isometrically isomorphic to M(G) for each p E [1. 00). The seminal study of the Fourier algebra A(r) = A 2 (r) for a locally compact group r is that of (Eymard 1964). It was proved by Eymard that
A(r) = {f * g: f,g E L2(rn· The definition of Ap(r) and the proof that it is an algebra are taken from (Herz 1973); 4.5.32 is (ibid., Theorem 6), extending (Leptin 1968). In the case where r is amenable, each maximal ideal of Ap(r) has a bounded approximate identity, and so Ap(r) is a strong Ditkin algebra. It is not known whether or not A(r) has an approximate identity for each locally compact group r; it is not even known whether A(r) always has approximate units. The algebra A(lF2) does not have the 7r-property, and so A(lF2) is not biprojective. It is not known when A(r) is weakly amenable; for partial results, see (Forrest 1994, 2001). The weak operator closure of LI(r) in 8(£P(r) is the space PMp(r) of p-psf'udomeasures on r; PMp(r) is the dual space Ap(r)' of Ap(r). In the case where P = 2, PM2(r) is denoted by VN(r): it is the von Neumann algebra generated by the left regular representations of ron L2(r). As we stated, in the case where r is abelian, A(r) coincides with the previously defined algebra F(LI(f», and so VN(r) = LOO(r). In the cases where r is arbitrary and p = 2 and where r is amenable and p is arbitrary, PMp(r) is the space of multipliers on LP(r): a map T E 8(£P(r) is a multiplzer if T(f * g) = Tf * 9 (f,g E Coo(r». Suppose that the Fourier algebra A(r) is amenable. Then, by 2.9.57, A(r) has a bounded approximate identity, and so r is an amenable group. However, the converse is not true: there are compact groups r such that A(r) is not even weakly amenable (Johnson 1994). See also (Lau et al. 1996). The example of 4.5.33 was introduced in (Mirkil 1960), and further examined in (Atzmon 1980) and (Bade and Dales 1992).
Banarh algebms of power' 4.6
.'IerU~8
BANACH ALGEBRAS OF POWER SERIES
So far, the examples of commutative Banach algebras that have be£'n given WelC'all semisimple. In the following sections. we shall discuss classes of ('onullutatiw Banach algebras that include important radical examples. The present section is mainly devoted to Banach subalgebras of C[[X]]. and to the structure of the fmuily of closed ideals of these algebras; in particular, we shall consider the algebra.'.i £1 (Z+ . w), where w is a weight sequence. '\Te shall also consider tbe Beurliug algebras £1 (Z, w), where (.v' is a weight on Z. In some respects. this section should be considered as a precursor to §4.7, where the related, but more complicated, algebras £l(l~+.w) and L1(JR..W) will be studied. In §1.6. we introcillC('d the algebras ~ = C[[X]] and ~n = C[[X j •••• , Xn]] of formal power series in one and n indeterminates over C. respectively. Let us recall the standard notation. The product in ~n will at first be denoted by juxtaposition; the identity of each ~n is denoted by 1. A generic element of ~Il has the form I:arX r , where Q r E C (1' E Z-t-"); the maps 7fr :
L asX"
f-+
ar, ~n ----
C,
defined for r E Z+l1. arc the coordinate projections on ~n, and the order of a E ~n is o(a). By 1.6.19(i), Inv~n = {a E ~n : o(a) = o}. For k E Z+, Mk = {a E ~n : o(a) ~ k}; by 1.6.19(ii), ~n is a local algebra with maximal ideal MI, and, by 1.6.20, {Mk : k E Z+} is the family of non-zero ideals in ~. As in 2.2.46(i), ~n is a Fnlchet algebra with respect to the topology of coordinatewise convergence Tn defined by the sequence (Pk : k E Z+) of beminorms, where Pk(a) = I:{la1 1 : r E z+n, Irl ::; k} (a = I:arX r E ~n) for k E Z+. Further. ~n is a Q-algebra. Theorem 4.6.1 For' each n E N. the topology Te is the unzque topology wzth respect to whzch~" zs an (F)-algebm. Proof Let T be a topology such that (~n' T) is an (F)-algebra. We shall first show that (~n.T) is a Q-algebra: we must show that Inv~n is open, i.e., that the maximal ideal M1 of ~n is T-closed. Indeed, assume that M1 = I:;=1 Xj~n is T-dense in ~n' Then, by 2.2.14(ii), n{.l\,[k : kEN} is dense in ~n. But this is a contradiction because n{Mk : kEN} = {o}. Let kEN. Then Mk (T) is a finitely generated ideal, and so, by 2.6.37, Mk is r-closed. Take r E z+n. Then 7fr I .lVllrl+l = 0, and so, by A.3.42(i), 7fr is continuous on (~n, T). It follows that the identity map (~n' r) ---- (~n, Te) has a closed graph. By A.3.26, T = Te. 0 Corollary 4.6.2 For each n E N, the algebm ~n zs not a Banach algebm with respect to any norm. Proof Assume towards a contradiction that (~n, 11·11) is a Banach algebra. By the theorem, each projection 7fk is continuous on (~," II '11). For each k E Z+, set ak = 117f(k,o, ... ,O) and set a = L:;::1 kakXf E ~n' Since 7f{k.O, .. ,O)(a) = kak, \\Te have k ~ IIail for all k E Z+, a contradiction. 0
II,
500
Commutatwe Banach algebms
This result also follows from the deep theorem 5.2.36: formal partial differentiation with respect to Xl is a derivation on ~n' but its range is not contained in Ah = rad ~1l' and so ~n cannot be a Banach algebra. The above corollary showt-l, for example, that (~. rc) is not the image of a Banach algf)bra by a continuous homomorphism. However, we shall see in 5..5.19 that ~ is the image of a commutative Banach algebra by a (necessarily discontinuous) homomorphism. Proposition 4.6.3 Let (A. r) be a Fnkhet algebm such that A c ~n and CCO[X 1 , ... , Xn] C A and such that the embeddw,g (A, r) ---+ (~n, rc) is cont~nuous. Then each de'l'ivatwn D : A ---+ A lS continuous. Proof Let D : A ---+ A be a derivation, and take 'I' E z+n. By using the derivation identity to expand D(xr), we see that (7r r· 0
D)
! (1\IlrH2 n A) = 0
('I' E z+n).
By A.3.42(i). 7r1' 0 D is continuous on (A. r), Af11'1 + 2 nA being closed and of finite codimension in (A, r). Since this holds for each r E z+n. it follows easily from the closed graph theorem that D is continuous. 0 On the other hand, it is immediate from l.8.18 that there are discontinuom, derivations from ~ into an (F)-~-module; a modification of this result will be given in 5.6.8l. We now turn to consideration of some sub algebras of the algebra ~. Definition 4.6.4 Let A be a subalgebra power series zJ:
of~.
Then A is a Banach algebra of
(i) A contains the indeterminate X.(ii) A
2S
a Banach space wzth respect to a norm
(iii) the projections
7rk :
(A,
!! ·11)
---+
!!.!!;
CC aTe continuous for each k E Z+.
It follows from (i) that A contains CCo[XJ, the algebra of polynomialt> with zero constant term. We first check that a Banach algebra of power series is indeed a Banach algebra. Proposition 4.6.5 A Banach algebm of power series is a Banach algebra with respect to a norm equivalent to the given norm. I Proof Let A be a Banach algebra of power t-leries. Suppose that bn La(bn ) ---+ C in A. Then, for each k E Z+, we have
---+
0 and
k
7rk(C) = lim 7rk(abn ) = lim'" 7rj(a)7rk_j(b n ) = 0, n-+cx> n-+CXJ L-t j=O
and so c = O. Thus, by the closed graph theorem A.3.25, La is continuous. By 2.2.6, multiplication in A is continuous, and this gives the result. 0
Banach algebras
0/ power senes
501
»
For example. let A be A(iij) or H'Xl(lDJ) or D(][j (Mk in the case where D(][j (Mk)) is a natural, quasi-analytic Banach function algebra. Then the monomorphism
identifies A as a Banach algebra of power series. Note that. in the case where A = A(ll}). the polynomials are dense in A, but, in the case where A = Hoo 'I7n/n = 00. For example, (O"n), where 0" E JR, gives a weight sequence w with Pw = e-O". and (n')') , where 'Y > 1, gives a radical weight sequence. Note that there a.re radical weight sequences w such that w~/n --+ 0 more slowly than any prescribed llequence in jR+-: take lIn = nXn (n E N), where (Xn) is an increasing sequence lVith Xn --+ 00 as n --+ 00. The convolution product * (now taken to be on /l) was defined in (2.1.7). 'rhe next result is easily verified.
w;/n
w
Cornrnutatwe Banach algebras
5U2
Theorem 4.6.7 Let w be a we1.ght on Z. Then.e 1 (Z. w) is a commutative, unital Banach algebra with resprct to the product *. and it i8 polynomzally generated by {61 ,LJ}. 0 The algebra.o.; t 1 (Z, w) for w a weight on Z are called Beurlmg algebras on Z. In the case where ",,' is a weight on Z+, we write flew) for Pl(Z+,L Ql,
is a splitting homomorphism whenever T : G --> F' is a linear map such that 0 T = la. However, in the case where t(E) is not complemented in F, the extension I: is not admissible. 0 q
504
Commutative Banach al.gebms
Now let w be a weight on Z, and set Pi
= inf{w;/"
: II EN}
P2
and
= sup{w=~/n
: n E N}.
· l/n 1)y A .1. 26 . S'mce W_n -l/n $ Wo-l/n Wnl/n ( n E 1M) so t h at P2 = IlIIl"_ .... _n
For example, suppose that W = (wn) is a weight on Z with Wn ~ 1 (n E Z). and set A = f 1 (Z. :...,). Then Ace 1 (Z), 1f c IP A. and the restriction a I'll' of the Gel'fand transform of a coincides with tbe Fourier transform of a.
Example 4.6.13 For a > 0, set wu(n) = (1 weight on Z, and X",,, = 'll'. Define
Ax
+ In!)"
(n E Z). Then w,.. is a
= Q(C 1 (Z,..v,,)).
so that, by 4.6.12, Au is a natural Banach function algebra on 1f. We continue to write f(B) for f(e iO ). First note that, in the Ca8(, where Q ~ k, where kEN, Aa C C(k)(1f), and so there is a continuom; higher point derivation of ord('r k on A, at each character of Au. Clearly. if k > a + 1, then C(l Q, and Ret a:~ = c50 + I::=I a/~,nc5n, where the sequence (aB,n) was specified in (2.1.5). Then la/:I,nl = O(l/nl+f3) aH 'f/ ~ 00, and so af3 E Pl(Z,Wo.)' We have af3(B) = (1 - e i8 )f3, and so 1li,ti(B) I / IBII:I ~ 1 as B ~ O. Define the maximal ideal Alex = {f E A", : f(O) = O}. If f EM,.., then If(B)1 = 0(101") as B ~ 0, and so If(B)1 = 0(/BI20.) as B ~ 0 for each f E M~. Thus the family {af3 + M~ : f3 E (0:,2a)} is linearly independent in Mo./M~, and hence JI.1; has infinite co dimension in Mo.; the maximal ideal 11,10. of Ao. does not have a bounded approximate i.dentity, and there are discontinuous point derivations on Ao. at the character given by z = 1 (and hence at every character). 0
Banach algebras of power senes Example 4.6.14 Take
0:
505
? O. In a similar way, we define
A~ = Q(pl(wu)) = {f = ~ i(n)zn : Ilfll
w"
= ~ li(n)1 (1 + nY < DC}
,
At = A, n A(ii'») and At = A t-(ii'»). By 4.6.9(i). the charact Aw is defined by
(PcI?)(n) = (cI?, Ln) and Rw = ker P. For ( E 1I', we regard function on 1I'. Then
(cI? E A~, n E Z), as an element of
C(
A~
and PcI? as a
(4.6.3) n=-oo
For cI? E A~ and n, k E Z, we have
(cI? . Ln)(k) = (8k' cI? . Ln) = (cI?, 8_ 11
•
8k ) = (cI?, Ln+k) = (PcI?)(n - k),
and so cI? . 8_ n = L~=_oo(PcI?)(n - k)8_k in A~. We clazm that P : (A~, 0) -> Aw is a homomorphism. Indeed, for each cI?, \]i E A~, we have 00
k=-oo
=
""
L
(p\]i)(n - k)(PcI?)(k) = (p\]i
* PcI?)(n)
(n
E
Z),
k=-oo
giving the claim. Similarly, P: (A~, 0) --> Aw is a homomorphism, and so Rw is a closed ideal in (A~, 0) and (A~, 0); the products in A~ corresponding to the above decomposition are
(a, cI?) 0 (b, \]i) = (a (a, cI?) 0 (b,\]i) = (a
* b, a
*
. \]i + cI? . b + cI? 0 \]i), } I b, a· \]i+cI? b+cI?O\]i) ,
(4.6.4)
where a, bE Aw and cI?, \]i E Rw. We see that
L(A~; Rw) : 0 ~ Rw ~ (A~, 0) ~ Aw ~ 0 is a Banach extension of Aw by Rw, and that the extension splits strongly. In the case where w == 1, the products cI? 0 I}! and cI? 0 I}! in (4.6.4) are necessarily distinct because, by 3.3.28, .e1(Z) is not Arens regUlar. However, this is not the case for certain Beurling algebras.
Banach algebms of power series
507
In the following results, we use the above notation; note that the weights We> (for a > 0) satisfy the condition imposed on w in 4.6.16, below.
Lemma 4.6.15 Let w be a weight on Z. Suppose that w(m + n)/w(m) Iml - 00 for each n E Z. Then
I(ep, A . Ln)1 :::; 211epliliAII for each A E
A~
1 as
(n E Z)
and ep E Rw.
Proof Let A E A~, cP E R w , and n E Z. Set F = {m E Z: w(m+n) 2: 2w(m)}; since w(m + n)/w(m) - 1 as Iml - 00, F is a finite set. Next define a(m) =
{~n-m
(m E F), (m E Z \ F).
Then a E A~, and
lIall = snp la(-m)1 = sup mEZ
w(m)
mEZ\F
IAm+nl . w(m + n) :::; 211A1I .
w(m + n)
w(m)
Since a - A . Ln E Ew , we have (ep, a) = (ep, A . Ln), from which the result follows. 0
Theorem 4.6.16 (Lamb) Let w be a wezght on Z with wen) and w(m + n)/w(m) - 1 as Iml- 00 for each n E Z. Then:
00
as Inl -
00
(i) Aw is Arens regular;
(ii) R! = 0 and Rw is the mdical of (A~, 0); (iii) A~ has the strong Wedderburn decomposition A~ =
Proof Let A E
A~
Aw E!1 Rw.
and ep E Rw. We have
I(ep . A)(n) I = I(L n, ep . A)I
= I(ep, A . Ln)1
:::; 211epliliAII
by 4.6.15,
and so ep . A E Ew because l/w(n) - 0 as Inl - 00. Now take CP, W E Rw. Then (ep 0 W, A) = (ep, W . A) = 0 (A E A~) because W . A E E w , and so CP 0 W = O. Thus formulae (4.6.4) become
(a,CP)O(b,w)=(a,ep)O(b,w)=(a*b,a·
w+ep· b). o
The result follows.
Let w be a weight on Z satisfying the conditions of 4.6.16, and let F be a closed subset of T. We set Iw(F) = {f E Aw : f I F = O}, now regarding Aw as a Banach function algebra on T, and
Aw(F)
= Aw/1w(F) = {f IF: f
E
Aw}.
By (2.6.29), we have (Aw(F)/, OJ ~ (A~, 0)/ Iw(F) 00; by 4.6.16(i) and 2.6.18. the Banach algebra Aw(F) is Arens regular. Now let P : A~ - Aw be the canonical projection, as above. For each ( E T, we have e( E Iw{F)O, and 80 (cp,e() = 0 for each cp E Iw(F)OO, whence pcp E Iw{F) by (4.6.3). Thus
Commutative Banach algebras
508
P induces a continnous projection PF : Aw(F)" ---> Aw(F), and PE-' is an epimorphism. Set Rw(F) = Rw/(Rw n Iw(F) 00). Then
2:)Aw(F)"; Rw(F» : 0 ~ Rw(F) ~ Aw(F)" ~ Aw(F) ~
°
is a Banach extension of Aw(F) by Rw(F), and the extension splits strongly. Since Aw(F) is semisimple and Rw(F)2 = 0 by 4.6.16(ii), it is clear that Rw(F) = radAwCF)". Thus A...,(F)" is semisimple if and only if the Banach space Aw(F) is reflexive. In the particular ca::;e where W = Wa for 0 < ex < 1, we have Aa(F) C liPaF, where we are writing Aa(F) forA w" (F). Corollary 4.6.17 (Lamb) Let F be a closed subset ofll, and take Then Aa(F) = liPuF zf and only if F i8 fimte.
0
E (0,1).
Proof Suppose that Ac,(F) = liPaF. By 4.4.34, (liPnF)" '::: LiPaF, and so Ac,(F)" is I:)emisimple. Thul:) Ac,(F) is reflexive, and so liPaF = LiPaF. This occurs only if F is finite. 0
We now study the family of closed ideals in certain radical Banach algebras of power series. We are again denoting the product in J' by juxtaposition. Let A be a subset of J'. We set
Mk(A)
=
Mk n A
=
{a
E
A : o(a) 2: k}
(k E Z+),
and we write Ah (w) for II, h (f 1 (w)). If A is a Banach algebra of power series, then Mk(A) is a closed ideal in A. Definition 4.6.18 Let A be a Banach algebra of power series, and let I be a cl08ed ideal in A. Then I i8 a standard ideal if 1= )\,h(A) fOT 80me k E z+ aT if I = 0, and I 'is a non-standard ideal if I zs not standard. The algebra A 'l8 unicellular if each closed zdeal zs standard. and a weight W on Z+ is unicellular if £ 1 (w) Z8 unicellular.
We shall show that weights satisfying various conditions are unicellular. First note that, if I is a closed ideal in f 1 (w) and if Xk E I for some k E Z+, then Ah(w) C I. For I:)uppose that a = 2:]:k ajX j E Mk(W), Then clearly a = limn -->00 Xkb n , where bn = 2:;=0 (Lj+kXJ E £l(W), and so a E I. Proposition 4.6.19 Let w be a radtcal weight on Z+. Then the following conditions are equivalent: (a) W is unicellular; (b) for each a E £l(W)·, there exist8 k E Z+ with Xk E a€l(w); (c) for each non-zero, closed ideal I in £ 1 (w), there exists k E Z+ with X k E I. Proof Clearly (a)==>(b){::}(c). (c)==>(a) Let I be a non-zero, closed ideal in £l(W). By (c), there exists k E Z+ with X k E I, and so Mk(W) C I. Set n = min{o(a) : a E I}. Then Ie Mn(w). If k = n, then I = Mn(w). If k > n, take a E I with o(a) = n, say
Banach algebras of power series
509
a = E7=n ajX j . Set b = E~:-~ ajX j - n and c = E}:k ajXj. Then a = Xnb+c and c E Mk(W) C I, and so Xnb E I. Also bE qXj C flew) and 7ro(b) = an -:10, so that b E Inv f 1 (w) because f 1 (w) is a local algebra. Thus xn E I and Mn(w) C I. We have shown that I is the standard ideal l\1n (w), and so (a) holds. 0 Before giving the next re..'lult, we introduce HOme further notation. Let w = (w(n)) be a weight on Z+. For k E Z+, set
S_k(W)
= {a E ~: Xka E flew)} and
Iiall- k
= jjXkallw
(a E S_k(W»).
Then, as a Banach space, (S-k(W), II· "-k) is flew), where w is the translated sequence given by wen) = wen + k) (n E Z+), and S-k(W) :::> flew). In general, S-k(W) is not an algebra because w may not be a weight. However, if w is a radical sequence, then w is a radical sequence. Let a = E:=o anxn E~. Then we define
n=O we say that lal :S Ibl in ~ if l7rn (a) I :S l7rn (b) I (n E Z+). Clearly, if a E S-k(W) and b E ~ with Ibl :S 14 then bE S-k(W). Lemma 4.6.20 Let W be a wetght on Z+, and let k E Z+. Suppose that bE Ml(W) and that (1 -Ib!)-l E S-k(W). Then bm E S_k(W) (rn EN). Proof Set (' = (l-Ibl)-l, so that c = 1 + E~l IW in ~ and c E S-k(W). Take mEN. Then l1l"n(b m )I :S l1l"n(lbl m )1 :S 7rn (c) (n EN), and so bm E S_k(W), as 0 required. Proposition 4.6.21 Let W be a wetght on Z+, and let k E Z+. Then the following are equivalent: (a) sup{w(rn + n + k)/w(rn + k)w(n + k) : rn, n E Z+} < 00; (b) S-k(W) is an algebra. Suppose that W is a radical weight. Then these condttions are also equivalent to: (c) X 2k E af1(w) for each a E flew) wzth o(a) = k. Proof Set A = S_k(W); the norm in A is II· II- k . (a)=:}(b) Let the supremum in (a) be C. Then, for a, bE A, we have oc
IIabll_ k :S
L L
lamllbnl w(rn + n
+ k)
r=Om+n=r 00
:S C
00
L laml w(m + k) L Ibnlw(n + k) = C lIall_ k IIblL k ,
m=O n=O that (b) holds. (b)=:}(a) The algebra A is a Banach algebra of power series, and so, by 4.6.5, there exists C > 0 such that lIabll_ k $ C lIall_k IIbll_ k (a, bE A). Applying this with a = xm and b = xn leads to (a).
80
510
Commutative Banach algebras
Now suppose that w is a radical weight. (b)=>(c) By (b), A is now a local algebra. Take a E fl(W) with o(a) = k, say a = Xkb, where b E (Inv3') n A = Inv A. Then X 2k = (Xkb)(Xkb- l ) E afl(w), and so (c) holds. (c)=>(b) To establish (b), it suffices to prove that a 2 E A for each a E MI(A). Take a E MI(A), and set b = X k (l-lal). Then b E fl(w) with o(b) = k, and so, by (c), there exists c E f I (w) with X k (l-laDc = X2k. In fact, c = X k (1-lal)-l, and so (1 - lal)-l E A. By 4.6.20, a 2 E A. Thus A is an algebra. 0 Again let w be a weight on Z+. Then we define
S-oc(w) = U{S-k(W) : k E Z+}. Clearly S-oo(w) is always a subalgebra of 3', but S_",,(w) may not be a local algebra even if fl(W) is; this holds if and only if 1 - a E InvS_oo(w) for each a E MI(S-oo(W». If each S_k(W) is a local algebra, then so is S-oo(w). Proposition 4.6.22 (Grabiner) Let w be a weight on Z+. Then the followmg conditions are equwalent: (a) S_CJO(w) is a local algebra; (b) for each non-zero ideal I zn flew), there extsts kEN with Xk E I; (c) for each non-zero ideal I zn fl(W) and each a E MI(w)·, there exists kENwithakEI; (d) the only pnme ideals zn flew) are .MI(w) and O. Proof (a)=>(b) Let I be a non-zero ideal, and choose a E I·, say a = Xnb, where b E Inv 3'. Then b E S-oo(w), and so, by (a), b E Inv S-oo(w). There exists mEN with Xmb- l E flew), and then xm+n = a(Xmb- l ) E I.
(b)=>(a) Take a E S_oc(w) n Inv3', say ab = 1 in 3'. There exists n E Z+ such that Xna E flew), and so, by (b), there exists c E flew) and m E Z+ with xm = (Xna)c. Then Xmb = Xnc E flew), and so b E S-oo(w). Hence a E IIlV S_oo(w). (b)=>(c) Let I be a non-zero ideal in fl(W), and take a E Ml(w)·, say o(a) = mEN and 1Tm(a) = 1. By (b), there exists kEN with X k E I. Now a = xm(l - b) for some b E lVh(S_m(w»). We have Ibl E S-m(w) C S-oo(w), and so, since (a) holds, (l-lbl)-1 E S-oo(w), say (l-Ibl)-l E S-i(w). For each r E N, br E S_j(w) by 4.6.20, and so Xib T E flew). Thus
ak+i = xm(k+i)
k+J (
~
k;
= X kX(m-l)(k+3)
J). (-br
~(-lr
e;
j)Xibr E Xk(l(w) c I.
(c)=>(d) Let P be a non-zero, prime ideal in (I(W), and take a E Ml(W)·. By (c), there exists kEN with a k E P. Since P is prime, necessarily a E P, and so P = Ml(W).
Banach algebras of power series
511
(d)=}(b) Let I be a non-zero ideal in £1(W), and assume towards a contradiction that Xk f/. I (k EN). By 1.3.44(ii), there is a prime ideal Pin £l(w) with I c P and X f/. P, and so P is a non-zero, prime ideal in £1(W) with P =j:.A1t(w), a contradiction of (d). Thus X k E I for some kEN. 0 We can now summarize some relations between properties of weights on Z+; ordinary algebras were defined in 1.6.24. Definition 4.6.23 A sequence w
= exp( -'TJ) on Z+ is:
(i) convex if'TJo = 0, if'TJl ;::: 0, and if ('TJn+! - 'TJn : n E Z+) is increasing; (ii) star-shaped zf'TJo = 0, if'TJ1 ;::: 0, and if ('TJn/n : n E N) zs increasing; (iii) a basis weight zf, for each kEN, sup{w(m + n + k)/w(m + k)w(n + k) : m, n E Z+} < 00; (iv) an ordinary weight if £ 1 (w) is an ordinary algebra. Clearly a convex sequence is star-shaped, and a star-shaped sequence is a weight sequence, for, if exp( -'TJ) is star-shaped, then
17m+n + n · 'TJm+n 'TJm+n=m·--m+n m+n ;::: m· -'TJm + n . -'TJn = 'TJm + 'TJn (m, n E Z +). m n Also a convex sequence is a basis weight: for each kEN, we have 'TJm+n+k - 'TJm+k ;::: 'TJn+k - 'TJk
(m, n E Z+) .
The nature of a star-shaped weight w = exp( -'TJ), and the reason for the name, can best be appreciated by drawing the graph of 'TJ: the graph is 'illuminated from the origin'. By 4.6.21, a weight sequence w is a basis weight if and only if S-k(W) is an algebra for each k E Z+. Suppose that w is a radical, basis weight. Then S-oo(w) is a local algebra, and so, by 4.6.22. the only prime ideal in the radical algebra Ml(W) is 0. It follows from 4.6.21 that a radical, basis weight is an ordinary weight, and from 4.6.19 that a radical, ordinary weight is unicellular. Thus we obtain the following result. Proposition 4.6.24 Take'TJ = (n"Y), where'Y > 1, or take 'TJ = (n(log(n+ 1»)'1'), where 'Y > 0, or take 'TJ = (n log log(n + 2». Then w = exp( -17) is a radical, convex wezght on Z+. Further, every closed ideal in £ 1 (w) is standar'd, £ 1 (w) is an ordinary algebra, and the only pnme zdeal m Ml(W) is O. 0 Proposition 4.6.25 Let (A, II· II) be a Banach algebra of power series. Then there is a convex wezght w such that A C £1(W) and lIallw ::; IIail (a E A). Proof Choose a sequence 'TJ = ('TJn) such that 'TJl 2:: 'TJ0 = 0, ('TJn+l - 'TJn : n E Z+) is increasing, and'TJn 2:: (n + 1) log 2 + log \l1rn ll (n EN), and set w = exp( -'TJ); clearly IIalL ::; IIail (a E A). 0
Commutatwe Banach algebms
512
Proposition 4.6.26 Let w be a r'adical weight on Z+, let a E f 1 (w). and let k E Z+. Suppose that
(Ilan/l", /w(kn))I/n: n E N) zs bounded. Then a weight.
E
1'vI,.,(w). The converse holds m the case where w zs a baszs
Proof Assume towards a contradiction that a rt. l\fdw), say o(a) = ], where < k, and set a = 7Cj(a) =I O. For each n E N, an E anX jn + Jvfjn +1(w), and so Ilanll w ~ o:nw(Jn). However, w(kn) :::; w(jn)w(n)k- i , and hence
j
(1laTlll w /w(kn))I/n
~ o:/w(n)(k-j)/n ---.
00
as n ---.
00,
a contradiction. Now consider the converse implication in the case where w is a basis weight. Take a E l\fk(W), say a = Xkb, where b E 8_k(W). By 4.6.21, there is constant C ~ 1 such that, for each n EN,
Ilanllw = IIXk(n-l)bntk :::;
c Ilx k(n-l) tk Ilbnll- k
:::; c IlxknlL Ilbll~k :::; Cnw(kn) Ilbll~k . and so (/laTlllw /w(kn))I/Ti :::; C
/lbll_k.
o
giving the result.
We shall finally show that weights in another wide class are unicellular. We require a preliminary result on radical weights. Lemma 4.6.21 Let w be a mdzcal weight on Z+. let ko E N, and set
,6.(k) = max {
W(ko+k)j/k. w(ko + j) : J
E N k- 1
} .
Then ,6.(k) < 1 for mfinitely many values of k. Proof Assume towards a contradiction that there exists ki E N such that ,6.(k) ~ 1 (k ~ kl), and set
= min{w(ko + J)I/ j : j E Nk,}. there exists Jl < k with w(ko + k)I/k c
For each k > k 1 , ~ w(ko + iI)I/J 1 • If ji > kI, there exists j2 < jl with w(ko + jI)I/j, ~ w(ko + h)l/h, and hO, continuing. we find that there exists j :::; kl with w(ko + j)l/j :::; w(ko + k)l/k. Thus 0 < C :::; w(ko + k)l/k (k EN). But w(ko + k)l/k ---. 0 as k ---. 00, a contradiction. 0
In the proof of the next theorem, we use the following notation: for an element a = E~oajXj E J and n E N, we set
n-l
8 na
= Xna,
Pna
= L ajXj,
00
and
Qna =
L ajXj ,
j=O j=n so that Pn +Qn is the identity operator on J. It is also now convenient to denote the product in ~ by * in the proof.
Banach algebras of power senes
513
Theorem 4.6.28 (Domar) Let w be a radical, star-shaped we~ght on Z+. Then w is 'lLnzcellular. Proof Take a E fl(W)- with o(a) = ko, say. We shall prove that there exists a sequence Uk) in flew) with liminfk~(X) * fk - X 2kollw = 0; by 4.6.19, this is sufficient for the result. We may suppose that 7fko(a) = 1. Set wen) = w(ko + n) (n E Z+). We first clazm that. for each j,p E Z+, the map 8 j is a bounded linear operator from Mp(w) into Mp+j(w), and that its norm satisfies 118j ll p ::; w(ko + J + p)j/(ko+J+p). (4.6.5)
lIa
To prove this. it is sufficient to show that
w(m + j + ko) ::; w(p + j w(m + k o) Since w is star-shaped, w(m + j
+ kO)j/(ko+J+p)
(m 2: p).
+ ko)l/(m+J+ko ) ::; w(m + ko)l/(mHo),
w(m + j + k o) ::; w(m + j + ko)l-(mHo)/(m+J+k o ) = w(m + j w(m + ko) ::; w(p + J + koF/(p+jHo) (m 2: p), giving (4.6.5), as claimed. Take b E ~ with a = Xko c E ~ with b * c = 1. We set
* b,
= Xko - a,
b=
+ ko)J!(m+j+ko)
so that b E flew) and 7fo(b) = 1, and take 00
00
a
and so
L bjX j = 1 - b,
c=
L cjX j =
1 + C,
j=O
j=O
so that a E flew), b E flew), C E ~, a = xko * b, o(b) 2: 1. and o(e) 2: 1. (However, we cannot assert that c E f 1 (w) because f 1 (w) may not be an algebra.) For each kEN, it is clear that PkC E qX) C flew), and so it is sufficient to show that
for, in this case, the result follows with fk = xko a * (Pk + Qk)c = Xko in ~. we have
o(a and so a
a
* Pk(' -
* Pkc -
liminf k~oo
XkO) = o(a
= Qko+k(a * Pkc - XkO). (XkO - a) * (1 + PkC) - Xko
* PkC) =
0 and IIQkoHUilw
PkC, Take kENo Since
* QkC) 2: ko + k,
Xko
Xko =
But QkoH(Xko to show that
* Pkc -
*
--+
Also = Xko
0 as k
--+
* PkC -
U - U * Pkc.
oc, and so it is sufficient
IIXko * Qko+k(a * PkC) I w = liminf IIQko+k(a * Pkc)lb = O. k~oc
Take k ~ 2 and r E Nk-l. Then
Qko+k(a
* Xr) = Qko+kSko+rb = Sko+rQk-rb,
Commutative Banach algebras
514
:s I/bl/""
and Qk-rb E Mk-r(W) with I/Qk-rbll", j = ko + rand p = k - r), we have
IIQ ko+k(a:
and so, by (4.6.5) (applied with
* Xr)lb :S w(2ko + k)(ko+r)/(2ko+k) Ilbll", :S uJ(ko + k)(ko+r)/(ko+k)
Ilbll", .
2:::;
Since Pkc = c.,.x r and w(ko + k)(ko+r)/(ko+k) :S w(ko + kr/ k for r < k, we see that it suffices to show that k-1 (4.6.6) liminf" Icrl w(ko + kr/ k = O. k->oc L...J r=l
We now estimate the sum in (4.6.6). Take k 2: 2 and temporarily write k - l Ibj I ~k' j Smce . -c = " " " b , we have ~k = w ( ko + k )1/k and (3k = " L.....,j=1 L.....,8=1
-*8
00
Ic! :S L
Ibl*8 = Ibl * (1 -lbl)-1
IPk C! :S Ipkbl
and
*
(1 -l p kb j)-1 .
8=1
We regard PkC and Pkb as polynomials, and evaluate the second term, above, at the point ~k to see that 2:~:; ICj! ~t :S (3k(1- (3k)-1 provided that (3k < 1. Thus we finally see that it suffices to show that lim infk--> 00 (3k = O. The sum for (3k may be written as
k-1 =" w(ko + k)J L...J (k j=1 + )
"/k
(3k
W
J
0
IbJ !-(') wJ .
By 4.6.27, there is an infinite subset, say S, of N such that
w(ko + k)j/k w(ko + j) < 1
2:j:1
(k
E S, j E N k-
1) .
Ilbl/",
Also Ibjl w(j) = < 00, and w(ko + k)J/k /w(ko + j) ---.. 0 as k ---.. 00 for each j E N. Hence it follows from the dominated convergence theorem that (3k ---.. 0 as k ---.. 00 with k E S, as required to complete the proof. 0 The question that we have left open so far is whether or not every radical weight is unicellular. This question was posed by Silov in 1940, and it was finally resolved by Thomas in 1984. Theorem 4.6.29 (Thomas) There zs a radical weight w on Z+ such that the 0 Banach algebra f 1(w) contains a non-standard closed ideal.
Unfortunately, we are not able to include a proof of this result here. The basic idea is as follows. One takes a sufficiently rapidly increasing sequence (nk) in N (for example, it is sufficient to take nk = 22k) and defines the weight w inductively so that w(nk + n) = w(nk)w(n) whenever nk + n < nk+l, and so that W(11k+1) is very much smaller than w(nk+l - 1). If certain inequalities are satisfied, one obtains a radical weight w such that the element a = 2- k x nk / W( nk), which is supported 'at the large drops in w', generates a non-standard, closed ideal in f 1 (w).
2:%:1
Banach algebras of lJOwer se7'1.CS
515
Let w be a weight on Z+ such that .e 1 (w) contains a non-standard. closed ideal I. Then the algebra e(w)/I is a local Banach algebra; it seems to be a particularly intractable example in this class, and essentially nothing is known about it. For example. it would be interesting to know if it could be an integral domain. We conclude' this section by identi(ying the mUltiplier algehra of an algehra M1(W); we again denote the product in ~ by juxtaposition. Let w be a weight on Z+ , and set M = 1\,h (w). Since 1\,J is an integral domain, each multiplier on M is continuous. and so the multiplier algebra M(M) is a closed subalgebra of (13(M).III·11D (where 111·111 now denotes the operator norm on M). As in 2.5.12(ii), the embedding of M as an ideal in M(M) is continuous. However, since M does not have a bounded approximate identity, there is no reason for M to be closed in M (M). For a = 2::;:"=0 anX n E ~, set La = 2:::'=0 an+lX n . so that SlL is the identity operator on 1\,h. Let w be a weight on Z+. Then ~w=
{
~
~
n
a=L-anX E~:lllalllw=supL-lajl nENj=o
n=O
w(n+j) } () o. It contradiction of the fact that w is radical. Hence pI (w) =f. S -1 (w). D For example, in th{' case where wen) closed in its multiplier algebra.
= cxp(-n2 ) (n EN). Ah(w) is not
Notes 4.6.32 Theorem 4.6.1 is due to Allan (1972) (in the casE' where n = 1) and to Becker and Zame (1979) in the general case; our proof is different. The algebras fl(Z.W) and fl(w) are discussed in §19 of the text (Gel'fand et al. 1964). and 4.6.9 and 4.6.12 are given there. The algebras flew) are also discussed in (Rickart 1960, A.2.12). The Beurling algebras Aw ~ f I (Z, w) are -'locally isomorphic' to certain Beurling algebras L I (JR, w), to be introduced in §4.7 (Reiter and Stegeman 2000, 6.3.13). It follows that Aw is regular on 1r if and only if ;:.. L.,
logw n
1 + n2
ILf described above is an isometric linear embedding of L1(I.w) into M(I.w). Let w be a positive. measurable function on 1 such that f K w < x for each compact subset K of 1. ThE:'n LOO( -1, :;;-1) is the set of complex-valued, measurable functions A on - 1 such that X/W is essentially bounded on I; L DC ( - 1, (;;-1 ) is a Banach space with respect to the norm given by
IIAlloc:..", =
esssup{I>'(t)l/w(t) : tEl}.
Cf'rtainly L=(-1.(;;-1) is the dual space of L1(I.w) with respect to the pairing (f. A) = .I~ I i In the case where w is a positive. continuous function on 1, we definE:' C o( -1. (;;-1) to be the set of complex-valued. continuous functions I on -1 such that ]/w E Co(I). Then Co(_l.:;;-l) is a clost'd subspace of L=(-1.(;;-I), so that l'vI(-I.w) is the dual space of Co (_1,':;:'-I) with respect to the pairing (cf. (4.5.4»
(f,IL)
=
If(-t)dlL(t)
(jECo (-I,(;;-I),/J.EM(I,w»).
For a measurable function w onjR+, set p", mdical if p", = O.
= inf{w(t)l/t : t
~
I}. Then w is
Definition 4.7.3 Let S be jR or jR+. A weight function on S is a real-valul'd, measumble luncizon w on S such that
,,-,'(0)
=
1,
w(S»O
(SES).
,,-,{~
+ t)
:::; w(s)w(t)
A wezght lunrt'ion w zs: (i) bounded zj ~mpw(S) < origin zj limsuPt---+o+ w(t) < 00.
00;
Let w be a weight function on jR+. and set T/(t) that w = exp (-1]). ThE:'n
=
·,,(0) =
o.
1](09)
+ 1](t)
:::; 17(8 + t)
(.'I, t E S).
(ii) bounded near the
-logw(t) (t E jR+). so (4.7.5)
(8, t E jR+):
if w is radical, then, as we shall see in 4.7.4(iii), 1](t)/t
---t
oc
llli t
---t
(4.7.6)
oc. I
Conversely. ifr/ is a measurable function on jR+ satisfying (4.7.5), then exp (-7/) is a weight function, and this function is radical if (4.7.6) also holds . .For examples of weight functions, first set wq(t) = e-O't (t E jR+), where a E JR. Then Wa is a weight function on jR+. and W q is bounded if a ~ O. Next, we say that a function W = exp (-1]) : jR+ ---t jR+- (with 1](0) = 0) is convex if 1](s + t) - 7}(s) is an increasing function of S for each t E jR+, and star-shaped if 1](t)/t is increllliing on jR+-; as in §4.6, a convex function is star-shaped, and a star-shaped function is a weight function on jR+. Finally, a weight function W is regulated if .,,(s + t) - .,,(s) ---t 00 as S ---t 00 for each t E jR+-; a convex,
Con'Uolutwn algebras on the real line
521
radical weight function is regulated. For example, set "1 : t t--+ fl, where I > ], or "1 : t t--+ t(log(l + t))"!, where I > 0, or 71 : t ...... tloglog(t + 3); we obtain continuous, radical. convex weight functions on JR.+. Again, set "1: t t--+ (t 3 -l)/t (with 17(0) = 0); in this case exp (-"1) is a radical weight function which is not bounded near the origin. Finally, we note that the fundiomi t t--+ ('xp(ltll') and t t--+ (1 + Itl)° are weight functions on JR. for I E IT and for 0' E JR.+, respectively.
Lemma 4.7.4 Let LV be a weight function on JR.-+. Then: (i) supw(K) 0 for each compact subset K of JR.+; (iii) Pw = limt~oo W(t)1/t; (iv) tf w zs bounded, then there 'tS an equwalent bounded wrzght function on JR.+ such that w zs decreasmg.
w
Proof Set w = exp (-"1). as ahove. (i) Take a, b with 0 < a < b < 00. Assume that for each n E N there exiHts tn E [a, b] with TI(t n ) ::; -no Set En = {t E [0, b] : ry(t) ::; -n/2}. a mea..'mrable set. Since ry(t) + 17(t n - t) ::; ry(t n ) (t E [0. tn]), we have En U (t n - En) :::) [0. t n ], and so rn(En) ?: t n/2 ?: a/2. Since En+1 c En. we have En) ?: a/2. Thus there ('xists t E En. But TI(t) ::; -n/2 for all n E N. a contradiction. Hence inf17([a, bJ) > -:xJ, and thb proves (i). (ii) Assume towards a contradiction that there exists a > 0 and (t n) in [0, a] with ry(t n ) ?: n (n EN). We may suppose that (tn) converges, say tn ~ to· Let M = inf17([1,3J). and set 8 n = to + 2 - tn. Then 8 n E [1.3] eventually, and so ry(to + 2) ?: 71(Sn) + ry(t n ) ?: AI + n for all large n, a contradiction. Hence sup "1([0, a]) < 00, and (ii) follows. (iii) By using (i), we see that this is essentially the same proof as that of A.1.26. (iv) Suppose that wet) ::; M (t E JR.+). Define
rn(n;::l
n:=1
wet) = inf{w(s) : 0 ::;
8 ::;
t}
(t E JR.+).
By (ii), wet) > 0, and it is easy to check that w is a decrea.."ing wf'ight function on JR.+ and that wet) ::; l\fw(t) ::; Mw(t) (t E JR.+). 0
Theorem 4.7.5 Let I be JR. or JR.+, and let w be a weight function on I. Then L1(I,w) is a subspace of Lfoc(I), and (L1(I,w), *, 1I·lI w ) is a commutative Banach algebra. Proof Let w be a weight function on JR.+, and let K be a compact subset of infw(K) > 0, and this implies t.hat L1(JR.+,W) is a subset of Ltoc(JR.+). Clearly £1 (JR.+, w) is a Banach space. For f,y. E L1(JR.+,w), we see immediately that f * g E L1(JR.+,w) with II! * giL::; IIfllw Ilgliw' Hence L1 (JR.+, w) is a Banach algebra. Now let w be a weight function on JR.. By 4.7.4(ii) applied to wand W, infw(K) > 0 for each compact subset K of JR., and so Ll(JR.,W) c Ll"oc(JR.). Again Ll (JR., w) is a Banach algebra. 0 ]R+. By 4.7.4(ii),
522
Commutative Banach algebras
Of course. if w = 1 on lR, or lR,+, then we ohtain the Banach algebras L 1 (lR,) and Ll(lR,+), respectively. We write U(w) for the Banach algebra L1 (lR,+ ,w). By 4.7.4(i). Coo(lR,+e) is deni:le in Ll(w). The algebras L1 (lR" w) for w a weight function on lR, arc called Beur'ling algebms. As before, the shift operator is denoted by Sa, so that (Saf)(s) = f(s - a). Note that, if a > 0 and f E LloJlR,+), then (Saf)(t) = 0 (t E [0, aD. If w is a weight function on lR,+ and a E lR,+, then Sa E B(Ll(w)) with
IISal1 ~ sup {w~(:)a)
: S E lR,+ }
~ w(a).
(4.7.7)
Clearly (Sa: a E lR,+) is a real semigroup in B(Ll(w)).
Lemma 4.7.6 (i) Let w be a wezght function on lR,+. Then, for each f E Ll(W). the map a I-> S,d, jR+ -> Ll(w), 2S continuous on lR,+e, and 2t 2S also continuous at a = 0 if w is bounded near the origin.
a
(ii) Let w be a weight functwn on R Saf, lR, -> Ll (lR" w), is contmuous.
Then, for each f
E
L1(lR"w), the map
I->
Proof (i) First take g E Coo(lR,+e). It followi:l from 4.7.4(ii) and 3.3.11 that we have IISag - Sa"gIL" = O(IISag - Saoglll) -> 0 as a --+ ao in lR,+. Now let f E Ll(w). Take t > O. and choose g E Coo(lR,+e) with IIf - gliw < t. Then
+ II Sao II) Ilf - gllw + IISag - Saogliw (11Sall + IISaoll)e + IISag - Saogllw . -> ao provided that liSa II ii:l bounded for a
IISaf - SaoflL ~ (IiSall ~
Thus Saf -> Saof as a in some neighbourhood of an. By (4.7.7) and 4.7.4(i), thii:l is always true for ao > 0, and it is also true for ao = 0 if W ii:l bounded near the origin. (ii) This is similar.
0
We define
M(lR,+e,w)
= {J-L E M(lR,+,w)
: IL({O})
= O}.
Let S be a subsemigroup of I, where I is lR, or lR,+. We have previously (in 2.1.13(v» defined the Banach algebra fl(S,W) when w is an arbitrary weight on S. In the case where w is a continuous weight function on I, we identify f 1 (S, w) with the subset of M(I.w) consisting of the discrete measures oni S. The following result is straightforward.
Theorem 4.7.7 Let I be lR, or lR,+, and let w be a continuous weight function on I. Then: (i) (M(I,w), 11·lIw) is a unital Banach algebra with identity 80 ; (ii) M(lR,+e,w) is a maximal ideal m M(lR,+,w); (iii) fl(S,W) is a closed subalgebra of M{I,w) for each subsemigroup S of Ii
(iv) L 1 (I,w) is a closed ideal in M(I,w).
0
Convolution algebras on the real line
523
Let I be lR. or lR.+, and let w be a weight function on I. For I E L 1 (I,w), ), E LOO(-I,w- 1 ), and t E I. we have
(f
* ),)( -t) =
1
l(s)X(s + t) ds =
1
I(s - t)X(s) ds
= (Bd,),)·
(4.7.8)
In the case where w is continuous, take f..L E M(I. w) and A E LOO( -I, w- 1 ). Then A . f..L E LOO( -I, w- 1 ) is defined by
(f, A . f..L) = (f
* f..L,
A)
(f E L 1 (I,w))
(cf. (3.3.8)). If IE Co(-I,w- 1 ) and t E I, then I * f..L exists at -t, and l(f * f..L)(-t) I < (1](s+t)ld1f..LI(s) wet) - if wet)
S
}
11 ~~:: :~ w(s) d 1f..L1 (s) s II/II"".w IIf..L1Iw .
(4.7.9)
1
It now follows easily from the dominated convergence theorem A.4.6 that (f * f..L) I-I E Co(-I,w- 1 ). Also, for each g E L 1 (w), we have
(g. (f
* f..L) I -I) = =
1 get) (1](."1 + t) df..L(S)) dt 00
1 (1 00
leu)
g(u - s) df..L(S)) du
and so we may identify I . f..L with the function (f * f..L)
= (g * f..L. f) ,
I -1.
Theorem 4.7.8 Let 1 be lR. or lR.+, and let w be a continuous weight functwn on 1. Then (Loo( -1, w- 1 ), 1I·lIoc,w) is a dual Banach M(l, w)-module wzth respect to the module operation (A,f..L) f-t A . f..L, and (Co(-1.w- 1 ), 1I'lIoo,w) zs a closed submodule 01 LOO(-1,w- 1 ), the module operation being (f,f..L) f-t (f * f..L) I -1. The dual module operatwns on M(I.w) = (CO(-1,w- 1 )) , comczde with the pmduct in M(I,w).
o
Proof This is essentially the same as the proof of 3.3.15.
Proposition 4.7.9 Let w be a continuous, regulated weight functwn on lR.+. Then, lor each I E Ll(W) and A E Ll(w)', I . A belongs to Co(lR.-,w- 1 ). Proof Essentially
a.. O. If / E L 1(w.,.), then Cj E Ao(ITq) by 4.5.3(iii) and A.2.25. 0 Thus C : M(lR+,w.,.) monomorphisms.
---+
Ab(IT.,.) and C : LI(Wq)
---+
Ao(IT.,.) are continuous
Corollary 4.7.13 Let Il E Mloc (lR). Suppose that fn~. t n dp,( t) = 0 (n E N) and that 0 E int VCw Then J.l = 0 in Mloc(lR). Proof Since 0 E int V Cp" it follows immediately from equation (4.7.11) that (.CIl)(n)(o) = 0 (n EN). Thus CM is constant on VCP,' whence CM = 0 and P~Q
0
Examples 4.7.14 (i) Set jet) = eO"t (t E lR+). Then VC! = IT q , and 1
(Cf)(z) = z-a
(ii) For
(z E IT.,.).
eE lR+, set
e)
€ exp ( - 4t Ce(t) = 21l"1/2t3/2
(~th Ce(O) = 0), so that Ce E L1(R+), and let He = CCe. Then VH E = II. 'l'he following calculation gives HF,. explicitly.
Commutative Banach algebms
526 Take x E JR+. Then
,; roo t 1 exp (e) } - 4t - xt dt ~ (~) 1/2 roo _1 exp (-s _ex) ds, 2 11" Jo 8 2 48
H~(x) = 211"1/2 Jo =
3/ 2
(4.7.12)
3/
where we are setting
8
cp(u) =
= xt. Define
roo _1_ exp (-s _u Jo 2
8 1/ 2
)
ds
8
Then cP is continuous on JR.+ with CP(O) = r(1/2) = 11"1/2. Also, cP is differentiable on JR.+. with cp'(u) = -2u
1
00
s31/2 exp ( -s - 1:2) d8
(u E JR.+.).
(4.7.13)
Setting 8 = u 2 It, we see that cp'(u) = -2
roo _1_ exp (-t - ut Jo t l/2
2
)
dt
= -2CP(u) (u E JR.+.).
Hence cp(u) = ke- 2u (u E JR.+.), where k is a constant. Since cP is continuous on JR.+, we have k = cp(O) = 11"1/2. From (4.7.12) and (4.7.13), we obtain 1 cP' (';Xl/2) H~(x) = 2;1/2 -2- = exp( _~xl/2)
(x E JR.+) .
Since H~ E Ao (IT), it follows that (.cC~)(z) = He(z) = exp(-,;z1/2)
(z E TI).
(4.7.14)
o We now discuss the inverse Laplace transform .c- l in the form in which we shall use it. Take G 2:: O. The Banach space (H1(TIu), 11.11 1) is defined in A.2.38: a function F belongs to Hl(TIu) if F E O(TIu) and 11F111 =
!~~ [: IF(x + iy)1 dy < 00.
For example, suppose that F E A(TIu) and F = O(Z-2) as z ---+ 00 in TIu. Then F I TIu E H1(TIu). (In fact, in our applications, these are the only functions F for which we shall define .c-lF; the results in 4.7.16 and 4.7.17; below, can be seen very easily for these functions.) Note also that, if C(z) = exp(-zO), where 0< 8 < 1, then ZnC E Hl(TIu) (n EN). Definition 4.7.15 Let F E H1(TIu), where G 2:: transform of F, denoted by .c- 1 F, 'lS defined by (.c-1F)(t)
where
T
>
G.
= 211" 1
1
00 -00
o.
Then the inverse Laplace
F(T+iy)e(r+iy)tdy
(tEJR.),
Convolution algebras on the real lme
527
By A.2.39(iii), IF(z)1 --> 0 as z --> 00 in II... for each T > (1', and so it follows from Cauchy's theorem that th£' value of (.C- l F)(t) is independent of the choice of T. Set F... (y) = F(T + iy) (y E JR) for T > (1'. Then
(.c-lF)(t)
= 2~ e...t(FF... )(-t) = e... t (F- 1 F... )(t)
(t
E
JR).
(4.7.15)
where F and F- l denote the Fourier and the inverse Fourier transforms, respectively, as defined in §4.5. Again note that .c- 1 0 £ is only known to be the identity operator on {J E Lfo("(JR) : £f E Hl(IIuH·
Theorem 4.7.16 Let F E Hl(IIu), tvhe1"(~
(1'
~ 0, and .set f = £-IF.
zs continuous on JR, and f IJR- = O. For each T > (1', f IJR+ E Ll(w... ) and IIfll"'7 ~ 11F111 j27r(T -
(i) The function f (ii)
(1').
(iii) Suppose that F = O(Z-lI) as z --> 00 in ITu for each n EN. Then f is infinitely differentiable on JR, and f(k)(O) = 0 (k E Z+).
Proof (i) By 4.5.3(iii), FF... E Co(JR) (where T > (1'), and so, by (4.7.15), f is continuous on R Take t ~ O. First suppose that G E Hl(IIu) is such that G = O(Z-2) as z -+ 00 in IIu. Then
lin G(z)e dzl zt
=
O(R- 1 )
as R
--> 00,
where r R is the semicircle in C from T + iR through T + R to T - iR, and so it follows from Cauchy's theorem that (£-lG)(t) = o. By A.2.40, such functions G form a dense subset of Hl(IIa), and the map G 1---+ (£-lG)(t) is continuous. and so f(t) = O. (ii) This follows because
If(t)1
~
11F1l1 eut j27r (t
E JR+).
zn
(iii) The condition implies that F E HI (IIu) (n E N), and so the result follows from (i) by the use of 4.5.11(ii). 0 Theorem 4.7.17 Let F,G E Hl(IIu). where (1' ~ 0, and take T > FG E HI (II... ) and £-1 (FG) = (£-1 F) * (£-1G).
(1'.
Then
Proof Clearly FG E HI (II... ). Take p > T, and also set f = (.c- 1F) IlR+, g:::: (£-1G) IJR+, and h = (£-1 (FG» IJR+. Then f, g. h E £1 (w p ), and we have £h:::: FG = £(J * g) in Ao(TIp). Thus h = f * g. 0 For example, if HI (z) = exp( _Z1/2) (z E IT), then it follows from equation (4.7.14) that
(.c- 1 Hd(t) =
{~"I;t3/' exp (~ ~t)
(t > 0), (t
~
(4.7.16)
0).
Our next objective is to prove Titchmarsh's convolution theorem. First, we require some preliminary facts which are themselves important.
528
Commutatzve Banach algebras
Definition 4.7.18 Let p E M1o(·(R)· and f E Ltoc(R)·. Then a(p)
= inf supp p
and
aU)
= inf supp f ,
and we set nCO) = oc. For We Af1o(·(R), define a(W) = inf{u(p) : pEW}.
Suppose that p E Mloc(R) and o:(p) > -oc. Then a(p)
= sup{o E R: p((-oc, o))
=
o}.
The fundamental link between the value of a(p) and the behaviour of £p is given in the following proposition. The proof includes a refercnce to functions of exponential type: these functions are defined in A.2.24.
Proposition 4.7.19 Let p E l\.1(R). Then the following conditwns on a E R ar'e equzvalent:
(a) a(p) (b)
~
1)C/1. :,)
a:
IT and 1(£p)(z)1
= O(e- UT ) as
z ~
')C
in
IT;
(c) there exzsts G E A(IT) su,ch that G(iy) = ji(y) (y IG(z)1 = O(e-aX) as z --+ 00 m II; (d) a(p) >
E
R) and such that
IT, and there eXl,sts cP E (-7r/2, 7r/2) such that as r ~ 00.
-00, 1)c/J. :,)
I(£p) (rei'P) I = O(e-arCOS'P)
Proof Suppose that (a) holds. Then, for z E II, we have
IJR[ e-
zt
dP(t)1
~
[ J[a,oc)
e- xt d Ipl (t) = e-a.r
r
ex(a-t)
d Ipi (t)
J[a.oc,)
~ e- ax Il/lll
'
and so (b) holds. Next supposc that (c) holds. We define Pl(W) = p(W n (-x,a)) and P2(W) = p(W n (a, oc») for a Borel subset W of R, so that Il = III + P2· Also set Fl = £Ill and F2 = £p2. We have shown that 1) F2 :,) IT and that IF2(z)1 = O(e-aX) on IT. Similarly, 1)F1 :,) oIl and IF1(z)1 = O(e-aT) on oIl. We now define
(z E oIT), (z E IT). Since G(iy) = /L(Y) = FI(iy) + F2 (iy) (y E R), H is well-defined, and H is continuous on C. Since H is analytic on oIl U II, it follows from Morera's theorem A.2.6 that H is entire. But H is bounded on C, and so' H is constant by Liouville's theorem, say H = k. Then Fl(iy) = ke- iay (y E R), whence PI = koa · It follows that suppp c [a,oo) and that a(p) ~ a, proving (a). Trivially (b) implies (c), and so (a), (b), and (c) are all equivalent. ClearlY (a) and (b) together imply (d). Finally, suppose that (d) holds. Since a(p) > -00, £p is of exponential type on II by the fact that (a) implies (b), and £p is bounded on V. By A.2.27, 1(£f)(z)1 = O(e-aX) on II, and so (b) holds. [J
529
Convolution algebras on the real hue
Corollary 4.7.20 Let 11 E M(lR). Suppose that there exzsts an enti1'e function G of exponentwl type a 8uch that G(iy) = /iCy) (y E lR). Then sUPPIt C [-a, a]. Proof We have \G(z)\ = 0 (e(a+c)lzl) as z -+ x. in O. Since G \ V is bounded, \G(z)\ = O(e aX ) on IT by A.2.27, and so, by 4.7.19. we have SUPPll C [-a, (0). Similarly, sUPPIl C (-00, a]. 0
Our proof of Titchmarsh's convolution theorem will be based on the Ahlfors· Heins theorem, and it is in the following proposition that that theorem is applied. Proposition 4.7.21 Let 11 E M(lR) wzth It
lim
r--+x
=I 0
and 0.(11)
>
-00.
Then
~r log I(£It)(re i8 ) I = -a(lt) cosO
for almost all 0 E (-7r /2, 7r /2).
Proof Set F = Cit. Since (}!(I1) > -00, Dp ::> IT and F is of exponential type on IT by 4.7.19. Also, F =I 0 and F is bounded on the imaginary axis V. By the Ahlfors-Heins theorem A.2.47, there is a constant c E lR such that
~10gIF(rei8)1-+ r
asr -+ 00
-ccosO
(4.7.17)
for almost all 0 E (-7r /2, 7r /2). Let a = a(I1)' By 4.7.19, IF(re iO ) I = O(e-arcos8) as r -+ ex; for each (), and so c ~ a. Now take 0 E (-7r/2,7r/2) such that (4.7.17) holds, and take E > O. Then IF(re i8 )1 = O(e-(c-e)rcos8) a..'l r -+ 00, and so, by 4.7.19, a ~ C - E. It follows that c = a, and this gives the result. 0 Theorem 4.7.22 (Titchmarsh's convolution theorem) with a(I1), a(v) > -00. Then 11 * v E Mloc(lR) and a(1t
*
v) = a(Jt)
Let I1,V E 1I11oc(lR)·
+ a(v).
(4.7.18)
Proof Take k = min{ a(It). a(v)}. For W C (-x, t), (11 * v)(W) only depends on the values of It and l/ on sets contained in (-00, t + \k\), and so we may suppose that It and v have compact support, and hence that 11. * v =I 0 and 'D.cll = D.cv = efinition 4.7.24 Let u(t) = 1 (t E JR.+).
Commutatzve Banach algebras
530
by
The function u is of particular importance because convolution multiplication iil the operation of indefinite integration:
1.1,
(1.1,
* f)(t) = lot f(s) ds
(t E JR.+, f E
L~c(JR.+))·
The following lemma iil proved by an easy induction.
Lemma 4.7.25 Wzth u as above, tn -
1
1.I,*n(t) = (n _ I)!
(t
E
JR.+, n
E
N).
o
\Ve now introduce some ilpecific notation. Let w be a weight function on JR.+. Then (4.7.19) a w = -logp", = lim ry(t)/t.
t_= Note that, if In{+ w
< 00, then a w > 0 because (We take a", = ::xJ if Pw = 0.) ::::·e-"'w t . and so intD.cg::J IL.,..., ::J II for each 9 E L=(w- 1 ). Recall from 2.2.7 that an element a of a commutative Banach algebra A is a polynomial generator of A if Cola) = A.
wet)
Theorem 4.7.26 Let w be a weight function on JR.+ such that In~.+ w
0
as n -;
(x).
Let It E M(lR,+,w). Then there exist a E C and v E M(lR,+.w) such that #L = a80 + v and v( {O}) = O. Since v is the limit of measures with compact support in jR+., v is quasi-nilpotent. and so the result follows. 0
Definition 4.7.29 Let A = U{Ll(wo-) : a ~ O}. Thus A is a 8ubalgebra of the integral domain Lfc)(' (lR,+), and A C L1(W) whenever w is a continuolls, radical weight on lR,+. The algebra A is a union of semi simple Banach algebras.
Proposition 4.7.30 The algebm A# is local and Henselian. Proof The norm in L1 (wo-) is denoted by 11·11.,.; clearly Ilfllo- -; 0 a.. 0 such that 2(1 + kn 2 ) IIfoll.,. < 1, Where k = max{II12Il", ... , \lfnll".}. By 2.4.38, the Henselian polynomial fo + X + 12X 2 + ... + fnxn has a root in Ll(wo-), and hence in A. Thus A# is Henselian. 0
532
Commutatwe Banach algebras
The following specific calculations will be wied in §4.9 and §5.7. Proposition 4.7.31 Let w be a decreasing weight functwn on JR+.
(i) Let (nn) be a 8equence in JR+. such that lim n-->ooCa n/w(8n))1/n = 00 for each £5 > O. Then there eX18ts f E Ll(w) n L1(JR+) wzth a(J) = 0 and Ilf*nll", : .: 0:71 (n EN). (ii) Let h E L1(W) be such that (Y(h) = O. h(t) ::::: 0 (t E JR+), and J;~ h = 1. Then
Ilh*"IIw::::: wen)
11(80 + h) *"11,.; : : :
and
t, (;)w(j)
(n EN).
Proof (i) By hypothesis. for each kEN, there exists 3k E (0,1) with 2 n(JkW
(n) +
(z E
+z
II (1
)
and
1
(£h)(z) = k _
Z
We see that g + h = 2kg * h by noting that both sides of the equation have the Same transform, and so cp(g) + cp(h) = 2kcp(g)cp(h). This gives 1
1 2k = -,----,--,,---,k - (2 (k + (l)(k - (2) .
-- + -k
+ (1
Whence (1 = (2. Thus (1
E
X and cpU) = (£f)«r) (f
E
A). as required.
0
We discuss further the Beurling algebras L1 (~, w o,) in the case where wQ(t) = (1 + ItDO and a > OJ we write A(wo ) for .r(Ll(~, wo». By 4.7.33,
Commutatwe Banach algebras
A(~,,) i~ a natural Banach function algebra OIl Y. By 4.5.11(i), C~~) c A(~().) for k > n + 1 and so A(w",) is regular. As in 4.6.13, A(w,,) IKe lipo:K for each compact subset K of y, maximal modular ideals of A(woJ do not haw bounded approximate identities, and there are dil;continuoub point derivations at ach character. For f E Ll(Rwo ) and p > 0, define
(Mpf)(t) = pf(pt)
(t
E
JR).
ThC'Il j,fpf ELl (R wa ); IIMpft" :::; 11/11",,, if p ~ 1 and IIMplt" :-s; p-o II/IL"
if 0 < P < 1. Also, ri;J(y)
= [(yip) (y
E
JR).
Lemma 4.7.34 (i) Let I,h E J}(Rw a ) with J~h = 1. Then Mph * f ~ f as p --+ x. (ii) The Banach algebra A(wo ) has a sequential bounded appr'OximlLte identdy mntained m Aoo(wn ).
Proof (i) Set 1/;(8) = IIB..I - fllw" (s E JR). Then l};(S) :-s; (1 + w{x(s» IIfIL" and, by 4.7.6(ii). 'i/J(s/p) ~ 'ljJ(O) = 0 as p ~ ex:: for each s E R \Ve have (Mph and
*f
1;0
II.'lph
* f - fLo
:-s;
- f)(t) =
1:
I:
(Mph)(s)(J(t - s) - f(t»d3,
I:
I(Mph)(s)IIISsf - fll", ds =
Ih(t)lv-,(t/ p) dt.
The relmlt follows from the dominated convergence theorem, which applies because the integrand is bounded by 211fllw", Ih(t)1 wa(t) for p > 1. (ii) There exists h E Ll(JR,waJ with .fa h = h(O) = 1 and h E Aoo(wo )· By (i), Mnh * f ~ f as n ~ 00 for each I E Ll(JR,Wa). Clearly we have IIMnhll"'a :-s; IlhlLn (n EN), and 1;0 (Mnh) is a sequential bounded approximate 0 identity. We now utilize the fUIlction r which was defined in (4.5.13); f is the trapezium function of (4.5.14). Set
ft(t) = 2r(2t),
3h(t) = r(t/2)
(t
E
JR),
and set fo = fth. Since r = O(Z-2) as It I --+ 00, we have ft,h E Ll(JR), fo E Ll(JR,wa) for a < 3, and ft~a,hwa E L2(JR) for a < 3/2. Clearly
h (Y) = f(y/2),
3];(y) = 2f(2y)
and so (supph) U (supp];) c [-4,4] and io(y) = for each p > 0, we have M;](y) = 1 (Iyl :-s; p). In the next lemma, we use the function
0 with <poet) :::; C 1 1tl (t and so 0 such that
IISaMplo - lIJplot" ~ Co lala rpo(pa) 1-0:
(a E R 0 < p < 1).
Proof Denote the left-hand side of (4.7.22) by
la,p = p
I: I: I: I:
Then
fa.!}'
I/o(pt - pa) - 10(pt)1 wa(t) dt
~ p-a where
(4.7.22)
I/o(·~ -
pa) - 10(s)1 wo:(s) ds ::; p-"'(Aa.p + Ba.p).
11(8)llh(s -
Aa.p =
Ih(s - pa) -
pa)lw",(s)ds.
Ba,p =
Ih(s)llh(s - pa) - h(s)lwa(s)ds.
By the Cauchy-Schwarz inequality, Aa.p ::; IISpah - hIl 2 11(Spah)wa Il 2 . Now
IISpah -
hll2 = 11s;;:h - hl12 =
(I:
Ih(y)12Ie-iPay - 112 d Y) 1/2 ,
and so IISpah - hl12 ::; rpo(pa) Ilhlb because supph C [-4,4J. Also,
II(Spah)w a Il 2 =
(I:
Ih(u)1 2
w~(u + pa) dU) 1/2 ::; wo:(pa) IIhwa ll
2 ,
and so
p-n Aa,p ::; C 3 P-arpo(pa)a wo (pa)rpo(pa)1-a ::; C 4 1alO: rpO(pa)l-a for some constants C3 and C 4 . A similar estimate holds for p-nBa,p, and so the result follows. 0
Lemma 4.7.36 Suppose that a < 1. For each
II! * Uploll"""
-t
0
I
E
L 1 (lR,w a ) with f(O) = 0,
as p - t 0 +
.
Proof Since J~ I(s) ds = 0, we have III
* Mplollw"
::;
~
I: I:
i: i:
~ Co
i:
I/(s)II(SsMplo)(t) - (Mplo)(t)1 wa(t) ds dt
II(s)IIISsM"lo - MploL" ds 11(8)1 Isla rpO(ps)l-a d8
for p < 1 by 4.7.35. For each s E lR, rpo(p8)1-a II(s)llsl rpo(ps) I-a ds
~ 21 - a
0 as p - t 0+, and also
I:
-t
II(s)1 WaCs) ds.
'rhus the result follows from the dominated convergence theorem.
o
536
Comrnutatwe Banarh algebms
Theorem 4.7.37 Let Q E lR.+. Then the follo11Jzng cOTld'ttwns are cqlL'tValcnt for the Brurlzng algebm L1 (lR.,.v,,): (a) (.}
O. ThtH; the nilpotentH arc dense in V, in C*,o. and in .I\1(1R+· n ll) with respect to the appropriate norms. 0
Let A he the set of functions f E C(ll) such that 0'(1 - p) > 0 for some (necessarily unique) polynomial p. Clcady A iH a dense sub algebra of (C*, I· In), and so A iH a commutative normed algebra. Set J = {J E A : a.U) > O}. The map R : f I-t p. A --t Co[uJ. (with the ahove notation) is an epimorphism with ker R = J, and so. by 1.5.3(ii), rad A c .1, Since J is a. nilpotent ideal, .J C rad A by 1.5.6(iii). Henc(' md A = J. Thus, in thb example. r ad A is not dosed in A; indeed. WE' see that radA = {J E A : f(O) = O}. We have previouHly (in §§2.1, 2.9) defined thE' notion of a bounded approximate identity and of various typer" of sernig! oups in a Banach algebra. \Ve now give some interesting examples of approximate identities and semigroups in the convolution algehras on IR and IR+ that we are studying. ThE' proof of the first theorem is essentially the same as that of 3.3.23; we now use 4.7,(j(i). Theorem 4.7.41 Let S be IR or IR+, and let w be a wezght function on S s11cll that w is bounded near the ongzn. Let (en) be a sequence in V (S.u.!) such that: (i) en(t) ~ 0 (t E S, n EN); (ii) J~ en(t) dt
--t
1 as n
(iii) !S\[-,;,,;I en (t)(1
--t 00;
+ wet») dt --t 0 as n
--t 00
for each 8 > O.
Then (en) is a sequentzal bounded approximate identity in Ll(S,w).
o
538
Commutative Banach algebras
For example. take en = nX[O,l/"j (n EN). The sequence (en) is a bounded approximate identity in L 1 (B.w) in the case where w is bounded near the origin, and so, by 2.9.30(i), L 1 (B.w) factors. Similarly. (en) is a bounded approximate identity in V. and V factors. In these cases (en) is called the standard approximate identity.
Proposition 4.7.42 Let A be V or L1 (B. w) Jar a continuous w(~ight Junction w on B, where B zs IR or IR+. Then A is not Arens regular. Proof The Banach algebra A has a bounded approximate identity, but it does not have an identity. Regarded as a Banach space, A is weakly sequentially D complete by A.4.4. I3y 2.9.39, A is not Arens regular. In the first part of the following result, we identify the multiplier algebra of L 1 (B,w).
Proposition 4.7.43 Let B be IR or IR+, and let w be a continuous weight Junction on B with w(O) = 1. (i) For each T E M(L 1(B,w», there zs a umque J-LT E M(8,w) such that TJ = J-LT * J (f E L 1(B,w)), and the map T f---7 J-LT, M(L1(B,w)) --) M(B.w), is an isometric isomorphzsm. (ii) The standard approximate identity (en) is also a bounded approximate identity Jar the module C o(-S,w- 1 ), and Co(-B,w- 1 ) zs an essential Banach Ll (B, w)-module. Proof (i) In this case (enl Ilenll) is an approximate identity for L1 (B, w) bounded by 1. The proof is now essentially the same as that of 3.3.40. (ii) Set K = sup{w(s): SEll}. Take A E C o(_B,W-l), and take e > O. Then _ (lin I(en . A)(t) - :\(t) 1:::; n io 1:\(8 + t) - :\(t) 1ds (n ~ N, t E B). Choose to E B with 1:\(t)l/w(t) < e (t tt. B n [-to. toD, and then choose N E N such that I:\(s + t) - ~(t)l/w(t) < e (t E B n [-to. to], S E [0, liND. Then 1:\(.'1 + t) - :\(t)l/w(t) < £(1 + K) for t tt. B n [-to, to) ami s E [0, liN), and so lien' A - All:::; e(l + K) (n ~ N). Thus (en) is a bounded approximate identity forCo(-B,w- 1 ). 0 The algebra C* does not have an approximate identity because C;
c {J
E C(ll) : J(t) = OCt) as t --) O+} ~ C*.o ,
and so C; C C*,Q, and C*.o does not have a bounded approximate identity because C;,Q c {J E C(ll) : J(t) = oCt) as t --) O+} ~ C*.o .
°
However, a sequence (en) in C*.o such that, for each n E N, en(t) ~ (t Ell), Jo1 en(t) dt = 1, and supp en C [0, lin) is easily seen to be an approximate identity for C*,o. It follows that C; and C;,o have infinite codimension in C.. and C*,Q, respectively, and so there are discontinuous point derivations on the Banach algebras C* and C*,Q.
Convolution algebms on the r-eal line
539
In 4.6.16, we examined the Banach algebra (e 1 (Z, w)", D), noting that e1 (Z, w) is Arens regular in the case where w = Wa for a> 0, for example. The situation for the analogous algebras L1 (~. w) is the following. Let w be a continuous weight function on ~, and write Aw = £1 (R w) and Mw = M(~,w). For I!> E A~, set P(I!» = I!> I CO(Rw- 1 ), and regard P(I!» as an element of Mw' Then P : A~ -+ M (~, w) is a continuous epimorphism. There is a mixed identity I!>o for A~ with 1I1l>01l = 1. By 2.9.49(iii), the map jL f--+ jL . I!>o, M"" -+ (A~. 0), is an isometric embedding. Let 11, E Mw and A E CO(~,w-1). Then
o:
(A, (P by (2.6.30), and so P
O)(jL»)
0 0
= (A, jL .
= (Il>o . A, jL)
I!>o)
= (A, jL)
0 is the identity on Mw' Define Rw = ker P. Then again
~~)A~; Rw): 0 - + Rw
-+
(A~,O) ~ Mw
-+
0
is a Banach extension of )\J(JR, w) by Rw; the extension splits strongly, and A~ = M(~,w) EEl Rw' Clearly Rw C radA~. We omit the proof of thE' following result.
Theorem 4.7.44 Let w be a contznuous weight functwn on~. Suppose that wet) -+ 00 as It I -+ 00 and w(s + t)/w(s) -+ 1 as lsi -+ 00 for each t E JR. Then: (i) R~ = 0 and Rw is the radical of (A~, D); (ii) A~ has the strong Wedderburn decomposition A~ = M(~,w) EEl Rw; (iii) R~ and Rw/ R~ are mJinite-dzmensional spaces. 0 Let S be JR or JR+, and let w be a continuous weight function on S. There are two topologies on A1 (S, w) other than the norm topology that will arise. First, we set 0" = O"(M(S,w), Co(-S,w- 1»; a net (jLa) converges to jL in the locally convex space (M(S,w).O"), written or
0"
-limjLa = jL,
* jLa)(O) = (A, jLa) -+ (A * jL)(O) = (A, jL) for each A E Co( -S, w- 1). Since Co(-S,w- 1) is separable, it follows from A.3.20 that M(S,W)[lj is compact and metrizable with respect to 0", and so each bounded net in (M(S,w), 1I·ll w ) has a convergent subsequence in (M(S, w), 0"). Second, we denote by so the strong operator topology on AJ(S,w); a net (jLQ) converges to jL in (M(S,w), so), written if (A
Pa
if jLQ
*
f
-+
jL
*
so -+
f for each f
or
P E
so -limjLQ
= jL,
Ll(S,W).
'l'heorem 4.7.45 (Ghahramalli) Let S be ~ or ~+, and let w be a continuous weight function on S. Then: (i) the algebra (M(S,w),O") has separately contznuous multzplication; (ii) L1(S,W) and lin{8 8 : s E S} are dense in (M(S,w),so); (iii) if jLn ~ jL and sup IIjLnllw
0, then
1 (CG~)(x) = 27rl/2~1/2
=
27r 1
foo exp (t2 ) - 4~ - xt dt -00
/;e/ 2 exp(~x2)
i:
exp ( -
(2~~/2 + t;.1/2x) 2) dt = exp(~x2) ,
and this is sufficient to imply (i). It follows that,
1--+
Q' is a semigroup on II.
Convolution algebras on the re..al line
543
Clearly Gl/n(t) ? 0 (t E JR), and J::'oo Gl/n(t) dt = (CGl/n)(o) kEN with 2k > a + 2. There if:! a constant Ga > 0 f:!uch that
IGl/n(t)1 ::; Ga n{l/2)-k C
2k
= 1.
Take
(t E JRe),
and so ~tl>.5 Gl/n(t)(l + wet»~ dt ~ 0 a.'l n ~ 00 for each 0 > O. Thus (Gl/n) is a bounded approximate identity in L 1 (JR, w). Clause (iii) follows from (ii) and 2.9.6, and (iv) is a simple calculation. 0 Let A be one of the algebras M(JR+e, w) or M(JR+enlI) or /1(S, w) or e 1 (SnlI), where S if:! a f:!ubsemigroup of JR+e. Then A does not have any approximate identity: for t E JR+ such that Ot E A, we have inf{lIot - Ot * all : a E A} > o. However, I do not know, in any of these cases, whether or not either A factors or A factors weakly. The next result shows that there is no vef:!tigial bounded approximate identity in various discrete semigroup algebras. Proposition 4.7.53 Let S be a subsemigroup of JR+e, let w be a weight on S. and set A = el(S,w). Then 9 rt- 9 * A[l] for each 9 E Ae. Proof Suppose that 9 E A and that (fn) is a sequence in A[l] with fn * 9 ~ g. Set T = U{suppfn : n EN}, a countable subset of S. Since el(T,w) is the dual of the separable space co(T, w- l ), the unit ball e 1 (T, W)[l] is compact and metrizable in the weak* topology by A.3.20. By passing to a subsequence, we may suppose that fn ~ f E fl(T,w)[l] in this topology. Set f(s) = 0 (s E S\T). Then lim n -+ oo fn(8) = f(8) (8 E S). It is easy to see that lim (fn
n-+oo
and so
* g)(s) =
(f
* g)(8)
(8 E S),
f * 9 = g. Since A# is an integral domain, it follows that 9 = O.
0
Let w be a weight on JR+. Clearly eI (Q+e, w) contains the rational semigroup (Ot : t E Q+e) and el(JR+e,w) contains the real semigroup (Ot : t E JR+ e). We shall now show that the separable Banach algebra e1 (Q+e , w) does not contain any real semigroup in the case where w is not a radical weight. (We shall prove in 5.7.29 the surprising fact that, in the case where w is a radical weight, eI(Q+e, w) does contain a real semigroup.) We shall require as a preliminary a result of Bohr on almost periodic functions. Lemma 4.7.54 Set F(z) = ~:=l anexp(-zt n ) (z E II), where {t n : n EN} is a subset ofJR+e and (an) Eel. Suppose that F(z) #- 0 (z E II). Then, for each r > 0, we have inf{IF(r + iY)1 : y E JR} > o. (4.7.25) Proof Take r > O. Assume towards a contradiction that there exists (Yk) in lR such that F(r + iYk) ~ 0 as k ~ 00. Set Gk«() = F«( + iYk) E II). Then the sequence (Gk) is uniformly bounded on II. By A.2.11, we may suppose, by passing to a subsequence, that Gk ~ Go, say, in O(II). Clearly Go(r) = 0, and so, by Hurwitz's theorem A.2.8, Go = O.
«(
Commutative Banach algebras
544
Take € > O. Then there exhlt N E Nand IF - Hln < €, where
S], ... , SN
E Ql+- slIch that
N
H(z) =
L
ct n
exp( -zs,,)
(z E II).
n=l
Take mEN such that {ms n : n E NN} eN. Since Gk ---+ 0 in O(II), there exists kEN with IGk(T + iy)1 < € (Iyl :::; 2m7r), i.e., slIch that
IF(T
+ iy)1 < €
(Iy -11kl < 2m7r) .
Take j E Z with 12jm7r - Ykl :::; 2m7r. Since {jms" : n E NN} c N, we see that H(T) = H(T + 2jl1mi), and hence WeT) - F(T + 2]m7ri)I < 2€. But IF(T + 2jm7ri)I < €, and tlO W(T)I < 3€. This holds for each € > 0, and so F(T) = O. again a contradiction. Thus (4.7.25) holds. 0 Theorem 4.7.55 (H. Bohr) Let f E J!1(1R+) be such that f((~(f) = O. Then £f has a zero on each half-plane IIO" with a > O. Proof Without loss of generality, we may suppose that rt(f) = O. Assume towards a contradiction that a > 0 and (£ f) (z) i= 0 (z E ITO"). and set F(z) = (£f)(z + T) (z E II), where T > a. Then F E Ab(IT). By 4.7.54, inf{W(it)I : t E 1R} ~ ",. say, where", > O. Since F(z) i= 0 (z E IT). it follows from Nevanlinna's theorem A.2.46 that there exists c E IR such that
1jOC
log IF(z)1 = 7r
By 4.7.19, for each (for otherwibe o:(f) x;;llog IF(xk)1 ---+ 0 IF(z)1 ~ '" (z E IT). contradiction.
-oc X
a.
2
x
+-
(
Y- t
)2 log W(it)1 dt
+ ex
(z E II).
(4.7.2G)
> 0 it is not true that IF(x)1 = O(e-nX) as x
---+ X>
~
a). and so there exists (Xk) in 1R+ with Xk ---+ 00 and as k ---+ 00. It follows that c = O. ad so. from (4.7.2G), However, by 4.7.12(ii), F(x) ---+ 0 as x ---+ 00, the required 0
The set Q(A) for a Banach algebra A was defined in 2.2.3 to be {a E A: aA = A}. Theorem 4.7.56 (Estcrle) Let S be a subsemzgroup of 1R+-, and let w be a non-radical weight on S. Then: (i) Q(fl(S.W» = 0; (ii) in the case where S is countable, fl(S,W) does not contain a non-zero, real semigroup. Proof Set A = f 1 (S, w). Since w is non-radical, there exists a ~ 0 such that £(A) C Ao (ITO"). (i) Assume that there exists f E Q(A). Clearly o:.(f) = O. Since f(O) = 0, it follows from 4.7.55 (applied to the function t 1-7 e -O"t f (t» that there exists Zo E IIO" with (£f)(zo) = O. By 2.2.8(i), I i= {O}, where I = * A: n EN}. Choose 9 E I-. Then £g has a zero of order n at Zo for each n EN, and so 9 = 0, a contradiction. Thus Q(A) = 0.
nu*n
Convolutwn algebras on the real line
545
(ii) Assume that (ft : t E JR+e) is a non-zero semigroup in A. Then therE' exists to E JR+- such that =F 0 (t E (O,to]). Take t E (O,tol. and set 9 = p. If g«(1(g») = 0, then it again follows from 4.7.55 that (Cg)(zo) = 0 for some zoo Since 9 = (ft/n)*n (n EN), again 9 = 0, a contradiction. Thus g(a(g» =F 0, and SO a(g) E S. The map t ~ er(r). (0. tol -+ S, is an injection. But this is not possible because S is countable. So A does not contain a non-zero semigroup 0 over JR+-.
r
Our final topic in this section is a study of the family of closed ideals of some of the algebras that we are considering.
Definition 4.7.57 Let w be a weight function on JR+, and take 8 > O. Then Mo(w)
= {J
E
Ll(W) : a(f) ~ 8}.
A closed zdeal I zn Ll(w) zs a standard ideal if 1= Mo(w) for' sorne 8 E JR.+, or if 1= 0, and I zs a non-standard ideal if I zs not standard. We have similar definitions for the algebras V and C*. Let A = V or C.. , and let 8 E [0.1). Then Mo(A) = {J E A : a(f) ~ 8}. A closed ideal I of V is standard if I = M8(V) for some 8 E [0,1), or if I = 0, and a closed ideal I of C* is standard if I = Mo(C*) for some 8 E [0,1), or if 1= 0, or if 1= C*,o. We wish to determine whether or not all closed ideab of V, of C*, and of the radical algebra...., £l(w). are standard. For V and C*. this is ea....,y.
Theorem 4.7.58 (i) (Dixmier) Each dosed ideal of the Volterra algebra V is a standard ideal. In partzcular, V contains no closed. prime ideals. (ii) Let (nil) be a sequence in JR.+-. Then there exzst. O. Define h 1 (t) = eio.tg(t) (t E JR), where R Then hI E A. and, by choo::>ing 0: appropriately, we may ::>uppo::>e that (£ht}(iyo) =I- 0 and that (£hl)(iy) = 0 (y < Yo - c). Set l/J = 'P. Theil BI/' = B.p, and so we may repeat this calculation with 'P replaced by '1,1] and the upper half-plane by the lower half-plane to obtain h2 E A with (£h 2 )(iyo) =I- 0 and (£h2)(iy) = 0 (y > Yo +6"). Set II, = hi * h 2 • Thm hE A, (£h)(iyo) =I- 0, and (£h)(iy) = 0 (Iy - yol > c). Thi::> implies that A is regular. 0:
E
1
(b ):::}( c) Sinc{' A is regular. tlH're exists f EA· such that supp is a compact subset of V. Since f = £-1 (1) I R we may suppose that f is the restriction to JR of an entire function. Take A E A' with A 1.. Aoo. Then fA E LI(JR). For each y E R we have
I:
f(t)A( _t)e- iyt dt = 0
(y E JR) .
Thus /5.. = 0 in U(JR), i.e., f(t)>.(t) = 0 for almost all t E R Since f has only ('ountably many zeros. A(t) = 0 for almost all t E R and so A = 0 in A'. Thi::> proves that is dense in .4, and so. by 4.1.24, spectral analysis holds furA. .
Aoo
(c):::}(a) Since c.p satisfies (4.7.27), it is dpar that there exist and M > 0 with the following properties: '1/] I and
if]"
~
M (t E JR) and are bounded on JR .
1'P(t) - ,¢(t)1
lj',
E CCoo)(JR)
1jJ(0) = 0:
By A.2.37. (ii) and (iii). there cxist::> Fo E A(U) with ~F() bounded on say by k, and with ~Fo(t) = l/,/(t) (t E JR). Now define Go(()
= exp (
-1e;
F(l(W) dW)
((
E
U) ,
where the integral is takcn over the straight line from 0 to Go E A(U). Set go = Go I JR. For t E R we have
Igo(t)l=exp(-fot(~Fo)(W)dW)
U,
=cxp(-fot l/J'(W)dW)
C,
say.
Clearly
= cxp(-1jJ(t»,
and so e- M ~ igol expc.p ~ eM. Also. for each a E JR, the function (SaGO)/GO : (I-> G o (( - a)/Go (()
is analytic on U and its modulus is bounded bye k1al . Define T : f I-> fgo, £l(JR) --+ A. Then e- M
IIflll
~ IITfll ~ eM
11/111
(f
E LI
(JR),
and so T is a linear homeomorphism. Now define K = T(Hl(U)), so that K is a closed linear subspace of A. Clearly K =I- A. Take I E Hl(U) and a E R Since Sal E Hl(U) and since (Sago)/gO is the restriction to JR of the function (SaGo)/Go, which belongs to HOO(U), the
Convolution algpbras on the real line
549
function (Su1)(Sa90)/90 belongs to Hl(U), and so Sa (f90) = (SaJ) (Sag) belongs to K. Thus K is translation-invariant: by 4.7.10. K is a closed ideal in A. Since spectral analysis holds for A. tilere exi~ts Yo E lR such that
K For each
(t
c {g
E
A: y(Yo) = O} .
~ 0, the function (f--> c i 1 -:00 +t
-00.
But 10gI91(t)1 = -1b(l) - 2loglt+il (t E lR). and so T3 •• < and A is non-quasi-analytic.
')C.
Thus T3
(c) of the above theorem in thc' cas(' where i.p = 0, is Wiener's Taubrrian theor-em: we restate this result for the weight ..va : t f--> (1 + ItW", and give an alternative proof. Let (G< : ( E II) be the Gaussian s('migroup in I}(lR). as in 4.7.51. and set G = G 1 .
Theorem 4.7.63 Let n E lR+, and If'f I be a closed %deal in L1 (R..va) such that ~(I) = 0. Then I = L 1 (lR,w,,). In par-twular-, lin {os * G: s E lR} is dense in L1 (lR, wa ). Proof (Esterle) Spt, ~ = Ll(R..vu)/I. By 4.1.1l(i). 2l is a radical Banach algebra. Define a< = G( +1 E II), so that (at; : ( E IT) is an analytic hcmigroup in 2l. By 4.7.52(iv). IIG 0 such that F(z) =1= 0 for Z E JI))(iyo; 28) n II. Next, choose E cg~) (JR.) with supp c (-8, J). By 4.5.1l(i), = r:p for some (() E Ll(JR.). Set
(Ill .
go in L1 (JR.). Then
--(gk * ({) * hZk )('T})
=
H(Zk + i1])(1]) F(Zk + i1])
because ('T}) = 0 for 11]1 ~ 8. Since gk inversion theorem that
i:
.
= W(Zk + 11])(IJ) ('T}
* (() * hzJ..
E JR)
E C(JR), it follows from the
* (() * hZk)(O) = \(I(Zk + i'T})('T}) d1] = ('11 . 0) and IA(x + iy)1 :S IIAII I Ixl (x < 0), and so (Ixl}i,,(:r + iy)1) is uniformly bounded on j[))(iyo: 8). Also wn(z) -> \ji(z) as n -+ oc: for Z E j[))(iyo; 8) \ V. Thus, by A.2.13, W has an analytic extension to ][)l(iyO; 8). It follows that the function A on on has an analytic extension W to C \ 1)(1). By hypothesis (i), W is an entire function. We next show that W is of exponential type, and to do this we call in aid Thcorem A.2.48. Let W have representation (4.7.28) for some f E I. Clearly, hypothesis (i) of A.2.48 is satisfied (for each (J > 0). Since IH(x + iy)1 :S 1111,11 00 Ix, we see that { log+ IH(z)1 dxdY2:S
ln
1+
Izi
IIhlloo
{I
10
log
(~) t
dtJx 1 dy 2 < -co + y
00.
and a similar estimate holds for the integral involving log+ IA(z)1 over on. By A.2.41,
In (IOg+ IF~Z) I) 1d:~~4
0).
(4.7.31)
Proof The integrand is exp(ua(ba +t) -lLfT(b a )). Since U u attains its maximum at bu , this integrand is at most 1. For t > aa+l, necessarily TJ(t) ~ (a + l)t. and so the integral in (4.7.31) is bounded by
l
a O"+l
-b"
dt
+
1""
eutw(t) dt :::; au+l + ba
+
1
a" tl
e- t dt.
0
Equation (4.7.31) follows because biT :::; aa
Lemma 4.7.70 There eXlsts and a 2: al.
00
(II
= O«(Ik')
as a
-> ClO.
D
> () such that e- ut :::; w(t)e- U /2 for 2:::; t:::; ba
Proof Take al 2: TJ(2)+log2 such that bu 2: 2 (a ~ al). The stated inequality follows because lLu(t) 2: u a (2) for 2 :::; t :::; bu. D Lemma 4.7.71 Let A E A'. Then LA is an entire functwn which is bounded on oIl and whzch has order at most k + 1 on II.
10
Proof Let Z = a + iy. Then I(LA)(z)1 :::; IIAII 00 w(t)e at dt. If a:::; 0, then this integral is bounded by IIR.+ W. If a > 0, then w(t)e ut = exp(uu(t)): if t 2: a2u, then rl(t) 2: 2at, and so u".(t) :::; -TJ(t)/2. Thus
as a
->
oc, and the result follow~ because
au =
O(a k ) as a
-> 00.
o
Theorem 4.7.72 (Damar) Let w be a weight flLnction on R+ such that w satisfies (4.7.29) and (4.7.30). Then each closed zdeal of Ll(W) is standard. Proof Set A = Ll(W). We shall prove that f * A = A for each f E Ll(W) with aU) = O. Assume that we have proved this result, and let w be as specified in the theorem. For each c > 0, the function We is a weight function on 1R+ which also satisfies (4.7.29) and (4.7.30). Take g E A·, and let f = 8_0.(g)g. Write LAo for the set of elements of Lfoc with compact support. By the stated result, f * LAo = £1 (Wo.(g»), and so 9 * A = Mo.(g) (w). The theorem follows.
Convolution algebras on the real line
555
We prove the stated result. Set I = J * A, and take A E Ah] with A ..1 1. We shall show that A = 0, which is sufficient. We may suppose that Ilfll = 1. For a> 0, set Ju = JX[O.bO'] , with bu as above, and set gO' = Ju * A. Then
(bO' glT(t) =gu(-t) = io J(s).(s+t)ds Since (Bd, A)
= 0 (t E JR+).
(4.7.32)
(tEJR).
it follows from (4.7.8) that
roo
J(s».(s + t) ds (t E JR+). ibO' It is clear from (4.7.32) that gu(t) = 0 for t ::; -bu' We claim that gu(t) = -
Igu(t)1 ::; w(bu + t)/w(bu ) (t ~ -bu ). First, suppose that t E [-bu.O]. From (4.7.29), the function s [a, (0) for each a > 0, and so
(4.7.33)
(4.7.34) 1--+
w(s - a)/w(s)
is increasing on
w(s + t)/w(s) ::; w(b u + t)/w(b u ) (s E [-t,b lT ]). Thus, from (4.7.32), we have
b e- a , and sO a + iy E E. We now have all the estimates that we require to finish the proof. Define H(z) = e- 2Z A(z). It foUows from (4.7.37) that, for each () E 8, there exists Ko such that IH(z)1 ::; Ko (z E Ro). This also holds for () = ±1r/2 because A is bounded on V. Choose ()j E e u {-1r /2.1r /2} so that
-1r/2 = and
()j - ()j-l
< 1r /(k + 1) (j
()o
00 as t -> 00. It is not known whether or not this eondition 18 necessary; indeed, the case in which wet) = 1/(1 + t)t is still open. A sophisticated analysis of the primary ideals at infinity for Beurling-type Frechet algebras in the Qllasianalytic case is given in ~Borichev 1996). See also (Esterle 1996a). Examples of radical ~ight functions w on 1R sueh that Ll(W) has non-standard closed ideals are obtained Ill. (Dales and MeClure 1987) by modifying Example 4.6.29 of Thoma.'). However, no Illtample of a radical weight function w such that Ll(W) contains a function f with Q(f) = 0 and f f/- O(Ll(W» is known. Other results on the family of closed ideals in algebras Ll(lR+,W) and Ll(W) are given in (Bade and Dales 1981), (Domar 1956, 1975), (Gurarii 1976), and (Hedenmalm 1985, 1986).
558 4.8
Commutative Banach algebms PRIME IDEALS
Our main objective in the present section is to study the families of prime ideals in the algebras Co(O), where 0 is a locally compact space. The results that we obtain will be used in §5.4, where we shall obtain some positive re~mlts about the automatic continuity of homomorphisms from the algebras Co(O), and they will be used more substantially in §5.7, where we shall give a variety of constructions of discontinuous homomorphisms from these algebras. Let 0 be a compact space, and let P be a prime ideal in C(O). Then the algebra Ap = C(O)/P is always algebraically closed. and its 'real part' Ap(JR) is a totally ordered algebra. We shall concentrate on the case where 0 = j3N, and we shall show that Ap is then always a valuation algebra. Throughout, we shall use results about the uniform algebras Co(O) that were given in §4.2. However, before turning to the algebras Co(O), we shall consider prime ideals in general commutative Banach algebra.'l, obtaining results to be used in 5.4.26. Let R be a commutative, radical algebra. The identity of R# is eR, and the unique character on R# is denoted by 'PR. Lemma 4.8.1 Let R be a commutative, mdzcal Banach algebm. (i) Suppose that there exists a E R \ IJ1(R) with a E aR. Then there zs a sequence (an) in InvR# such that aan ~ a as n ~ oc, Ilaka;;:lll ~ 00 as n ~ 00 lor each kEN, and I'PR(a 71 ) I 2: 1 (n EN). (ii) Suppose that there eX'tsts a E R- wzth a E aR. Then there zs a sequence (an) m InvR# such that aa n ~ a and lIaa;;:!1I ~ 00 as n ~ 00.
Proof (i) We may suppose that ~n
Iiall =
1. For n E N. set
= {( E C : 0 < 1(1 < I/4n}.
Iia - abnll < I/4n. Since an =f 0, there exists + bn)-l) =f o. Define· = ).(an(zeR + bn)-l) (z E ee).
For n E N, choose bn E R with ). E R' with 11).11 = 1 and ).(an(eR I(z)
Then I E O(C-), I =f 0, and I(z) ~ 0 as z ~ 00, and so I has a non-remoV'dble singularity at O. Thus I is unboundt.,'-> A (a(zeR + bn)-l), c· ---+ IC. We obtain an E InvR1F with lIaa~lll2: nand lIa - aanll < lin. 0 Theorem 4.8.2 (Esterle) Let R be a commutatzve, mdical Banach algebra. (i) Suppose that there exists a E R \ IJt(R) with a E aR. Then there exist xo, Yo E R such that x~ ~ yoR# and y~ ~ xoR# for each kEN. The set of prime ~deals m R, ordered by incluswn, zs not a cham. (ii) Suppose that there exists a E R· with a E aR. ThPn there exzst Xo, Yo E R such that Xo ¢ yoR# and Yo ~ xoR#. Proof (i) Set I = aR, and let (an) be the sequence specified in 4.8.1(i). For k, n E N, define
Un.k = {(x, y)
E
I x I: inf{Jlxu -
xII + liuykll-1
:u
E
R#}
0, there exists (x, y) E Un.k with Ilx - abll < c and lIy - Gcll < e. By 4.8.1(i), there exists Tn E N such that lIa - aamlillbil < min{e, 1/2n}, set x = abam , y = ac. and u = a;;"t. Then Ilxu - xii = Ilab - abamll < 1/271 and IIuyk 2: lIaka;;.lll / Iic-k > 2n. and so (x, y) E Un,k' Also Ilx - abll < e, and so the claim is established. Set Vn.k = {(x,y) E I x I : (y,x) E Un,d. Then Vn,k is also a dense, open SUbset of I x I. By the category theorem A.1.21, there exists (xo, Yo) E I x I such that (xo, Yo) E Un,k n Vn,k for each n, kEN. Take kEN. Then there exist (Un) and (v n ) in R# such that, for each n E N.
II
Ilxoun - xoll
II
+ IIUny~lr-l < lin
and
Ilyovn - yoll
+ IIVnX~rl
No. Since is a cone over QH. the result follows from 1.2.21
Aft-
Mt-.
Alt-
Mt-
(b)::::}(c) Let Sand T be countable subsets of KI'(JR.) with S First. suppose that 0 « S u T. We may suppose that
« T.
S U T = {bnC;;-l : n E N}.
ut-
where {b n , en : n E N} c Mt-. Take c E with c < C n (n EN). By 4.8.17(iii). (:2 E cnAp, and so c2 S « c2T in By (b), there exists a E with c2 S « a « c2 T. and then S « ac- 2 « T in Kp(JR.). Second, suppose that S U T « O. Then there exists a E Kp(JR.) such that -T« a« -S, and now S« -a« T. In the general case, first take b E KI'(JR.) with b « S U T, and then take c E Kp(JR.) with S - b« c« T - b: we have S « b + c« T.
Aft-.
Mt-
Mt-. By (c), there Mt-, the sequence
(c)::::}(a) Let (an) be a strictly decreasing sequence in exists a E K p (JR.) with 0 < a < < {an : n E N}. Since a E (an) is not coinitial in
ut-.
0
We modify the above notation in an important special case. Recall from §4.2 that, for p E .8N\N, Mp is a maximal ideal in Gum) = too, MP is a maximal ideal in eN, and J p is a minimal prime ideal in Gum).
Definition 4.8.22 Let P E pN \ N. Then Ap = G({3N)/ J p, 1rp : G({3N) ~ Ap is the quotient map, Ap(JR.) = 1rp(C(fJN, JR.», and Kp and Kp(JR.) are the quotient fields of Ap and Ap (JR.). respectively. The qnotient fields Kp can also be defined directly in terms of ultrafilters. Indeed, set U = Up, the ult.rafilter corresponding t.o p, and let f, 9 E eN. Then
f
"'U
9
if {n EN: fen)
= g(n)}
E U,
so that f "'U 9 if and only if f - 9 E MP. The corresponding field eN /U was described in ~1.3. Clearly eN/u ~ eN /MP ~ Kp, and (JR.N/U, ::;). tht' ultrapower of R by U, is an ordered field which is order-isomorphic to (Kp(JR.). S).
Theorem 4.8.23 Let P E {3N\N. Then (Kp(JR.), S) /(p is an algebmzcally closed field of cardinality c.
'/,S
a real-closed Til-field, and
Proof We show directly that Kp is algebraieally closed, and hence that Kp(JR.) is real-closed (although this is a special case of 4.8.14). Let p = ao
+ a1X + ... + anX"
E
Kp [Xl
with an =1= 0, and take fo, ... , in E eN such that [i1 ] = aj (j E z;t). Set = {k EN: in(k) =1= O}. Then a E Up, and, for each k E a, there exists 9(k) E e with io(k) + ... + in(k)g(k)n = O. Set a = [gJ. Then pea) = 0 in Kp. Thus Kp is algebraically closed. U
Comrrmtatwe Banach algebm8
568
To establish that Kp(JR) is an Til-set, it suffices, by 4.8.21, to show that coi > ~o. Let (an) be a strictly decreasing sequcnce in .Up+-. and O. By induction, there exists a sequence (an) in A l:luch that IIx - al . x/l < c:/2 and /Ix - an . xII < c:/2n(1 + Ilalll)'" (1 + lIan-lll) (n;::: 2). For kEN and n ;::: k, set Yk,n = ak ... an . x, so that IIYk,n - Yk,n+111 :::; Ilak'" anll /Ix - an+1 . xII < c:/2n+1 .
Thus (Yk,n : 11 ;::: k) is a Cauchy, and hence convergent, sequence in E, say limn-->oo Yk,n = Yk· Then Yk = ak . Yk+l (k EN), so that YI E ~al ... an . E. and Ilx - uI11 :::; Ilx - al .
xii + E~IIIYI,n+1
- YI,nll < c:.
0
Proposition 4.9.5 Let A be a commutative Banach algebra, let E be a Banach A-module, and take a E A and x E E w~th x E aA . x. Then x E lim an . E. Proof Set B = aA, and let 111·111 be the quotient norm on B, so that (B, III·IID is a Banach algebra and E is a Banach B-module. Also x E B . x, and so, by 4.9.4, for each c: > 0, there exist (b n ) C Band (Yn) C E with IIx - YIII < c: and Yn = bn . Yn+1 (n EN). There exists (an) in A such that bn = aa" (n EN). Set Zl = YI and z" = al'" an-I . Yn (n ;::: 2). For each n E N, we have
Zn
= al"
·an-Iaan . Yn+1
=a
. Zn+l. and so YI E lima n . E. It follows that +--
o
x E lim an . E.
Proposition 4.9.6 Let R be a commutative, radzcal Banach algebra, and let E be a Banach R-module. Suppose that (x n ) zs a sequence in E such that Xn E R . Xn+1 (n EN), and set S= {XE R· x:R· x=U{R. xm :mEN}}. Then S is dense in R . x. Further, if Xl =F 0, then S =F {O}.
Proof Set G = Inv R# and F = R-module. For n, kEN, define
UR
Un,k = {x E F: inf{llx n
. x m , so that G = R# and F is a Banach -
a . xII: a E R}
< 11k} .
Then each Un,k is an open set in F, and Xm E n{Un,k : kEN} (m;::: 11 + 1). Take x E Un,k and u E G. Then there exil:lts a E R with /lx n - a . xII < 11k, and then Ilx n - (au)-l . (u . x)1I < 11k. so that u . x E Un,k. Thus G . Un,k C Un,k, and R . Xm C Un.k (m;::: n + 1). It follows that R . Xm C Un,k (m EN), and So F = U.. ,k. Set U = n{Un,k : n, kEN}. By the category theorem A.1.2I, U = F. For each x E U, we have Xm E R . x (n EN), and so F = R . x. Thus U c S, and SO S = F, as required. The final clause follows because Xl E R . Xm (m EN), and so Xl E S. 0
Commutatwe Banach algebras
572
Recall from 4.7.56(i) that O(t'l(S,w)) = 0 for each subsemigroup S of JR+and each non-radical weight function w on JR+-. In contrast, we obtain the following corollary to 4.9.6. Corollary 4.9.7 Let S be a dense, difference subsemzgro7l,p of JR+-, and let w be a radical weight function on S. Then OW(S,w)) zs dense m t'l(S,W). Proof Set R = t'l(S,w). Since S is dense in JR+., there is a decreasing sequence (sn) in S such that 8 n --7 O. Since S is a difference semigroup, (sn - sn+d C S, and so 8sn E 88 7 * R: mEN} = R, and so, by 4.9.6, O(R)
= R.
0
Corollary 4.9.8 Let R be a commutatzve, radical Banach algebra. following condztions on R are equivalent: (a) there eX'lsts a E R- such that a E aR;
(b) there eX'lsts (an) in R such that
~ al ... an R
(c) there exists (an) in R such that
al
Then the
-I 0.-
-10
and an E an+lR (n EN); (d) there zs a strictly mcreasing sequence of prznczpal ideals zn R#. Proof The equivalence of (b) and (c) is immediate from the definition of ~al'" anR. By 4.9.4, (a)=>(b), and, by 4.9.6, (c)=>(a). (d)=>(c) By (d), there is a Htrictly increaHing sequence (anR#) with al -10. Clearly (an) C R. for otherwise there exists kEN with anR# = R# (n ~ k). For each n E N, an E an+!R# \ an+! . Inv R#, and so an E an+1R. (c)=>(d) Take (an) aH specified in (c). Then anR# C an+lR# (n EN). AHHume that ak+l E akR# for some kEN. Then ak+l E ak+!R, and so ak+! = O. By (c), we have al = 0, a contradiction. Thus anR# ~ an+lR# (n EN). and 0 (d) holds. Corollary 4.9.9 Let A be a commutative, unital Banach algebra such that the set of prznczpal ideaL., forms a chain. Then A is finite-dimensional. Proof As in 4.8.3, A = R# for a radical Banach algebra R. By 4.8.2(ii). a rJ- aR for each a E R-, and so R does not sath;fy any of the conditionH in 4.9.8. Set N = 'J1(R), so that, by 1.5.26(iii), N iH a prime ideal in A. SuppoHe that N = O. Then A is a valuation algebra, and HO A = CeA by 4.8.3. SuppoHe that N -I 0, and asHume first that R \ N -I 0. Take' al E N- and bE R \ N. Then al E bA because b rJ- alA, and so al = ba2 for some a2 E N-. Continuing, we obtain a sequence (an) in N- with an E: an+lN- (n EN), and so R satiHfieH 4.9.8(c), a contradiction. Thus N = R. Take al E N-. If alA -I R, take a2 EN \ alA, HO that alA ~ a2A. Continue in this way. Since it is not the case that there is a strictly increasing sequence of principal ideals in A, there exists ao E N· with aoA = R, say ao has index k + 1Then A = CeA EB Cao EB Ca5 EB··· EB Ca~, and A is finite-dimensional. 0
Classification of commutatzve, radzcal Banach algebras
573
Corollary 4.9.10 Let A be a commutative Banach algebra, let E be a Banach A-module, and let a E rad A. Then the following conditwns on a are equzvalent: (a) there exzsts x E E e with x E aA . x: (b)
~a?l .
E
=f. 0;
(c) E contains a non-zero, a-dwisible submodule; (d) there exist y E E e and a sequence (b n ) m Coral 'Unth y = limn-->:x. bn Proof (a)=}(b) If x E aA . x, then, by 4.9.5, x E
~a"
.
y.
. E.
(b)¢:}(c) This follows because ~ an . E is the maximum a-divisible submodule of E. (b)=?(d) Set R = Coral. so that R is a commutative, radical Banach algebra. By (b), there exists (Xn) in R with Xl =f. 0 and Xn = a . Xn+l (n EN). By 4.9.6. there exists y E Ee with y E ~, and now (d) follows. (d):::}(a) By (d), y E aA# . y. and so y E a 2 A# . y C aA . y.
0
Corollary 4.9.11 Let R be a commutative, radical Banach algebra, and let a E R. Then the following conditwns are equzvalent:
(a) there eX1-sts X E Re with x E axR; =f. 0; (c) R has a non-ze1"O, a-dwzsible subspace.
-
(b) lima 1t R
o
Proof This is immediate from 4.9.10.
We now investigate when there are various semigroups in a commutative, radical Banach algebra. Recall from (2.2.1) that Ia = n{an . Q(A) : n EN}.
Theorem 4.9.12 (Esterle) Let R be a commutative, radzcal Banach algebra. and let a E Q(R). Then there is a ratwnal semigroup contamed m Ia. Proof Set Rl = aR. Then Rl is a commutative, radical Banach algebra for the quotient norm, say 111·111, from (R, 11·11). Set R2 = a 2 R}, taking the closure in (R 1 , 111·111), so that a 2R2 = a4 Rl = a5 R in R 1 • There is a sequence (un) in 5 n - a 2 111 = O. so that R with limn-+ oo 4 n - all = 0, and then lim n -+ oc a2 E a5 R. Thus a 2 E Q(R2) by 2.2.11(i). Set S = (Q(R2)' .) and b = a2. By 2.2.4(i), S is an (F)-scmigroup. Let n E N, and define On : S f--t 05"+1, S ~ S, SO that On is continuous. Sinc'(> bn . lnv Rf c Sand lnv Rf is a divisible group with Inv Rf = R2, we have ~(S) = bn + l R2 = R 2, and so On has dense range. By the Mittag-LefRer theorem A.1.25, limproj {S; On} =f. 0, and so, by 1.2.20, there is a rational semigroup, 8ay (cO< : Q E Q+e), in S. Let Q E Q+.. For each n E N, we have co from (Esterlp 1998). Morp general result5 related to 4.9.1\) are given in (Sinclair] 9H2, Chapter V): they arc based OIl work of Estprle. In fact. Esterle giveh a somewhat more detailed claslSification than ours For pxample. Esterle introdu('('1S a clasH which Wf' call ('lass VIvo this class consisti:i of the comlllUtat.ivp. radical Banach algebras which contain a nOIl-7.pro. analytic semigronp over the hector S,;, (whNP l,J E (0,71"/2]). Clearly each Class VI" is ('ontaillPd in Class VI and contains Cla..,s VII Examples givpn in (Esterlp 19H:Ja) ::ihow that th('f(' are: (i) algebras in Cla..% VI. hilt not in Class VI" for any 1.' E (0.71"/2); (ii) algpbra..., in Class VI11'/2' bllt not in Clab::i VII: (iii) algebra::i in Class VI,,!, hut not. in Cla.'!::i VI,". whenever 0 < 'It)l < ~'2 ~ 71"/2. An example of a commutative. radical I3anaeh algf'bra R which contains a divisihle element in (R- . . ). but is such that R dom, not E'ven helong to Class III. is giVPIl ill (Esterie 1983a). It is proved in (lbzd., §6) that each algehra in Class VI ('ontains a nonzcro, infinitely diffcrt'ntiable real sCllligroup. An exampll' of a (ommutative. radical Banaeh algebra which is an intpgral domain and whi n 2 2).
Clearly X Ru = n. and, for pach n 2 3. X" is nilpotent, so that Ro is a nilpotent algphra. A generic element of Ro is uniquely expressiblc as
a
= oX + L
(liXi
+L
t:J,X{X 2
+
L
"fi,j
"x~n-lXi71'
(-l.9.5)
j~O
wit('re each a.(liJ1j, and li.j,,, hplong to C. and thew are only finitely many lion-zero terms in pach sum (and we intprpret xl) Xj as X7)' A norm 11·11 on the abovE' algebra Ro is defined by the equation
lIall = Ir~1 + L 10;1 Ti 2 +- 2 L l.6iI 2 ·- i " +
L
hi,j
,,12 1 ,
(4.9.6)
where a is a..'> in (4.9.5). It is easily checked that 11·11 is an algebra norm on Ro. The completion of Ro with respect to II . II is the required algebra R; pach element of R has the form (4.9.5), but now infinite sums are allowed. subject to the condition that the norm specified by (4.9.6) be finite. It is clear that X E R n for each n E N, and so {R" : n E N} =I- {OJ. However, a calculation shows that n{R[n] : n E N} = {OJ, so that R belongs to Class II.
n
Commutatwe topological algebms 4.10
581
COMMUTATIVE TOPOLOGICAL ALGEBRAS
We shall discuss in this final section of the chapter some commutativE:' topological algebras and, in particular, UvlC and Frechf't algebra..'i, being eoncerned with the identification of the spaces (If continllous characters of these algebras, and with the determination whether or not each charader is continuous. The main tool that is used is a theorem of Arens about Frechet algd)ras whidl is a conseqlH'nce of the Mittag-Leffler theorem. \Vf' shall also describe some Ll\IC algebra.s of continuous functions and of analytic functions, rliscussing their character spaces, and we shall conclude with some specIfic examples which show the limits of our general theorems on LMC algebras. Definition 4.10.1 Let A be a topologzcal algebra. Then th(' continuous character space of A zs the set ~A of cordznuo'/Ls characteTs Or! A. The algpbm A Z8 functionally continuous 'If each rhamcteT on A 'lS mntmn01l8. The set ~A is a subset of IP A, the character space of A. and it is always taken to have the rPlative a(A x. A)-topology. Of course. 4> A depends only on the algebraic structure of A. but ~A depends also on its topology. One of the aims of this section is to dC'terrnine conditions on the topologieal algebra A which ensure that A is functionally continuous. Certainly. each Banach algebJ a has this property. but it is not known whether or not each commutative Frechet algebra is functionally continuolls: this is a famous old question called lv!zchael's problem. Let A be a complete Li\IC algebra. As in 2.2.47, A = limproj{Ap;7rpq;P}, where {Ap: 7rpq: P} is a projective system of Banach algebras and norm-decreasing homomorphisms. Throughout we write 7rp : A --+ Ap for thf' coordinate projections, and we suppose that 7r]>(A) = Ap for t'a.ch pEP. We now systematically write 4>p for the character space of Al" Set
.p(a) = cp is an embedding of IPp in IP,\. and so wp may regard cI>p as a subspace of cI>A. Clearly LA = U{cI>p : pEP}. In the case where A is a Frechet algebra, A = limproj{A,,: 7rmn }. where {An: 7r mn } is a projective HequclIce: Pn(a) ::; Pn+l(a) (a E A. n EN). as in (2.2.10), and EA = U{cI>n : TI EN}, where we wnte " = cI>A" (n EN). The structure theorem for complete Ll\IC algebrilli (Theorem 2.2.47) and standard facts about comJJmtative Banach algebras allow u~ to give a test which involves only the continuous characters for the invertibility of an elernellt in a complete, commutative LMC algebra. Proposition 4.10.2 Let A be a complete, commutatzue, 'anital LMe algebm, and let a E A. Then: (i) (T(a) = a(~A) = a(cI>A); (ii) a E Inv A if and only if l)) = {O} if and only if 1l"p(a) E rad Ap. the last two sets are equal. 0 Corollary 4.10.4 Let A be a complete, commutative, semisimple LMC algebra. Then the Gel'fand transform 9 : A ---> C('E A) is a monomorphism. 0 Proposition 4.10.5 Let A be a f1rechet algebra. Then LA is a hemicompact space. Proof The family {q>n : n E N} is a countable family of compact subsets of 'EA; we shall show that each compact. subset of 'E A is contained in some q>n. Let K be a compact subset of 'EA, and set Kp is closed in q>p; by 2.4.33. there is a ~niquc clement J(Ap) such that IIp(cI>p n K) c {I} and ap(q>p \ K) c {O}. Suppose that P ~ q. Then dearly 1l"pq(aq) E J(Ap) and 1l"pq(aq) = p, and so 1l"pq(aq) = ap· Set a = (a p) E J(A). Then II = XK; by 4.10.3, a is uniquely specified by this conciition. 0 a,) E
--
a
We noted in 4.10.2 that t e : A -+ B be a homomorphism. Then ()(k) : A(k) -+ n(h) is also a homomorphism, where
e(k)((aj .... ,ak» = (O(ad, ... ,B(ak»
((ab .... ak)
E
A(k).
Unimodular elements of A(k) were defined after 1.5.33.
Lemma 4.10.7 Let A and B be umtal Banach algebras. and let () : A -+ B be a unital homomorphzsm wzth dense Tnnge. Suppose that a E .4 (k) zs unimodular in A, set C = {c E A(k) : a· (' = eA}. and set D = {d E B(k) : B(k)(a)· d = eB}' Then O(k)(C) is a dense subset oj D. Proof Clearly e(k)(C) c D. Take c E C and d E D. For each j E Nk. there is a sequence (Cj.n: n E N) in A such that e(Cj,n) -+ dj as n --+ 00. Define en = (r.l,n .... ' Ck.n) (n EN). We have a . (c + en - c(a . en» = eA, and so c+ Cn - c(a . en) E C. Also
e(k)((. + c" - e(a . er,»
-+
Thus e(k)(C) is dense in D.
(}(k) (c)
+d-
(}(k) (c) ((}(k) (a) . d)
=d
as n
---+ :)().
o
Theorem 4.10.8 (Arens) Let A = limproj{An;7l"mn} be a umtal Fh§chet algebra, let kEN, and let a E A(k). Suppose that 7l"~k)(a) 'is unimod'ular in An JOT' each n E N. Then a i.~ unzmodulaT' in A. Proof Clearly A(k) = limproj{A~k): 7l"~~1,}. For n E N, define En = {:r E A~k) : 7l"~,k)(a) . X = 7l"n(eA)}. Then En is a non-empty. dosed subset of A~J, and so En h; a complete metric space. Sint:e 7l"mn : An -+ Am has dense range, 7l"~~(En) is dense in Em whenever m ::; n by 4.10.7. By the Mittag-Leffler theorem A.1.25, limproj{En: 7l"~~} =10. Take b E limproj{En:7l"~~~}. For each n E N, we have 7l",\kJ(b) E En, and so 1!"n(a . b) = 7l"n(eA). Thus a . b = ('A. and a if' unimodular in A. 0 Theorem 4.10.9 Let A be a commutative, umtal Frechet algebra. (i) Let kEN. Suppose that a E A(k). Then a(a) = a(EA) = a(.4.)' (ii) Let kEN. Suppose that aI, ... ,ak E.: A and 'P E A. Then theT'e e:r.zsts 'I/J E EA with VJ(ai) = 'P(ai) (i E N k ). (iii) The space E.4 is dense in (UIIle towards a contradiction that K" = 0. Since ~A is hemicompact, and hence Lindel6f, there exists {an: 1£ E N} c A with n{K"" : 1£ E N} = 0. By hypothesis, there exists {bn : n E N} c A with Z(b n ) = Z(a n - cp(an)eA) and a(b n ) C II U {OJ (1/ EN); note that cp(b n ) = O. Now eA + bn E Inv A, and !:oo. rpplacillg bn by b,,(eA +bH)-I, we may suppose that lI(bnJ )::::: 1. Finally, let Cn = bn /2n(1 + Pn(b,,)). Then Pk(C,,) ::::: Pn(cn ) ::::: 2- n (k ::::: 1£), and so 2:~=1 C n converges in A, say C = 2::'=1 cn. Since Cn(~A) C IIU {O} (1£ E N) and n{Z(cn ) : n E N} = 0, necessarily C(~A) C n. Now cp(cm) = 0 (m EN), and so
n
I .•. , x qn + 1 )
1-+
(Fen) (a) + (xq,,+l> .. . , x qn+ 1 ), Xl,' .• , X q,,) , En+l - En·
Com:nmtatille Banach algeb1'U.8
586
Since A!", has the discrete topology and F(II) : A(Pnt 1 ) -> A(p,) is continuous, en is c-:ontinuous. Since lvI«J is dense in A, we have (~n (En+d = Bn (n EN). By the l\fittag-Leffier thcorC'm A.1.25, E = limproj{En;8n} =1= 0. say (Un) E E. Set 1/,n = (a(n),V(n»)' where a(n) E A(p") and v(TI) E lVI~q,,). For each n E N, we have e,,(un+d = Un, and so a(n) - F(n+l) (a(n+1») E AI(p,,); this shows that 'PPn(a(n») = 'PP,,(F(n+I)(a(n+I»))' By (4.10.1), we have
'P1J" (.f(n+l) (a(n+l))) Set z(n)
= 'PPn (a(n»)
E
= .f(n+l) ('PPn+1 (a(n+1)))
(n E N).
CP". Then (Z(n») E Jimproj{C(Pn); F(n)}.
o
Corollary 4.10.16 Assume that there exzsts p ;::: 2 and a sequence (F(n») m O(Cp, CP) slLch that n:'=l (F(l) 0... 0 F(n»)(CP) = 0. Then all commutatwe Prechet algebras are functionally contwuous. Proof Assume' that there is a commutative F'rechet algt'hra which is not func-:tionally continuous. Then limproj{CP;F(n)} =1= 0, a contradiction of the fact that 1T1(limproj{CP; F(n)}) C n{F(1) 0·" 0 F(n»)(CP) = 0. 0 It is a fascinating open question whether or not there is a sequence (F(n») satisfying the assumption of the above corollary. We saw in 2.3.4 that each commutative, scmisimple Banach algehra ha.'l a unique complete norm. \Ve shall now prove the analogous result for commutative, semisimple F'rechet algebras. If we knew that each (F)-algebra were fUIlctionally continuous, then the argument of 2.3.4 would give this result..
Theorem 4.10.17 (Carpenter) Ltt A be a commutative, semzsimple Prechet algebra. Then A has a umque topology as an (F)-algebra. Proof We continue to write A = lim proj An, so that EA = U n. Let T he the given topology on A, and suppose that d is a complete, invariant metric on A. We first claim that, for each compact subset K of EA, all but finitely many characters of K are continuous on (A, d). For assume towards a contradiction that ('¢k) is a sequence of distinct characters in K. each discontinuous on (A. d). By 2.2.26, for each kEN, there exists ik E A such that 'ljJi(ik) = 0 (j E Nk) and 1/JAik) =1= 0 (j > k); define O'.k = 1/Jk+l(h'" fk), so that Ok =1= O. Since t,Jk+1 is discontinuous at 0, we can successively choose gk E A such that: (i) dUj ... fkgk, 0) < 2- k (j E N k ); (ii) 1000k'¢k+I(gk)1
> k + I'¢k+l ('L7::
it··· figj) I
(k;::: 2).
Let hj = 'L~=j fj ... hg,.· (j EN); by (i), each of these serirs converges in (A, d). We have k
hi
= Lit· .. f 3 gj + it ... fk+1 (gk+1 + h k+2 ) (k
E N) .
j=1 But 1/Jk+ I (fk+ d = 0, and so
..
"H' (h,) ~ "'+> (~h /;9,) + "''''+> (9')
(k E N).
Commutati'll€ topolog%cal algebras
587
By (ii), l.,pk+l(hdl 2 k (k 2 2). However, K is compact, and so hI is bounded on K, and hence we have a contradiction. Thus the claim is established. Take a E A such that a = limk---->oo ak in (A, d) for a sequence (ak) in A with in (A, r), and set U = N E EA : .,p(a) ::j:. OJ. Then U is an open set in ~A, and, by the claim, un H(,(, and p : X ...... C" is a local homeomorphism. A function f on (X,p) is analytzc if. for each x E X, f 0 p-I is analytic- on a neighbourhood of p(x) in 1C11 • The algebra of analytic functions on (X,p) is defiupd to he Ox. Let U he an open set in e ' , let E = Eo(u), and let p = (ZI'.'" Z,,). Then it is proved in (Gunning and Rossi 19B!'>. J.G.ll and LH.15) that (E,p) is a Ripmann domain and that the Gel'fand transform is an isomorphism from O(U) onto 0>.:. (The set E is thf' envelope of holom011Jhy of U.) The Riemann domain (E,p) is, in particular, a Stein manifold of dimension n, and so. by (ibid., VII.C.R), there exist /J, ... , hn+l E O(U) such that the map 1/J ...... (1f;Ut), ... , 'IjJ (hn+l)), E ...... 2n + 1, is injective. By 4.10.10, O( U) is fun(,tionally continuous. Examples 4.10.29 and 4.10.30 are from (Williamson 1954), and Example 4.10.31 is from (Arens 1946); see also (Zelazko 1965, Chapter 12). We know from 2.2A1(i) that a complete, metrizable, locally convpx algebra which is a divi5ion algebra is isomorphic to C. It seems to be an open question whether or not an (F)-algebra which is a field is necessarily isomorphic to C. Example 4.1O.:i2 is from (Prankiewicz and Plebanek 1995): every algebra has a linear space topology with respect to which it is a semi-topological algebra.
c
5
Automatic continuity theory
In this final chapter. we turn to the substantive automatic continuity theory of homomorphiHms and derivations from Banach algebras. We shall obtain many positive results. and we shall also construct various discontinuolls homomorphism." and derivationH that show the limits of our positive theoreml:>. In §5.1. we shall introduee our subject by discussing which Banac'h algebras have a unique complete norm. It was this 'uniqueness-of-norm' quest.ion that initiated the study of automatic continuity theory. We have beautiful result.s in bot.h directions, but we still await a full charact.erization of which Banach algebras do have a unique complete norm; this remains true if we restrict ourselves to t.he class of commutative Banach algebras or to the class of Banach algebras wit.h a finitc-dimenHional radical. In §5.2. we HhaII build t he theory of the separating space of a linear map betwecn topological linear spaces, and in ~5.3 we shall st.udy the continuity ideal of Huch a map: this iH the annihilator of the separating space. Study of the possible form of such a continuity ideal will lead to a variety of positive automatic continuity results. The theme of §5.4 is the main boundedness t.heorem of Bade and Curtil:>. We are led to a full deHcription of the Ht.ructure of an arbitrary homomorphism from the uniform algebras C(n) and various related algebras. Linear functionals--positive functionals and higher point derivations- are the topic of §5.5, and. in !i5.5. we shall produce a medley of reHults on the automatic continuity of derivationl:> and intertwining maps. We shall also determine the form of continuous derivations from various Banach algebras. In particular, we shall characterize the amenable and weakly amenable algebras in varioUS classes of Banach algebras. The Hection concludes with the conHtruction of Home discontinuouH derivationl:>. , The final section, §5. 7. iH the climax of our work: we shall construct (with CH) discontinuouH homomorphil:>ms from essentially every infinite-dimensional, commutative Banach algebra. In particular we shall prove that there are discontinuous homomorphiHms and algebra norms not equivalent to the uniform norIll for the algebras C(Q) whenever Q is an infinite, compact space. We Hhall aiso prove that many commutat.ive algebras, including all th-valuation algebras, are normable. In this chapter, we shall denote the family of open neighbourhoods of the origin in a topological linear space E by N E.
Uniqueness of norm 5.1
597
UNIQUENESS OF NORM
We have explained that a major concern of this book is to study the relation..-.;hip between the algebraic and topological structureH of a Banach algebra. The nature of this relationship is brought into sharp focus by the 'uniqu('ness-of-norm' problem. which was descrihed in §2.1: we wish to know which Banach algebras have a unique complete norm (see Definition 2.1.6). We have already proved in 2.3.3 that paeh commutative. semisimple Banach algebra has a unique complete norm. It wa."> a historically important question, rabed by Rickart in 1950, whether or not each semisimple Bana.ch algebra has a unique complete norm; we shall resolve this question in the present section. The solution to Rickart's problem was obtained by Johnson in 1967. and we shall give Johnson's original proof in 5.1.5. The ideas contained in Johnson's paper were the seed from which much automatic continuity theory has grown, as will become apparent in !i5.2 and §5.3. In 1982, Aupetit gave a dramatically different proof of Johnson's result. This proof used the theory of subharmonic function& and Vesentini's theorem 2.3.32. Finally, in 1989, Ransford gave a beautiful simplification of Aupetit's proof. and it is this proof that we shall present in 5.1.9. A third proof of a more general version of Johnson's theorem will be given in 5.2.28(iii). The fact that an algebra 8(E) has a unique complete norm was published by Eidelheit more than 60 years ago in 1940. (This theorem preceded GeI'fand's theorem for commutative, semisimple algebras, which was published in 1941.) We shall give a generalization of Eidelheit's result in 5.1.14. Further results exhibiting Banach algebras which have a unique complete norm will be given in 5.2.18 -5.2.21. A particularly important notion in automatic continuity theory is that of the separating space of a linear map. vVe introduce this space here; its properties will be explored further in §5.2. Definition 5.1.1 Let E and F be topological linear spaces, and lpt T be a linear map from E into F. The separating space of T zs 6(T), where
(5.1.1) It is clE'ar that y E 6(T) if and only if there is a net (xv) in E such that 0 and Tx" ~ y in F. Suppose that E and Fare metrizable. Then Y E 6(T) if and only if there is a sequence (xn) in E such that Xn ~ 0 and :1:" ~
TX n
~ y.
The separating space is already implicit in this work: for example, the proof of 2.3.3 essentially shows that, if A is a Banach algebra. if B is a commutative Banach algebra, and if () : A ~ B b a homomorphism, then 6(()) c md B. Proposition 5.1.2 Let E and F be topologzcallznear spaces, and let T : E ~ F be a linear map. (i) The separating space 6(T) is a closed linear subspace of F. (ii) Suppose that E and F are (F)-spaces. Then T is continuous if and only iJ6(T) = O.
Automatzc rO'TIt'tnltZly theory
Proof (i) Certainly 6(T) is closed ill F, and 6(T) is a linear subspace of F \)('C11USP addition and scalar multiplication are continuous in E and F. Oi) This is a restatement of the closed graph theorem A.3.25. Proposition 5.1.3 Let A and B be Banach algebras, and let () : A homomorphism with O(A) = B. Then:
0 -+
B be a
(i) 6 (fJ) zs a closed ideal m B; (ii) for each bE 6(0), O"B(b) is a connected sttbset ofe conta'inmg the ()T'tgin; (iii) 6(0) n J(B)
= {O}.
Proof (i) This is immediat('. (ii) Lpt bE 6(0). There is a sequence (an) in A with an -+ 0 and fJ(a n ) in B. By 1.5.28, VB«()(a n )) -+ 0, and so the result follows from 2.4.8. (iii) For p E J(B) \ {O}, O"B(P)
= {O, I}, and so this follows from (ii).
-+
b 0
\i\Te now give Johnson's original proof that each semisimple Banach algebra 11a:. a unique complete norm. Let A be a Banach algebra. and let E be a simple left A.-module. It was proved in 2.6.26(i) that there is a unique norm II . II snch that (E, 11·11) is a Banach left A-module. The key point of Johnson's proof is that E has a unique norm with respect to which it is a weak Banach left A-module, in the terminology of 2.6.1. Theorem 5.1.4 Let A be a Banach algebra. and let p be a simple representatwn of A on a normed space E l1nth peA) c B(E). Then p : A -+ B(E) 1,8 contmuous. Proof "We begin with a preliminary remark. Let x E E, and let S be a subset of E with S.L . x =I=- O. Suppose that A[l] . x is unbounded in (E. II· II)· We dazm that (S.L )[lJ . x is unbounded in (E, II ,11). By 1.4.29(iii), x.L is a maximal modular left ideal in A and so, by 2.2.28(i), x.L is closed. Also, S.L is a closed left ideal in A. Since S.L x.L. we have S.L +x.L = A, and so, by A.3.43. there exists n > 0 such that, for each a E A, there exist b E S.L and c E ,r.L with a = b+c and 11bll + IleII :::; a Ilall· For each n E N, there exists an E A[lJ with Ila n . xii> an; set a" = b" +C n , where bn E S.L, Cn E x.L, and Ilbnll + Ilenll :::; allanll :::; a. Then a-Ibn E (S.L )[1] and Ila- 1bn • xii = 0:- 1 Ilan • xii> n, and so (S.L )[1] . x is unbounded in (E, 11·11), as claimed. We now prove that p : A -+ B(E) is continuous. 1f E is finitl,>..dimensional, then ker p is closed in A, and so, by I A.3.42(i). pis cOlltinuous. Thus we may suppose that E is infinite-dimensional. For x E E, define Px : a f---? p(a)(x). A -+ E. and set
rt
F = {x E E: Px E B(A, En. Then F is a submodule of E, and so F = 0 or F = E because E is simple. Assume towards a contradiction that F = O. Since E is infinite-dimensional, there exists a linearly independent set {x n : n E N} in E with Ilxnll = 1 (n EN). For E N, set Mn = and In = M1 n ... n Mn. Then Mn is a closed maximal ideal in A, and, by 2.6.26(iii), In . Xn+l =1= O. Since X n +1 ¢ F, the set A[l] . Xn+l
n
x;
599
Uniqueness of norm
is unbounded in (E, /I. I/), and so, by the claim, (1n)[IJ . X l1 +1 is unbounded in (E, 11·/1)· Thus there is a sequence (an) in A such that an E In and Ilonll S 2- n for n E N. and such that Ilan . x n+lll ~ /I(al
Set bn =
E%:n ak
+ ... + a -d . X n+lll + n
= Q.l + ... + an + bn+l
Since ak E Mn+l (k ~ n + 1),
~
IIp(bdll
(5.1.2)
(n EN); each series converges in A, and
b1
Thus, for each n
(n ~ 2).
l1
WE'
(n E N).
have bn+1 E 1\,[n+l, and so
2, we have ~ /Ian' Xn+l/l-/l(al
+ ... + an-I)
.
X n+l/1
~ n.
by (5.1.2). But this is impossible. Thus F =I- 0, and so F = E. For each a E A, pea) E B(E), and, for each x E E, /lp(a)xlI S IIPxll (a E A[l])' By A.3.37(i), there exists M such that IIp(a)xll S 1\f IIxll (x E E. a E A[l])' Thus p : A --+ B(E) is continuous. 0 Theorem 5.1.5 (Johnson) Let A and B be Banach algebras, and let () : A --+ B be an epimorphism Jrom A onto B. Then S(()) c radB. Suppose, Jurther, that B is semis~mple. Then () is a1Ltomat~cally continuous. Proof Let bo E S(()), sayan ---+ 0 in A and ()(a n ) --+ bo in B. Let M be a maximal modular left ideal in B with right modular identity u, and set E = BIM. Define p(a)(b +]\f) = ()(a)b + M (a E A, bE B). Then E is a left A-module with respect to the representation p. Since () ill an epimorphism, p is a simple representation, and, since IIp(a)(b + M)II S 1I(}(a) II lib + Mil
(a E A, b E B),
we have peA) c B(E). By 5.1.4, p ill continuoull, and so p(a n ) particular, p(an)(u + 1\1) --+ O. But p(an)(u + At) = (}(an)u
+ !vI --+ bou + M
---7
0 in B(E). In
= bo + M.
Thus bo EM. It followll that S«()) c M for each maximal modular left ideal M. By 1.5.2(ii), 6(0) c radB. If B is Ilemisimple, then S(O) = 0 and 0 is continuous. 0 The first corollary of this result is Johnson's uniqueness-oJ-norm theorem. Corollary 5.1.6 Let A be a semisimple Banach algebra. Then A has a unique complete norm.
0
Corollary 5.1.7 Let A be a semisimple Banach algebra with an involution. Then there is an equivalent norm on A such that (A, *) is a Banach *-algebra. Proof This is immediate from 3.1.2.
o
600
AlLtomatu: continlLity theor'Y
Our second proof of the uniqueness-of-norm theorem is quite different.
Lemma 5.1.8 Let A be a Banach algebra, let P E A[X], and take R > 1. ThrTt
(ZlA(p(1)))2 S sup lIA(p(Z))' Izl=R
(5.1.:3)
:mp lIA(p(Z)). Izl=l/ R
Proof Let q E A[X], and take A E A' with IIAII = 1 and (q(l), A) = Ilq(l)il. Set F = A 0 q. By t.he maximum modulus theorem applied to tIl(' function z f-+ F(z)F(l/z) on the annulus {z E C: l/R S Izl R}. Wl' obtain
s
IF(1)1 2 S
sup Izl=R
IF(z)I'
sup Izl=l/ R
IF(z)l·
It follows that
Ilq(1)11 2 Apply (5.1.4) with q 2.3.8(iii).
(1I p2 (z)11
1 / 2 ")
n
Izj=R
= p2n,
Ilp2" (z)11 2" ----> 1/
S sup
Ilq(z)II'
where
lIA(p(Z))
11
a1:>
sup Izl=l/ Ii
E N.
n
is decreasing, the map z
----> f-+
IIq(z)ll·
(5.1.4)
By the spectral radim; formula
ex:: for each z E C. The sequence
Ilp2"(z)11 1/ 2n
is continuous on C,
and the map Z f-+ lIA(P(Z)) is U.S.c. by 2.3.24(ii). Thus it follows from Dini's theorem A.1.lO(ii) that.. for each r E jR+. we have sup II p2 "(z) 111/2"
---->
Izl='
sup lIA(p(z))
as n---->oc.
Izl=r
o
Hence (5.1.3) follows from (5.1.4).
Theorem 5.1.9 (Aupetit) Let A and B be Banach algebras. and let T: A be a linear map su.ch that lIu(Ta) S 1I.4(a) (a E A). (i) Su.ppose that b E 6(T). Then
(lIB(Ta))2 S lIA(a)ZlR(Ta - b)
(a
E
A).
->
B
(5.1.5)
(ii) T(A) n 6(T) c D(B).
(iii) Suppose that T is a .mr'jectlOn. Then 6(T) c rad B. Proof (Ransford) (i) Choose (an) in A with an -> 0 and Tan ----> b as n ----> 00, and take a E A. For each c > 0, we may, by 2.3.20, choose norms on A and B which are equivalent to the given Horms and which are such that lIall S 4 (a) + c and IITa - bll S ZlB(Ta - b) + c. We apply 5.1.8 in the case where p = (Ta - Tan) + (Tan)X E B[X], where n E N. We have p(l) = To., and so, for each R > L
r.
(lIB(Ta))2 S sup ZlB(p(z:))· Izl=R
Now liB (p(z)) S liTo. - Tan II + by hypothesis,
IzlliTanll.
ZlB(P(Z)) S ZlA(a - an
sup ZlB(P(Z)).
(5.1.6)
Izl=l/ R
Also p(z)
+ zan) S Iia -
=
an II
T(a - an + zan), and so,
+ Izillanil .
Uniqueness oj norm
601
Thus, from (5.1.6), we have (vB(Ta))2 5 This
hold~
(lia - anll + R 110.,)11)(IITa -
for each n E N. and so, letting n (vn(Ta»2S
+ II Ta' n II / R).
we see that
110.11 (11Ta - bll + Ilbll /R).
But this holdb for each R > 1, and (vB(Ta)2S
----> ::x:>.
Tan II
~o,
IlalillTa - bll
letting R
5 (/JA(a)
----> 00.
we obtain
+ e)(lIB(Ta -
b)
+ e).
Finally, this holds for each e > 0, and so (5.1.5) followb. (ii) This is immediate from (i). (iii) Take b E 6(T) and c E D(B). Since T is a surjection, there exists a E A with Ta = c + b. By (5.1.5). vB(b + c) = O. and so, by 2.6.31. bE rad B. 0 We obtain the ~econd proof of 5.1.5 a~ follows. Let A, B, and 0 be as in 5.1.5. By 1.5.28, v}J(O(a») 5 vA(a) (a E A). and so, by 5.1.9(iii). 6(0) c radB. One might seek to generalize 5.1.9 by proving the following statement: zJ A is a Banach algebra, zJ B is a sern-lszmple Banach algebra, and iJ T : A -+ B is a lznear map such that vB(Ta) 5 vA(a) (a E A) and T(A) is dense 'in B, then T zs automatu:ally continuous. However. this statement is false. For let A be any infinite-dimensional Banach algebra. and let B be the example of Dixon described in 2.3.15, so that B is semisimple. bllt B contains a countable subset {'W" : n E N} such that Bo = lin{ Wn : n E N} h; a dense subalgebra of B con~isting of non-zero nilpotent ('lement~. Let {an : n E N} be a linearly independent set in A. (kfine Tan = (n 110.,,11 / Ilwnll)wn (n EN). and extend T to be a linear map from A into Bo. Then vu(1'a) = 0 (a E A), and T(A) = Bo is dense in B, but T is di~continuou~ because IITanl1 = n II(1nll (n EN). However, the linear map T of the above example is not a homomorphism. We find the following to be a significant open question.
Question 5.1.A Let A be a Banach algebra, let B be a semzsimple Banach algebra, and let 0 : A ----> B be a homomorphzsm with O(A) = B. Is 0 automatically contmuo'us'? This question is equivalent to the following.
Question 5.1.A bis Let A and B be Banach algebras, and let (j : A ----> B be a homomorphism. Take b E 6(0). Is it necessarily the case that a(b) = {O}? First, assume that the answer to Question 5.1.A i~ positive and that A ----> B is a homomorphism. Let 7f : O(A) ----> O(A)/radO(A) be the quotient map. Then 7f 0 0 is a homomorphism with dense range in a semisimple . Banach algebra, and so 7f 0 0 is continuous. Thu~ 6(0) C rad8(A) C D(B). Conversely, assume that the answer to Question 5.1.A bis is positive and that (J : A ----> B is as in Question 5.1.A. Now 6(0) is an ideal in B with 6(9) c D(B), and so 6(0) c radB by 1.5.32(iii). But B is semisimple, and so 6(0) = 0 and 0 is continuous. (J :
A utomatzc continuity theory
602
There is an extension of 5.1.6 which applies to some Banach algebras which are not semisimple. For the thf'ory of analytic spaces, see Appendix 5. Let A be an algebra. and let I be an ideal in A. As in 1.4.4. we define I.!.
=
{a E A : aI
= O},
IT
=
{a E A: Ia
= O}.
In the case where A is a Banach algebra, I.!. and IT are closed ideals in A. Theorem 5.1.10 (Dales and Loy) Let (A, 11·11) be a separable Banach algebra 'Unth radzcal R such that R.1. RT has fimte codimension zn A. Then A has a
umque complete norm. Proof Set I = R.1. RT. let 11·11' be a second Banach algebra norm on A, and define the projective norms II . 11.71" and II . II~ on I corresponding to II . II and II '11', respectively, as in (2.1.12). Then 11·11 7r and II'II~ are algebra norms on I with Iiall :::; Ilall 7r and Iiall' :S Ilall~ for each a E A. The radical of R.1. is R.1. n R. Denote the quotient norms on R.1. / (R.1. n R) corresponding to 11·11 and 11·11' by 111·111 and 111'111', respectively. By 5.1.6, there is a constant C such that
Ilia + R.1. n Rill' :::; Cilia + R.1. n Rill
(a
R.1.).
E
Similarly. we may suppose that
IIla+RTnRIII' :::;Cllla+RTnRIlI
(aERT).
Take a E I and c > O. Then there exist a1, ... , an E R.1. and bl with a = r:.j=l aJbj and r:.j=l IlaJIIllbj II < Ilall7r + c. We have
IIlaJ + R.1. and so
n RllllllbJ + RT n Rill:::;
Ilaillllb j II
(j
E
, ...•
bn
E
RT
N n ),
n
E IIlaj + RJ.. n Rill' IIlb) + RT n Rill' < C
2
(llall7r + c).
j=l
Now choose r·1 .... , rn E R.1.
n Rand .'11,""
Sn E RT
nR
with
n
E lIaj + rjll' Ilb
j
j=l
+ Sjll' < C 2 (llall 7r + c).
We have
+ ajsJ + rJsJ E RRT + R.1. R + RRT = 0 and this implies that r:.j=l(aj + rj)(bj + 8j) = a. Thus This holds for all c > 0, and hence lIall~ :::; C2 Ilall7r. rjbj
(j E Nn ). lIall~
< C 2 (lIall7r + c).
By hypothesis, (A, 11·11) is separable and I has finite codimension in A, and so, by 2.2.16(ii), I is closed in (A, II '11) and there is a constant ]I.,[ such that lIall 7r :S M Iiall (a E I). For a E I, we have Iiall' C 2 .l\,f lIall, and thus, by A.3}3, the identity map (A, 11·11) _ (A, 11,11') is continuous. Hence !I'II and II . II are equivalent on A. 0
:s
We now give a result in the opposite direction to the above theorem.
Uniqueness of norm
603
Lemma 5.1.11 Let (A, 11·11) be a Banach algebra, and let I and K be closed ideals in A with K c I. Suppose that {Tl, ... , Tk} is a lineaTly independent set in KnI1. nIT with TiTj = 0 (i, j E N k ), and that AI, ... , Ak are lineaT functionals on A with Ai I (12 + K) = 0 (i E N k ) and such that Al I I is dzscontmuous. Define
IIlalli ~ inf {ila + Then
III . III
t II t (,r,
+
I>,(a) - (,I ' (" ... , (. E
is a Banach algebra nOTm on I and
III . III
c}
(a El).
is not equivalent to
II '11.
Proof Define E = I EB C k as a linear space, and set k
p(a, (1)'''' (k)
= lIall + L IAi(a) i=l
((a, (1, .... (k) E E).
- (i!
Then p is a norm on E, and (E,p) is a Banach space. Define a product on E by
(a, (1, ... , (k)(b, 'TIl,···, 'TIk) = (ab, 0, ... ,0). Then E is an algebra for this product. We have p(ab, 0, ... ,0) Ai ! 12 = 0 (i E Nk)' and so p is an algebra norm on E. The map
=
lIabll because
k
() : (a, (1, ... , (k) ~ a -
L (iTi,
E
---->
I,
i=l
is a homomorphism because TiTj = 0 and Til = ITi = 0 for i,j E Nk, and () is a surjection. We have kE'r() = { (t(iTi,(l, ... ,(k)
: (ll'" ,(k E
c} ,
a finite-dimensional subspace of E, and so ker () is closed. Thus I is a Banach algebra with respect to the quotient norm, say q, from E. :For each a E I, we have q(a)
= inf {P(b, (1,"" = inf {ila
+
(k) : a
t t (iTill
+
tt=l
(iTi}
IAi(a) -
(il}
= b-
=
IIlalil
because Ai I K = 0 (i E Nk). Thus 111·111 is a Banach algebra norm on I. Take (an) in I with lIanll ----> 0 and IAl(an)1 ----> 00 as n ----> 00. Assume that II Ian III ----> 0 as n ----> 00. Then, for each sufficiently large n EN, there
IIL7=1
I L7=1
(In,"', (kn E C with (inTi + IAi(an) - (in I ~ 1. But then klnlliTlll + IAI (an) - (lnl ~ 1 because {Tl, ... , Tk} is linearly independent. This is a contradiction of the fact that IAI(an)1 ----> 00 as n ----> 00. Thus 111·111 is not equivalent to II . II on I. 0 exist
604
A'afomatic continuity theory
Theorem 5.1.12 (Dales and Loy) Let (A.II·II) be a unital Banach algebra with a non-zero, finite-dzmensional mdu:al. Snppose that A contain., a minimal radical zdml K Buch that K.1. KT has infinite codimen.'0 E E' such that (1)'(S,,xo)1 : n E N) is unbounded. Set Zn = (8nxo, >'o) (n EN). Define T = Xo ® >'0 E~. Now
(T8nT)(x)
=
(x. >'0)(T8n )(xo)
=
z,,(x, >'o)xo
=
znTx
(x E E, n EN),
and HO znT = T8nT (n EN). Thus IZnllllTll1 ~ IIITI1I2. But this iH a contradiction b~ause limsuPn-+00 IZnl = 00 and IIITIlI =1= O. 0
Theorem 5.1.15 Let E be a Banach space, and let ~ be an operator algebra on E. Then any two complete algebra norms on ~ are equwalent. Proof Supposc that '2t is a Banach algebra with rCHpcct to III ·1111 and III . 111 2 , Take (T..) in ~ such that Tn -+ 0 in (~, 111·1111) and Tn -+ T in (~, III· lib)· By 5.1.14, Tn -+ 0 and Tn -+ T in (B(E), II·ID, and so T = O. Titus 111·1111 and III . II b are equivalent norms. 0 Corollary 5.1.16 (Eidelheit) Let E be a Banach space. Then each of B(E), A(E), K(E), N(E), I(E), and S(E) has a unique complete norm. 0 Let ~ be a Banach operator algebra on E. It followH from 2.1.7 that there is a discontinuouH homomorphism from ~ into a Banach algebra if and only if there is an algebra norm on ~ not equivalent to the given norm. We remarked in the introduction to §2.1 that, if A is an algebra with A 2 == 0 and if A is a Banach space with respect to two inequivalent normH, then A is a Banach algebra with respect to each of these normH, and HO A does not have a unique complete norm. We now give two examples (in 5.1.18 and 5.1.20) which show that, even if A is a commutative Banach algebra with a o;le-dimensional radical, then A does not necessarily have a unique complete norm. Let A be a commutative Banach algebra, and let E be a Banach A-module. Set ~ = A EB E, where (a,x)(b,y) = (ab, a . y +b . x) for a,b E A and x,y E E· Then, aH in 1.8.14, ~ is a commutative algebra with rad~ = (radA) 0 E. Let D : A -+ E be a derivation, and set
lI(a,x)1I1
= lIall + IIxll,
lI(a.x)lb
= lIall + IIDa - xl!
(a E A, x E E).
Uniqueness of norm
607
Theorem 5.1.17 The algebra Qt. 1S a Banach algebra w'£th respect to both and II . 112' The two norms are equivalent 1f and only 1f D 1S continuolLs.
II . III
proof Certainly (Qt., II· lit) is a Banach algebra and 11·112 is a norm on 21. We verify that 11·112 is a complete norm. Let «an' xn)) be a Cauchy sequence in (Qt.,11·11 2)· Then (an) and (Dan - xn) are Cauchy sequences in A and E, respectively. Since A and E are Banach spaces. there exist a E A and x E E such that an ----+ a and Dan - Xn ----+ x. Then (an' xn) ----+ (a, Da - x) in (Qt., 11.11 2) and so 11·112 is a complete norm. For (a, x), (b,y) E Qt., we have
II (a, x)(b, y)11 2 = lIabll + Iia . (Db - y) + b . (Da - x)11 :::; (ilall + IIDa - xll)(lIbll + IIDb - yin = lI(a,x)11211(b.y)11 2 , and so 11·112 is an algebra norm on Qt.. Suppose that D is continuous. Then
II(a,x)11 2:::; Iiall + IIDllllal1 + Ilxll :::; (1 + IIDI\) II(a,x)11 1 «a,x) E Qt.), and so 1I·ll t and 11.11 2 are equivalent norms. Conversely, suppose that 11·111 and 11·112 are equivalent norms, say II(a,x)1I2 :::; c II(a,x)llt «a,x) E Qt.). Then I\Dall :::; II(a, 0)112:::; c lI(a, 0)11 1 = c Iiall (a E A), and so D is continuous. 0 Theorem 5.1.18 There 1S a comrnutatwe algebra w1th a one-dimensional radical which is a Banach algebra with respect to two inequzvalent norms. Proof Let A be a Banach function algebra with a discontinuous point derivation d at a character cp. As in 2.6.2(ii), C is a Banach A-module with respect to the operation (1, z) t--+ cp(1)z, A x C ----+ C, and so we are in a situation where 5.1.17 applies: Qt. = AEBC is a Banach algebra with respect to the product (J, z)(g, w) = (1g, cp(1)w + cp(g)w) and each of the norms 11·111 and 11.11 2, where 11(1, z)lI l = Ilfll + Izl and 11(1, z)11 2= IIfll + Id(1) - z\. As above, rad Qt. = {O} EB C. and so rad Qt. is one-dimensional. 0 Proposition 5.1.19 Let (A, II-II) be a Banach algebra w1th radical R. Suppose that R2 = {O} and that A has a Wedderburn decomposition A = B 0 R which UI not a strong Wedderburn decompos1tion. Then there 18 a complete algebra norm 111'111 on A, not equivalent to 11·11, with Illblll = lib + RII (b E B). Proof Each a E A can be uniquely expressed in the form a = b+r, where bE B and r E R; in this case, set Illalll = lib + RII + IIrli. It is clear that (A, III-liD is a Banach space, and that Illblll = lib + RII (b E B). Take al = bl + rt, a2 = b2 + r2 EA. For each Sl, S2 E R, we have ala2 = bl b2 + (bl
because R2
+ 8l)r2 + (b 2 + s2)rl
= {O}, and now 111·111 is an algebra norm on
A because
III ala2111 :::; IIbl + Rllllb2 + RII + IIbl + Rlllhil + IIb2+ Rlilirill ::; III allllllla2111 . The norm 111·111 is not equivalent to 11·11 because B is closed in CA,III'III), but not in CA,II·II). 0
608
Automatzc contmuity theory
Theorem 5.1.20 (Bade and Dales) There
Z8
a st1'Ongly regular Banach functzon
algebra A wzth a closed subset E ofiPA such that dimrad(AIJ(E)) AI J(E)
Z8
=
1 and
a Banach algebra for two mequivalent norms.
Proof Let A = /1.1# be as in Example 4.5.33. and take F = 2Z U {oo} and 21. = AI J(F), so that 21. is a Banach algebra with respect to the quotient norm 11·11. Set 9l = rad21j by 4.5.33(vi), dim9l = 1 and 91 2 = {o}. Now definE' 9)1 = AI I J( F), so that, by 4.5.33(viii), 9)1 = 9l~. By 4.5.33(ix), }.f2 + J(F) has infinite codimension in AI, and so 9)12 ha..c; infinite codimewiion ill 9)1. By 4.5.33(x). 21. is decomposable, but not strongly decomposable. The result follows from 5.1.19. 0 A further example of a commutative algebra with a oIlL'-dimem;ional radical that is a Banach algebra for two inequivalent norms will be given in 5.4.6. Despite the substantial amount of work on the uniqueness-of-norm problem for Banach algebras, much remains unknown.
Question 5.1.B Let A be a Banach algebra wInch is a przme algebra, a 8emipr't1TlC algebra, or an mtegral doma,m. Does A necessarzly have a unique complete norm? It will be shown in 5.3.12 that, if the answer to any of these questions be negative, then t.here exists a topologically simple, radical Banach algebra; if it Banach algebra which is an integral domain fails to have a unique complete norm, then there is a commutative, topologically simple Banach algebra. Notes 5.1.21 The first result showing that a Banach algebra A has a unique c:omplete norm was EidclhE'it's proof (1940) of 5.1.14 in the casE' where A = B(E); thE' Il1('thod is close to all even earliE'r Ol1e of Mazur from (1938). An argument in the'proof of 0.1.14 shows that th(' operator norm on A is mmimal' IIITIII = IITI! (T E A) whenever 111·111 is an algebra norm on A such that IIITIII ~ IITII (T E A) (Bonsall 19.'54). It was Rickart who focused on thE' uniqueness-of-norm question for semisimple Banach algebras in (1950). RE'sults obtained up to 1960 are summarized in (Ric-kart 1960, II, §5). For example, Rkkart proved (ibzd., 2.5.9) that each primitive Banach algebra with mmimal ollc-sided ideals has a unique complf'tc norm. Extensions of Rickart's results were given by Yood (1954, 195fl). Johnson's solution to the uniqueness-of-norm problem is in (1967a); Se€' also (Bonsall and Duncan 1973, §25) A rf'lated result is givcn in (Ptlik 1968). Aupetit"s proof of 5.1.9 was given in (lD82); see also (Aupetit 1991, V, §5) Ransford's proof is in (1989). In the case where T is a homomorphism, the appeal to 2.6.31 in the proof of (iii) is unnecessary becausc in this casE' SeT) is an idcal in B contained in fl(B), and so SeT) c rad B. Aupdit's proof also established thE' uniqueness result for 'Jordan-Banach algebras', and this result was extended by Rodriguez-Palacios (1985): each complete, normed, nOll-associative algebra with zero 'weal{ radical" h~ a unique complete algebra norm. Theorems 5.1.10 and 5.1.12 are taken from (Dalcs and Loy 1997); results stronger than 5.1.10 are proved, and it is conjectured that t.he converse to 5.1 12 holds. An example in (ibid.) exhibits a separable Banach algebra A with a non-zero, finite-dimensional radical R such that {R.l. n RT? has infinite codimentiion in A. but R.l. RT has finite codimension, and so A has a unique complete norm. Other examples indicate that difficulties can arise even when dim R = 2. Proposition 5.1.19 and Theorem 5.1.19 are from (Bade and Dales 1993); in this paper, other examples of mutually inequivalent norms are constructed.
The separatmg space and the stability lemma
609
5.2 THE SEPARATING SPACE AND THE STABILITY LEMMA
We begin our BtUrly of automatic continuity with somE' general resultB about the separating space 6(T) of a linE'ar map T between topological linear spaces E and F. The separating space 6(T) WaB defined in 5.1.1, and we noted in 5.1.2 that 6(T) is a closed linear subspace of F and that, in the case where E and F are (F)-spaces, T is continuoUl; if and only if 6(T) = O. The fundamental result about the separating space is the stability lemma 5.2.5(ii); a more general version of essentially the same result is calk>d the gliding hump theorem 5.2.6. We shall then give a number of applications of these results, establishing that various linear maps are indeed automatically continuolL. F be a
linear map. and let Q : F -> F/6(T) be the quotient map. Then 6(QT) = O. Suppose that E and F are both (F) -!ipace8. Then QT is continuous, and the linear subspace T-l(6(T») is closed in E. Proof Let y E F \ 6(7'). Then there exists U E NE with Y E NR with V + V c U, and set R = 6(T) + T(V). We have
tI- T(U). Take
V
R C T(V)
+ T(V)
c T(U) ,
and so y tI- R. Now Q(R) is closed in F/6(T) because Q is an open map. and so Qy tI- Q(R) = (QT)(V). Thus Qy tI- 6(QT), and hence 6(QT) = O. The fact that QT is contiuuoUB in the case where E and F are (F)-spaces iB the closed graph t.heorem A.3.25. and then T-l(6(T» = (QT)-I({O}) is closed ~R
0
Proposition 5.2.2 Let E and F be (F)-Sl)Q.ces, and let T : E -> F be lmm'l·. (i) Suppose that E1 is an (F)-space and that R : EI -> E is a contmuoU8 linmr map. Then 6(TR) C 6(T); if R is a surjection. then S(TR) = S(T). (ii) Suppose that Fl is a topological linmr space and that S : F -> Fl is a contmuous linmr map. Then S(S(T» = 6(ST). If F1 is an (F)-space, then
ST is continuous if and only if S(6(T» = o. (iii) Suppose that R E B(E) and S E B(F) are such that TR-ST E B(E, F). Then S(S(T» c SeT).
610
A utornatic contmuzty theory
(iv) Suppose that El and Fl are lmear subspaces of E and F, respectwely, such that T(E l ) C Fl and 6(T) C Fl' Then T (:E0 C FI, and the lmear map x + El ~ Tx + F l , E/ El --t F / F l , zs contmuous. (v) Suppose that E and El, ... , En are (F)-spaces, that R j E T3(Ej, E) fo7' j E Nn , and that E = Rl(Ed + .,. + Rn(En). Then
6(T) = 6(TRd
+ ... + 6(TRn).
(5.2.1)
(vi) Suppose that F is an (F)-algebr'a, and that 8 zs a non-empty subset oj F such that 6(T) . 8 = 0. Then {:l: E E : Tx . 8 = O} is closed m E.
Proof In (i) and (ii), we are considering the following linear maps: R
El
--+
(i) Let y E 6(TR) and U E
E
T
--+
F
S
--+
Fl .
N E . Then R-l(U)
E
NEll and
y E TR(R-l(U» c T(U). Thus y E 6(T), and so 6(TR) C 6(T). Now suppose that R is a surjection. Let y E 6(T) and V E N E1 . Then R(V) ENE by the open mapping theorem A.3.23, and so y E (TR) (V). Thul:> y E 6(TR), and hence 6(T) C 6(TR). (ii) Let y E 6(T) and U ENE' Then y E T(U) and 8y E (8T)(U), so that 8(6(T)) C 6(8T). Since 6(8T) is closed, 8(6(T» c 6(8T). Let Q : F --t FI6(T) and Ql : Fl --t F/S(6(T» be the quotient maps. By 5.2.1. QT is continuous. Since ker Q = 6(T) c ker Q 1 8, it follows from A.3.1O(i) that there is a continuous linear map S: FI6(T) --t Fl/S(6(T» with SQ = Q1 8, and so the map Q 1 8T = SQT is continuous. Thus 6(Q 1 8T) = O. But Ql(6(8T» c 6(Q 1 8T), and hence 6(8T) C ker Ql = 8(6(T». (iii) Take y E 6(T), say Xn --t 0 in E and TX n --t Y in F. Then RX n --t 0 and TRx n = (TR - 8T)xn + 8Tx n --t 8y, and hence 8y E 6(T). (iv) Let Q : F --t FIFI be the quotient map. By (ii), QT is continuous, and so (QT) (:E0 = O. Thus T (:E0 C F l , and the map x + El ~ Tx + Fl is continuous by A.3.1O(i). ~==~------~==~
(v) Set G = 6(TRl) + ... + 6(TRn), and let Q : F --t FIG be the quotient map. For j E NT!) we have 6(TRj) C 6(T). so that G C 6(T), and Q(6(TRj» = 0, which proves that QTRj is continuous. The map
(Xl, ... , Xn)
I--t
RjX1
+ ... + Rnx n ,
II" E
/
j --t
E,
j=1 is a continuous linear surjection, and hence an open map. Thus QT : E --t FIG is continuous. By (ii), 6(T) c G, and so 6(T) = G. (vi) Set G = {a E F : a . S = O}, a closed linear subspace of F, and let Q : F --t FIG be the quotient map. Since 6(T) C G, the map QT is continuous and so {x E E : Tx . S = O} = ker QT is closed. 0
The sepamtzng space and the stabiltty lemma
611
Proposition 5.2.3 Let E and F be topolo.Qu:allinear space .... and let T J and T2 be linear maps from E mto F such that T1 - T2 is continuo'/l'>. Then S(Tl) = 6(T2)' 0 Proposition 5.2.4 Let E J , E 2 , F 1, and F2 be (F)-spaces, and let T1 : E1 ----> F1 and T2 : E2 ----> F2 be linear maps. Supp08e that R E B(E1. E 2) is a suryectwn, that S E B(F1.F2), and that T2R-STl zs continuou.>. Then 6(n) = S(6(Td). Proof By 5.2.2(i). 6(T2R) = 6(T2)' I3y 5.2.2(ii), S(6(TJ)) 5.2.3, 6(T2R) = 6(STJ). Hence 6(T2) = S(6(TJ )).
=
6(STJ). By 0
We now corne to an important result, the stability lemma, that lies at the heart of many later theorem,,>. The method of proof in clause (i). below, is called a gliding hump argument; it is a development of that given in Theorem 5.1.4. A yet more general ven;ion will be given shortly, in Theorem 5.2.6.
Theorem 5.2.5 Let E and F be Banach spaces, let T : E ----> F be a linear map, let (En: n E Z+) be a sequence of Banach spaces with Eo = E, and let Rn E B(En' En-J) (n EN).
(i) Let (Fn) be a sequence of Banach "'paces, and let Sn E B(F, Fn) (n EN). Suppose that SmT Rl ... R" zs contmU01lS whenever m < n m N. Then there exists N E N such that SnTRl'" Rn is continuous whenever n 2:: N m N. (ii) (Stability lemma) (6(TRl'" Rn) : n E N) is a nest m F which stabzlzzes. Proof (i) We may suppose that IIRnll = IISnll = 1 (n EN). Assume towards a contradiction that the result fails. By grouping the maps R n , we may suppose that SnTUn : En ----> Fn is discontinuous for each n E N. where Un = R l ·· ·Rn . For each n E N, choose Yn E En such that 2" IIYnl1 ~ ] and 2n IISmTUnlllly.,11 ~ 1 whenever m < n, and so that IIS1TUlYlll2:: 2 and n-l
IISnTUnYnl1 2:: n
+ ] + I) TUj Yj I
(n 2:: 2),
j=l
and then define Xn = L:j:n UjYj in E; the series converge because IIUJYj II ~ 2- j . For each n E N, we have Xl = U1Y1 + ... + UnYn + Xn+1, and so n-1 II S n TUny,,1I ~ IITxJII + II T UJYjl1 + II S n Tx n+111 . j=l
L
But II S.,TXn+1 II ~ L:j:n+1I1SnTUjIIIIYJII ~ 1, and son ~ IITxd foreachn E N, a contradiction. Thus the result holds. (ii) Set 6 n = 6(TR1 · ··Rn ) (n EN), and let Qn: F ----> FI6 n = Fn be the quotient map. By 5.2.2(i), 6 n+1 C 6 n (n EN), and so (6 n ) is a nest in F. We apply (i) with Sn = Qn+1: by (i), there exists N E N such that Qn+1TRl .,. Rn is continuous whenever n > N, and this implies that 6 n = 6N whenever n > N. Thus (6(TR1' "Rn)) stabilizes. 0
612
A lltom atzc contzrmity theory
We now give a form of clam;c (i) of the above theorem for more genen'll topological linear spaces. The unit ball of a normed space has a dual role: it is a neighbourhood of zero, and it. is a bounded set. \VhPll working wit h more general spaces, one realizes that the two roles of the 1Init ball appear at different stClges of the argument, and the following formulation of the proof clarifies thi~ distinction. The definition of a cOlllltably boundedly generatpd ~pace and some ('xarnpks Clre given in Appendix 3.
Theorem 5.2.6 (Gliding hump) Let (En : rl E Z+) be a sequenee of (F)spacf'S, let (Fn . n E Z+) be a sequence of topological lmear space8 such that Fo zs c01171tably bo'undedly generated, let T : Eo --+ Fa be a lmeaT map. and let Rn E B(En , En-I) and S71 E B(Fa, Fn) for n E N. Suppose that each map Sn T R 1 ... R n : En ----> F1I 1.S continuous. Then there exzsts 1\T E N such that
SnTR1'" RN : EN
--+
Fn
is cvrdmuo'u8 f()r each n EN.
Proof Let (Em,) be a sequence of hounded subsets of Fa with Fo = U:=l Bm: may suppm,e that Ern C Bm+l (m EN). Let. d n be a complete, invariant metric defining the topology of En. and set Ixln = dn(x.O) (x E En). Again ~et Un = RJ ... Rn (n EN). Assume towards a contradiction that the result is false. Since t.he maps SpTUn are continuous wllPnever p ::; 'T/, there is a strict ly increasing sequence (nk) in N such that Snk+1 TUnk : Enh ----> F"k+l is discontinnous for each kEN. For convenience, set no = O. \Ve shall construct for each kEN integers mk with 1Tl~ 2:: k, neighbourhoods Vk of 0 in Fn~. clements Yk E E"J.._l' and c,.. > 0 sueh that Yi = 0 and the following four conditions are ~atisfied for each k ~ 2. (Condition 1 is vacuous in the case where k = 2, and we set co = 1.) The conditions are: WE'
1. T
(L~:i U"j_1 Yj)
E Bm.;
2. IYkl nk _1 < E:k-d2 and IUnk :~. Sn~ TU"._J y~ (j. Sn. (B1fJk - Bm~) - Vi,.; 4. SnkTUn~ '11' E Vk for allw E Enk with Iwlnk
L q(M) be a linear map which commute8 with a shift operator Sa where a > O. Then T is automatically continuous. For operators which commute with the shift operator S-a where a > 0, see 5.3.42 and 5.3.48. Corollary 5.2.10 (Semigroup stability) Let E be an (F)-space, let F be a locally bounded (F)-space, let T : E --> F be a linear map, let P be a dense subgroup of (JR, +), and let a f-> ROI. and a r SOl. be morphisms fTOm p+- into the multiplicative semigroups of B(E) and B(F), respectively.
615
The separating space and the 8tabthty lemma (i) Suppose that TRo - SaT E B(E. F) (0 E P+·). Then
S,,6(T) = Sf/6(T)
(o:.{3 E P+·).
(ii) Suppose that TR" = SQT (a E p+.) and R,,(E) = Rr:J(E) (0. t3 E P+·). Then SaT(E) = SRT(E) (a, (j E P+·). Proof Set G = 6(T) in case (i) and G = T(E) in case (ii). and define «~ E P+·).
G n = Sa(G)
Take 0', fJ E P with 0 < 0' < {3. say (3 - 0' = ,. In case (i), S"f(G) C G by 5.2.2(iii). In case (ii), S"f(G) = S1T(E) = TR"f(E) C G. Thus, in either case, G{3 = S"S1(G) C S,,(G) = Go.. Suppose now that there exist 0', (j E P with 0 < a < f3 and Gn = G{3. We clazm that G') = Gn for each, E P with, ;::: (x. Since P is dense in JR, there exists 0 E P n (0, {3 - 0:). Consider the equations (5.2.2)
Since a < a+o < {3, equation (5.2.2) holds for n = 1. Assume that (5.2.2) holds for n = k. Then Gn+(k+l)O = S.,+(k+l)o(G) = SoSQH6(G) = So(Go+kO)
= SIi(G,,) = So(Sa(G»
= Ga+o
= Go. ,
and so (5.2.2) holds for rI = k+ 1. By induction, (5.2.2) holds. Now choose ~( E P with "I ;::: 0:, and take n E N with a + 110 > "I. Then G"f C Go. = GQ+n.s C G"f and so G') = G()" establishing the cla.im. To prove the two results, take a. /1 E P with 0 < ex ::; {3. Since p+. is a dense semigroup in JR+., there exists a sequence (on) in p+. such that "I" < 0: (n EN), where we a.re setting 1n = L~=l llk· "Vole apply 5.2.7, taking Rn = Ran and Sn = S"" for n E N: in each case. it follows that there exists N E N with G"fN+l = G"fN' But now it follows from the claim that GOt = G"fN = GfJ. The result is proved. 0 Corollary 5.2.11 Let A and B be Banach algebras, and let 0 : A homomorphism. (i) For e.ach a E A, there exists N E N such that ~O(an)6(O)
= O(a N )6(O).
(ii) Suppose that (ao. : 0: E JR+.) is a semigroup in A. Then
8(a )6(8) = O(a B )6(8) Q
Suppose, further, that ao::4 =
a{3 A
E
JR+.).
(0:, {3 E JR+.). Then
O(ao.A) = O(a.BA) Proof Set 6 = 6(0).
(a,{3
(a,/3 E JR+.).
--+
B be a
616
Automatzc continuity theof'U
(i) Define R(:r) = ax (x E A) and S(y) = B(a)y (ll E B). Then BR = 80, and so, by 5.2.7(i), there exists N E N such that B(a n )6 = O(a N )6 (n ;::: N). Sct E = 0(11\'6). Then O(a)E = E, and so limO(a")E = E by A.3.27(ii). E be an mtertwinmg map. Then 6(T) i.~ a neo-umtal Banach A-bimodule. Proof (i) Suppose that the bound of the left approximate identity in A iH 11l. Take :1' E 6(T). say (an) E co(N, A) with Tan -> :1: in E. By 2.9.30(ii), there exist b E A and (en) E co(N. A) such that an = ben (It EN). Clearly b· TCn -> ,1'. Take c > O. Then there exist n E Nand II E A[m] with IIx - b . Tcnll < c aud lib - ubll < c. We have
Ilx -
U •
xii
Ilx - b . Tenll + lib - ubllllTcnl1 + Ilullllx < (rn + l)c + (11xll + c)c. ~
b.
Tenll
and so x EA· 6(T). Thus 6(T) is essential. and hence, by 2.9.29(i). neo-unital. (ii) This iH similar.
0
Corollary 5.2.17 (i) Let A be a commutatwe Banach algebra w1,th a bonnde,d approxzmate zdentity. Suppose that each derivation [zntertwining map] from A into A' is contmuous. Then each derivatzon [mtertwmmg map] from A into a Banach A-module ·tS continuous. (ii) Let A be a Banach algebm until. a bounded approximate identzty. Suppose that eaeh derivation from A mto (A.§A)' tS continuous. Then each zntertwming map from A into a Banach A-bimodule 'is continuous.
Proof (i) Let E be a Banach A-module. and assume that T : A -> E is a discontinuous intertwining map. Then 6(T) i= O. By 5.2.16(i), there exist ao E A and Xo E 6(T) with ao . Xo i= O. Take A E E' with (ao . xo, A) = 1. and let R), E .4T3(E, A') be as defined in 2.6.6(i). Then (ao. R),xo) = 1, and sO R),xo ::J O. But R), 0 T : A -> A' is an intertwining map, and R),xo E 6( R), 0 T), and so R)., 0 T is discontinuous, a contradiction. Thus T is continuouH. In the case where T is a derivation, HO iH R)., 0 T, and so the same conclusion follows for derivations. (ii) This is similar; we use 5.2.16(ii) and 2.6.6(ii). By 2.7.7, each intertwining map from A is continuous whenever each derivation from A is continuous. 0 ,
We now apply Theorem 5.2.15 to obtain some positive automatic continuity ret'mlts. First, we prove some theorems about some specific commutative Banach algebras: for a discussion of the algebras Ll(w) = Ll(lR.+. w), M(lR.+. w), C*, and V, see §4.7. Theorem (i) Let p, E O(A)· Ll(w) and
5.2.18 Letw be a weight/unction on lR.+, and let A be an (F)-algebra. 0 : A -> M (lR. + , w) be a homomorphism such that there exzsts with a(p,) > O. Then 0 is automatically continuous. In particular, M(lR.+,w) each have a unique topology as an (F)-algebra.
The sepamting space and the stabilzty lemma
619
(ii) Let A be a subalgebm of l\I(JR.+.w) such that there eX'tsts J.L E A· with a(p,) > 0, and let (Dn) : A ---7 M(JR.+, w) be a hzgher derivation for which Do is the cont'lnUOUS embedding. Then (Dn) is automatically continuous.
Proof In each case, set E = A and F = M(JR.+,w). Then E is an (F)-Amodule. and F is a weak Banach left A-module for the operation a . Jl = (J(a) * J.L (in case (i)) and a . J.L = a * p, (in case (ii». In case (i). set T = (J. In case (ii), either set T = D 1 , or assume that D 1 •... , Dk-l are continuous, and set T = Dk' In each case, T : E ---7 F is a left.-intertwining map. By 5.2.15(iii), 6(T) is a separating module in F. Suppose that S is a non-zero subset of F and that p, E O(A)- (in case (i» or that p, E A- (in case (ii» with a(J.l) > O. By Titchmarsh's convolution theorem
4.7.22, a(p,*(n) * S) = na(J) +n(S), and so (i*(n) * S) does not stabilize. Thus SeT) = 0, and so T is continuous. The result follows (by the use of induction on k in case (ii». 0 The algebras L1(w), in the case where w is a radical weight, are examples of Banach algebras with a unique complete norm which are not semisimple. We shall see in 5.7.26 that there do exist dh:lcontinuous homomorphisms 0 from certain Banach algebra.'l A into £l(W) for each continuous, radical weight. function w on JR.+; part (i), above, shows that necessarily a(J) = 0 for each f E O(A)-. Part (ii) shows, in particular, that. in the case where w is a weight function on JR.+, each derivation on £l(W) and on M(JR.+,w) is automatically continuous; we shall give a representation of these derivations in 5.6.21 (ii). Theorem 5.2.19 (Jewell and Sinclair) Let A be an (F)-al.qebm, and let B be V or C*(JI) or M(JI) or Mloc (JR.+). (i) Let 0 : A ---7 B be a homomorphzsm such that, for each c > 0, there exists f E O(A) with a(J) E (0. c). Then 0 zs automatzcally continuous. In particular, V, C*, 1\I(JI), and Mloc (JR.+) have unique topologzes as (F)-algebms. (ii) Each intertw'ln'lng map on B is automatically cont'ln1LOus. Proof In case (i), set E = A, F = B, and T = 0; in case (ii), set E = F = B and take T to be an intertwining map on B. By 5.2.15(iii), 6(T) is a separating module in F. For each non-zero subset S of B, there exists f E (J(A)- (in case (i» or fEB(in case (ii» with a(J * S) E (a(S),1), and so S is not a separating module. Thus 6(T) = 0, and T is continuous, proving (i) and (ii). 0 A generalization of 5.2.19(ii) in the case where B = V will be given in 5.3.23.
Theorem 5.2.20 (Loy) Let B be a Banach algebm of power series. (i) Let A be a functionally mntinuous (F)-algebm, and let 0 : A ---7 B be a homomorphism. Then 0 zs automatically continuous. (ii) The algebm B has a unique topology as an (F)-algebm. (iii) Let A be an (F)-algebm which is a subalgebm of B, and let (Dn) : A ---7 B be a higher derivation. Then (Dn) is automatically continuous.
Automatic continuzty theory
620
Proof If O(A) C tel in C8.'le (i), then 0 E q> A U {O}, and so 0 is continuous. If A C tel in case (iii), then Dn = 0 for all Tl E N. If these situations do not arise, then there exists 1 E O(A) \ tel in case (i). 1 E B \ tel in case (ii), and 1 E A \ tel in case (iii). By replacing 1 by 'ffo(f)1 - P (where 'ffo denotes the projection onto the constant term), we may suppose that 'ffo(f) = 0, and now the argument of 5.2.18 gives the result. 0 Recall that we have already proved in 4.6.1 that the algebra J itself has a unique' topology as an (F)-algebra, and in 4.6.3 that each derivation D : J -+ J" is automatically continuous. However. it is not true that each epimorphism froll! a Banach algebra onto J is automatically continuous; see 5.5.19. Theorem 5.2.21 (Bade and Dales) Let A be a Dttkin algebra, and let I be a closed ideal in A. Then each epimorphzsm Imm a Banach algebra onto AII ~s automatzcally continuous. Proof Write 2l = AI I. and let 0 : ~ -+ 2l be an epimorphism from a Banach algebra~. Set E = ~(I), so that J(E) c I c I(E) and rad 2l = I(E)I I. By 5.1.9(iii), 6(0) c rad2l. But 6(0) is a separating module. and so, by 4.1.40, 6(0) = O. 0 The above theorem applies, for example. in the case where r is a non-discrete LCA group, E is a compact subset of r of non-synthesis, and 1= J(E), so that AII is not semisimple. \Ve now present a variant of Definition 5.2.14. Definition 5.2.22 Let A be a topological algebra, and let .1 be a closed ideal m A. Then .1 is a separating ideal zl the sequences (a1'" anJ) and (Jan'" ad both stabihze f01' each st'AJuence (an) tTL A. Trivially. a closed ideal I in a commutative topological algebra such that J is a topologically simple algebra is a separating ideal. Let .1 be a separating ideal in A, and suppose that I is a closed ideal in A. Then clearly .1/ (I n .1) is a separating ideal in AI I. Proposition 5.2.23 Let .1 be a separatzng ideal (i) There exzsts no E N such that
tTL
n{ .In : n E N}
an (F)-algebra. is dense
(ii) Suppo8e that (an) is an orthogonal sequence such that a~o+2 = 0 (n ~ no).
tTL
J.
tTL
Jno.
Th~n
there eXl,sts
TIo EN
n
Proof (i) Set I = In and A = J / I, so that A is au (F)-algebra. Let a E J. Since J is a separating ideal, there exists kEN such that ak+1J = akJ. By A.3.27(ii), nanJ = akJ. Thus ak+1 E nanJ c I. This shows that A is a nil algebra. By 2.6.34, A is nilpotent: there exists no E N such that Jno C I. Clearly 1= Jn o, and so no has the required property. (ii) We may suppose that bn = ~:'n ak exists in J for each n E N. Then b1 ••• bn = ~:'n a k (n EN). Since J is a separating ideal, there exists no E N
621
The separatmg space and the stabiizty lemma such that b1 ••• bnJ
= bi
...
bnoJ (n
~
n~o+2.= a"b 1 •·· bnoan E a n ( bI
because anb 1 ... bn +1 = O. Thus a~o+2
no). For n ···
bnoJ)
= 0 (n
~
no. we have
= an (bl ··· bn+1J) = 0
~ no).
0
Theorem 5.2.24 Let B be a Banach algebra.
(i) Let A be an (F)-algebra, and let () : A ~ B be an epimorphzsm. Then 6(19) is a separatmg ideal in B. (ii) Let E be a Banach B -bzmodule, and lpt T : E ~ B be an mtertwinin,q map over B. Then 6(T) is a separatmg uleal in B. Proof Set T = () (in case (i)). Then T is an intertwining map in both cases. It follows from 5.2.15 that 6(T) is a closed left ideal in B and that the nest
(b 1 •.• bn 6(T») stabilizes for each sequence (b n ) in B: to obtain these results in case (i), we use the fact that r9(A) = B. Similarly, 6(T) is a dosed right ideal in Band (6(T)b n
·· .
ill ) stabilizes for
each (b n ) in B. Thus 6(T) is a separating ideal in B. Lemma 5.2.25 Let J be a Banach algebra whzch is a separating ideal in Then J /rad.J is finite-dimenswnal.
0 it.~elf.
Proof Let E be a simple left J-module, so that, by 2.6.26(i), E is a Banach left J-module. Assume towards a contradiction that {xn : n E N} is a linearly independent set in E. By 2.6.27(ii). there exists (an) in J such that an' .. al . x" = 0 and an' .. al . Xn+l #- 0 for each n E No Set I n = JUn'" al (n EN). For each kEN, we have J . (ak'" al . Xk+l) = E by 1.4.29(i), and so Jk . ;l'k+! = E. whereas ak+! ... 01 . Xk+1 = 0, and so Jk+l . XA-+I = O. Thus J k +1 E be a discontinuous derzvatzorz. Then there are a non-empty, jinzte subset {CP1, ... , CPn} of A, linear functionals d 1, ... , dn on A such that dj zs a point de1'ivatzon at CPj for j E N n . and a contmuous de1'ivatzon D : A ---> E such that n
Da =
Da + L
dj(a)xj
(a E A),
J=l
where Xj zs the chamcterzstzc functzon of 'Pj for J E N n .
Proof Take {'P1, ... ,'Pn} C A to be as specified in 5.2.30(i) (with T = D). For j E N n . define d j : a r-> (Da) (cpj); clearly dj is a point derivation at 'Pi' Define D : a r-> Da -_'£7=1 dj(a)xj, A ---> E, so that D is a derivation. It remains to prove that D is continuous. Take (an) in A with an ---> 0 and Dan ---> go E E. For j E Nn , we have (Dan)(cpj) = 0, and so 90(CPj) = O. Now take cP E A \ {cp1 .... 'cpn}. There exists b E A with C(lI), is a non-zero derivation. However, it is the case that each derivation from A into itself is zero. Theorem 5.2.32 Let A be a Banach function algebm. Then there are no nonzero derzvations on A. Proof Let D: A ---> A be a derivation. By 5.2.30(iii), Dis conti?uous, and so, by 2.7.20, D(A) C radA = O. 0 Corollary 5.2.33 (i) The algebm C(oo) (lI) is not a Banach algebm with respect to any norm. (ii) For each non-empty, open set U in cn, the algebm O(U) is not a Banach algebm with respect to any norm.
Proof The algebras C(oo)(lI) and O(U) are semisimple, and they have obvious non-zero derivations. 0
The sepamtmg space and the stability lemma
625
The property that each derivation from A into itself is zero does not characterize semisimple algebras A in the class of commutative Banach algebras: we shall note after 5.6.17 that there is a local Banach algebra with this property. In the next results, we write g for the Gel'fand transform and Q] for the quotient map from an algebra A onto AI I when I is an ideal in A. Lemma 5.2.34 Let A be a Banach algebm, and let D be a derivatwn on A. (i) Let I be a closed zdeal in A. Suppose that, for each n E N, the map Q[Dn is continuous. Then there is a constant C > such that IIQ]Dnll ::; C n (n EN).
°
(ii) Suppose that A zs commutative and umtal, and that is contmuous for each n E N. Then D(A) C radA.
go Dn
:A
----+
C( II> A)
Proof (i) Set J = {a E I : Dna E I (n EN)}. By 1.8.5(ii), J is an ideal in A, and, by hypothesis, J is closed in A. Let bE SeD), say (ad E co(N, A) is such that Dak ----+ bin A. Take n E N. Then Q]Dnb = limk-+oc Q]Dn+l ak = 0, and so Drib E I. Thus S(D) c J, and so QJ D is continuous. Define D J : a + J I-t Da + J, a derivation on AIJ. By 5.2.2(iv). DJ is continuous. Clearly Q]D n = Q],JDJQJ (n EN), where Q],J : AjJ ----+ All is the canonical map, and so the result follows with C = IIDJII. (ii) By the argument in (i) (with 1= radA), we obtain a continuous derivation D J : a + J I-t Da + J on AIJ. By the Singer-Wermer theorem 2.7.20, DJ(AI J) c rad (AjJ), and so, by 1.5.4(ii), D(A) c rad A. 0 Theorem 5.2.35 (Johnson) Let D be a denvation on a commutative, umtal Banach algebm A. Then there is a finite, orthogonal set {Po, PI, ... ,Pn} of idempatents in A such that eA = Po + PI + ... + Pn, such that D(poA) C rad (poA) , and such that PIA, ... , PnA are local algebms. Proof Let P be the set of non-zero idempotents P in A such that pA is a local algebra and D I pA is discontinuous. Since J(pA) = {p}, P is orthogonal. We claim that P is finite. For assume that P is infinite, and let {Pj : j E N} be a subset of distinct elements of P. For each J E N, take Xj E pjA with IIxjll < 2- j and IIDxjl1 ~ j Ilpjll, and set x = Ej:1 Xj' Since PjPk = 0 (j =j:. k), we have PjX = Xj (j EN). Thus j IlpJ11 ::; IID(pjx)11 = IlpJDx11 ::; IlpJ1111Dx11 and IIDxl1 ~ j for each j E N. a contradiction. Hence the claim holds. Set P = {Pl .... ,Pn} and Po = eA - (PI + ... + Pn), so that {PO,PI,'" ,Pn} is an orthogonal subset of J(A). It remains to prove that D(PoA) C rad (poA). By 5.2.34(ii), it is sufficient for this to prove that (g 0 Dk) I PoA is continuous for each kEN. Notc that, since Dpo = 0, D(PoA) c PoA. Assume either that k = 1 or that k ~ 2 and (g 0 Dj) I PoA is continuous for j E Nk-I. Then (g 0 Dk) I PoA is an intertwining map, and so, by 5.2.30, either (g 0 Dk) I PoA is continuous or there exists q E J(PoA) with qA a local algebra and (g 0 Dk) I qA discontinuous. But in the latter case, D I qA is discontinuous, and so q E P, a contradiction. By induction, (g 0 Dk) I PoA is continuous for each kEN, as required. 0
A utomatzc continuity theory
626
In Corollary 2.7.20, we proved a theorem of Singer and "\\-'ermer that. in the case where A is a commutative Banach algebra and D : A ~ A is a contmuous derivation, D(A) c rad A. Singer and Wermer conjectured that D(A} C rad A for each (possibly discont.inuous) derivation D on A. We now present Thoma.c;;'s remarkable confirmation of this conjecture. It is clear from 5.2.35 that we must examine derivations on local Banach algebras. Theorem 5.2.36 (Thomas) Let R be a commutatzve, radical Banach algebra, and let D : R# ~ R# be a derivatzon. Then D(R#) c R.
The proof of this theorem will proceed through several lemmas, in which we maintain the same notation. We denote the identity of R# bye (so that R# = CeEBR) and the separating space of D by 6. For a E R#, we write aD for the map b 1---7 aD(b). Assume towards a contradiction that D(R#) Xl. (Note that We! cannot suppose that ((' m) is hounded.) This in turn implies that
(e
+ (1 -
()em)s
---> S
as m
--->
(5.2.8)
ex::
fur each 0 such that (aj)1 E Vj whenever aI, ... , ak E Rl with Ila) - SUI,) II < Cl (j E Nk)' Now assume that Cl and 'Uf.,l,' .• , 'lLe.k have been chos('n so that condition!> Ok (ii)f, and (iii)e hold for t ::; n. We choose Tnn+l.l, ... ,1rtn +1.k E N such that (i)n+l holds, where tLn+J.j = (n+J e + (1 - (n+J )em,,+l.j (j E N k ). By 5.2.44 again, we see that (su 1.j ... u n + 1.j)i E Vn+l' Since the set Vn+l is open. there exists Cn+l E (0.10 11 /3) such that (aj)t E Vn+l whenever a1.···, ak E R1 with lIaj - sU'1.j ... u n +1.j II < 2En+l (j E Nk). The induetive constructioll continueH. For each el, the sequence (SVl,n ... Vn.t~ : n E N) is Cauchy in RJ by (i)n, and so it eonverges, say 8V1,Q'" vn,a - t Va: as n - t oc. For j E Nk, set 8) = v{j}. We claim that {.'II,.'" .'Ik} is an s-recalcitrant system. For each n E N. we have IIsj - SU1.J .. , un.) II < 2.:~n Ci < 2clI (j E Nk), and so (8 J )T E Vn . Thus (Sj)~ E V. Now take R E {2 .... ,k}. and let {tl .... ,tt} be a subset of {sl, .... sd corresponding t.o the subset 0: of Nk. say. Clearly the products are such tl!at tJ ... tt = .'Ie-Iv(~ E st-1 R#, and so 5.2.42(i) is Hatisfied. \Ve have constructed the required s-reeakit.rant system. 0 By (5.2.7). there exists 1710 E N such that z7n O E zmoR. Set k = 1710 + 1 and s = zTno, so that, by 5.2.46, there is an s-recalcitrant system. say {sl, .... sd,
of k elements in R.
.
Lemma 5.2.47 Ther-e eJ:ist
a1, . ... ak E
R# such that
07=1 (Si -
sai) E .I.
Proof For i EN/,.. let the representation of '~i that wa.." specified in 5.2.41 be '~i = 2.:;0 ijil,;,). when' (0,.) : j E Z+) C 2(, and let ai be 'any element of g# !ouch that il,.1 = L...-J=mo "'~ ij-m0il,. Then I.) mu- 1
1;-l - imoil,-1-- ~ ijiI.·t.)· . ~
(5.2.9)
)=0
Set b = 0:=1 (Si - S(li). A term in the product for b belongs to tl ... tj.'lk-t R#. where each oftl, .... te belongs to {Sl ..... Sk}. By 5.2.42(i). tl ... tt E sf-1l!#, and so b E . 0 in E 1 • Then there exist elements al, ... , ak E K and sequences (Yl,n),"" (Yk.n) E co(N. E) such that Xn = 2:;=1 aj . Yj,n (n EN). By 5.3.2, T x a1, ... ,T x ak are continuous, and 80 T(aj . YJ.n) -> 0 as n -> 00 for j E Nk. Thus TX n -> 0 as n -> 00. a.nd so T I E1 is continuous. It follows from A.3.42(i) that T is continuous. 0 The following corollary concerns strong Ditkin algebras, defined in 4.1.31(iv). The main examples of these algebra.,; are Co(n) for n a locally compact space (see 4.2.1(v»), A(r) for r a locally compact abelian group (see 4.5.18), and AC(l[) and BVC(l[) (see 4.4.35(iii». We shall see in 5.3.6, below, that. all derivations
638
Automatzc contznUlty theory
from A = Co(O) and from A = A(f) in the case where f is compact into a Banach A-module are zero; in 5.6.43, we shall prove the latter result for each LeA group f. We shall also note (in 5.6.8) that there are non-zero derivations from AC(II) and BVC(II).
Corollary 5.3.5 Let A be a strong Ditkzn algebr·a. Then each zntertwinzng ma.p from A znto a Banach A-bimodule is contznuous. Proof By 4.1.36(i), A is pliable. Let I be closed ideal of finite co dimension in A. Then I has a bounded approximate identity, and so, by 2.9.29(i). null sequences in I factor. Thus the result follows from the theorem. 0 In contrast, we pointed out in 4.1.42 and 4.1.46, respectively, that there are examples of strongly regular Banach function algebra..'l and of Banach function algebras with bounded relative units on which there are discontinuous point derivations.
Corollary 5.3.6 Let A be ezther Co(O), where 0 zs a locally compact spacc,.or [l(G), where G is an abelzan group, and let E be a Banach A-module. Then Zl(A, E) = {O}. Proof Let DE Zl(A. E). By 5.3.5, D is continuous; by 2.7.16, Zl(A, E) = {O}, andsoD=O. 0 Corollary 5.3.7 (Ringrose) Let A be a C* -algebra. Then each intertwining map from A znto a Banach A-bzmodule zs continuous. Proof By 3.2.25(i), A is pliable, and, by 3.2.21. each closed ideal in A has a bounded approximate identity. The result follows, as before. 0 Corollary 5.3.8 Let A be an amenable Banach algebra. (i) Suppose that A zs plzable. Then each intertwining map from A into a Banach A-bimodule is continuo1ls. (ii) Each intertwzning map from A into a finite-dimensional Banach A-bimodule is contznuous. Proof Let I be a closed ideal of finite co dimension in A. Since A is amenable, I has a bounded approximate identit.y by 2.9.59, and so null sequences in I fact.or. The result in (i) follows from 5.3.4. In case (ii), it. is only necessary t.o consider the case where the left continuity ideal is a closed ideal of finite to dimension in A, and so the same argument applies. 0 It may be that the assumpt.ion of pliabilit.y in (i), above, is redundant. Question 5.3.A Let A be an amenable Banach algebra. Is every derivation from A into a Banach A-bimodule automatically continuous'? The definitions of the approximation properties BCAP, BAP, and AP are given in A.3.59.
The contmuzty ideal and the singularity set
639
Corollary 5.3.9 Let E be a Banach space with BAP. Then each mtertwming map from K(E) into a Banach K(E)-bimodule is contmuous. Proof We may suppose that E is infinite-dimensional. Since E has AP, it follows from 2.5.2(iii), 2.5.9(i). and 2.5.10 that K(E) is pliable and has no proper, closed ideal of finite codimension. Since E has BCAP, it follows from 2.9.37(ii) that null sequences in K(E) factor. The result now follows from 5.3.4. 0 We now turn to the circle of ideas associated with the prime ideal theorem and the prime kernel theorem. The first result is the basic 'dichotomy theorem' that is the key to the theory. Theorem 5.3.10 (Dichotomy theorem) Let A be an algebra, and let E be a topologzcallmear space 'Which is a left A-module. Then E zs a separatmg module if and only zJ, for each subset S of A such that S . E =1= 0, there exists ao E S with ao . E =1= 0 and such that, for each b E A with aob E S, either aob . E = 0 or aob . E = ao . E. Proof Suppose that E is a separating module, and let S be a subset of A such that S . E =1= O. Choose al E S with 01 . E =1= O. If alb· E E {a. al . E} for each b E A with alb E S, take ao = al; if not, choose a2 E A with ala2 E S and 0 S;; ala2 . E S;; al . E. Continue in this way: either we obtain an element ao E S which satisfies the given condition, or we obtain a sequence (an) in A with al ... an+! . E S;; al ... an . E for each n E N. But this latter alternative cannot occur, for it would contradict the fact that each sequence ( al ... an . E) stabilizes. Conversely, suppose that the given condition is satisfied, and let (an) be a sequence in A. Set En = al···an . E (n EN). and set S = {al" ·a n : n EN}. If EN = 0 for some N E N, then certainly (En) stabilizes. Otherwise, there exists N E N with al ... aNb . E = EN for each b E A with al ... a;vb E S. In particular. En = EN (n 2: N), and so (En) stabilizes. 0 Before we apply 5.3.10 in the proofs of the prime ideal and prime kernel theorems, we use it to establish a consequence of the existence of certain discontinuous maps into a semi prime Banach algebra. Recall that, if I is a non-zero ideal in a semiprime algebra and a E Ie, then (aI) n (Ia) =1= o. Proposition 5.3.11 Let A be a Banach algebra. Assume that A contams a non-zero, semiprime separating ideal.!. Then A contains an zdeal I with I c .I such that I zs minimal m the family of non-zero, closed ideals of A. Proof We have A . .I =1= 0, and so, by the dichotomy theorem 5.3.10. there exists ao E A with aoJ =1= 0 and such that, for each b E A, either aobJ = 0 or --2 aobJ = aoJ· By 1.5.25, (ao.!) =1= O. and so JaoJ =1= O. Set I = JaoJ. Then I =1= 0 and Ie.!. Take K to be a non-zero, closed ideal of A with K c I, and take a E K e . Then I a =1= 0, and so there exists bE .I with aoba =1= O. But aoba E J, and so aobaJ =1= O. Thus aobaJ = aoJ, and 1= JaobaJ C K. Hence I is a minimal non-zero, closed ideal. 0
640
A utomatic continuity theory
Theorem 5.3.12 (Cusack) Let A be a Banach algebra, and let B be a semiprzme Banach algebra. Assume that there 'lS a discontmuous ep'lmorph'lsm from A onto B or a d'lscontmuous interlwimng map T : B -+ B. Then B contams a topologically 8'lmple, radical Banach algebra. Proof Let () : A -+ B be a discontinuous epimorphism, and set J = S(()), or let T : B -+ B be a discontinuous intertwining map, and set J = SeT). Then J is a non-zero, semi prime separating ideal in B. By 5.3.11, there is a minimal non-zero, closed ideal I in B Huch that I c J. Since B is semiprime, I is nonnilpotent by 1.5.25, and so, by 2.2.34, I is a topologically simple Banach algebra: by 2.2.33(ii). I is either primitive or radical. Assume that I is a primitive algebra. Then I is semisimple, and so, by 5.2.26, I is finite-dimensional and contains a non-zero idempotent, a contradiction of 5.1.3(iii) in the case of (). ThiH is abo a contradiction in the case of T by the argument in 5.2.28. Thus I is a topologically simple. radical Banach algebra. 0 Corollary 5.3.13 Assume that there is a commutatwe Banach algebm wh1ch ~s an mtegral domam, but wh'lch does not have a umque complete norm. Then there is a commutative, topolog'lcally s'lmple Banach algebra. Proof This is immediate from the theorem.
o
Of course. as we remarked in §2.2, no topologically simple, radical Banach algebra is known. Theorem 5.3.14 (Prime ideal theorem)(Bade and Curtis) Let A be a commutative algebm, let E and F be weak Banach left A-modules, and let l' : E -+ F be a discontinuous left-intertwmmg map. Then either IdT) = A 01' ther-e eX'lsts ao E A such that T x ao : E -+ F is (L discontinuous left-tntert'Unnmg map w1t,h S(T x ao) C SeT) and IdT x ao) J IdT) and such that IdT x ao) is e1,ther a prime ideal m A or IdT x ao) = A. Proof If A . SeT) = O. then IdT) = A. Now suppose that A . SeT) i- O. By 5.2.15(iii), SeT) is a separating module, and so, by the dichotomy theorem 5.3.10, there exists ao E A with ao . SeT) i- 0 and such that, for each b E A. either aob . SeT) = 0 or aob . SeT) = ao . SeT). Set Tl = T x ao and K = IL(TI). Then Tl is a left-intertwining map and S(TI) = ao . SeT) c SeT), so that K J IL(T). Since S'(Td i- 0, Tl is discontinuous. Suppose that there exist a, b E A with ab E K and b f/. K. Since aob = bao, we have aob . SeT) = bao . SeT) = b . S(TI) i- 0, and so aob . SeT) i- O. Thus b . S(Tl ) = S(Tl)' and a . S(Td = ab . S(Tt} = O. This shows that a E K, and so either K = A or K is a prime ideal. 0 Before giving the general form of the prime kernel theorem, we prove separately the commutative version of the result.
The contmuity zdeal and the singularzty set
641
Theorem 5.3.15 (Sinclair) Let A and B be commutatzve Banach algebras, and let () : A ---- B be a discontznuous homomorphism. Suppose that ao E A zs such that ()(ao) E 6«()) \ I)1(B). Then the7'e is a closed zdeal Kin B such that the map
O:al--.()(a)+K,
A----B/K,
is a discontznuous homomorphzsm wzth O(ao)
i= 0
and ker
8 a prime zdeal in A.
Proof Set 6., = ()(a(j)6«()) = a(j . 6«()) (n EN). and take N E N such that 6 n = 6N (n ~ N). Set T = (} x a~, so that T h; an intertwining map and 6(T) = 6N. Since O(ao) E 6(0), we have ()(ao)N+l E 6 N . Also O(ao)N+1 i= 0, and so 6(T) i= 0 and T is discontinuousj ao r:t I(T) because 6N+I = 6 N , and so I(T) i= A. By the prime ideal theorem 5.3.14, there exists al E A such that T x al is a discontinuous intertwining map and either I(T x al) is a prime ideal in A or I(T x al) = A. Define K = {b E B : b()(adO(a~)6«()) = O} , a closed ideal in B. Since aIa~+1 . 6«()) = 6(T x ad i= 0, necessarily O(ao) r:t K. Define 8: a 1--+ ()(a) + K, A ---- B/K. Then ker 8 = I(T x ad, 8(ao) i= 0, and 8(ao} E 6(8), and so ker 8 is a prime ideal in A and 8 is discontinuous. 0 The general form of the prime kernel theorem requires a preliminary result that follows from the dichotomy theorem. Theorem 5.3.16 (Cusack) Let A be a dense subalgebra of a Banach algebra B, and let J be a closed zdeal in B whzch zs a separatmg module for A. Suppose that P is a prime ideal in A such that J n A ct P. Then there is a closed ideal Kin B such that K n A is a prime ideal m A with K n A c P. Suppose, further, that J n A is semiprime and that P is a minimal przme zn A. Then there exists a non-zero, closed ideal I in B such that: I c J; I n A = I; P = IT; for each closed zdeal L of B wzth LeI, either L n A = 0 or L = I. Proof We first claim that there exist ao E A \ P and bo E A such that the following properties hold: (i) aoaJ
= aoJ i= 0;
for each a E A with aoa r:t Pj
(ii) boaoJ
(iii) for each a E A, either boaoaaoJ (iv) aoJ = ao(J n A).
= 0 or boaoaaoJ = boaoJj
First note that, for each a E A \ P, aJ i= o. For suppose that a E A and and take b E (J n A) \ P. Then aAb = 0, and a E P by 1.3.42. By the dichotomy theorem 5.3.10 (applied with S = A \P and E = .I), there exists ao E A \ P such that aoaJ = aoJ for each a E A with aoa r:t P. By 1.3.42, aoAao ct P and Aao(JnA) ct Pj in particular, ao(JnA) ct P and AaoJ i= o. By 5.3.10 (applied with S = Aao and E = J), there exists bl E Aao with bd i= 0 and such that, for each b E A with bIb E Aao, either bIbJ = 0 or bIbJ = bIJ. Take bo E A with bl = boao. Then boaoJ i= o. Also, for each a E A, bIaao E Aao, and so either boaoaaoJ = 0 or boaoaaoJ = boaoJ·
aJ
= 0,
642
A utornatzc continuity theory
Since ao(J n A) ~ P, there exists al E J n A with aoal (j. P. Set I = J Since A = B. I is an ideal in B, and so alJ C I. Thus aoJ
= aoad C
aoI
= ao(J n A) c
n A.
ao.!.
and so (iv) holds. We have established the claim. We now define an important closed ideal K in B by the formula: K
= {b E
B : boaoJbBaoJ
= o} .
Take a E A \ K. Since A = B and ao(J n A) = a;;:J, there exist c E J n A and dE A such that boaocadaoJ ¥ 0. By (iii), above, boaocadaoJ = boaoJ. and so boaoJ = boaoJaBaoJ. To prove that K n A is a prime ideal in A, take a, b E A with aAb c K n A and a (j. K. Then boaoJbBaoJ
= boaoJaBaoJbBaoJ = boaoJaAao(J n A)bBao,]
c boaoJaAbBaoJ.
But aAb c K. so boaoJaAbBaoJ = 0, and hence boaoJbBaoJ = 0, i.e .. bE K. Thus K n A is a prime ideal. We finally prove that KnA c P. If possible, choose al E (KnA)\p. Assume that ao(JnA)alAaO c Pj then we see successively that ao(JnA)al C P because ao (j. P, that aoA(J n A)al C P, that (J n A)al C P again because ao (j. P, that (J n A)Aal C P, and that J n A c P because al (j. P, a contradiction. So ao(J n A)alAao ~ P. and there exist b E J n A and c E A with aObalcaO (j. P. By (i), we have aoJ = bObalcaOJ, and so boaoJ
= boaobalcaOJ C
bOaOJalBaoJ
=
°
because al E K. Hence boaoJ = 0, contradicting (ii). Thus K n A c P. Now suppose that J n A is semiprime and that P is a minimal prime. Then K n A = P, and so P = {a E A : boaoJaBaoJ = OJ. Suppose that a E A \ P. Then, by (iv) , boao(J n A)aAaoJ ¥ 0, and so there exists x E (J n A)aA with boaoxaoJ ¥ 0. By (iii), boaoxaoJ = boaoJ, and so we have shown that (5.3.3) boaoJaBaoJ = boaoJ (a E A \ P). Define I = JboaoJ, so that I is a closed ideal in B with I c J. Clearly I n A is a closed ideal in B. By (ii) and (iv), boao(JnA) ¥ 0. Since JnA is semiprime. (boao(J n A))2 ¥ 0. and so I ¥ 0. Also, I n A = I because I
nA
~
ao(J n A)boao(J n A) = aoJboaoJ ~ JboaoJ = I.
Take a E A \ P. Then I = Jboao.!aBaoJ by (5.3.3), and so I = IaBaoJ and la ¥ O. Now take a E P. Then JboaoJaJboaoJ = 0, and so IaI = O. Thus «(InA)af = 0, and so (InA)a = because JnA is semiprime. Hence Ia = 0. We have shown that P = IT. Finally, let L be a closed ideal of A with L e I and L n A ¥ O. Since (L n A)2 ¥ 0, there exists a E L n A with (L n A)a ¥ 0. But now I a ¥ 0, and so a (j. P. By (5.3.3), we have I = JboaoJ = J(boaoJaBaoJ) C L, and so L = I. This completes the proof. 0
°
The contznuity ideal and the smgulartty set
643
Corollary 5.3.17 Let J b(~ a separating 1,deal in (L Banach algebra A. Then the prime radical !.p(J) is a dosed and nilpotent 1,deal in A. Proof By 1.5.23, ~(J) = J n !.p(A). Thus !.p(J) = J n I, where I b the intersection of the minimal prime ideals P of A such that J ct P. By 5.3.16, each such ideal P is closed, and so !.p( J) is a closed ideal in A. By 1.5.26(i). !.p(J) is a nil ideal, and so. by 2.6.34. !.p(J) is nilpotent. 0 We now give details concf'rning the structure of certain separating ideals.
Theorem 5.3.18 (Cusack) Let A be a Banach algebra. Assume that A contams a non-zero, semiprime separatzng 1,deal J. Then there ex?st n E N, minzmal p'T"tme ideals PI, ... ,Pn in A, and non-zero, closed ideals It .... ,In m A such that the following properties hold: (i) for each i E N n , Ii C J, Ii is not nilpotent. Ii is minimal in the family of non-zero, closed ideals of A, and Ii 1,S a topologically simple Banach algebra wh1,ch 1,s a separatzng tdeal and is either finite-dl,mensional or radtcal;
(ii) for each i,j E N n with i:f. j, IiIj = Ii n I J (iii) for each ~
1,
E
Nil, Pi = Il
1,S
= 0;
a closed ideal zn A wtth J
ct
Pi and
n Ii = 0; (iv) for each prtme tdeal P in A. etther J C P or Pi C P for some i E N n ;
(v) J n PI n· .. n P n
= 0;
(vi) for each non-zero, closed tdeal L zn A with L such that Ii C L.
c J,
there eXtsts i E N n
Proof We denote by :F the family of non-zero, closed ideals in A. Let S = {I E :F : I C J}. By 5.3.11. the set S is not empty. Clearly 1112 = It n 12 = 0 whenever It and h are distinct members of S. Let I E S. Then I is semiprime and not nilpotent because J is semiprime, and so, by 2.2.34, I is topologically simple. Since I = J I = I J, I is a separating ideal in A. By 2.2.33(ii), I is either primitive or radical. But, if I is primitive. then I is finitedimensional by 5.2.26. By 2.6.34, each I E S contains an element which is not nilpotent. Assume that S is infinite. Then there is an orthogonal sequence (an) in J such that each an is not nilpotent, a contradiction of 5.2.23(ii). Thus S is finite, say S = {It, ... ,In}. For i E N n , set Pi = Il. so that Pi is a closed ideal in A. By 2.2.34, Pi is a minimal prime ideal with J ct Pi and Pi n Ii = O. Let P be a prime ideal of A with J ct P, and let Q be a minimal prime ideal with Q C P. Then J ct Q. By 5.3.16, Q is a closed ideal and there exists I E S with Q = IT; necessarily I = h and hence Q = Pi, for some Z E N n . It follows that J n PI n ... n Pn = 0 because !.p(A) n J = O. Let L be a non-zero, closed ideal in A with L C J. Then there exists i E N n such that L ct Pi' But now IiL:/:- 0, and so Ii = IiL c L. 0
Av,tomatzc contmv,ity theory
644
Corollary 5.3.19 Let A be a Banach algebra. Assv,me that there is a separating zdeal .1 in A such that rad .1 Ztl not mlpotent. Then there zs a topolog1cally simpl(o, radical Banach algebra.
Proof By 5.3.17, !.p(.1) i~ a clmled and nilpotent ideal in A. Since rad.1 is not nilpotent, !.p(.1) ~ rad .1. By replacing A by A/'.l3(.1), we may suppose that .1 i~ semi prime and that rad.1 =J: O. Let P l , ... ,Pn be as in .5.3.18. Assume that each of P l , ... ,Pn is primitive. Then rad.1 C .1 n P l n ... n Pn = 0, a contradiction. Thus one of the ideals Pi is not primitive; denote it by p, and denote the corresponding ideal Ii by I, so that I is a topologically simple I3anach algebra. By 1.4.38, P has infinite codimensioll in A, and so I is infinite-dimensional, and hence radicaL 0 Theorem 5.3.20 (Cusack) Assume tha,t the1'e is a derwatwn D on a Banach
rt
algebra A such that D(Po ) Po for some prirmtwe ideal Po of A. Then there zs a topologically simple, radzcal Banach algebra.
to
Proof Set .1 = 6(D) and L = !.p(A), and denote the quotient map from A A/L by 7T. Then .1 is a separating ideal in A. Assume that .1 is nilpotent, so that .1 eLand 'If 0 D is continuous. By 1.8.6, D(L) c L, and so D(L) c L. It follows that there is a continuous derivation DL : a+L - Da+L on A/L. Now Poll is a primitive ideal of A/L, and so, hy 2.7.22(ii), DL(Po/L) c Poll, i.e., D(Po ) C Po, a contradiction. Thus J is not nil potent. By 5.2.27, rad.1 is not nilpotent, and so, by 5.3.19, there is a topologically simple, radical Banach algebra. 0 Theorem 5.3.21 (Prime kernel thcorem)(Cusack) Let A be a Banach algebra, let B be a Banach algebra, let B : A - B be a homomorphism with B(A) = B, and let ao E A. Suppose that B(ao) E 6(B) and B(ao) zs not n/, the pnme rad,tca,l of 6(B) n B(A). Then there zs a closed ideal K m B such that the map (f:aI--*B(a)+K,
A-B/K,
is a dl,scontinuous homomorphism with (f(an) =J: 0 and ker (f a prime zdeal in A.
Proof By 1.5.23, B(ao) f/-!.p(B(A» and so there is a prime ideal P in B(A) such that B(ao) f/- P. Since 6(B) is a separating module for B(A) and since B(ao) E (6(p)nB(A» \P, it follows from 5.3.16 that there is a closed ideal K in B such that K n B(A) is a prime ideal in B(A) with KnB(A) c P. Define (f: a 1--* B(a)+K. Since B(ao) f/- K, necessarily (f(ao) =J: O. By 5.2.2(ii), (f(ao) E 6((f), and so (f is discontinuous. Since ker (f = B-l(K n B(A», ker (f is a prime ideal in A. 0 The force of the above results is that the exh;tence of some discontinuous maps associated with an algebra A would imply the existence of certain prime ideals in A; if A has no such prime ideals, then there can be no such discontinuous maps.
645
The continuzty ideal and the szngulanty set
Theorem 5.3.22 Let A be a cornmutatzve Banach algebra whzch has no closed, prime zdeal of infinite codimenswn zn A. Suppose that A satzsfies either: (i) null Sf'.1}'uences in Mop factor for each
F be a left-interlunning map over A. Then: (i) A(T) is a jinzte set; (ii) J(A(T)) c IL(T); (iii) T IS contznuous if and only if A(T) = 0; (iv) A(T) = ~(ILCT)). Proof (i) Define £E : FWA ----> SE and £F : FWA ----> SF a.. EjG be the quotient map, and take T E B(EjG) with T07l" = 7I"oT. Clearly
IITnl1 ::;
Q
II(T I FY'II
(n EN), and so v(T) ::; v(T I F). Thus a(T) c]IJ);
take P2 E (PI, 1) with a(T) C ]IJ)(O: P2). By A.3.78, there exist sequences (9k) in O(]IJ), F) and (h k ) in O(]IJ), G) such that fk = 9k + hk (k EN). We have
7I"«(IE - T)/k«()) = (OEIG - T)«7I" 0 9k)«())
(k
E
N, ( E ]IJ)).
(5.3.7)
It follows from (5.3.7) that (71"
0 9k)«() = «(IEIG - T)-l 0 71") «(IE - T)fk«())
«( E 'Jl'pJ.
Since «(IE - T)/k«() ----> 0, (71" 09k)«() ----> 0 in EjG uniformly on 'Jl'P2' By the lllaximum modulm; theorem A.3.77(i), 71" 0 9k ----> 0 in O(]IJ)(O; P2), EjG). By A.3.79, there exists a sequence ('9k) in O(]IJ)(O; P2), E) such that gk ----> 0 and 7rogk = 7I"09k (k EN). We have 71" 0 Uk - gk) = 71" 0 hk = 0, and so Uk - gk) c O(]IJ)(O; P2), G). Also (IG - T I G E Inv8(G) for and so it follows from the fact that «(IE - T)(fk - gk)«() ----> 0 uniformly on '['PI that (fk - gk)«() ----> 0 uniformly on 'lI'Pl> and hence that II/k - 9kliK ----> O. Since 119kliK ----> 0, we have IIfkllK ----> 0, and so /k ----> 0 in O(]IJ), E), as required. D
652
A utomatzc contmuity theory
Let T be a decomposable operator on E. By an easy special case of 5.3.35, T has the single-valued extension property: if ~ is an open disc in C, and the function f E O(~, E) is such that «(h - T)f«() = 0 E ~), then necessarily f = O. It follows that, if x E E, then there exists f E O(p(T, x), E) such that «(h - T)f«() = x E p(T, x)), and this in turn implies that X T (0) = o.
«(
«(
Theorem 5.3.36 Let E be a Banach space, and let T be a decomposable op_ erator on E. Then XT(W) is closed in E for each W E Fe, and the map XT : Fe ~ SE zs decomposable and stable. Proof Let W E Fe, and take (xn) in XT(W) and x E E such that Xn For each disc ~ C C \ W, there exists Un) in O(~, E) such that «(IE - T)fn«() = Xn
~
x.
«( E ~, n E N).
Let «mk, nk) : kEN) be a sequence in N2 with mk ~ 00 and nk ~ 00, and set gk = fmk - fnk' Then «(IE -T)gk«() ~ a in E, uniformly on compact subsets of ~. By 5.3.35, gk ~ a in O(~, E), and so Un) is a Cauchy sequence in O(~, E). Since O(~, E) is complete, there exists f E O(~, E) such that fn ~ f. Clearly «(IE - T)f«() = x E ~), and so x E XT(W), Thus XT(W) is closed in E. By (5.3.5) and the above remark, X T is stable, and it follows from (5.3.6) that X T is decomposable. 0
«(
The algebraic spectral spaces ET(W) were defined in 1.4.21. We now investigate the relation between the spaces XT(W) and ET(W), and whether or not ET(W) is closed, in the case where E is a Banach space and T E B(E). Note that Er(0) is only closed in the case where Er(0) = O. Theorem 5.3.37 Let E be a Banach space, and let T E B(E). Then, for each Wee, we have XT(W) C ET(W), Proof Let x E XT(W), and take z E C \ W. Then there exist V E N z and f E O(V, E) such that «(IE - T)f«() = x E V). Since To f E O(V, E), we have Tx E XT(W). Define 9 on V by setting
«(
g«()=f«()-f(z)
(-z
«(EV\{Z}),
g(z)=J'(z).
Then 9 E O(V, E). For ( E V, «(h - T)f«() = (zh - T)f(z), and so the equa.lity «(h - T)g«() = - fez) holds for each ( E V \ {z}, and hence for each ( E V. Thus fez) E XT(W) and x E (ZIE - T)(XT(W). This establishes that (zh - T)(XT(W)
= XT(W)
(z E C \ W).
By the maximality of ET(W), necessarily XT(W) C ET(W),
I
o
In fact, if w e e and ET(W) is closed, then XT(W) = ET(W). For take Xo E ET(W) and z E C \ W. Then there exists k > a and (xn) in ET(W) such that (ZIE - T)(xn) = Xn-l and Ilxnll ~ k IIxn-lii for each n E N. Set V = Jl)(z; 11k) and f«() = 2:~=o(z - ()nXn+l E V). Then f E O(V, E) and «(IE - T)f«() = Xo E V), and so z E peT, xo). Thus aCT, xo) C W, Xo E XT(W), and XT(W) = ET(W).
«(
«(
The contznuity zdeal and the szngularzty set
653
'l'beorem 5.3.38 Let E be a Banach space, let T be a super-decomposable operator on E wzth ET(0) = O. Then XT(W) = ET(W) (W E Fe). proof Let WE Fe. By 5.3.37, XT(W) c ET(W). Let U E N w . Since T is super-decomposable. there exist R, S E B(E) such that RT
= TR, R+S = IE, a (T I R(E)) c U, and a (T I S(E)) c C\ W. For
,E W, we have S(E)
= ((IE - T) (S(E)), and so S(E)
ST = TS, we have S(ET(W))
C ET(C \ W). Since
C ET(W) by 1.4.24. Thus
S(ET(W)) c ET(W) n ET(C \ W) = ET(0)
= O.
Also R(En(W)) C XT(U) by (5.3.6), and so ET(W) C XT(U), By (5.3.5). ET(W) c XT(W), 0 We omit the proof of the following theorem.
Theorem 5.3.39 (Curtis and Neumann) Let E be a Banach space. let T E B(E), and let k E Z+. Suppose that T has a C(kLfunctzonal calwlus. Then T is superdecomposable, and ET(0)
=
n
{((IE; - T)k+3(E) : (
JUrlher, for each W E Fe, XT(W)
= ET(W) =
n
E C}
= O.
{((IE - T)k+3(E) : (E C \
w} .
o
For example, suppose that T E Inv B(E) and that IITnll + liT-nil = O(nA-) 00. Then T ha..., a C(k+2)-functiollal calculus by 4.5.13, and so it follows from the theorem that ET(0) = {((IE - T)k+5 : ( E C} = O. A normal operator T on a Hilbert space has a continuous functional calculus, and so T is super-decomposable with ET(0) = O.
as n ~
n
In Theorem 5.3.41 and the preliminary lemma we shall use the following notation. Let A = 8qX] bE' the space of all sequences of polynomials in qX] such that each sequence has only finitely many non-zero terms. For p = (Pj) E A, set ap = max8pj. suppp = {J EN: Pj -I a}. and c(p) = Lj.A- Ill'j.kl. where Pj = Lk Uj,k Xk . Let E be a Banach space, and let R E B(E). As before, E is a unital qX)-module for the product p . x = p(R)x (p E qX), x E E); for p = (Pj) E A and x = (Xj) E EN, define p . x = LPj . Xj E E and set 3;.1. = {p E A : p . x = O}.
Lemma 5.3.40 Suppose that. for each x E EN, there exists p E Ae s1Lch that = O. Then R zs algebraic.
p . x
~roof For each n EN, set
En
= {x
E
EN : there exists p E x.1. with 8p ::; n, with suppp C N n , and with Tn ::; c(p) ::; 2"} .
654
Automatic contznuity theory
Thus each En is closed in the Frechet space EN. By hypothesis, U:=l En = EN, and so there exists mEN and y = (Yj) E EN such that y E int Em. Define
W = lin {RiYJ : i
E z~, j E
Nm
},
so that dim W:::; .71,1, where M = m 2 + m. Set N = M2 + A1. Take x E EN. Then there exists f3 > 0 such that y + 8x E Em. and so there exists p E Ae with 8p :::; Tn and :,mppp C Nm such that p . (y + /3x) = 0; since p . yEW, necessarily p . x E W. Now take x E E and k E Z+, and apply this to the sequence Xk
=
(Rkx, R k+(m+1)x, R k+ 2 (rn+1).r .... ) :
we obtain Pk,b'" ,Pk,m E qX], not all zero, with 8Pk,j :::; m (j E Nm ) such that qk . x E W, where qk. = Pk.l Xk
+ 1>k,2 xk+(m+1) + P k,3 X k+ 2 (m+l) + ... + Pk,m xk+(m-l)(rr>+l) .
The only powers of X whose coefficients in the polynomial Pk.jXk+j(m+1) are non-zero lie in the intervalIj = [k+j(m+1), k+ j(m+1)+m], and I j nlj + 1 =.0. Hence qk =F O. Further, the powers of X with non-zero coefficients in qk lie in the interval [k, k + m 2 + m - 1]. Since dim W :::; M, it follows that the set {qo . x, qm . x, q2m . x, ... , qA!2 . x} is linearly dependent in W. We have 8qj AI :::; j AI + M - 1, whilst the minimum power of X whose coefficient in %+1)M is non-zero is (J + 1)M, and so we conclude that there exists q E qXj" with 8q :::; N - 1 such that q . x = O. This shows, in particular, that E is a torsion module over qX). For ( E C, define 00
Ee, =
U {x
E
E: (X - ()n .
X
=
O} .
n=l
By 1.6.15, E = 8{ Ee, : ( E C}. Assume that there exist distinct points (1, ... ,(N in C such that Ee,j =F 0 (] E NN), and take Xj E E(] (j E NN)' Then there exists q E qx)e with 8q :::; N - 1 and q . X(X1 + ... + XN) = O. For j E NN, q . .rj = 0 and so (X - (j) I q, whence 8q ;::: N, a contradiction. Thus there 1 E(]. For each x E E(" there exists exist (1, ... , (N -1 E C such that E = q E qXje with 8q :::; N -1 and q . x = 0; this remains true if we delete from q any factor X - ( with ( =F (j, and so (X - (j)N-1 . X = O. Set P = I1j:-;,l(X - (J)N. Then P . x = 0 for all x E E, and so peR) = O. This establishes that R is algebraic. 0
of=-;,
Theorem 5.3.41 (Sinclair) Let E and F be Banach spaces, let R E B(E), and let S E B( F). Suppose that either: (i) (R, S) has a critical ezgenvalue; or (ii) R is not algebraic and Es(0) =F O. Then there is a discontinuous linear map which intertwines (R, S). Proof First, suppose that (R, S) has a critical eigenvalue z. Then there exists Yo E Fe such that SYo = ZYo and a discontinuous linear functional oX on E such
655
The contw'UZty ideal and the smgulanty set
that oX I (zh - R)(E) = O. Set Tx disContinuous linear map, and (TR - ST)(x)
= oX(Rx)yo
=
oX(x)Yo (x E E). Then T : E ~ F i:, a
- oX(x)SyO = oX(R - zIE)(x»yo
=0
(x E E),
SO that T intertwines (R, S).
Second. suppose that R is not algebrak and that E8(0) 1= O. By 5.3.40, there exists x = (xi) E EN such that p . .C 1= 0 for each pEA·; we may suppose that Ilxjll = 1 (J EN). Set G = 0~1 qX] . Xj, so that G is a qX]-subrnodule of E. Take y E Es(0) with lIyll = 1 and define T: LPi .
Xj
~ LjPj . y,
G
---->
Es(0).
Then T is a qX]-modnle homomorphism. By 1.4.19, the unital, qX]-divisible module Es(0) is injective, and so T can be extended to a qX]-module homomorphism T : E ~ Es(0) c F. The map T intertwines (R, S). and T is discontinuous becausc IITxj II = j U EN). 0 The following result should bp compared with the remark after 5.2.9; a stronger result will be given in 5.3.48.
Corollary 5.3.42 Let P, q E [1, oc]. and take a E lR.+.. Then there is a discontinuous lmear rna.p T : V(lR.+) ~ Lq(lR.+) such that TS_ a = S-aT.
Proof Certainly S-a is not algebraic on V(lR.+). By 1.4.22(i). the algebraic spectral space E.'L" (0) contains the linear subspace of Lq(lR.+) consisting of the functions of compact support, and so E!:L" (0) 1= O. The result follows from the theorem. 0 Theorem 5.3.43 (Laursen and Neumann) Let E and F be Banach spaces, let R be a decompo.~able operator on E, and let S be a super-decomposable opemtor on F. Then the following are equivalent: (a) each linear map which intertwines (R, S) is automatically continuous; (b) (R, S) has no crit%cal eigenvalue, and either R %s algebrmc or Es(0) = O. Proof By 5.3.41, it suffices to prove that (b)~(a). Let T intertwine (R, S), and set 6 = 6(T). We first claim that. to prove that T is continuous, it suffices to show that there exists p E qX]· with p(S)6 = O. For suppose that p is such a polynomial. By cancelling from p a.ny factors X - ( for which (IF - S is injective, we may sUppose that each root of p is an eigenva.lue of S. Since (R, S) has no critical eigenvalue, p(R)E has finite codimension in E, and so, by A.3.24, p(R)E is closed in E. Thus p(R) : E ----> p(R)E is an open map. By 5.2.2(ii), p(S)T is continuous, and 80 Tp(R) is continuous. Hence T I p(R)E is continuous. By A.3.42(i), T is Continuous, as claimed. Suppose that R is algebraic, say p(R) = 0 for some p E qXj·. Then p(S)6
C
p(S)T(S)
c Tp(R)(E)
and so p is the required polynomial in this case.
= 0,
A utomatic continuity theory
656
Suppose now that Es(0) = O. By 5.3.36, the maps X R : Fe ---+ SE and Xs : Fe ---+ SF are decomposable and stable. For W E Fe, Xs(W) = E..,(W) by 5.3.38, and so, by 1.4.24, T(XR(W» c Xs(W), whence
6(T I XR(W»
c T(XR(W» c Xs(W). there is a finite set A = {All"" Ad c
Thus Theorem 5.3.28 applies: C Such that 6 C Es(A). Set q = Il;=l (X - Aj). By 5.2.15(iii), 6 is a separating module, and so there exists no EN such that qn . 6 = qno . 6 (n;::: no). Set
G
= qno . 6 = qn °(S)(6) and
Sj
= A)Ia - S I G (j E N k ).
For j E Nk, Sj(G) c G because Sj(6) C 6, and so SJ E B(G), and SJ(G) = G because q(s)no+l(6) C Sjq(s)no(6) c Sj(G). By A.3.27(i), there is a dense linear subspace H of G such that Sj(H) = H (j E N k ). and hence such that H C Es(C \ A). Since H C 6 c Es(A), we have
H C Es(A) n Es(C \ A) Thus G
= Es(0) = O.
= 0 and p(S)6 = 0, where p = qnO is the required polynomial.
. 0
general~zed scalar operators on a Banach space. Then each linear map which intertwines (R, S) is automatically contmuous zf and only if (R, S) has no critzcal e~genvalues.
Corollary 5.3.44 Let Rand S be
Proof By 5.3.39, each generalized scalar operator T is super-decomposable and ET(0) = 0, and so this is immediate from the theorem. 0 Corollary 5.3.45 Let G be a locally compact group, let E and F be translation-
mvanant Banach function {liSnail : n E IE} is bounded E into F whzch commutes and only zf (Sa, Sa) has no
spaces on G, and let a E G. Suppose that the set in both B(E) and B(F). Then each linear map from w~th translation by a is automatzcally continuous if critical ezgenvalue.
Proof Since IIS~II + IIs;nll = 0(1) as n ---+ 00, Sa is a super-decomposable operator on both E and F and EsJ0) = O. 0
For example, let G = JR k , and take a =I- O. First take q E [1,00). Then it is easily seen that Sa has no eigenvalues as an operator on Lq(JR k ), and so each linear map T from a translation-invariant Banach function space into Lq(JR k ) such that T commutes with translation by a is automatically continuous. On the other hand, if E = Co(JR k ) or U(JR k ), where p E [1,00], and if F = LOO(JR k ), then z = 1 is a critical eigenvalue of (Sa, Sa), and so there is a discontinuouS linear map from E into F which commutes with translation by a. In Theorem 5.3.43, we proved the equivalence of conditions (a) and (b) when R is decomposable on E and S is super-decomposable on F; we suspect that these requirements on Rand S could be weakened. Question 5.3.A Let Rand S be bounded linear operators on Banach spaces
E
and F, respectively. Under what conditions on Rand S is it true that each linear map which intertwines CR, S) is automatically continuous?
The
continuity ideal and the singularity set
657
We conclude this section with an extension of Corollary 5.3.42: we obtain a discontinuous linear operator commuting with all shifts S-a.
Definition 5.3.46 Let E be a linear space, and let S be a subset of L(E). A linear subspace F of E is S-invariant if S(F) c F (S E S) and is strongly divisible for S if, further', S I FE Inv L(F) for each S E S \ {o}. proposition 5.3.47 Let E be a linear space contaming linear subspaces El and
£h, and let S be a unital integml domain contained in L(E). Suppose that: (i) Go is an S-invariant subspace of E l , and Xo E El \ Go is such that Sxo ri. Go (S E S \ {o}); (ii) F is a subspace of E 2, F tS strongly divistble for S, and Yo E F. Then there is a linear map T : El -- E2 such that TS = ST (S E S), T and Txo = Yo·
I Go =
0,
Proof Let 9 be the family of pairs (G, T), where G is an S-invariant subspace of E1 and T E L(G, F) is such that TS = ST (S E S). Set (G l , Td :::5 (G 2, T 2) in 9 if G l c G 2 and T2 I G l = T I · Clearly (9,:::5) is a partially ordered set and (Go, 0) E g. Suppose that (G, T) E 9 with G -=I- E l . Take Uo E EI \ G, and define U
= {S
E S : Suo E G} .
For each S E U \ {O}, TSuo E F: set Vs = (S I F)-lTSuo E F. Now take 8 11 S 2 E U \ {O}. Then SlS2VS2 = SlTS2uo = S2TSlUO = S2SlVS1 = S1S2VSl' where we are using the facts that TS I = SIT, TS2 = S2T, and SlS2 = S2S1. Since S is an integral domain, SlS2 E S \ {O}, and so VS 2 = vs l • Denote this common value by vo, so that Svo = TSuo (S E U). In the case where U = {O}, choose Vo arbitrarily. Define H = G + {Suo: S E S}. Then H is an S-invariant subspace of El with G c H. Consider the map V :x
+ Suo
1---+
Tx
+ SVo,
H -- F.
We clatm that V is well-defined. For suppose that Xl + SIUO = X2 + S2UO in H. Then S2 - Sl E U, and so T(XI - X2) = T(S2 - Sd(uo) = (S2 - Sl)(VO). Thus V is well-defined. Clearly V is linear and VS = SV (S E S), and so (H, V) E 9 with (G, T) ::S (H, V) and Uo E H. The above argument shows that there exists (G b Td E 9 such that (Go,O):::5 (Gl,Td, with Xo E G!, and with TlXO = Yo. By Zorn's lemma, there is a maximal element of g, say (0, T), with (G I , Td :::5 (0, T). Again by the above argument, necessarily 0 = E l , and so T is the required map. 0 'l'heorem 5.3.48 (Dales and Millington) Let p, q E (0,00]. Then there is a discontinuous linear map T : LP(IR+) -- Lq(IR+) such that TS_ a = S-aT for each a E IR+. Proof We apply the above proposition, taking EI = LP(IR+), E2 = Lq(JR.+), E = El +E2, S = lin {S-a : a E JR.+}, Go = {J E EI : supp! is compact},
658
Autvmat1c contznuity the()ry
and Xo to be any element of El \ Go. Clearly S is a unital integral domain contained in C(E), and condition (i) of 5.3.47 is satisfied: we shall define a space F satisfying condition (ii) of 5.3.47. For n E Z+. we denote by Rn the interval tn, n + 1] of R regarded as a suhset of C: the' corresponding uniform norm on R" is denoted by 1·l n . For N E N and IE C(OC) (lR.+). set
=L oc
PN.71(J)
Nj
I/U) I I
n
(n E Z+),
J.
j=o
and consider functions I with PN,n(f) < DC. Clearly each such an analytic function on the open set {z E C : d(z. Rn) < N}. For N, kEN and I E C(OO) (lR.+). set
II/IIN.I.: =
sup {exp ((n
+ k)2)PN.n(J)
I
extends to be
: n E Z+} •
and set
FN.k = {J
E C(:X;) (lR.+)
: 11/11N.k < x}.
so that (FN .k , 11·IIN.k) is a Banach space, and each I E FN,I.: extends to be an analytic function on the open set {z E C : d(z, lR.+) < N} = {x + iy : Iyl < N}. Now set F = n{F"l.k : N, kEN}. so that F is a linear subspace of C(OO) (IR+) and each I E F is the restriction to lR.+ of an entire function, also denoted by f. For each I E F. we have III" = O(exp( _n 2 )) as n ~ x, and so Fe E 2 . Set lo(t) = exp( _ti) (t E lR.+), and take N. kEN. By applying the Cauchy estimates to the entire function z ~ exp( -Z4) on {z E C : d(z, Rn) ::; 2N}, we see that there exists (AIn : fI E Z+) C lR.+ such that Afn = O(exp(-n:i)) as n ~ 00 and Nj JO < 1'11n. ( n.j E Z + ). j! ,,- 23
I/U)I
We have PN,,,(JO) = O(cxp( -n 3 )) as n ~ 00, and so Il/ollN.k < :xl. Thus 10 E F, and hence F =I- O. Let a E lR.+, and set m = [a], the integral part of a. Take I E F, where w(' are regarding I as being defined on R and define 9 = S-al 1lR.+ and h = Sal 1lR.+, so that g.h E C(OC) (1R+). Fix N,k E N. Sinre PN,n(g) ::; PN.n+m (f)
+ PN,TI+m+l (f)
(n E N)
I E FN k' we have 9 E F.N k. Similarly, since I E F N.k + 171l we have hE F".k' Thus y. h E F, and so S_a(F) = F. Now take I E F with S-al = O. Thm I I [a. (0) = 0, and so I = 0 because I is entire. Thus S-a I F is injective, and so S-a I FEIn" C(F). and since
I
The operator norm in B(FN . k ) is abo denoted by 11·II N .k. Take b E lR.+ e and N, we have
E Fn,k' For each j E
PN,,,(S-3bf) ::; PN,n+bb](f)
and so
+ PN,n+[jb]+l(f)
(n E N),
The contmlL'lty ideal and the 8lf1gulllT'tty set
55!)
An arbitrary elemcnt of S\ {O} ha~ til{' form8_ a (01E+8_ b T). \dH'l'c a E jR-r-. a E C·, bE R+·. and T E S. \Ve kuow that .'i-a I FE Inv £(F). As an olwrator on FN,k, wc hayc
11(S'-bT)jll~:'~::; IIS-J"II~.~ IITIIN.~ -> 0 as j -> ')C. and so (nh; + .'i-br) I FN.k E IIlV £(F'tv.k) for caeh N. kEN. It. (alE + S-bT ) I FE lnv £(F). and so F is strongly divisible for S.
follows t.hat
We conclude from 5.3..!7 that there is a lineCU' map T : E1 -> E2 sneh that = S'-a T (a E R+). 1'1 Go = O. and Txo -# O. Since Go is dense iu £1. T is necessarily diseontinuolls. 0
TS-a
Notes 5.3.49 The notion of the mntinuity ideal for various maps can be found in such early papprs al'> (.Johnson 1969a), (Rillgro&e 1972), and (Stein 1969b). Theorem 5.3.7 on the continuity of derivations from C·-algebras into modules WH.'l first prowd by Ringrose in (1972) (cf. (Kadisoll and Ringrose 1983. 4.6.66)). Propositioll ii.3.2 it< given in (I3ade and Curtis 1974) (for rlerivation~) and (Laurs('ll 1981) (for intertwining maps). The importance of pliability in questions of automatic continuity was r('cognized by Jewell (] 977). where 5.3 8(i) wa& proved. The prime ideal th('oH'm 5.3.14 is due to Bade and Cllrtis (1978b). The commlltative prime kernel theorem 5.3.15 is due to Sindair (1B75). (1B76. 11.4): thp general case 5.3.21 is due to Cusack (1976). The devory of both classes is sllb~tantially developed in the monograph (Laursf'n and Neumann 2000). Definitioll 3.3.:n(i) is equivalent. to t.he original definition by a result of Albrecht which is es~ential1y 5.3.35; tltis re.')ult show~ that a decomposable operator has 'I3ishop's property (13)' Let T E 8(E). By (LaurseIl and N('umann 2000, 1.2.16(f)), the following are equivalent: (a) T has the single-valuerl extension property; (b) XT(0) = 0; (e) XT(0) is closed. Thf' operator T has DU7Iford's p1'Opertll (C) if X·r(lV) is elosed in E for each dosed set ~V in C; sl'(~ (Dunford and Schwartz ln71. Chapter XVI) and (Laursen and NeulllanIl 2000, §1.2). PropC'rty ((3) implies property (C) (ih·td .. 1.2.19). but the ('onverse is not true (ibid .. 1.6.17). Let A be a I3anach function algehra, and let f E A. By a theorem of Neumann (IB92). the following arC' equivalf'nt: (a) the map L, : 9 1-+ fg is decomposable; (b) the map L] . 9 1-+ fg is super-decomposable: (c) the function f is continuous on if> A with respect to t he hull k('rnel topology. For further equivalences, see (Laursen and Neumann 2000. 4.4 5 and 4.5.4). Theorem 5.3.:m is from (Curtis and Neumann 19H9); a proof is obtained by combining 1.4.15 and 1.5.4 of (Laursen and Neumann 20(0). For a Ilormal operator T on a Hilbert space. ill fact XT(W) = Br(W) = n{((TE - T)(E) : (E C \ W} (~bid., 1.5.7) To some extent, the theory of the algebraic spectral spaecs WA~ developed because the analogue of 1.4.24 rlo('s not hold for the analytic spectral spaces, even when R and S are both super-decomposablf': there is a Banach space E, a super-decomposable operator Ron E, and a discontinuous map T E C(E) such that RT = TR, but such that T(Xn(W)) ct. Xn(W) for some W E :Fe (Laursen and Neumann 2000, 5.4.10),
A utomatzc contmuity throry
660
The general problem of the automatic continuity of a linear map which intertwines a pair of bounded linf'ar operators originates with (Johnson 1967c) and Johnson and Sinclair (1969). Theorem 5.3.41 is due to Sinclair (1974a), extending (.Johnson and Sinclair 1969), and 5 :3.43 is (Laursen and Neumann 1986. 4.3). Ther are many other cases (some elsewhere in this book) where the requirements that R be decomposahle and that S be super-decomposable can be weakpned. but the ('xact conditions on Rand S which are required are not known. Extensions of 5.3.43 an> given in (Laursen and Neumann 1991 and 2000, §5.4)j for example, 5.3.43 holds for a pair (R, S) whenever O'(R) is countable (ibid .. 5.4.6). It is conceivable that conditions (a) and (b) of 5.;~.43 are equivalpnt for eaeh R E B(E) and S E B(E); progress on this quest.ion would be made if it could be proved that ET(W) = X1'(W) for each closed subset W of C and ('ach bounded lin('ar operator T on a Banach space such that E1'(0) = O. For a discussion of these open questions, see (ibid., §6.3). For results related to 5.3.43 in the case where E and Fare Frechet spaces. see (Thomas 1978a). The following result is shown in (Sinclair 1974a. Theorem 3.3). Let E he a Banach space, and let R E B(E), so that E is a unital qX]-module. Suppose that ER(0) =1= O. Then there is a non-zero, torsion-free, divisible submodule in E. For 5.3.48, see (Dales and I\lillington 199:~): the argument of 5.3.48 also produces a discontinuous linear operator T : Co{lR+) -> Co{lR+) sueh that T8- a = S-u T for each a E 1R+. Let E be a translation-invariant space of funetions or measures on a locally compact group G. A tmnslation-invariant functional on E is a linear functional A such that (Sal. A) = (f, A) (a E G, I E E). Suppose that E and F are translation-invariant topologieallinear spac('l) on G, that A is a discontinuous translation-invariant funetional on E. and that 1 E F. Then the map T : I f-7 (f, A)1. E -> F, is a discontinuous linear operator such that TSa = SaT (a E G). The question which spaces E have discontinuous translation-invariant functionals has been much studied: see (Mcisters 1983) and (Nillsen 1994) for surveys. For example. it was proved by Mcisters (1973) that there are discontinuous translation-invariant functionals on gOO(Z), Ll(IR). and L2(IR). Let G be a eompact abdian group. Then every translation-invariant functional on L 2 (G) is continuous if and only if G / Go has a finitely generated. dense suhgroup: see (Meisters 1983) and (.1ohnson 1983). Let G be an infinite, compact group. Then there are discontinuous translation-invariant functionals on L 1 (G) and, in the casf' where Cd is amenable, on C(G) (Saeki 1984): for eaeh p E (I, (0), every translation-invariant functional on LP (G) is continuolls for each connect('d, mE'trizable, compact abelian group G (Bourgain 1986). Now suppose that G is a locally compact group which is not compact. Then it is proved in (Woodward 1974) that there are discontinuous translation-invariant funGtionals on Co(G), C"(G), and LP(G) for p E [1,ac] whenever Gis O'-compact and amenable, and on Ll (G) whenever G is O'-compact or abelian. A different behaviour oecurs if G is not amenable: in this case, every translation-invariant functional on Co(G) and on £P(G) for p E (1,00] is 0 (Willis 1988, 1990). Continuous linear opf'rators which commute with shifts often have a special form. For example, suppose that E = Co(IR), Ch(IR), or £P(IR), where p E (1,00), and that F = Ll(IR), or that E = LP(IR) and F = U(IR), where 1 :$ q < p < ac. Thf'n each linear map T : E -> F such that TS I = SIT is zero (Dales and Millington 1993), (Hormander 1960): for a general theory, seE' (Larsen 1971).
5.4
THE MAIN BOUNDED NESS THEOREM
We come in this section to the final major principle of automatic continuity theory, the main boundedness theorem of Bade and Curtis from 1960, given as 5.4.2. This result shows that an intertwining map must be bounded on certain orthogonal sequences; the theorem has immediate applications to maps from algebras with many idempotents. For example, we shall prove in 5.4.12(i) that
The main bonndeAiness theorem
661
each algebra norm on B(E)
is equivalent to the operator norm (i.e., B(E) h& N, such that (5.4.2) Zn(i.j), Vi,j = Xn(i,j), a.nd M = liT I HII. For each i E N, define vi,t/2 e; the series for Vi converges in E, and IIvili :s 1. Now choose j(i) EN such that !(Vi) :s 2j(i). Finally, define Z = 2:~lnk,j(k)/2k E G.
where
Ui,j
Vi = 2:~1
=
A7d.ornatu; contmmty theory
(j(j2
For t'ach lEN. we have' -xc, 1 B(Z.'I'i) = 2~+lB(llk.j(~).Vi.f)
L
k.f=1
1
= 2i +j(I)B(lli.)(i).1· i .j(i»)
1 2k+fB(Uk.)(k).I'U).
L
+
(k.l)#(i.jU»)
All tIl(' tenns in the second SUIll on the right-hand side of this equation lwlong to H. and so it follows from (5. t2) that IIT(B(z. v,))!1
~ (2,+j(i) + it!) -.\1
t
2/+ 1
~ 2 i +)(;) > 2i f(1'i).
k,l=l
HI)WcV('r, IIT(B(z, v,))I! ~ J(v')g(z) (i EN). and hence g(z) > 2; for allz E N. a contradiction. Thus the rc~ult is true. 0
As a special case ofthe above theorem (taking G = A. B to be the continuous left Ulodule map. and H = 0). we obtain the classic theorem. . Theorem 5.4.2 (l\Iain boundedness theorcm) (Bade and Curt b) Let A lw a Banach algebm, let E be a Banach left A-module, let F be a weak Banach left A-module, and let T : E ~ F fJC a left-inter·twining map. Suppose that (a,,) lLnd (XII) a1'(~ srquence.s zn A and E. Te8pcctwely, such that am . J'n = 0 (TIl -:f 1/). Thrn th('1'(~ 1,8 a constant C > 0 8uch that
IIT(an
.
x,.) II ~ C
ilo"llllxnli
(n E N).
o
Corollary 5.4.3 Lft A and B be. Banach olgebm8. and let () : A homornorphzsm. (i) FaT PIlch oTthogollfLl set {Pn : 11 E N} in J(A), theTe .mch that 11()(PTl)11 ~ C ilPnl1 2 (11 EN).
(ii) Suppose that (an) and (bTl) are ,~equences Then anb" E I(O) e~wnhtally.
1.11
(iii) In the case whef'e. A is commutative, thcre
110(1')11
~
l,S
1,,'1
~
B be a
a constant C
A unth a",b" = 0 (m
a cort.8tant C
>0
i= II).
> 0 .mcl! that
(' IIpl12 (p E J(A).
Proof (i) This is immediate from the theorem.
Oi) ASSllIllC towards a contradiction that a 1l b1l (j. Id()) for infiilitely mallY 1/. For each sHch n. there exists en E A- with 11()(a"b"rTl)11 > nllallllllb.,iI lie" II· a cout.radiction of the theorem. So anb., E It-rburn decomposition A = \E(A) (iJ R. and A = CO(cI>A)' Suppose that A = B ~ R for a closed subalgebra B of A. Then. by 2.8.8(i). \E(A) c B. and so \E(A) = B. Thus the strong decomposition is unique. Clause (d) implies that J(A/R) is bounded (with respect to the quotient norm). In the case where rad A is uniformly radical, it now follows from 2.4.24(iii) that J(A) is also a bounded set.. and so (a) holds. 0 It follows easily from 5.4.4 that. in the cas!' where n is a totally diseonnected. compact space. every commutative. uniformly radical extension of C(n) splits strongly: this rE'sult will be established for an arbitrary compact space n in 5.6.4. We now describe some examples which confirm the limits of 5.4.4.
Examples 5.4.5 (Bade and Cu'rtZb) (i) We exhibit a COllllIlutatiYe Banach algebra A with A = Co (so that
1J be a hOlllomorphism. We dw.m that {llfJ(p)1I : p E J(C(n))} is bounded. For a.'lsume t.hat. this is not the casco Then. by 2.4.37(iii) (in the case where n = 0), there is an orthogonal set {Pn : 'fI E N} in J(C(O)) such that. 1I()(Pn)ll -> x. a~ n -> :>c. and so there i~ an injective lllap (1,)) f---> 11(1,)). N X N -> N. huch that IIO(qi.j)1I ~ 2ii-j (i,J EN), where qi.j = P,,(i.j)· Set U i •i = {.r EO: qi.j(.r) = I}. By the above r F be a left-znte1twwzng map. Set
K = {b
E
B : AbA
C
IdT)} .
SIlP1JOse that (b n ) and (c n ) are seq'uenc('s m B buch that bmc lI b"c n = bn (n EN). Then bn E K eventually.
= 0 (m
-=I- 11) and
Proof Assume towards a contradiction that the result i~ false. \V(' may suppose that bn t/: K (n EN), and so then' exist sequences (sn) and (tn) in A such that (s"bnt,,) C A \ IdT). By 5.3.2, each T x snbT/ n is dh;continuous.,l:uHI so there exists (Yn) C E- such that (5.4.3) Apply the maiu boundedness theOlem with an = sT!bn and 3: n = cnt.,. . YII: we have am . Xn = 0 if Tn -=I- n and an . Xn = snbrJn . Yn for n E N. and so. by 5.4.2, there is a constant C > 0 such that
IIT(snbntn . Yn)1I ::; C IIsnbnllllcntnllllYnll
(n
E
But this is a contradiction of (5.4.3), and so the result holds.
N).
o
The main boundednes8 theorem
Theorem 5.4.11 Let of the identzty. let Al identzty, and let A2 be AI@A2 into a Banach
667
B be a umtal Banach algebm 1Vzth a contznued bzs('(;tion be a closed zdeal in B wzth a bounded left approximate a untfal Banach algebm. Then each homomorphism from algebm 1,05 automatzcaliy continuous.
Proof Set E = A 1 @A 2 , and let () : E -+ C be a homomorphism into a Banach algebra C. As in Example 2.7.3(vi), E is a Banach left A)-module and () is left-intertwining ovcr At. Set K = {b E B : AlbA l C IdO)}. By hypothesis, there exist (Pn) and (qn) in J(B) such that eB = PI + qt and. for each n E N. Pn = Pn+l + qn+l and Bp"B = Bq"B. the latter implying that AIPnAl = A1qnAI' By a remark after 1.3.24, q", 1- qn (m =f n). and so, by 5.4.10. qn E K evcntually. say qk E K. 'We see successively that qk.Pk.Pk-),qk-I ..... PI,q],eB belong to K. Thus A~ c IdB). Let (xn) E co(N, E). By 2.9.29(i), there exist a E Al and (YII) E co(N. E) such that Xn = a . y" (n EN). Since () x a is continuous. (}(:r 1l ) = (B x a)(Yn) -+ 0 in C as 11 -+ x. and so () is continuous. 0 Corollary 5.4.12 Let E be a Banach space such that E a Banach algebm.
~
E 8:l E. and let A be
(i) Each homomorphzsm from B( E) 0 A and from B( E) / K;( E) mto a Banach algebm zs autornatzcaliy contznuous, and each algebm nonn on B(E) zs equzvalent to the opemtoT norm. (ii) Suppose that E has BCAP. Then each homomorphism from K(E) 0 A into a Banach algebm is automatzraliy continuous, and each algebm nonn on K(E) is equivalent to the opemtoT norm.
Proof By 2.5.11. B(E) has a continued bisection of the identity. In case (ii), K(E) has a bounded left approximate identity by 2.9.37(i). So the results for B{E)@A, B(E), and K(E)0A follow from th(' th(,'Orem and 5.1.14. The result for B(E)/K;(E) then follows from 2.1.5. 0 It follows that each extension of B(E) which splits also splits strongly in the case where E is a Banach space such that E ~ E CB E. Corollary 5.4.13 Let H be a Hzlbert space. Then each homomorphism fmm B(H), f1'Ofn K;(JI). and from B(H)/K;(H) znto a Banach algebm is automatically contznuous. Proof If H is finite-dimensional, this is trivial, and, if H is infinite-dimensional, the result follows from 5.4.12. 0
Theorem 5.4.10 also leads to the following result about C*-algebras. For the sake of later applications, it is formulated in terms of bimodules and intertwining maps.
668
A utomatzc continuzty theory
Theorem 5.4.14 (Johnson) Let A be a closed zdeal in a C*-algebra B, let E be a Banach A-bzmodule, let F be a weak Banach A-bzmodule, and let T : E ---+ p be an intertwznzng map. Set
K = {b
E
B : AbA
C
I(T)} .
Then K zs a closed zdeal of jinzte codzmension in B. Proof Assume towards a contradiction that K has infinite co dimension in B. Then, by 3.2.25(ii), there exist (b n ) and (c n ) in B \ K with bmcn = 0 (m f:. n) and bnCn = Cnbn = bn (n EN). By 5.4.10, AbnA C IL(T) eventually. Similarly. AbnA C IR(T) eventually, and so Ab.. A C I(T) eventually, i.e., bn E K eventually, a contradiction. 0 Corollary 5.4.15 Let A be a closed ideal in a unital C*-algebra B. Suppose that each closed zdeal I of jinzte codzmension in B has an identzty for I. Then each homomorphism from A into a Banach algebra zs automatzcally continuous. Proof Let () be a homomorphism from A, and set K = {b E B : AbA C I(())}. By 5.4.14, K has finite co dimension in B, and so, by hypothesis, K has an identity for K. Thus K is a closed ideal in B, and K n A is a closed ideal of finite codimension in A. By 3.2.21(ii), K n A has a bounded approximate identity. and so it follows as in 5.4.11 that () I K n A is continuous. Hence () is continuous. 0 Let H be an infinite-dimensional Hilbert space. By 2.5.11, B(H) has no proper ideal of finite codimension, and so Corollary 5.4.13 for B(H) also follows from 5.4.15. Let B = f 00 (N, M n ), as in Example 3.2.30, so that B has exactly one simple representation of dimension n for each n E N. Then we proved that each closed ideal of finite codimension in B has an identity, and so it follows from the above corollary that each homomorphism from a closed ideal in B is continuous. Despite the intensive study of C* -algebras over the last decades, the following automatic continuity problem for C* -algebras is still open. Question 5.4.A Let A be a C* -algebra. Suppose that, for each n E N, A has only jinitely many simple representations of dimension n. Does it follow that each homomorphzsm from A into a Banach algebra is automatically contznuous? We shall see in 5.7.35 that (with CH) there is a discontinuous homomorphism from A whenever A is a C* -algebra with infinitely many distinct simple representations of dimension n for some n E N. Thus a positive answer to Question 5.4.A would give a characterization of those C* -algebras A for which there is a discontinuous homomorphism from A. Our main application of the main boundedness theorem in this section is in the study of the continuity of homomorphisms and other intertwining maps from regular Banach function algebras. The following theorem constitutes the first step.
669
The main boundedness theorem
Theorem 5.4.16 (Laursen) Let A be a regular, unital Banach functwn algebra on A, let E and F be unital Banach left A-modules, and let T : E ---+ F be a discontmuous left-intertwmmg map over A. Then the singularity set A(T) is a non-empty, finzte subset of A, and there exists a constant G > 0 such that
liT x
fgll:S Gllfll IIgil
(! E J(A(T)) n:h(T), 9 E J(A(T)))
.
(5.4.4)
Proof Set A = A(T), .I = J(A), and K = IL(T). By 5.3.29, A(T) is a nonempty, finite subset of A, and .I c K. We first claim that there is an open neighbourhood U of A and a constant G 1 such that fg E K and liT x fgll :S C 1 IIfllllgll whenever f, 9 E .I with supp f c U and supp 9 C U. For assume that this is false. Then we may choose inductively a sequence (Vn ) of open neighbourhoods of A, sequences (fn) and (gn) in .I and (Yn) in E such that, for each n E N, Wn C V nand Vn+ 1 n (WI U··· U W n ) = 0, where Wn = (supp fn) U (supp gn), and such that II(T x fngn)(Yn) II ~ n IlfnllllYnllllYnl1
(n
E
N).
Since f mgn = 0 (m f=. n), this is a contradiction of the main boundedness theorem 5.4.2 (applied with an = fn and Xn = gn . Yn). Thus the claim holds. Fix ho E A with ko = 1 near A and with supp ho C U, and set hI = 1 - k~, so that hJ E J. Take f,g E A. Then fh 1 E K, T X fh 1 : E ---+ F is continuous, and liT x fghlll :S liT x fhlllllgil. This shows that the bilinear map
- (f,g) T:
~
T x fgh!,
A (~
---+
B(E,F),
is separately continuow:;. By A.3.39, if is continuous: there is a constant C 2 such that liT x fghlll :S G2 11fllllgil (f, 9 E 1). Set G = G1 Ilhol1 2 + C 2 • Now take f.g E.1. Then fho,gh o E .1 with suppfho C U and suppgho C U, and so, by the claim, x fgh~1I :S G1 Ilfhollligholi. Thus
liT
liT x fgll :S liT x fg h 611
+ liT x
fghIil :S G 1I/IIIIgii .
Finally, consider the general case, where f E JnK and 9 E J. Take sequences Un) and (gn) in .J with In ---+ I and gn ---+ g. We have T x Ign ---+ T x Ig as n ---+ 00 because f E K, and also T x I mgn ---+ T x I gn as m ---+ 00 for each n E N because (gn) C K. We have shown that liT x Imgnll :S G Il/mllllgnll (m,n EN), and so liT x fgll :S G 11/1111gll. Thus (5.4.4) holds. 0 The following corollary is a special C8.'le of the theorem.
Corollary 5.4.17 Let A be a regular, unital Banach function algebra on A, let e : A ---+ B be a discontinuous homomorphzsm into a Banach algebra B, and set
A( e) = {cp
E A :
for each U E NIP there exists I E A with supp I C U and e x I discontinuous} .
Then A(e) is a non-empty, finite subset 01 CPA, J(A(e)) C I(e), and there is a constant G > 0 with lIe(fg) II :S Gll/llllgl1 (f,g E J(A(e))). 0
Automatic continllity theory
670
Let A, 0, and A = A(e) be as in the above corollary, and take K to be a compact subset of A with K n A = 0. Set LK = {J E A : supp f C K}. a clm;ed ideal in A. Then, by 4.1.18(i), there exists ho E J(A) with ho = 1 on K, and so !IO(J)II = Ile(Jho)ll:::; Cllfliliholl (J ELK)' showing that e I LK if:; continuous. Thus 0 is 'continuous on functions with support fixed away from A(e)'. To obtain the continuity of eon J(A(O)) itself require::; stronger hypotheses on A than regularity: see 5.4.20 and 5.4.22. Our first application of the above result is to obtain a result about the nonexistence of Wedderburn decompositions that extends 4.5.25. Lemma 5.4.18 Let A be a Ditkm algebm, and .suppo.se that E ~s a closed subset of A whzch zs of non-synthe.s~s and zs such that AI J(E) i.s decomposable. Then there i.s a compact .subset K of E which i.s of non-.synthe.s~s and is such that AI J(K) ha.~ a decomposztwn Q: EB J jor a closed .subalgebm Q: and a non-zem,
closrd ~deal J of AI J(K) wzth J C rad (AI J(K)).
Proof Set ~ = AI J(E), ~ = rad~, and let ~ have a Wedderburn decomposition ~ = 23 0~. Set A(E) = AII(E), so that 9 : ~ -+ A(E) is the Gel'fand tran::;form. The map 9 I ~ : ~ -+ A(E) is a continuous isomorphism; its (possibly discontinuous) inverse is () : A(E) -+ ~ C ~, and P = eo 9 if> the projection of A onto 23 with kernel~. Set A = A«(}), the singularity set of (), so that A is a finite subset of E. By 4.1.38, there is a compact set K of E \ A such that K is of non-synthesis and I(E) \ J(K) =I- 0. Take ho E Aoo such that ho = 1 on a neighbourhood of K in A and supp hll is a compact ::;ubset of A \ A, and set F = En supp ho, ao = ho + J(E), bo = Pao = eCho I E), ~o = AI.J(K), and Q: = 7l"(23) and J = 7l"(~), where 7l" : ~ -+ ~o is the natural epimorphism. We have llln = K. \Vf' shall show in a number of step::; that Q: and J have the required properties. Certainly, Q: is a subalgebra of ~o, J is an ideal of ~o contained in rad ~o, and ~ = Q: + J. Also J =I- 0 because I(E) \ J(K) =I- 0. First note that 7l"(bo) E Inv~o. Indeed 9(bo ) = g(ao) = ho I E, and so we have g(7l"(bo )) = ho I K = 1, whence 7l"(bo ) E Inv~o. Set L = {J
E
A(E) : supp Ie F} ,
a closed ideal ill A(E). By the remark aft£'r 5.4.17, () I L: L and so O(L) is closed in ~.
-+ ~
is continuous,
Take I E J(K). By 4.1.25, B(hol I E) = P (hoI + J(E)) E J(K)IJ(E). Since ho lEE Land B I L is continuous, it follows that
P (hoI
+ J(E))
E
J(K)I J(E)
(I
E
J(K)) .
(5.4.5)
For each I E Aoo, we have j - Iho E J(K), and so A(eA - h o) C J(K). By (5.4.5), P (hoU - Iho) + J(E)) E J(K)I J(E) U E A). Hence, for each b E 23, 7l"(bo )(7l"(b) - 7l"(bbo)) = 0, and so 7l"(b) = 7l"(bbo) E 7l"(8(L)), whence Q: = 7l"(8(L)).
The main boundedness the01'Cm
671
We clazm that 0 such that
liT x
fgll ~
Oll/lillyll
(J
E
J n K, 9 E J) .
(i) Take f E J. Since A is normal, there exists 9 E J with fg = f. and so T x f is continuous and f E K. The ideal J has an approximate identity of bound m, say, and so there exists h E J with Ilg - ghll < 1/(11911 + 1) alld IIhll ~ m. Since 1= fg(g - gh) + Ih, we have
liT x III
(11/IIIIg(g - yh)11 + 1I/IIIIhll) ::; 0(1 + m) 11III . taking C 1 = 0(1 + m).
~ 0
Thus (5.4.6) follows,
(ii) Since A is finite, it follows from the hypothesis that J has finite cois a dense subspace of the finite-dimensional space dimension in A. Since K K /J, we have K /J = K /J, and so there exist elements hI, . .. , h n E K such that K = ChI ffi·· . ffiCh n ffiJ. There exist Aj E (K)' for j E N n and a projection
/J
Automatic continuzty theory
672
P of K onto J such that f = P f + L:7=1 >"j (f)hj E J n K (f P(K) C J. Now take f, 9 E K. Since Pf, Pg E J n K, we have
liT x
Note that
(Pf)(Pg)1I :s; C IIPfllllPglI ,
and so it is easily checked that
liT x
fgll :s; C2 l1fllllgll, where
n
C2= C
E K).
11P1I2 + 211P1I L lI>"jllliT x
n
hJII
+
j=l
The general case of (5.4.7), in which
L
lI>"jll lI>"kll liT x
hJhkll .
j.k=l
f
E K and 9 E K, now follows.
0
The corollary below is a special case of part (i) the above theorem. Corollary 5.4.21 Let A be a r'egular, unztal Banach functwn algebra on A, and let 0: A -) B be a d~scontinuous homomorphism into a Banach algebra B. Suppose that J
.(fg). Thus>. I At = () I At. Since At is dense in A, >. is uniquely specified by this condition. (iii) Take f E A and 9 E .I. Then ()(f)>.(g) = ()(fg) = >.(fg) = >.(f)>.(g) , and so p,(f)>.(g) = O. Since>. is continuous, it follows that Il(A)>.(I) = 0, and so ",(A) . >'(1) = O. In particular, >.(ef)Il(f) = >.(ei)p,(f) (i E Nn , f E A). Now take f, gEl. Then
p,(fg) = ()(f)O(g) - >.(f)>.(g) = (>.(f) + p,(f»)(>.(g) + 11,(g» - >.(f)>.(g) = J.L(f)Il(g) , and so p, I I is a homomorphism. (iv) For i E N n , define Ili(f) = J.Lhf) (f E A). Then P,l, ... ,J.Ln : A --+ B are linear maps. Since 1 - L~l ei E .I and J.L I J = 0, we have p, = P,l + ... + J.Ln. Let t E Nn , and take f, 9 E lI-I"'i. Then ed, eig E I, and so
Jl(ed)p,(eig)
= p,(e~fg) = 11,(edg)·
Thus J.Li I M.pi is a homomorphism. There exists fa E A with supp fa C Wi such that p, x fa is discontinuous. Since fa - edo E .I, we have P,i x fa = p, x fa, and so J.Li =I- o. If f E .Iipi' then ed E .I, and so J.Li I .Iipi = o. Thus J.Li I M.p; is a radical homomorphism. Take i E N n and f E A. For j =I- i, we have >.(ej)>.(ed) = 0 because €iej = 0, and so >.(eJ)J.Li(f) = >.(ej)(>.(ed) + p,(ed» = O(eJ )8(ed) = O. Also >. (1 -
LJ=l ej) P,i(f) E >.(I)J.L(A) =
0 by (iii), and hence
n
P,i(f) = >.(l)P,i(f) =
L j=1
>.(ej)p,i(f) = >'(ei)P,i(f) = >.(ei)JL(J).
(5.4.9)
Automatic continu1ty theory
674
It follows that b = )..(ei)b (b E J.li(A)) , and hence (5.4.8) holds. In fact, we have
J.li(A) n
(ENi J.lj(A))
=
0, and so Ji(A) = EB~=l J.li(A).
(v) Since).. = ()-J.l is continuom.;, 6(0) = 6(J.l). Clearly J.l(A) = Ji(1) c 6(Ji}. Take f E I. say f = limfk where (Ik) C J. Then IL(f) = limIL(f - fk) E 6(Ji), and so IL(A) C IL(A) = 6(Ji). Similarly, Jii(A) C 6(IL;) = Ili(A) (i E N n ), and 1>0 6(Ji.) = EB7=1 6(IL,)· Since It = III + ... + /l'n, IL(A) C E~=l Jii(A) C radB, and hence IL(A) C rad B. ily (5.4.9). ker J.l = n~=l ker ILi. (vi) Let zEN... Assume towards a contradiction that IL;(M.",) C I)1(B). Take f E AIcp;. The ideal AIcp; has a bounded approximate identity, and so, by 2.9.24, there exist gEl and (hk) C I such that f = gkhk (k EN). Since ILi(g) E I)1(B), we have l1"i (f) = 0, and so ILi I M'P; = O. a contradiction. The result follows. This completes the proof of the theorem. 0 Corollary 5.4.23 Let A be a llnital strong Ditkin algebra, and let B be a (OTnmutatzve Banach algebra s1tch that rad B is nilpotent. Then each homomorphtsm from A into B is contmuous. Proof Assume that 0 : A ~ B iH a diHcontinuouH hOIllomorphiHm. ily 5.4.22. (v) and (vi), 6(0) C rad Band 6(0) \ I)1(B) #- 0. But I)1(B) = rad B. a 0 contradiction. 'vVe continue to use the notation introduced in the statement of Theorem 5.4.22, taking A = 0. At = A, and ).. = 0 in the case where (} iH continuous. Definition 5.4.24 Let A be a unital strong Dztkm algebra, and let 0 be a homomorphism from A into a commutative Banach algebm B. Then the unique ('ontznuo'us homomorphzsm. ).. : A ~ B ,mch that).. I At = 0 I At is the continuous part of and the map /l = () - ).. zs the Hingular part.
e,
It follows from 5.4.22 that there is a discontinuous homomorphism from A into a Banach algebra if and only if there iH a non-zero radical hompmorphism from a maximal ideal of A. Let us suppose merdy that A is a strongly regular. unital Banach function algebra on A, and that 0 : A ~ B is a discontinuouH homomorphiHm. Then A( 0) iH a non-empty. finite set. but the following example show::, that it is not necessarily the case that there iH a denHe subalgebra of A on which 0 is continuous.
Example 5.4.25 Let A = (£2.11·lb) and (2(, Iii· liD be a.."i in Example 5...1.6. TIH'1l A # iH a unital Ditkin algebra on N U {x}, and the natural embedding () : A# ~ 2(# is a discontinuouH homomorphism with Hingularity set {oo}. We claim. that 0 is very discontinuous. Indeed. assume towards a contradiction that B is an infinite-dimensional subalgebra of A with 0 I B continuouH. Then>. I B is 11·1I2-continuous, and so there exists a = (an) E [2 such that >.(b) = E~l f3n(~n for each b = (f3n) E B. In particular, 00
00
Lf3~an = Lf3~ n=l
n=l
(b= (f3n) E B, k ~ 2).
(5.4.10)
The main b01LudedTlcss theon'TTI
675
Since B is an infiuitp-dirneIlHioual suhalgphra. it iH easy to construct inductively pairwise disjoint spqncIlce (Sj) of finite subsets of N and a sequence (b j ) in B such that ,iij n = 1 (n E Sj) and 1.1),,.1 < 1 (11 E N \ Sj) for each .J E N. where bJ = (OJ.II)' Let j E N. By the dominat.ed convergence theorem applipd to (5.4.10), E{nn : n E S1} = E{l : It E Sj}. and so, by Hulder's irH'quality. E{lan I2 : n E S/} 2: ISjl. But this contradicts the fa.ct that (1 E £2. and so tlw claim is establbhed. 0 3
The information contained in 5.4.22(vi) is sufficient to allow us to deduce a striking t.heorem of Esterle on the automatic continuity of epimorphisms from the algebras C(n).
Theorem 5.4.26 Let A be a ltmtal strong Dztkm algebra. Suppose that there is a dt8conlwuolts epZrn071Jhism Imm A onto a Banach algebm. Thm there is a non-maxzmal. pnme ideal P In A such that thf set 01 prime 1,deals zn AlP zs not a cham with mspect to mclnswn.
Proof Lpt () : A ~ B be a discontinuous epimorphism from A onto a Banach algebra B. By 5.4.22(vi), there exists 10 E A with (}(fo) E 6(0) \ I)1(B). By the commutative prime kprnel theorem 5.3.15. there is a closed ickal K in B such that the map (f : I 1-+ ()(f) + K. A -+ B I K, is a diocontiunouo epimorphism with (f(fo) =1= 0 and with ker (f a. primp ideal in A. Set P = ker (f, 6 = 6«(f). and R = rad (D I K). and takp l\I to be t h{' unique maximal ideal in A with Pc AI. Since AlP ~ BIK. we have }lflP ~ R. Set 00 = (f(fo), so that flO E R\ I)1(R). By the stability IpIlllna 5.2.5(ii), there £'xists N E N such that 0 06 = a{'f 6 for each n 2: N. Take a = a~+J. Then a tJ- I)1(R), and a E (L~' 6 c (16 c aR. By 4,8.2(i), the set of prim£' ideals in R, ordered by inclusion. is not a chain, and so the same is tmp of lUI P. and }H'nce of AlP. 0 Theorem 5.4.27 (Esterle) Let n be a non-empty, compact space. Then mcl! epimorphz8rn from c(n) onto a ilana('h algebra lS a1Ltomatu:ally contut'ltous. Proof By 4.8.18, the set of prime ideals in C(n)1 P is a chain for each prime ideal Pin C(n). Thus the result follows from 5.4.26. 0 Theorem 5A.22 also Ipads to a condition involving divisible subspacps which ensureo the automatic continuity of a homomorphism from A. Let B be a commutatiw, unital Banach algebra, and let bED. H£'('all that Lb : .r. 1-+ hJ; is the left regular repreo('ntation of b on B: by 1.5.29(iv), O'(b) = O'(L/,). The algebraic spectral space ELb(0) was defined in 1.4.21. A subspace D of B is C[bJ-dwiszble if (zeB - b)D = D
(z E
q,
(5.4.11)
and EL/,(0) is the maximum ouch C[bJ-divisible subopacej it is now denoted by Db. Note that, if Db is closed in B, then Db = O. Aloo, D satisfies (5.4.11) if (zen - b)D = D (z E O'(b»; in particular, if bE rad D, then Db is the maximum linear subspace D of B with bD = D. Clearly Db is an ideal in B for each b E B, and Db C radB. Indeed, let x E Db. For each
a discontinuous homomorphism such that ker Ii is a prime ideal in AI. Spt P = ker ji. TlwIl P ha.s the required properties. 0
To further analyse discontinuous homomorphisms froUl algebras CeO). it is now convenient to consider a locally compact, non-compact space n and a nonzero radical homomorphism from CoCO). As in §4.2. Co(n) is a closed ideal in C(80): for a closed subspace F of (3n, J(F) and I(F) are ideals in C(f:lO).
Theorem 5.4.33 Let 0 be a locally compact. non-compact space, and let II be a non-zero homomorphism from Co(n) mto a radical Banach algebra R.
(i) T1U'.7'e is a non-empty. finzte subset
A = {'ttl!, ... , 'Ibm}
in j30 \ 0 such that
It I Co(O) n J(A) = O. (ii) There exist non-zero radical homomorphisms JLl, ... , JLm : CoCO) ~ R such that JL = /11 + ... + Ilm, such that p,j I CoCO) n J.p, = 0 (j E N m ), and such that ker /-L = n~:l ker /-L3 . Proof Set A = cCj3n), 1= CoCO), and K = {f E A : II
c IC/-L)}.
679
The mazn boundf;d1u'ss theorem
(i) By 5.4.14, K has finitE' co dimension in A. DefinE' Ato be the hull of Kin pO, so that A is a fillite set. say..::\ = {l/Jl ... .. 'If'm}. Take f E 1 n .J(A). Thell / == /li2 for some fl. h E In .J(A), and so
[n .J(A) c
I.J(A) c
[K c I(/I).
(5.4.12)
By 5.4.31(v), III I(I1) is continuous. and so, by (5.4.12), 11 I In.J(A) is continuolls. By 4.1.37, III Coo(O) = 0, and so 11 I £ (U) = 0 for each U c n (we note thE' definition of £ (U) given in 5.3.26). Thus A c nand /11 In .J(A) = O.
an \
(ii) Take Cl, •..• em E Cum) such that Ci = 1 near Wi in rm for i E N m and eiej = 0 (i -=I j). Define Il,(f) = l1(ed) (f E J). Then 11 = I1t + ... + 11m. and each /1i : J -> R is a non-zero homomorphism with /1i I [n .Jw; = O. For f E I. we have /1i(f)I1J(f) = 0 (i -=I j), and so /1i(f),l(f) = ILi(f)2. Thus it follows that ker /1 =
n;:1 ker ILi, as required.
0
We conclude our description of an arbitrary homomorphism e from C(O) hy describing more prech:iely the ideal I(O), which we recall is the maximum ideal L in C(O) such that 0 I L is continuous. The result follows from the above theorems. Theorem 5.4.34 (Johnson) Let 0 be a non-empty. compact space, and let () be a discontmllous homomorphzsm from C(O) into a Banach algebra. Then there are a non-empty, finzte subset At of n and a non-empty. finite subset A2 of (3(0. \ At) \ (0 \ At) such that I(O) contams the l,deal
{f where
f
E C(O) :
f = 0 on a neighbourhood of A2 in B(n \ AI)}.
denotes the Stone extenszon of f
I (0 \ Ad
to {3(0 \ AI)'
o
As yet, we do not know that there is a discontinuous homomorphh;m from any C(O); the original aim of Bade and Curtis was to prove that all sueh homomorphisms arc continuous b,\' following the conclusions of the above theorems to a contradiction. In fact, w(' shall prove in §5.7 that (with CH) then' are discontinuous homomorphisms from C(O) for each infinite. compact space 0, and that Theorem 5.4.34 is best-possible. It is a natural conjecture that the ideal I(O) is always a finite intersection of prime ideals. This is an open question for an arbitrary space O. hut it is true for some special spaces O. Theorem 5.4.35 (Esterle) Let n be a non-empty, r:ompact space, and let 0 be a discontzntwus homomorphism from C(O) into a Banach algebra. Suppose pit her that n zs an F -space. or that (J(O \ {'P}) zs an F -space for each non-P-point 'P of n. Then I(O) i'J a finite intersection of prime ideals zn C(H). Proof First, suppose that 0 is an F-space, so that, by 4.2.18(ii), J
. + /Ll + ... + /1n, with the notation of 5.4.22, so that Iti I M 1/2 eventually, we have .B(n) < a(n)1/2 eventually, and so {3 E cjj. Also. a/ /p2 {3 = 0 in Co, and so aflp2 Q. Now take [fl E N(p}/V, and set v([f]) = [of /9 2 ] E co/Q.
tt
Clearly v([f]) is well-defined and v([f]) E (co/Q)-. Take [hl T(b)(n) eventually. Thus T(b) F, is continuous. Clearly L is an ideal in A. and K c IdT) c L. Let C be a unital, commutative C* -subalgebra of A. By 5.4.16 and 5.4.20(i), there is a finite SUbSl-t A of c1?c and a constant AI such that
II(T
x
f)(J:)II ::::; M IIfil 11:1;11
(f E J(A), :1" E E).
Now take g E J(A) and;]' E G D, say.r: = L~=l al ..... ak E K. For each c > 0, there exists have
For j
E
where .Tl,"" J(A) with IIf -
Xj • oJ'
f
E
Xk
E E and
gil
T(a . x), L x Go ---4 F. is contiuuous, and so there is a continuous, bilinear map T : (a, x) f---> T( a . x). L x G ---4 F. We note that T(ab,x) = T(a, b . x) (a,b E L, x E G). (5.4.14)
f
E B(L i§ G, F) and S E B(L ®G, E) such that r(a ® x) = T(a,x) and Sea '& x) = a . x for a ELand x E G. Since kerS is a. closed left L-submodule of L ®G, each Z E ker S can he written in the form z == a . y for some a ELand y E kerS, say y = E~1 bj 0Yj. where (b j ) C L and (Yj) C G with E~l lib) lilly) II < oc. We have
By A.3.69. there exist maps
f(.)
~f
(t,
ab; 0 Y;)
~ ~ flab;, Y;) ~ = T (a.
t,
f(a, b; . Yj)
by (5.4.14)
fbj . Yi) = T(a.Sy) = 0, )=1
and so ker S C ker f. Since L has a bounded approximate identity. S is an open surjection onto L . G by 2.9.30(vi), and so it follows from A.3.1O(i) that there exists R E B(L . G. F) with R 0 S = f. Let a E L. x E E, and bE K. Then .1: • bE Go and T(a . x . b)
= T(a.
x . b)
= f(a
~ x . b)
= RS(a ®:r
. b)
= R(a
.
T .
b).
Thus T agrees with Ron LEK ::) KEK. If a E K. then T x a is continuous. If also x E E and b E K, say b = limn_co bn with (b n ) C K, then T(a . x . b)
=
lim (T x a)(x . bll ) = lim R(a . x . bn ) n~rx::
n-+oo
= R(a
. x . b),
and so T agrees with Ron K EK. We conclude that T agrees with the continuous linear map R on the specified linear subspace of E. 0 The following theorem is a non-commutative analogue of 5.4.22 and 5.4.31. Theorem 5.4.40 (Sinclair) Let A be a umtal C* -algebra, let B be a 'umtal Banach algebra. and let 0 : A ---4 B be a nnttal homomorphism 'Wzth O(A) = B. Then the follo'Wzng results hold. (i) Y(O) is an ideal m A, and Y(O) has finite codm~ension in A. (ii) For each a E A and n E N, O(a) E O(a*a)nB and B(a)B = """OT(a::-*-a")1::-tB:;::::; if 9(a*a) is nilpotent, then 9(0.) = O. (iii) There is a finite-dimensional subspace F of A and a continuous homomorphism A : A ---4 B such that A = Y(O) ffi F, Y(O) 0 F is a dense subalgebra of A, and A I I(9) 0 F = 0 I I(O) 0 F.
A utomatic continuity theory
684 (iv) Set I-"
= () -
A. Then: 11,(A) . A (I«(}))
= A (I(O))
. I-"(A)
= 0; I-" I I(9)
is a homomorphism; I-"(A) c 6«(}) = 6(1-") = I-"(A); and there exists a E A such that I-"(a) E 6«(}) \ ')1(B). (v) A(A) ~s closed ~n B, and B = A(A) EB 6«(}). (vi) radB C 6«(}). (vii) ker(} is a *-~deal which ~s an mtersectwn of pnme ~deals in A. (viii) I«(}) ~s the maximum ideal L m A such that () I L zs continuous. (ix) I(O) = I(O) n ker 1-", and I«(}) ~s an intersection of prime ideals m I«(J). Proof Set K = I«(}), 1= K, 6 = 6«(}), and R = radB. (i) As noted in 5.4.39, this follows from 5.4.14. (ii) Take a E A and n E N. Thcn «a*a)'" : a E ~+.) is a semigroup in A, and (a*a)'" A = (a*a),6 A (a, f3 E ~+.) by (3.2.4), and so, by 5.2.11(ii). we have O«a*a)l/3)B = B(a*a)nB. By the generalized polar decompo!)ition 3.2.19, a E (a*a)1/3A, and so O(a) E (}(a*a)nB and B[OJli = O(a*a)nB. (iii) For each a E A. we have (J«a*a)o.)6 = O(a*a)6 (a E ~+.) by 5.2.11(ii). Thus, if a E K, then (a*a)1/3 E K and a E I(a*a)1/3 elK, so that K = 1K. By 5.4.39, () I K is continuous. By 4.2.5, there is a finite-dimensional subspace F of A such that A = I EB F and K 0 F is a dense subalgebra of A. Clearly () I K 0 F is continuous; take ,\ to be its continuous extension to A. (iv) Thc!)e results are proved as in 5.4.22, (iii), (v), and (vi). (v) By 4.2.4(ii), A(A) is closed in B. By 5.4.30(i), B = (}(A) + 6, and so
B = A(A) + I-"(A) + 6 = A(A) + 6. Since A(I) is a closed ideal in A(A) and since A(I) . 6 = 6 . ,\(1) = 0 by (iv), we see that A(I) i::; a closed ideal in B. Let n : B ~ BIA(I) be the quotient map. We claim that A(A) n 6 c A(I). For otherwise 6(n 0
(})
n n(A(A))
=
n(6) n n(A(A))
is a non-zero ideal in the finite-dimensional C*-algebra n(A(A)). Since n(A(A)) is semisimple, 6(n 0 (}) n n(A(A)) is semisimple, and so contains a non-zero idempotent p by 1.5.1O(i), a contradiction of 5.1.3(iii). Thus A(A) n 6 C A(I), as claimed. Set L = A (K) n 6. Then L i::; a C*-algebra, and !)o L = L[2] by 3.2.21(iii). But A (K) . 6 = O. and so L = O. Thus A(A) n 6 = 0, and B = A(A) ED 6. as required. (vi) Since BI6 ~ A(A) and A(A) is semisimple, R C 6. (vii) That ker () is a *- idcal is immediate from (ii). Let L be a nil ideal in (}(A), and take a E A with (}(a) E L. Then (}(a*a) E L, and !)o O(a*a)n = 0 for some n E N. By (ii), (}(a) = 0, and so L = O. By 1.5.25, {O} is an intersection of prime ideals in (}(A), and !)o kerB is an intersection of prime ideals in A. (viii) By (iii), () I K is continuous. (ix) This follows essentially as in 5.4.31(vi). 0
The main boundedness theorem
685
Again, A and J1. are, respectively, the contznuous and szngular parts of ().
Theorem 5.4.41 Let A be a strong Dztkin algebra or a C" -algebra. Then every Banach extension of A which splits also splits strongly.
Proof We may suppose that A is unital. Let L:O-I-Q(~A-O be a Banach extension of A which splits: there is a homomorphism () : A ...... Q( with 7r 0 () = 1.4. By 5.4.22 (for A a strong Ditkin algebra) or 5.4.40 (for A a. C*-algebra), there is a continuous homomorphism A : A ...... Q( such that (0 - A) (A) c 6(()). Take b E 6(()), sayan ...... 0 in A and O(an ) ...... b in Q(. Then 7r(b) = limn---+oca n = 0, and so 6(()) c ker7r. Thus 7r 0 A = lA, and L splits strongly. 0
Thu.c; a Banach algebra A such that A/rad A is a strong Ditkin algebra or a
C" -algebra has a strong Wedderburn decomposition whenever it has a Wedderburn decomposition. One part of Theorem 5.4.22 that has not been generalized to an arbitrary C*-algebra is the fact that 6(()) c radB, in the above notation. To establish whether or not this is necessarily the case is equivalent to resolving the following question, which is a special case of Question 5.l.A.
Question 5.4.C Let A be a C* -algebra, let B be a semzszmple Banach algebra, and let 0 : A ...... B be a homomorphism with O(A) = B. Is () automatically continuous?
A further special case of the above question is the following.
Question 5.4.D Let A and B be C* -algebras, and let () : A ...... B be a homomorphzsm wzth ()(A)
=
B. Is () automatzcally continuous?
Assume that there is a discontinuous homomorphism () : A ...... B such that O(A) = B, where A and B are both C*-algebras. Then, in the notation of 5.4.40, there is a homomorphism p, : I ...... 6(p,) such that J1,(I) = 6(p,). By 5.1.9(ii), p,(I) c .Q(6(p,)). Thus 6(p,) is a non-zero C*-algebra with the bizarre property that it contains a dense subalgebra consisting entirely of quasi-nilpotent elements. No such C*-algebra is known. A further part of 5.4.22 that has not been generalized to an arbitrary C*algebra concerns the splitting of the singular part of a homomorphism, as in 5.4.22(iv). We explore whether this is possible. Let A be a unital C* -algebra, let B be a unital Banach algebra, and let 0: A ...... B be a unital homomorphism with O(A) = B, and again set K = I(()) and I = K and take>.. and p, to be the continuous and singular parts of (), respectively. Then p, : A ...... B is a linear map such that p, I I is a homomorphism and p,1 K = O. Since I has finite codimensioD in A, there exist distinct maximal ideals M I , ... ,Mk in A such that 1= MI n··· n Mk. Set Qi = nf;6i P(Mj ) (i E Nk).
686
A utornatzc continmty theory
(Here P(Mi ) denotes the P('del'sen ideal oftho C*-algehra lIfi') By 4.2.5(i), there exiHt P.1, ..•. ell E Asa such that f' A - (e1 + ... + en) E K, such that Ci('J ::::: 0 for 1.) E Nk with i #- J, and snch that ei - e? E P(.U.), (';A - f'i E: P(M;). and. ei E Qi for i E Nk. For i E N k , cif'fine fJ'i(a)
= f.l(e;a)
(a E A).
Then each f.l; : A ...... B iH a linear map, and JLI + .,. + II" = f.l. Let 1 E N k . For each a E A. we have eia E Qi and eA - ei E P(}.Ii ), and so eia - ciaei E Q;P(M;) C P(Mi ) n Qi. Let us make the special hypothesis that (5.4.15) Then eia - e,ae; E P(I) fJi(ab)
c
K. It is always true that e;lIf; C Qillfi C I, and so
= fJ(e;ab) = fJ(ciaeib) = f.l(e;a)fJ({';b) = f.li(a)f.l.(b)
(a, bE Mi).
This shows that f.li l.Hi is indeed a homomorphism Huch that fJi(P(M i )) = {O}, and so, hy 3.2.27(iii). 11i(lIfi) c O(B). It follows aH in 5.4.22(iv) that f.li(A) . f.lj(A) = Ili(A) n Ilj(A) = {O} (i #-.j) and 11(A) = E9:~1 f.li(A}. Thus we have tIl(' required splitting of f.l. An example in which tho spC'cial hypotheHis (5.4.15) fails and there iH no analogous Hplitting of f.l will be given in 5.7.36. It if! also not known whether the analogue of 5.4.27 holds for all C* -algebras. Question 5.4.E Is every elnrnorphism fT'Orn a C* -algebra onto a Banach algebm alltomatically continuolls? The non-cornmutativp analogue of 5.4.32 is the following. Theorem 5.4.42 (Cusack) Let A be a C*-a.lgebm. Assllme that there is a dzs('ontzn1tOUS homomorphzsm (J from A mto a Banach algebra B. Then there zs a closed zdeal I of fimte codimcnsion in A and a dense prime ·tdml P in I such that there is an embedding of liP mto 6((J)IL for some closed idf'.al L zn 6((J). In partzculaT", liP is n07mable. Proof We may suppose that A, B. and (J arc unital. Set I = K, and consider the homomorphism f.l : I ...... B, in the notation of 5.4.40. By 5.4.40(vii), kefJl is an intersf'ction of prime ideals in I. Take ao E 1\ ker f.l. Then f.l(ao) E f.l(A) c 6(0) = 6(IL) = IL(A), and so the result follows from the prime kernel theorem 5.3.21. 0 Theorem 5.4.43 (Sinclair) Let A be a C* -algebm, and let (J be a homomorphism from A into a Banach algebra B. Suppose that (J I Cci(a) is continuous for each a E: Asa. Then 0 is continuous. Proof We may suppose that A, B, and 0 are unital, and that O(A) = B. We adopt the notation of the proof of 5.4.40, so that we have a closed ideal I = K of finite codimension in A and a homomorphism f.l : 1 ...... B with K C ker f.l. Take a E 1&&, and set C = C*(a). Then A and B are Banach left C-modules and 0 : A ...... B is a left-intertwining map over C, and so, by 5.4.20(i), there is a
'the main boundedness the01-em
687
finite subset A1 of cI>c such that J(Ad C Id(}). Similarly, there is a finite set
A2 in cI>c such that J(A2) c IR«(})' It follows that
I e). JL I e is continuous, and so I(A1 U A2 ) c ker(JL I e).
J(A1 U A2 )
c I L «(}) nIR«(}) n e = K
nee ker(JL
By hypothesis, This shows that ker JL n C* (a) is closed and has finite codimension in C* (a) for each a Elsa' By 3.2.24, ker JL is closed in I, and so JL I I = O. Thus () is continuous on the closed ideal I in A. Since I ha." finite codimcnsion in A. () is continuous. 0 Corollary 5.4.44 Assume that ther-e is a dzscontzn1lous homomorphism from some e* -algebra into some Banach algebra. Then there is a discontinuous homomorphzsm from Co znto some Banach algebra. Proof This follows from 5.4.37(i) and the theorem.
o
We now turn to the application of the above theory to the algebras E N. It was shown in 4.4.1 that c(n) is a natural, regular, unital Banach function algebra on 1I, and so Theorem 5.4.16 applies with ern) for A. Let to E 1I. The family {Mn.o(to), ... , Mn,n(to)} of all the closed ideals .J of e(n) with ~(J) = {to} was defint.' n, and refer to continuity with respect to the norm II· Ilk on e(k). To simplify some proofs we shall suppose that the singularity set A(T) of a map T is a singleton. This involves no loss of generality, for, in the case where A(T) = {t1, .... t m }, choose h, ... ,fm E e(oo) with fJ = 1 near tj and fiij = 0 (i =I- j), and set fo = 1 - 2:7=1 k Then T is continuous on Subalgebras containing e(oo) n J(A(T» if and only if each T x fi is continuous, and A(T x f) = {t)}. For convenience, we shall suppose that A(T) = {OJ. We recall that the algebra (An' III· II In) was defined in 4.4.3, that the map / f-+ zn /, An -; !vIn,n, is a linear homeomorphism, and that An contains the specific bounded approximate identity (1/;" : c > 0) defined in §4.4. We shall frequently use the results on the structure of ern) given in 4.4.5. We first obtain an analogue of 5.4.20(i).
688
Automatic contznuity theory
Theorem 5.4.45 Let n E N, let E be a unital, weak Banach c(n) -module, and let T : c(n) --+ E be an intertwining map. Then there is a constant C 1 > 0 with
liT x III ~ C1 1I/I12n and T
I c(2n) n J(A(T))
(f E c(2n) n J(A(T))) ,
is 11·1121l-continuous.
Proof We suppose that A(T) = {O}. Take f E c(2n) nJn(O). Then there exists > 0 such that 'l/Jd = f. Set g = f/Z n . By 4.4.7(ii), there is a constant mn,n such that Ilglln ~ m n .n IIfll 2n , and so it follows from (5.4.16) that C
liT x III ~ C Ilglln Ilzn"pclln ~ Cmn,n IIzn"pclln IIfl12n .
o
By 4.4.4, (zn"pc) is bounded in c(n), and so the result follows.
Let () : c(n) --+ B be a discontinuous homomorphism into a commutative Banach algebra B with A«()) = {O}. The above theorem implies that () I M 2n ,2n has a splitting analogous to the one described in 5.4.22. Indeed, there is a unique 11·1I2n-continuous homomorphism A : M 2n ,2n --+ B such that
A I M 2n,2n n In(O) Set Jl = 8 - A. Then Jl : M 2n .2n
6(8)
--+
= 8 I M 2n,2n n In(O) .
rad B is a homomorphism,
= 6(Jl) = Jl(M2n,2n) C
radB,
and 6(8) . A(M2n ,2n) = O. We shall now analyse further the automatic continuity of maps from c(n) when restricted to subalgebras C(k). The next lemma is the key tool; parts (i) and (ii) are related to the semigroup stability of 5.2.10. Lemma 5.4.46 Let n E N, and let E be a umtal, weak Banach C(n)-module. Suppose that T : c(n) --+ E zs an intertwining map with A(T) = {O}, and take k? n. (i) Suppose that 10 E Mn,n. Then Za fo . 6(T) = Z 810 . 6(T) (a,/3 > 0). (ii) For each a,/3 > 0, we have Zll+o . 6(T) (iii) Suppose that T
I Afk,n
=
zn+fj . 6(T).
is 11·llk-contznuous. Then Zk+l E I(T).
(iv) Suppose that Zk E I(T). Then T
I Mn+k,n
is 11·lln+k-continuous.
Proof (i) For a> 0 and I E Afn,n, set Ral = Zo 10 . TI, Sal 6 0 = 6(Ro). For each a > 0 and 'Y ? 0, we have Z'Ilo . So - Ro+"Y = Z"Y . (fo . T - T x fo) - Z"Y . (Zo: 10
.T
= T(ZO I), - T x
and
zo: 10),
and the right-hand side is continuous because T is an intertwining map. Thus 6 0 +"Y = 6(Z"Y 10 . So)· By 5.2.2(ii), 6 0 +"Y = ZO+"Y fo . 6(T) = Z"Y fo . 6(So)' Take a > 0, and choose a sequence (Cj) in R+· such that ~~1 Cj < a. By the stability lemma 5.2.5(ii), there exists mEN such that 6(SoJ = 6(S02)' where al = Cl + ... + Cm and a2 = Cl + ... + Cm+l, so that 0 < al < a2 < a. It follows that 6 01 +"Y = 6 02 +"Y (-y ~ 0).
689
'the main boundedness theorem
Consider the claim that, for each mEN, 6",,+ m ("'2-"")+1' = 6""+1 b ~ 0). The claim holds for m = 1. Assume that it holds for m = k. Then, for each '1 ;?: 0, we have
6",,+(k+l)("'2-"'!l+1' = 6""+"'2-""+1' = 6"'2+1' = 6",,+1" and so the claim holds for m = k
+ 1.
Thus the claim is established for each
mEN. It follows from the claim that 6", = 6f3 for each (3 ~ Q, and this implies (i). (ii) Take Q, (3 > 0, and choose c > 0 with c < Q. Then zn+e E Mn,n, and so (ii) follows from (i). (iii) We first claim that Mk,n n I(T) is closed in (Mk.n, II· Ilk)' For let (fJ) be a sequence in I(T) with Ii ---+ fo in Mk,n' By (5.4.16) and the fact that T x 9 - 9 . T is continuous, we have II(T x fo)(g)ll::; II(T x 9 - 9 . T)(fo -
Ii) II + IIg
. T(fo -
::; liT x 9 - 9 . Til lifo - fJlln + IIglin liT I Mk,nllilfo C lifo lin IIglin as j ---+ 00,
Ii) I + IIT(lig) II fJllk + C II Ii lin IIglin
---+
and so fo E I(T), giving the claim. It follows that Mk,n n I(T) :) Mu, and hence Zk+l E I(T). (iv) Let mn.k be the constant specified in 4.4.7(ii). Take f E Mn+k,k' and set g = f /Zk. By 4.4.7(ii), IIglin ::; mn,k IIfll n+k , and so
IITfll
=
II(T x Zk)(g)11 ::; liT x Zkllllgil n ::; mn,k liT x Zklillflln+k ,
showing that T I Mn+k,k is 1I·lIn+k-continuous. Since Mn+k,k is closed and has finite co dimension in Mn+k,n, the result follows. 0 Definition 5.4.47 Let E be a Banach space. A linear map T : c(n)(JI) ---+ E is eventually continuous zf there eX'tsts k ~ n such that T I C(k)(JI) is II· IIkcontinuous, and T is permanently discontinuous if no such k exists. Suppose that T is an intertwining map with A(T) = {a}. Then, by 5.4.46, T is eventually continuous if and only if there exists kEN such that Zk E I(T). Theorem 5.4.48 (Bade, Curtis, and Laursen) Let n E N, let E be a unital, weak Banach C(nLmodule, and let T : c(n) ---+ E be an mtertwtning map with A(T) = {h, ... , t m }. Then the following conditions are equivalent: (a) T is eventually continuous; (b) T I c(2n+l) is 1I·lbn+l-continuous; (c) I(T) contains the polynomial I1j:l(Z - tjl)n+l. Proof We suppose thatA(T) = {a}. The implication (b)=>(a) is immediate. (a)=>(c) Suppose that T I C(k) is II· Ilk-continuous, where k ~ n. By 5.4.46(iii), Zk+1 E I(T), and so, by 5.4.46(ii), zn+l E I(T). (c)=>(b) By 5.4.46(iv), T I M2n+1 ,n is 1I'112n+ccontinuous; it follows that T I c(2n+1) is 1I·1I2n+Ccontinuous. 0
690
A utomatzc contmuzty theory
Let B be a commutative Banach algebra, and let b E B. Recall from that We write Db = ELb(0) for the maximum C(b]-divisible subspace of B. Theorem 5.4.49 (Bade, Curtis, and Laursen) Let n EN, and let B : c(n) -+ B be a discontmuous homomorphism mto a commutative Banach algebra B with A( B) = {h, ... , t m }. Then the Jollowmg condztions are equivalent: (a) B is eventually continuous; (b) B I c(2n+1) is II· 112n+ccontinuous; (c) I(B) contams the polynormal (Z - tj 1)n+1;
n;:l
(d) Do(z) = 0; (e) 6(B)3 = O. Proof We may suppose that Band B are unital. We suppose that A(B) = {O}, and set 6 = 6(B). By 5.4.48, (a){:}(b){:}(c). (c)=}(d) Set D = Do(z) and E = B(zn+1)B. Then DeE. We claim that the map J I-t LoU) I E, c(n) ~ B(E) ,
e:
is continuous. For let Jk ~ 0 in c(n) and 8(fk) ~ R in B(E). For each b E B, we have 8(fk)B(zn+l)b = (B x zn+1)(fk)b ~ 0 because Zn+l E I(B) by (c), and so R = 0 and e is continuous. But now e is a C(nLfnnctional calculus for e(Z), and so D = 0 by 5.3.39. (c)::::}(e) By (c), zn+lJ1,1n ,n C I(B), and so znMn,n c I(B) by 5.4.46(i), whence M;,n C I(B) by 4.4.5(iv). Take bl , b2, b3 E 6, and choose sequences (fk) and (gk) in co(N, Mn,n) with B(ik) ~ b1 and B(gk) -> b2. Since (fkgk) C T(O), we have B(fkgk)b3 ~ 0, and so b1b2b3 = O. Thus 6 3 = O. (d)::::}(a) and (e)::::}(a) Set () I M 2n .2n = ,\ + p" in the notation above 5.4.46. Since A(M2n ,2n) . 6 = 0, we have L o(z2"+1) I 6 = Lf.1.(Z2n+l) I 6. We first show that both (d) and (e) imply that D = 0 for each C(()(Z2n+l)]divisible subspace D of B for which B(Z)D cDc 6. This is immediate from (d), and (e) impli(>s that D
= ()(z2n+l)D = Il(z2n+l)D C
6 . Dc 6
3 •
D
= O.
Now define
D = ~{p,(Zk)6 : k ~ 2n + I}. Then ()(Z)D cDc 6, and D = p,(Z2n+1 )D, so that D is C(J1,(z2n+l )]-divisible. and hence C(B(z2n+l )]-divisible. Thus D = o. But, by 5.2.11(i). there exists N ~ 2n+ 1 with D = p,(ZN)6 = ()(ZN)6, and so ZN E I(O), giving (a). 0 We now investigate when the exponent 2n + 1 in 5.4.48(b) and 5.4.49(b) can be replaced by 2n. Theorem 5.4.50 (Bade, Curtis, and Laursen) Let n E N, let E be a weak Banach c(n) -module, and let T : c(n) -> E be an intertwining map such that A(T) = {tl,.'" t m }. Then the following conditions are equivalent: (a) I(T) is closed in (c(n), II . lin);
The maw, boundedness theoTern
G!)]
(b) I(T) ha..~ finite wdtmenswn in e(n);
(c) I(T) contazns the polynomial rr~:l (Z - tjl)n. These conditions imply that T I e(2n) zs II . 112" -continuous.
proof We suppose that A(T) = to}. By 4.4.5(iii), (a)::::}(b). (b)::::}(c) Set L = I(T) n M n .". We first clazm that AnL C L. Certainly AnMn,n C M".n' Now take f E An and gEL. Then, for hE 111n.", we have II(T x fg)(h)11
=
I!(T x g)(fh)11 :::; liT x glillfhlin :::; I H --> HIE --> 0 is admissible. Since A is amenable, it follows from 2.8.60 that L splits strongly, and so there exists P E AB(H) with p2 = P and P(ll) = E. Let f E A. We have PL f = LfP. Since Lj = Ly , L f is a normal operator in B(ll) and, by 3.2.5, PL7 = LyP. Thus {f E C(n) : PLf = LfP} is a closed, self-adjoint subalgebra of C(n) containing A, and so, by the Stone-Weierstrass thporem, it is equal to C(n). Let f E C(n). Then P(f) = (PLf )(1) = (L f P)(l) = Lt(l) = j. and so fEE. It follows that there is a sequence (gn) in A with gn -> f in H as n ~ x. But now
lin
f d/{
=
lin
as 71 ~ r:x::, and so This proves that A
(f - gn) d/{
If) f dIL = = C(O).
Corollary 5.6.3 Let bu.t not bzprojective.
n
::;
(In
If - gnl 2 d IILI) = Ilf -
O. This holds for each
f
E
gnll~ ~ 0
C(n), and so II = O. 0
be an infinite, compact space. Then C(n) is amenable,
Proof This follows from 2.8.42 and the above theorem.
0
Recall from 2.8.74(i) that, for a commutative, amenable Banach algebra A, 1{l(A, E) = 1{2(A, E) = {O} for each Banach A-module E. Thus, for the algebras A = Co(n) and A = (l(G) specified in 5.3.6, we obtain the further fact that 1t 2(A, E) = {O} for each Banach A-module E. Thus commutative, singular, admissible extensions of Corn) split strongly; the following theorem gives a stronger reslllt. We recall from 2.8.49 that, in diHtinction to these facts, for each infinite-dimensional Banach function algebra A, there is a Banach Ahimodule E with 1{1 (A, E) =I {O}, In Example 5.4.5(i), we exhibited a commutative extension of Co which does not split; by 5.4.41, every extension of Co(n) which splits also splits strongly. Theorem 5.6.4 (Bade, Curtis, and Sinclair) Let 0 be a locally compact space. Then each commutative, nniformly radzcal extension of CoCO) splits strongly. Proof Let A be a commutative Banach algebra with radical R such that R is uniformly radical and AIR = CoCO). vVe may suppose that 0 is a compact space and that A is unital. The proof of this theorem is in two parts: in the first part we prove that R is complemented in A. There exists c > 0 such that Ilfllq :S c If If) (x E AIR), where 11·ll q denotes the quotient norm on AI R. Let K be the constant specified in 2.4.32 corresponding to c + 1. For each f E C(n, lR), we have lexp(itJ)ln = 1 (t E lR), and so,
Continuou,s and discontinuous deril'otwns
711
a
by 2.4.32, there exists a E A with = / and SUPtElR IIexp(ita)II < K. It follows from 2.4.23(ii) that a i::, unique in the sense that, if also b E A with b = f and SUPtEIR Ilexp(itb)11 < DC, then b = a. Define T(J) = a (J E C(rt ~)). Then T(J) is well-defined. Let o:JJ E ~ and 1,g E C(rt~). By the above uniqueness statement, T(0:1 + {jg) = o:T(J) + 8T(g), and so T : C(n,~) --> A if.; real-linear. Further, Ilexp(iT(J))II < K (J E C(O,lR.». By 2.1.10, there is a Banach algebra norm 111·111 on A equiva.lpnt to the given norm such that Illexp(iT(J»111 = 1
(J E C(n,~)),
Let 1 E C(0.lR.). Then T(J) is hermitian in (A, 111·111), and so, by 2.4.14, vA(T(J» = IIIT(J)III, whence IIIT(J)III = 111n· Thus T is continuous. Set X IR = T(C(O,~». so that XIR is a closed, real-linear subspace of A. and set X = X:R EB iXlR , so that X is a closed, linear subspace of A. Clearly A = X EB R, and so R is a complemented subspace in A. In the second part of the proof, we exhibit a strong \Vedderburn decomposition of A. Define G = {J E C(O) : 11(x)1 = 1 (.c E On. the unitary group of C(O). so that G is a bounded subgroup of Inv C(O). By 2.4.24(ii), there is a bounded subgroup H of IIlV A such that aHa. H --> G, is an isomorphism. Let .,p : G --> H be the inverse of this isomorphism. Then 'IjJ induces a continuous epimorphism 1fi : (leG) --> fl(JI) with 7jj(8.. ) = 81/-'(8) (s E G). Now set B = ( l (H), taking the closure in A. Since G is abelian. G is amenable, and so, by 5.6.1(i), (1(G) h; an amenable Banach algebra; by 2.8.64(ii), B is also an amenable ilanach algebra. Since C(O) = linG, we have B + R = A. Clearly radB = B n Rand B = (B n X) EB radB, and so radB is complemented in B. By 2.9.59, rad B has a bounded approximate identity, and so, by 2.9.30(i), rad B factors. But rad B is uniformly radical, and so, by 2.3.12, rad B = O. Thus A = B EB R, and this is the required strong Wedderburn decomposition. 0 A typical uniform algebra not equal to CoCO) is the disc algebra A(iD) , and a related a.lgebra is A+ (iD), the algebra of absolutely convergent Taylor series. Let A be either of these algebras. Then all point derivations on A arc continuous, but we shall see in 5.6.81 that there are discontinuous derivations from A into some Banach A-modules. Since there are nOll-zero point derivations on A. it follows from 2.8.63(ii) that A is not weakly amenahle. and hence not (2n - I)-weakly amenable for any 11 EN. Theorem 5.6.5 Let A be a um10rm algebra. Then A is 2n-weakly amenable /01' eachnEN. Proof Set n = . E (MMn,n)X with f.L(f,g)
Extend
= f.L(g, f) = ->.(fg)
(5.6.5)
(f EM, 9 E Mn,n)'
>. to a linear functional on !v! by setting ... ,
(5.6.6)
this is possible because zn+1 ¢. lvIMn,n' We claim that >'(Zi+j)
=
-f.L(Zi, zj)
(i,j EN).
(5.6.7)
This identity follows from the cocycle identity and (5.6.6) if i + J E Nn+l and from (5.6.5) if i + j ~ n + 2. We now claim that f.L(f,g) = ->.(fg) (f,g EM). By (4.4.2), it suffices to verify this in the cases where 1 and 9 have the form Zi for z E N n or belong to Mn,n' But this is now immediate in each case from either (5.6.5) or (5.6.7). (iii) Define a linear functional>' on M by requiring that
= 0,
>'1 ZMn,n
>'(Z)
= .,. =
>.(zn)
= 0,
>.(zn+l)
=
1;
this is possible because zn+1 ¢. ZMn.n . Note that >.(Zi) = 0 (i ~ n so it follows from (4.4.5) (in the case where k = n) and (4.4.9) that
+ 2),
and
n
>.(fg) =
L 8j (f)8n+1-j (g) = 8n+ (fg) 1
(f,g E M).
j=1
We define continuous, symmetric bilinear functionals f.Ll, ... ,f.Ln+l on A! x !v! as follows: for l,g E M, define n
f.Ll (f, g)
= 0,
f.Ln-j(f,g)
=-
f.Ln+l(f,g)
=
L
8r (f)8n+J+2-r(g) r=J+2 -8n+1(fg) = ->.(fg).
(j E Z!_2) ,
Next, define T = (f.Ll,"" f.Ln+1) : M x M --+ E, so that T is a continuous, symmetric bilinear operator. We verify that T is a cocycle. The equation for the first coordinate requires the following to hold for all j, g, hEM: 81 (f)f.L2 (g, h)
+ ... + 8n (f)f.Ln+1 (g, h) + f.Ll (f, gh) = f.L2(f, g)81 (h) + ... + f.Ln+l (f, g)8n (h)
}
+ f.Ll(fg, h) .
(5.6.8)
Contznuous and discontmuous derivations
717
On substitution and change of signs, the left-hand side of (5.6.8) becomes
81(/)8n(g)8n(h)
+ 82(/)(8n- 1(g)8n(h) + 8n (g)8n- l (h» n
n
+ ... + 8n- j - l (1)
L 8r(g)8n+j+2-r(h) + ... + 8n (/) L8r (g)8n+1- r (h). r=l r=J+2 This expression consists of a finite sum of terms of the form 8; (/)8 j (g)8 k (h), where i + j + k = 2n + 1 and each such term occurs exactly once in the sum. Clearly the right-hand side of (5.6.8) reduces to exactly the same sum, and so (5.6.8) is confirmed. The equations for the other coordinates in the cocycle identity reduce in a similar way. Thus T is a cocycle. We clazm that T is not a coboundary. To see this, assume towards a contradiction that there are linear functionals )\1, ... , ).n+ 1 on M such that T = 81 S, where S = ().l,"" ).n+d. Then
).n+1(zn+1) = -/-In+l(Z, zn)
= ).(zn+1) = 1.
The equation for the first coordinate implies that, for each
l,g EM.
we have
).l(1g) = 81(/».2 (g) + ... + 8n (1».n+1(o9) + ).2(/)81(g) + ... + ).n+1(/)8n (g). Apply this with 1 = zn and 9 = Zn+l to see that ).1 (z2n+1) = ).n+1 (zn+1) = 1, and with 1 = 9 = z(2n+l)/2 to see that ).1(z2n+1) = 0. This is the required contradiction. Thus T defines a continuous co cycle which is not a coboundary. The result follows from the commutative version of 1.9.5. 0 In 5.6.10. we exhibited a derivation with finite-dimensional range such that the derivation is discontinuous on every dense subalgebra of C(1) = C(1)(IT); we now give such an example with infinite-dimensional range. Note that C(l) satisfies the conditions on A in 5.3.30, and so every derivation from C(l) into a Banach Cell-module E is continuous in the case where p(Z) has no eigenvalue; here p is the representation. We take the module E to be the Banach space LP = U(IT), where p E [1,00), and we temporarily denote the norm in LP by 1·l p (whereas 11.11 1 is the norm in c(1). For 9 E LP, define
(Mg)(t) = tg(t) ,
(Jg)(t) =
lot g(s) ds
(t
E
IT),
So that M, J E B(LP). We shall use the representation p, where p(Z)
=T =M
- J.
For u E [0,1), denote the characteristic function of the interval [u, 1] by XU' Then it is immediately checked that TXu = ux" (u E [0,1», and so the operator T does have eigenvalues, and IT c aCT). In fact, we note that the eigenspace associated with the eigenvalue u is just CX". For suppose that 9 E LP and that Tg = ug. Then t-ug(t) = J~g(s)ds (t E IT\ {u}), and so 9 is absolutely continuous on [O,u) and on (u,I], and g'(t) = 0 for almost all t E lI. Thus 9 = ax" + f3 for some a,!3 E Co Since g(O) = 0, necessarily !3 = 0, and so 9 E Cx,,·
Automatic continuity theory
718
We establish a formula for p(f), where J E C(l). Finit note that, for 9 E LP and t E H, we have
1t (l = 1t
(J2g)(t) =
U
g(s)
dS) dv, =
1t (it dU)
= ((M J
(t - s)g(s) cis
g(s) cis
- J M)(g»(t),
and ~o M J = .J JI.f +]2. Since Tk+l = (M - J)Tk (k EN), it follows easily by induction on n that Tn = Mn - nJ Mn-l (n EN). Define p(f) = J(T) for a polynomial J. Then we see that the equality
(p(f)g)(t) = J(t)g(t)
-It
1'(8)g(S) ds
(5.6.9)
zn,
holds whenever 9 E LP, t E H, and J = where n E N, and hence whenever J is a polynomial. It is immediate from (5.6.9) that Ip(f)(g)l p S IIJlll1gl p whenever 9 E LP and f is a polynomial, and so p extends to a continuous, unital homomorphism p : C(1) ~ LP given by (5.6.9), where now f E C(1). The map p is a monomorphism: if p(f) = 0, then feu) = p(f)(xu) = 0 (u E [0,1»), and so f = O. Also, p(f)g is continuous at to whenever 9 is continuous at to. Since a(Z) = TI, it follows from 1.5.28 that aCT) C H, and so aCT) = H. Theorem 5.6.13 (Bade and Curtis) Let E = £p(H) be a Banach C(1)(H)-module for the representation p. Then there is a denvatwn D : C(1)(H) ~ E s7Lch that D I Ql. tS discontmuous for' every dense subalgebm Ql. of C(l) (H). Proof For
f
E C(1), define
(V f)(t) Then
IV flp
=
(l f'(s) ds (t
it
E
(0,1]).
s
I(Vf)(t)1 S IIflll 10g(l/t) (t E (0,1]), and this implies that Vf S Cp IIJll1 for some constant Cpo Thus V E B (C(l), £P).
Define
{f
K =
E MI,o :
11 If'.~s)1
E LP with
ds < oo} .
Then it is easily seen that K is an ideal in C U ) with Ml,o s;.; K C l'vh,l' Let >.. E C(1)x be such that
>..(f) =
r f'(s)s ds l
io
(f E K),
>"(Z) = >"(1) = O.
Then >.. is a discontinuous linear functional. Define
(Df)(t) = >..(f) - (l f'(s) ds
it
(t E H, f E C(l»,
s
so that D f = >..(f)I- V f· Then D : e(1) --- E is certainly a linear map, and D f is continuous on (0,1]. The separating space 6 of Dis C1; the continuity ideal I of D is the maximal ideal MI,o of e(l) because p(f)l = f(0)1 (f E C(l».
Continuous and discontznuous derwal'ions
719
Take f E J 1,0, say f I [0, e] = 0, where e t E (e, 1]. Since p(f)l = 0, we have
it
(p(f)Dg)(t) =
f'(u) (
E
t
> 0, take
9'(8)) dlL - f(t)
1".'3
9 E Al1 ,0, and take
11
g'(8) ds.
t
S
We apply Fubini's theorem AA.12 to the iterated integral to see that
(p(f)Dg)(t) = Since J 1 ,0
= ]l.f1 ,1,
[t f(8)g'(8)
, E:.'3
ds
=
t
10
f(8)g'(8) ds. s
a continuity argument shows that the formula
(p(f)Dg)(t) =
lt
f(8):'(8) d8
(t E (0,1])
(5.6.10)
holds for f E M1.1 and 9 E M 1 ,0. The formula is easily ch(~cked for f = Z and 9 E M1.o, and so (5.6.10) holds whPIlever f, g E M1.o, For f, 9 E lvho, we have /g E K, so that
D(fg)(t) = ).(fg)
-11
(fg)'(s) ds
=
s
t
{t (fg)'(s)
10
d8
(t E (0,1]),
s
and hence D(fg) = p(f)Dg + p(g)Df. This also holds whenever f = 1 or 9 = 1 because Dl = and p(l) = IE. Thus D is a derivation. Assume towards a contradiction that D is continuous on a dense subalgebra Ql of a(l); we may suppose that 1 E 2L Then Q( n 1111.1 is dense in 1111,1, and so ~ = (Q( n 2\11.1)2 is dense in Mr.1' and hence inllJl.l' Since 113 c K, there is a constant m > 0 such that
°
f'.~S) dsl =(ff~)(t) = g(t}x,,(t) (t E IT). The function If belongs to L1(dtjt) because f E 1If1,1, and so. by the dominated convergence theorem A.4.6,
+
dt g(t)." t [
1
=
1
I + 1f'1
get) dp,(t) .
(u.1]
Let a be the common value of these two integrals. Then
la-l1~tl a bounded approximatc idclltity awl, in particular. is eSbential. By 4.7.43(i). M(A) = Al(JR,w). Let D E Z(A. A') By 2.9.53, then' is a continuous derivation i5: l\f(Rw) ....., A' extending D, and jj: (l\1(JR,w), so)....., (A',a(A',A» is continuous. For each .'I E JR-, inf flEN 1i8n.,II", IllLnsliw In = 0 by (5.6.14). and so, h.\2.7.15(ii), D(86 ) = O. It follows that D I L = O. where L = lin{8s : s E JR}. By 4.7.45(ii). L is dense in (M(lR,w), so), and so jj = O. Thus D = 0, and A j:-, weakly amenable. 0 Corollary 5.6.19 Let i/o: < 1/2.
C\'
E JR+. Then £1 (JR, we»
'tS
weakly amenable il and only
Proof Suppose that n < 1/2. Then Ll(JR.Wa ) is weakly amenable by 5.6.18. Now suppose that w is a continuous weight. function on JR, and define the Banach space
-, {1 ILloc(JR) : 11111 Loo r= II(t)1+wet) It I
E.... =
E
=
1
dt
<x
}
.
In the C8.'>e where
""'(8 + t) ( sup { w(s)",. . (t)
1 + It I ) l+ls+tl
:s,tEJR
}
(5.6.15)
1 * g. The lIlap D : 1 I---> ZJ. Ll(Rw)....., Ew, is a non-zero. COIltinuous derivation. In the case where w = W a , condition (5.6.15) is satisfied if and only if 0: ~ 1/2. Thu.o.;. hy 2.8.63(iii), Ll (JR, :Va,) is not weakly amenable when 0' ~ 1/2. 0 Our next topic is the repre~entation of (continuous) derivations from some convolution algebras: we concentrate on the algebr&'l £l(w) = U(JR+,w). which were discussed in §4.7. By 5.2.18(ii), all derivations on Ll(w), on fl(JR+,W), and on M(JR+,w) are continuous. We write a for the weak* topololl,Y on AI(JR+ ,w). Recall the definition 0:(J1) = infsuppJL for JL E Mloc(JR+) from 4.7.18. Let w be a continuous weight function on JR+. For each JL E 1I-Jloc(JR+), set KJl = sup
so that KJl E [0,00].
{w~s) 1a+ w(s + t) d IJLI (t) : s E jR+ }
,
A utomatic continuity theory
724
Proposition 5.6.20 Let D be a denvalzon on M(JR.+,w). Then there ex/'sts a
Radon meas'ure J1. E Mloc(JR.+) with K/l- = IIDII such that D(I5.~) = s158 * J1.
(s E JR.+).
(5.6.16)
Proof Let s E JR.+ •. For each n E N. we have
D(Ds) = D(t58*i'~) = nD:/
(M, (T) is continuous. Let v
E
= IIDfll1
and such that
1'1'1. Then, by 4.7.45(iv),
* en) * Il - V * (Zen * Il)) = (T- lim (Z(v * en) - IJ * Zen) * Il = (T- n------?oc lim en * Z// * p, = Zv * Il.
D,,(v) = (T- n----+oo lim (Z(v n----+(XJ
Thus
IIDI-' II 2> sup {IIDI-'(Os) 11/w(s)
: s E JR+ }
(ii) Again by 2.9.53, there is a derivation
IIDII =
IIDII
and
D : (M,so)
->
= KJl'
i5
on M extending D such that
(M,(T) is continuous. By 5.6.20, there exists
Jl. E Mloc(JR+) with Kf1. = IIDII and D(os) = SOb * Il (s E JR+). Let Df1. be the derivation extending DJ.I to M, as specified in (i). Then D and Df1. are socontinuous maps which agree on lin{os : s E JR+}, a subspace which is so-dense 0 in M by 4.7.45(ii). Thus D = D/J' whence D = DI-' and IIDII = KJ.I'
Theorem 5.6.22 (Ghahramani) Let w be a continuous weight Junction on JR+. Then there is a non-zer'o derzvation on Ll(w) zJ and only if thcTe exists a E JR+. such that sw(s+a) } (5.6.17) sup { w(s) : s E JR+. < 00. Proof Suppose that there exists such an a E JR+.. Then K 8" < 00, and so D : f f - ; Z f * oa is a non-zero derivation on L1 (w). Conversely, suppose that D is a non-zero derivation on L 1 (w), and take Jl. E Mloc(JR+) with KI-' = IIDII and D = Dw Then Il -=1= 0 and Il -=1= 00, and so there exists a E ]R'+-. such that c = Il-ll«O. a]) > O. Let m = sup{w(t) : t E [0, a]}, so that w(s + a):::; mw(s + t) (8 E JR+, t E [0, a]). For each s E JR+, we have m c
IIDII 2: ~
and so (5.6.17) holds.
r
cw(s) J(o,a]
w(s
+ t) d IIlI (t) 2: sw(s + a) , w(s)
o
Certainly, for there to be non-zero derivations on L 1 (w), w = e -1) must be a radical weight function. However, there are also continuous radical weight functions w such that Ll(W) has no non-zero derivations: for example, this is
A utornaizc continuity theory
726
the case for 1J: t t---+ tloglog(t + 3). Condition (5.6.17) is sati::;fied for.,,: t I---? (whl'rc "y> 1) for each a E lR+·. and for 1/: t i-+ tlog(l + t) for each a ~ l.
n
The following result. which applies to the Volterra algebra V. can be proved by a slight variation of the above arguments. \Ve now dpfine 1(/1 =
sup
{s 1 1
-
S
d
1111 (t)
for II E JI(R); the value of
: ,r; E [0.1) } = :-mp{s IILI ([0,1 -
IIDI! holds because Kit
= sup{l!fl8b
.'In : sE [0. I)} * Itlll
: s E [0. I)}.
Theorem 5.6.23 (Kamowitz and Scheinberg) Let D be a linear operator on V. Then D zs a derwatwn 2f and only 2f there eTtsts IL E At (R) wzth K I' < ex: and Df = Zf * 11 (f E V). In this case, D zs contmuous and IIDII = K IL • 0 now consider the form of derivations from A = Ll (w) into it::; dual module P'-(lR-.":;;-l); we rpcall that A' is also a Banach 1\I(lR+,w)-module for the product (IL, A) i-+ 11 . A. where \\,p
.4'
-.=
(,1 . A)(t) = (IL
* A)(t) =
10roo X(s + t) d/L(S)
(t E lR-).
Note that we do not know that all derivations from A into A' arc continuous. It follows from 4.7.9 and t.he fact that A factors that, ill the ('a:;c wherO f(8)(C
l
f(8)(g, D(e n
= =
°
x
1
00
* 8.. ) dS) )
n
* 8s )) d8
f(8)(g, Dp n . 88 ) ds +
1
00
f(8) (g, en . D(8.. )) ds.
By 2.9.7, Den in (A', a) as n - 00, and so, taking limits on both sides of the above equation 8.0 Sf(8)1jJ(t - .'I) d8 = (Zf * 'I/,)(t)
whence
D""f =
10
00
f(8)0(8) ds = Zf
* 'I/J.
(t
E
JR-),
(5.6.21)
Automatic contmnzty theory
728
Theorem 5.6.25 (Gnmbock) Let w be a cOT/,tinuo'l1s wr.zght junction on lR+.
(i) Suppose that ~I is a measurable jU7u:tIOn on lR - e wzth K'P < oc. Then Ll(w) --+ Ll(w)' is a cont'inuO'Its derivatIOn with liD,;, II = K'lj;, and D~, has an extenswn to a eontm1to1t8 df'.rivatwn D1L, : }\I(lR+,w) --+ Ll(w)' surh that
ntj) :
D",(Os) = (sos
* 1jJ)
IJR-
(05 E JR+) .
(ii) Suppose that D : Ll(W) --+ Ll(W)' is a cont'l7l,1to'l18 derivatwn. Then there is a measnrable junction '1/1 on JR-e with K'i} = IIDII and D = D~" Proof Set A
= Ll(W)
and AI
= M(JR+,w).
(i) We have seen that D,} : A ---+ A' is a linear operator with IID~; II ::; K ",. Essentially as in 5.6.38, below, j) is a derivation. By 2.9.53, there is a derivation
n.
D.p : M have
--+
A' extending D1fJ with
1
'XJ
o
//i\,//
j(s)D,;, (8s ) ds =
=
IID,pIl. By (5.G.19) and (5.6.21). we
roo f(s)l.J(.5) ds
./0
(J E A).
Thus. for each g E A, (g, D';J(Os)) = (g. l.J(s)) for almost all .5 E JR+: since both sides of thb equation are continuous from the right, the equation holds for all s E JR+, and so DlI,(os) = l.J(s) in A' for cach S E JR+. Hcnce IID",II ~ sup{IIl.J(s)II')(),w /w(s) : 8 E JR+} = K",. (ii) Again by 2.9.53. then' is a derivation
I/DI/
= IIDII and such that
is a measurable function
D : ;\1
D: (M,so) --> (A'. 0')
--+
A' extending D with
is continuous. By 3.G.24, there
1/' on JR-e such that (5.6.22)
Let s E JR+, and take (N n ) in Q+ with Sn --+ s. Thm 8sn ~ 0,. and so (5.6.22) holds for .', E JR+. Thus K," ::; IIDII. Define l.J(8) for s E JR+ ab in (5.6.20). As in (i), D,p(s) = l.J(s) (s E JR+). a.nd so jj and D", agree onlin{os : s E JR+}, whence D = D", and liD II = K,;.. 0 Set A = Ll(w). We denote by W the set of (equivalence classes of) measurable functions on lR- e for which K,;, < oc; for 1/) E W, set 1111/-111 = K~,. It follows from 5.6.25 that (W. III . III) is a Banach space whirl! is isometrically isomorphic to Zl(A,A ' ) = '}-{l(A.A'). Proposition 5.6.26 Let '1/' be a measurable j1tnction on JR-e. Then 1jJ E W ij and only ij Z1/) E A'. In this case. III~)III = IIZ1jJll oo .w' Proof Suppose that 1jJ E W. For each rn EN, set Am = (-00, -1/ rn). Then 4)XA m E L1 (lR). It follows from 3.3.11 that there is a strictly increasing seqnence (Sk) in lR- e with Sk --+ 0 as k --> 00 such that, for each s E (Sk'O), there is a subset Mk,s of Am with measure at most 1/2 k and such that 11jJ(s
+ t) -1jJ(t)1
::; 1/2 k
(t
E
Am \ Mk,s)'
Continuous and dzscontmuous derwations
Fix tEAm and c > O. There is a set Nt
-t Iw(s + t)1
729
C jR -
of measure 0 s',lch that
:::: (1111/)111 + c)w( -t)w(--s) (s E jR \
(5.6.23)
Nt).
Select an increasing sequence (Uk) with Uk E (Sk, 0) \ Nt (k EN). The set AIA,uk has measure at most 1/2k. Set Bk = U{Mj,uj : J 2 k} and B = n{B k : kEN}. Then Bk+l C Bk and m(Bk) :::: 2-k+J for kEN, and so m(B) = O. Moreover, limk~:)C 1./J(Uk + t) = 1/J(t) (t E Am \ B). Set s = 'Uk in (5.6.23). and then let k -+ x; we see that
-t 11/J(t) I ::::
(IIIV!III + c)w( -t) (t E Am \ It follows that -tf'l,u(t) I :::: 1117,;lllw(-t) for almost all t E with IIZvJlloc,w ::; 111'1,1;111· Conversely, suppose that Z'/j} E A'. For each
I(ZI * '/jJ)(t) I < (Xl 811(8)1 ,,-,( -t) - ./0
.
1E
jR-,
A and
tE
and so ZI/) E: A' jR-,
we have
10(t - s)1 ds w( -t)
< (XJ I(Z7/))(t - s)l. w(s - t)
- ./0
B).
w(,.; - t)
.
If(s)lw(s) ds
w(s)w( -t)
because s :::: It - 81 = s + (-t) for s E jR+. This implies that ZI * 1/) E A' with IIZI * ~11"",w :::: IIZ~II"",w Ilfll",· Thm; the map D : 1 1-+ (ZI * 1/J) I ~- is a continuous linear operator with IIDII :::: II z,ifJ 11.:>o.w , Clearly D is a derivation. By 5.6.25(ii), there iH a meaHurable function X on ~-. with Ky :::: IIZ'If'lloo '" such that D = Dx. Let [a, b] be a cloHed interval in jR+., and take 1 E A with ZI = X[a,bJ. Then
I
t a -
l'lf' - x(s)1 ds
O. Then an element b E B is a metric approximate unit for A with constant 8 ~f Ilbll 1 and Ilab - all 811all (a E A).
s
s
In the following three lemma'l, we shall consider an FDNC algebra A: the identity of A# is denoted bye, and we fix dEN such that Ad = O. Lemma 5.6.29 Let (A, 11·11) be a Banach algebm wh~ch lS an FDNC algebm. and take 8 > O. Then there zs a Banach algebra extension (B, III . III) of A s'llch that B ~s also an FDNC algebm and B contnms a metric appronmate umt for A with constant 8. Proof \\Te may suppose that 8 < 1. Let C be the Arens- Hoffman extension of A # with respect to the polynomial Xl\"+l, where N E N is chosen so that N8 > 1 and N > d(2/8)d. Then C is a commutative, unital Banach algebra for the norm II· lie specified by setting
IIL;:'O aiXillc = L~~o lIadl· Let B be the subalgebra of C consisting of elements
with constant term in A (rather than A#-). Then (B, II· lie) is a Banach algebra which is an FDNC algebra. and A is a I:mba1gebra of B. "Ve now define a different norm all C. First, we define a subsemigroup S of (C, . ). For kEN, set
Sk =
{8- k rr(ajX -
aj) : oJ E
A. II0JII = 1 (j E Nk)} .
J=l
and then set S = {e} U U{Sk : kEN}. Clearly S is a unital subsemigroup of (C, .). Also Ilslle: (2/8)d (s E S) because Sk = {O} (k > d). and so S is bounded in (C,II·lld. Let 111·111 be the norm on C dt-fined ill terms of S in (2.1.3). Then (C, 111·111> is a Banach algebra such that 11109111 S 1 (09 E S) and lileill S lIell e (c E C). Clearly IIIXIII S 1 and IllaX - 0.111 S 8110.11 (a E A). We claim that the embedding of (A, 11·11) in (C, 111·111) is isometric. Let x E A with Ilxll = 1. and take ,\ E A' with 11,\11 = '\(x) = 1. Extend'\ to (A#), by setting ,\(e) = 1, and then define ,\ E by setting
s
ex
(5.6.24) We wish to show that 1'\(e)1 :::; 1 whenever C E (C, 111·IID[11' It follows from the definition of 111·111 in (2.1.3) that it suffices to show that I'\(cs)I :::; 1 whenever C E (C, 11·110>[1] and s E S. By the definition of 1I·ll e , it suffices to suppose that c has the form aXi, where a E A~l and i E ZJt;.. The condition that I'\(cs)I :::; 1
731
Contznuo11.s and discontin11.ous derivations
is immediate in the case where s = e, and so we may suppose that s E Sk. where k ENd, and so es has the form 8- k a(X - l)k Xi for some a EAt]' The first case is where k + i ::; N. Consider the subcase where k = 1. Then
IA(es)1 = 8- 1 1(1 - (i + 1)/N) - (1- i/N)IIA(a)1 = IA(a)1 /N8, and so IA(es)1 ::; I/N8 < 1. Next consider the sub case where k 2:: 2. Then
{jkA(CS)
= A(~(-lY G)aX k - r +i ) = ~(-lY (~)
(1- k - ; + i)A(a) = 0
e)
because L:~=o( -It (:) = L:!=o s( -1)" = 0, and so certainly IA(es)1 < 1. The second ca..c;e is where k + i > N. The coefficients 1 - i/N that arise in (5.6.24) are bounded above by kiN because i 2:: N - k in each case, and so
IA(es)1 ::;
}k ~ (~) ~ ::;
(~) k~ ::; (~) d~ < 1.
Thus IA(es)1 ::; 1 whenever e E (e, 11·lId[l] and s E S, and so A E (e,III'III)[1]' Hence Illxlll 2:: IA(x)1 = 1, and so the embedding of (A. II· II) in (e, 111·111) is indeed isometric. Thus (B, 111·111) is a Banach algebra extension of A. The element X is a metric approximate unit for A with constant 8. 0 Definition 5.6.30 Let (B, 11·11) be a Banach algebra. and take 11, E B[l], bo E B, and T], 8 > O. Then: (i) b. E B0B is a weak metric approximate commutant for bo with image 'U and constants (q,8) zf 1Ib.117r ::; 1. if 1l'B(b.) = 11" and if there exists b E B with lib - boll ::; 1/ IIboll and lib . b. - b. . bll7r ::; 8l1boll; (ii) b. E B®B is a metric approximate commutant for bo with image 11, and constant 8 zf 1Ib.1I7r ::; 1, zf 1l'B(b.) = 11" and if IIbo . b. - b. . boll7r ::; 8l1boll.
The heart of Read's construction lies in the following lemma. Note that the proof does require a lower bound on the value of T]. such as 'T] > 9/10'; it does not work with an arbitrarily small 'Tf > O. We make the following preliminary remark. Let (Qt, 11·11) be a Banach algebra. and take a E Qt with lIall = 1. Take N 2:: 2, and consider the element 1
N-I.
.
b. = - - - ' " aJ ® aN -J N-l ~
E Qt ® Qt.
j=l
Then 1l'21(b.) = aN and a . b. - b. . a = (aN (j9 a - a 0a N )/(N -1) because most terms in the expansion cancel in pairs. and so lIa . b. - b. . all7r ::; 2/(N - 1), whereas 11b.1I7r is probably around 1. This idea is used in the following proof. Now suppose that Qt is commutative, and take b.!, b. 2 E 21. i8l 21.. Then we have a . b. 1 b. 2 - b. l b. 2 • a = (a . b. l - b. l . a)b. 2 = (a . ~2 - b. 2 . a)b. 1 , and so
lIa . b. 1 b. 2
-
b.1~2
whenever 11b. 1 1i7r' 1i~2117r ::; 1.
. all7r ::; }~i~2"a . b.j
-
b. j
.
all7r
Automatzc contin'U'lty theory
732
Lemma 5.6.31 Let (A, 11·11) be a Banach algebra which v; an FDNC algebm, and take 'U E A[l], ao E A, 7/ E [9/10,1]' and 0 > O. Then, fOT each rI EN, there is a Banach algebra extenswn (Bn, 111·111) of A such that Bn is also an FDNG algebra and Bn 0 Bn contazns a weak metric approxzmate commutant faT ao with image un and constants ('fJ n , no). Proof The proof is by induction on n; most of the work is contained in the case where n = L and this is our first objective. We may suppose that 0 < 1 and that lIaoll = L We first fix a = d log 4/ log(6/5), and then choose N E N such that
(N - 1)0 > 1,
N> (1 + a)d,
and
Nr/," > [a]!,
where [a] is the integral part of a. Finally, set ko = N 2 d. We set D = {E ar,xr E A# [Xl, ... , X N ] : 11'1 ::; ko}, so that D is a quotient of the algebra A#[X] = A#[XI, ... ,XN], where we are using the notation of §1.6 and Example 2.1.18(v): here, xr = Xr' ... Xr;r and xr X· = 0 whenever IT + sl > ko. The algebra D is a finite-dimensional, commutative algebra containing A as a subalgebra, and it is a Banach algebra for the norm giWIl by IIE1rl:'Oko arXrllD = Elrl:'Oko lIarll· For n
E
N, we write n for the element
(n, ... , n) of Z+N, so that xn = Xi"'" XiV. Let J be the (necessarily closed) principal ideal in D generated by the element X N - 'U, and set C = D / J, HO that C is a Banach algebra for the quotient norm, which is denoted by 1I·lI e . The subalgehra of C formed from elements of D which
have constant term in A is the algebra B. Certainly B is an FDNC algebra, with Bq = 0, say. For convenience, we write c for the element c+ J of C; in particular. we regard each Xi as an element of B. We shall define an algebra norm III . III 011 C such that II !ell I ::; I!eli 0 (c E C) and Huch that (B, III· liD is the required extension of A. In fact. first define b = (Xl + ... + X N )/2N E B, and conHidcr the uuital semigroup S in (C, . ) consisting of e and clements of the form Tj-h' (b - ao)'" x r for k E N q and l' E Z+N with 11'1 ::; ko. Then S iH finite, and HO is bounded in (C, II· lid; the algebra norm 111·111 is defined on C in terms of S as in (2.1.3). so that 1118111 ::; 1 (8 E S) and IIlelll ::; IIcllo (e E C). Since III Xi II I ::; 1 (i E NN), we have IIlb - aolll ::; Tj and IIlblll ::; 1/2. For each i E NN. we define N-I
AlL x· = -N-1
LJ.i
j i
®
x iN - j
E B ®B ,
)=1
so that
,6. =
III,6.illl71" ::;
,6.1 ... ,6.N,
1 and IIiXi . ,6.i -,6.i . Xi II 171" so that 111,6.11171" ::; 1 and
Illb . ,6. -,6. . bll 71" ::;
::;
2/(N - 1). Now we define
1 N 2N
L IllXi . ,6. -,6. . Xilll71" i=l
1 ::; 2N
N
L IllX
i .
i=l
,6.i -,6.i . Xilll71" ::; lieN -
1) < 8.
733
Continu01L8 and discontinuou8 derivations
We have 7rB(~) = X{ ... xfJ = 7L because X N - u E J. Thus we have verified that B has all the required properties. save for the fact that the embedding of (A, II . II) in (B, III . III) is isometric. Let x E A with Ilxll = 1. and take>. E A' with 11>'11 = >.(x) = 1. Extend>' to (A#), by Hetting A(e) = ]. As in 5.6.29, it suffices to show there is an extenHion of A to C such that 1>.(e)1 ~ 1 whenever e E (C. ill·lll)[l)' \Ve first define>. on elementH of D of the form aXT. Take k E ZJ-I' The clusteT at kN consists of the vectors l' = kN + s, where 8 E Z+N and lsi < d. For a vector l' in this cluster. kN 2 ~ Itl ~ kN 2 +d -1, and so the clusters do not overlap because N 2 > d. Also, for each l' in a cluster. 11'1 ~ (d - 1)(}\T2 + 1) ~ ko. The addition of the vector N to a point in one cluster gives a point in the next cluster. In the case where l' belongs to a cluster, we set (5.6.25) We also set >.(aXT) = 0 when T does not belong to any of the clusters. Extend .A to be a linear functional on D. Our first claim is that >. I J = O. To see this, it suffices to show that >'((XN - u)aXT) = 0 for each 7" E Z+N and (l E A#. Suppose that l' = kN + 8 belongs to a cluster. If k < d - 1. then >'(1WXT) = >.(au k+ 1 ag l) = >.(aX N+T). and, if k = d - 1. then >.(uaXT) = >.(aX N+T ) = 0 because u d = O. The claim now follows. ThuH we may regard A as a linear functional on C. Essentially as in 5.6.30, it suffices to show that
I>'((b - ao)kaXT)1 ~ r/
(k
E
Z:' a
E
ArrJ'
l' E
Z+N, 11'1 ~ ko).
(5.6.26)
\Ve fix k. a. and T' as in (5.6.26). In the remainder of the proof, the element s = (81 ..... SN) varies through the index set Z+N. \Ve first make a preliminary calculation. \Ve have k
(b - ao)k =
L J=O
= '" L
k
(~) l? (-ao)k- i = L (2~)j (~) (Xl + ... + XN):i (-ao)k-- j
Isisk
J
J
)=0
1
(2N)lsl
(k) (_l)k- siXsak-lsl i
8
0
'
where (:) is the multinomial coefficient k!/.'ll!··· 8N!(k -lsi)!. We now establish equation (5.6.26) by considering three separate cases.
(1) The case wheTe k ~ a. Since ag = 0, the norm of the coefficient of X" in the above sum is bounded above by 4d times the norm of the corresponding coefficient in the multinomial expansion of (b - ao/4)". Since Ilaoll = 1 and Illblll ~ 1/2, the sum of these norms is at most (3/4)k, and so the sum of the norms in the original expression is at most 4'} . (3/4)k, a number which is at most (9/1O)k because k 2 a. Hence the SUIll is bounded by 'fl. Since lIall ~ 1, we can write (b - ao)kaxr as a sum L: 1s 19 asXs with L: 1s19 11 a sll ~ ",k. But we have shown that 1>.(asXS)1 S lIa s ll in each case, and so I>'«b - ao)kaXT)1 S ",k, as required for (5.6.26).
Automatic contmuzty theory
734
(2) The case where r = jN + t, where j E Z!_l and t E Z+N with
It I < d.
We first see that A(a~-lslaXjN+s+t) = A(a~+ltlauj). If lsi + ItI < d, then this follows from the definition (5.6.25). The equation also holds if d :::; lsi + It I < N2 because both sides of the equation are O. Since N 2 > (1 + 0: )d, the case where 1.'11 + It I 2: N 2 does not occur. It now follows that
A«b-ao)kaXT) =
L
(2~)181
e} -1)k-18IA(a~+ltlauJ)
=
(-1/2)k A(a~+ltla11j),
Isl::;k and so IA(b - ao)kaxr)1 :::; (1/2)k < ",k.
and r is not of the form zn (2). The result we are seeking is trivial if every term in the multinomial expansion of A«b - ao)kaXT) is O. If this is not so, then there is an index s E: Z+lV with lsi :::; k such that r + s = jN + t for some j E ZJ_l and t E Z+N with It I < d. The possible values of Ir + sl cannot Hpan an interval of width N2 - d in Z+, and so only one value of j arises as s varies. Further, there is a lea. E' extendzng D such that
(J . x, DJt) = (x, D(Jt * f)) - (x . Jt, DJ) (f E LI(G), Jt E M(G). x
} E
E).
In the case where D 'lS continuous, D zs also contzrmOU8 with ----> (E',a) is contmuous.
(5.6.28)
IIDII = IIDII
and
i5: (M(G),so)
(iii) 1{n(LI(G), E') ~ 1{n(J.. f(G). et) (n EN).
0
Throughout, we shall regard an essential Banach L1 (G)-bimodule as a unital Banach M(G)-bimodule, and hence as an f1(G)-bimodule, in this way. Let E be a unital Banach f 1 (G)-himodule. Then 0.. . x and x . OR are defined in E for each s E G and x E E; they are often denoted by 8 . x and x . s, respectively. Similarly, we define 8 x oX and oX x s in E' for each s E G and oX E E'. Note that, for each s E G and x E E, we have 118 . :1:11 = IIx . 0911 = Ilxll; also, the maps oX ~ 09 X oX and >. ~ >. x 09 are continuous on (E', a). It is convenient to use a certain language of G-derivations. Definition 5.6.35 Let G be a group, and let E be a unital Banach £I(G)bimodule.
(i) A map
i) :
G
---->
E is a G-derivation zJ
"O(st) = "0(09) • t
+ 09 • "O(t) (s, t
E
G)
and "O(G) is bounded in (E, 11·11); "0 is inner if there exists x E E such that "0(09) = s . x - x . S (s E G), and then "0 is implemented by x.
Continuous and dzscontzmwu8 derivatwns
(ii) A map \Ii : G
->
737
E is a crossed homomorphism
\Ii(st) = s . \Ii(t) .
09- 1
+ \Ii(s)
~f
(8, t E G)
and \Ii(G) is bounded in (E, 11·11); \Ii is principal if there e:cists x E E such that ilI(s) = s . x . s-1 - X (s E G), and then \Ii zs implemented by x.
Let D : G -> E be a G-derivation, and set \Ii(s) = D(s) . S-l (s E G). Then ill is a crossed homomorphism, and \Ii is principal if D is inner. Let \Ii : G -> E be a crossed homomorphism, and set D(s) = w(s) . s (s E G). Then D is a G-derivation, and D is inner if \Ii is principal. Proposition 5.6.36 Let G be a locally compact group, let E be an essential Banach L1(G)-birnodule, and let D : G -> E' be a G-derwatwn. Then the map lJ : G -> (E', 0') zs cont'inuous. Proof The range of D is contained in the compact space (Elm]' 0') for some m> O. We apply A.1.S. First, let Se> -> ee in G with D(sa) -> .-\ in (E', 0'). Take x . fEE· L1. Then
I(x . f,
Sa
X
D(sa) - D(sa))! :::; m Ilxllllf
* 8sa
-
fill
->
0
by 3.3.11, and so Sa X D(sa) -> .-\ in (E'. 0'). Similarly, D(sa) X Sa -> .-\ in (E', 0'). Hence D(s~) -> 2,-\, and, by an immediate induction, D(s~:J -> n.-\ for each n E N. But n 11.-\11 :::; m (n EN). and so .-\ = 0 = D(ee). By A.l.8, D is continuous at ee. Now let Sa -> 8 in G. Then s-lsa -> ea and
D(sa) = D(s) x
8- 1 S"
+,
D(s). o
Thus D is continuous on G.
We establish a correspondence between continuous derivations from Ll (G) and G-derivations into dual modules. Thus, let G be a locally compact group, and let E be an essential Banach £l(G)-bimodule. First, let D : Ll(G) -> E' be a continuous derivation, and extend D to M(G) as in 5.6.34(ii). Set (5.6.29) D(s) = D(8 s ) (s E G). Then D : G -> E' is a G-derivation. Clearly, if D is an inner derivation implemented by .-\. then D is also implemented by .-\. The following lemma shows how D can be recovered from D. Lemma 5.6.37 Let D and D be as above. Then
(x, Df) = ( f(s)(x, D(s)) dm(s)
le
(f E L1(G), x E E).
(5.6.30)
Proof The integrals are defined because the function s 1-7 (x, D( s)) is bounded and continuous OIl G for each x E E. We first claim that
i
f(s)(8 t
.
y, ll(s)) dm(s) = (8 t . y, Df)
(f E L1, t E G, y E E).
(5.6.31)
738
Automatic contmudy theory
Indeed, this holds because, for each fELl, t E G, and y E E, we have
1;
f(s)(8 t
.
y, il(s)) dm(8)
= ( f(s)(y, D(8st )} dm(8)
Je
-
( f(s) (y . 88 , D(8t )) dm(s)
Jc
= \Y, D ( i f (s)8 st dm(S))) - \Y' if(S)8sdm(s), D(8 t ))
Now
= (y, D(f * 8t )) - (y . 1, D(8t )) by 3.3.10 = (y, Df x 8t ) = (8/ . y. D1). take x E E. Then x = 9 . Y E E, where 9 E LI and
y E
E. We have
, f(s){x, il(s)) dm(s) = ( f(05) / ( g(t)8 t dm(t) . y, il(S)) dm(s)
.[ (,
\-Ic
Je
= 1;g(t) (iJ(S)(8t . g(t)(8t
y, il(S))drn(s)) drn(t)
=
Je
.
y, D1) dm(t)
=
(g . y, D1)
by 3.3.10,
(
by 3.3.10
by (5.6.31)
and so the lemma follow::;.
0
Second. let il : G --> E' be a G-derivation, and define D J for fELl (G) by equation (5.6.30): Df E E', and D: (LI(G), 11.11 1 ) --> (E', 11·11) is continuous. Lemma 5.6.38 The above map D zs a derivatwn.
Proof Take
(x, D(f
f. 9
E Ll (G)
* g)) =
i (i
and x E E. Then
f(t)g(C 1 s) dm(t)) (x. D(s)) dTn(s)
= { {(:I;, il(ts))J(t)g(s) drn(.s) drn(t)
JcJc
.L [( = 1;
=
\Ve also have (x, DJ x g)
(8 8
•
x, il(t))
+ (x
. 8t , ()(s)) )J(t)g(s) drn(s) dm(t) .
J(t)(g . x, D(t)) dm(t)
= [J(t) ( [ g(.~)8s dm(s) . :I:, D(t)) dm(t) = { [(8
JcJc
and similarly for (x,
f x
(x, D(f Thus D is a derivation.
8
,
;1;,
by 3.3.10
il(t))J(t)g(s) rim(s) dm(t),
Dg), and so it follows that.
* g))
=
(x, Df x g)
+ (x, f
x Dg).
o
Contzn:uous and discont1,1wous derivu.t1Ons
739
From the G-derivation (l, we obtained the derivation D; df'finc D' from D a..-; before. By 5.ti.:H, for each f E U(G) and x E E, we have
1
f(s)(x, D(s») dm(s)
G
1
= (x, Df) =
f(s)(x. D'(s)) dm(s),
G
and so the functions .'I H (.r. D(.s)) and s J-+ (:r, D' (8)) are equal a..c.; clementH of LOO(G). However, by 5.6.36. both of the fUIlctionH are continlloU~ on G, and so (x, D(s)) = (x, D'(s)) (s E G). Hence D' = D. SUPPOSE' that D : G ~ E' iH an inner G-derivation. and take A E E' with lI(s) = S x A - A x S (8 E G). Let D be defined from D. Then, for f E Ll(G)
and x
E E.
w('
haye
(x, DJ) =
1
f(s)(x,
,S
x A - /\ x .s) dm(s)
(;
= =
(x '!a f(8)8,;
dm(s).
A) - (!a f(s)8 dm(s) . A) 8
(J; . f. A) - (f . x'. A) f x A) - (:1'. A x J).
;1;.
by 3.3.10
= (x,
and so Df = f x A - A x f and D iH inner. "Ve have cstabliHh('d the following result.
Theorem 5.6.39 (Johnson) Let G be a locally compact g1"OUp. and let E be an essent1al Banach Ll(G)-bzmodule. Then the map D H D. where we define b(s) = D(8.• ) (s E G), ebtablishes a bijective COr7Y'spondenc(' between contmnous derivations D : Ll (G) ---t E' and G-derivatwns D : G ~ E'. Further. inner derivations correspond to inner G-denvatwns, and 1-£1(L1 (G), E") = {O} if and only 'tj every G-denvation mto E' is inner. 0 Thus 1-£1 (U(G). E') = {O} if and only if every crosHed homoIllorphibm from G into E' is principal.
There iH a further remark which iH useful. Let G be a locally compact group, and let E be an ('~sential Banach Ll(G)-bimodule, so that E iH a unital Banach M(G)-bimodule. Let F be the Hame Banach space as E. but now define 11,
l> };' such that: ; (i) F : G
--+
(E', aCE'. E» is coutumons:
(ii) sup{ilF(s)!! : 09 E G}
O.
This is an admissible short exact s A . 8.. in L oo , and bO (A, A . (8 8 * en)) --> (A, A . 8s ). Thus (A, A . 8.. ) = (A, A). This holds for each A E Loo, and so 8,. • A = A. It now follows from 3.3.53 that the group G is amenable. 0
742
Automatic cOllt'l'f),n1,ly thp-(I "
Corollary 5.6.43 Lrt G be nn LeA group. and mod'ule. Then Zl(U(G). E) = {OJ,
[P.f
E be a BaT/arlt [1 (
I ,
Proof The abelian group G is amenable, and so L I is amenable. By 2.8.u3(iii), Zl(£I.E) = {OJ. By 4.5.18. A(f) is a ~trong Ditkin algebra. and so. by 5.3.5. all derivations from L1 art' cont.inuoui->. Thus ZJ(I}.E) = {OJ. [ Corollary 5.6.44 Let G be an LeA group. Then eaclt Segal algebra on G 'i8 weakly amenable. Proof The algebra F(B) of Fourier transforms of 5 is an abstract Segal algebra with wspect to A(r). where r is the dual group of G. By 5.6.42, A(f) b amenable, and so, by 4.1.10, S ~ F(B) is weakly anlf>nable. 0 Theorem 5.6.42 allows us to present the following two further interesting examples of radical Banach algebras. Corollary 5.6.45 (Curtis) TheTc is a closcd ,mbset E of'][' 8'uch that the 8ubalgf~bra. I(E)/J(E) of A ('][')/ J(E) is a non-zero, lreakl.1J I1mennble, m(itcal Banach nlgebm. Proof By 4.5.23(iv), there is a dosed subset E of '][' such that E is a Helson set. but. E is not of synthesis for A(,][,). By 4.5.23(iii), E is totally &,connected. Set 1= I(E), K = 12. and 12( = A(,][,)/K. Then cI>21 = E. Qi = G(E), 3! rad 12( = 1/ K is nilpotent. By 5.4.4 (or 5.6.4), 12( is strongly decomposable . . follows from 4.5.25 that K = I. By 5.6.42. A(,][,) is amenable. and hence weakly' amenable. By 2.8.6a(i). I is weakl~; amenable, and so. by 2.8.64(iii). I/J(F) is wpakly amPl1ablc. Thus I / J(E) has the required properties. 0 Corollary 5.6.46 (Runde) Ther'e t8 locally compact g'f'OUp G and a closed zdea.l I in. Ll(G) .'iltcTt that L1(G)/I 28 an amenable, md1cal Banach algebra. Proof Let G 1 be th(' amenahk group that is not weakly \Viellcr whose existellce was noted in 3.3.62(xiii). Since Ll (G 1) is not weakly Wiener. there is a closed icleal I in Ll(GJ) snch that 1}(Gd/l is a radkal Banach algehw. By 5.GA2. Ll(Gd is ameuable, and so Ll(Gd/1 if> amenable by 2.8.64(ii). U Theorem 5.6.47 (Dales, Ghahramani. and Hl'lemskii) Let G be a locally compart gmup. Then the meaSUTe algebr-a l\I(G) 105 an amenable Banach algebm if and only if G '/05 a discn:tc flnd amenable gTOUp. Proof Suppose that G iH discrete. Then 1\.l(C) = £1 (G), and so. by 3.6.42 (or 5.6.1), M(G) is amenable if and only if Gis am0nable. For the converse. we must show that JI.1 (G) is not amenable whenever G is not discrete. We do this under the extra assumption that G is metrizable. Set I = Mc(G), so that I is a dosed ideal in M(G) and I is complemented in .M(G) as a Banach space (see 3.3.36). By 3.3.39, ]2 'I I, and so I does not have a bounded approximate identity. By 2.9.59, AJ(G) is not amenable. 0
Continuous and dZSCOllttnuous derivatIOns
Theorem 5.6.48 (.Johnson) Lft G be weakly amenable.
743
locally rornpa.ct
(J
,Q1'OUp.
Then £1 (G) is
proof (Despic and Ghahmmarbl) We apply 5.G.41 in the case where E = L 1. 'So that £' = L'X. ; Let E 1)(' the family of bounded sets in (L~ .11 'II-xJ We claim that it is sufficient to construct a map
we have Pa(.T; - Yi)
->
0 as i
->
M~;;;
(I:
1ll(ti Sj)
+ 2W(t i S n ))
.
]=2
+ Yi
= (2jMa)w(ti) for each i E 1":1, so that 2 as i -> 00. By the uniform convexity of Pa, 00. But
2
P",(Xi - Yi) = ManPo:(w(tisd - W(tis n ))
~,2 Pa(ti x (w(sd - W(8 n )) X til) 1v1an 2 = ManPcx(W(sd - w(sn)) by (5.6.36). =
and so Po.(W(Sl) - 'It(s71)) = O. Since the family {Pu : 0 E A} is separating, 1ll(st) = W(sn). Similarly W(8k) = w(sn) (k E 1":In ), and so W(81) = ... = w(sn) = 0, as required. (ii) Assume towards a contradiction that there exist + ... + w(sn) = 0 and W(Sj) =J 0 (j E 1":171)' Set
SI,""
Sn
E C with
W(81)
E
= min{llw(sj)ll:
j E 1":I n }
> O.
The set w(G) in bounded in E', and so W(C) is dentablc: there exists to E C such that w(t o) ~ (w(C) \ BEJ. where Be = {A E E' : 11,\ - w(to)11 < s}. For each j E 1":In , we have Ilw(tosj) - w(to)11 = Ilw(8j)11 ~ E, and so w(tOSj) E w(C) \ Be· Thus w(to) = (L::~l W(to8j)) In E (W(G) \ BJ. a contradiction. Thus E' has the crossed homomorphism property. D The Banach L 1 ( G)- bimod ules (LP (G). * p) and (LP (G), . p ) were defined in §3.3; by 3.3.23, they are essential LI(G)-bimodules.
Corollary 5.6.52 Let P E (1,00). Then every contin1lO71S derivatwn from Ll (C) into (LP(G), *p) and mto (£P(G) . . p) is inner. Proof The modules are reflexive as Banach spaces, and hence have the Radon Nikodym property. Thus the result follows from 5.6.51(ii). D We now seek to prove that, for each locally compact group C, every derivation from Ll(G) into l\l(G) is inner. Note that the range of such a derivation is contained in Ll(G), and that, by 5.2.28(iii), all such derivations are continuous. We apply the above theory in the case where E = Co(G), so that E' = M(C) and a = a(M, Co); also, s x jJ, = 8s * jJ, and jJ, x s = jJ, * 8s for jJ, E M(C) and s E G.
7,16
Allt01TWtU: ("()ntinuity theo/'y
Theorem 5.6.53 (.Johm,on) Let C be a locally mmpact gmup which g7'Ol1.p. Then Hl(U(C). M(G)) = {a}.
1"S
a SIN
Proof Since G is a SIN group. there is a central hounded approximat (' icknt ity (e" : n E A) in Jj, and we may suppm,e that. ea,ch (' EO crio c I}. For (\' E A. dl'tiue Pa(") = II" * ca l1 2 Cit E kJ); by 3.3.19. Po is a continuous s('minorm on (.11.11'11). Clnd, by 3.3.24. the family {Pu : n E A} is separating. For each a E A. Pc.. is a uniforml? convex selllinorm and {>
11/1
* (0,,11 2 =
lis xIt *
Cn X
s-
1
1I2
(8
E G)
because G is modular, and so the Raar measure i~ left- and right-invariant. Abo en X ~-1 = ..,-1 X e" by (1.4.18) because en E 3(L1). and so (5.6.36) holds. The result follows from :).6.51 (i) . 0 The above results do not resolve the following natural question. Question 5.6.B Let G be a lorally ('ompact g'f"OUp. Is every denvatwll. fWIn LI(G) mto M(G) necessarily znner? We know that HI (LI(G), U(G)) = {O} in the cases whf'r(' G is amenable and where G is a SIN group: the latter case giv('s the result for all (liscretp groups C. We now turn to the question whether all intertwining maps, and. in particlllar, all derivations. [lOm LI(G) into a Banach Ll(G)-bimodule E are automatically continuous. In fact. this is an open qll('stion: we shall obtain positive lesults for various cla....;ses of locally compact groups aud various modules E, and libi some other positive results in the not('s. R('call that it follows from 5.3.5 that all intertwining maps from Ll (G) an' continuous in the ca....,e where G is a locally compact abelian group. The first result is an immediate eonsequ('Uce of 5.2.17(ii) and 3.3.21(ii): it shows t.hat it suffices to consider derivations into one specific L 1 (G)-bimodule. Theorem 5.6.54 Let G be a locally compact group. Assume that all derwation8 f1'01n L 1 (G) znto l L Xl (G X G) . . ) are continuous. Then each intertwming map from I} (G) mto a [Janach L II G) -bimorlule is corltimwuli. 0 Theorem 5.6.55 Let G be a compact group. Then each intertwzning mnp from L 1 (G) mto a Banach L 1 (G) -bz1nod'Ule zs contznuol1.s. Proof First note that G
X
G is compact, and hence unimodular by 3.3.4, and
1;-:-P = 1/ * F and ~ = F * 1/
for F E C(G x G) and 1/ E M(G x G). Consider a continllouH homomorphism P : Al (G x G) ~ A, where A is a Banach algebra, and define T" : F f--+ pCP), LI(G x G) ~ A. so that Tp is a continuous lineal map. For Jt E M(G), let OeJl and ()rlt be as in (3.3.15). Then A is a Banach M(G)-bimodule for the operations so
(/1, a)
f--+
p(Otlt)a,
For FE C(G x G) and It Tp(1t . F)
E
(It,a)
f--+
ap«()rlt),
M(G) x A ~ A,
M(G), we have
= p(O,1t * F) =
p(Oe/1)p(F)
= It
. Tp(F) ,
Continuo'lts and discont,nLuo'llS dC1'ivatzons
747
and, similarly, Tp(F . /l) = Tp(F) . /l, and so Tp : (L1 (G x G) . . ) ~ A iH an M(G)-bimodule homomorphism. By 5.6.54, it suffices to consider derivations into (LOC(G x G), .). ThuH. let D : £1(G) ~ (LOO(G x G), .) c (£I(G x G) . . ) be a derivation. Then Tp 0 D : I} (G) ~ A is a derivation. Since G is a compaC't group, G is amenable by 3.3.52(i). and so. by 5.6.42, Ll(G) is an amenable Banach algebra. By 5.3.8(ii), each derivation from £1 (G) into a finite-dimensional Banach I.} (G)-bimodule is continuous. By 3.3.47, there iH a family, say F, of continuous hOIIlomorphismH p from M(G x G) into finite-dimensional algebras such that n{ker p : p E F} = {a}. For each such p, neceHsarily 6(D) c ker T p , and so SeD) = {O} and D is continuous. 0 Theorem 5.6.56 (Willis) Let G be a locally finite group. Then each intf:r-iwining map from fl(G) into a Banach £1(G)-bimodule is continuous. Proof By 3.3.58. G iH amenahle and pI (G) is pliable, and so the result follow:-; from 5.3,8(i). 0 \,ye shall llext establish the continuity of intertwining maps from Ll (G) in the case where G is a soluble group; for example, the affine group of JR described in 3.3.62(i) is soluble. The rebult follows from a general theorem. Theorem 5.6.57 Let G be a locally compact group such that f 1 (G) 18 both plzable and amenable. and let E be a Banach left pI (G)-mod-ule. Then each linear map T : [1 (G) ~ E 'lL'hzch i8 left-inter-iwining over £ 1 (G) 18 contin'U(}'ll8. Proof We shall apply 5.3.4. Thus, let I be a closed ideal of finite codimension in £. 1. Since I! 1 is amenable, I has a bounded approximate ideutity, and so ILl is an essential. closed I-submodule of £1 and T I 1£1 is continuous. By 3.3.41 (ii). ILl has finite codimension in Ll. and HO T is continuous. 0 Theorem 5.6.58 (Willis) Let G be a soluble gwltp. Then each intertwznmg map from L I (G) mto a Banach Ll (G)-birnodl1le 'is continuous. Proof Since Cd is soluble. and hence amenable. Ji 1 (G) is amenable. By 4.5.22, f! 1 (G) is pliable. By 5.6.54. it suffices to consider derivations D : L1 (G) ~ E', where E is an essential Banach A-bimodule. By 5.6.34(ii), there is a derivation D : ]\[(C) ~ E' which extends D. By 5.6.57, D is continuous. 0 The group SL(2, lR) was desCTibed in 3.3.62(x), and three subgroups K, A, and I'll such that SL(2.lR) = K . A . N were specified there. In the following, we write c5" and (,. and an irreducible na x nO' matrix group G (\" and such that:
(i) (1),,,(EnJ : 0: E A) 1S a bounded approxzmate identzty for A(E); (ii) sup{ll1>o.(x)1I : x EGa, 0: E A} < x. We set Pa = 1>a(EnJ (a E A). Condition (i), above, is equivalent to the conditions that P,,(x) -> x (x E E) and P:.(>') -> >. (>. E E'). If thiH condition be satiHfied. then E has BAP. Theorem 5.6.63 (GnlIlbrek, Johnson, and Willis) Let E be a Banach space havmg property (A.). Then I(E) is amenable. Proof Set 2( = qE) ;g qE) and approximate diagonal in 2(. Define d(~ E 2( for (Y E A by dn =
Idal L
7f
= 7fK(E).
We Hhall show that qE) haH an
{1>a(X) ® 1>a(:r-l) : x E Gn}
in the notation of 5.6.62. By 5.6.62(ii). (do. : n: E A) is a bounded net in 2(. Take T E K(E). By 5.G.62(i), limn 7f(da )T = T because 7f(doJ = Pal and also lima P",TP", = T. By 1.9.20, d a is a diagonal of Mn a • and so we have ~,TPa . dn = dn . PaTPa . Thus lim(T . d a u
-
dn
.
T)
= lim((T - PaTPo.) . d a a
-
dn
.
(T - PcrTPa ))
= 0,
and thiR shows that (d u : (Y E A) is an approximate diagonal for I(E). By 2.9.65, the Banach algebra I(E) is amenable. 0 Corollary 5.6.64 Let /-L be (l positzve measure on a set S, and let p Then the algebra I(LP(/-L)) zs amenable.
E
[1, x).
Proof First SUPPOHC that p > 1: we shall Hhow that LV = £1'(M) has property (A.). The conjugate index to p is denoted by q. ConHider the collection of families S of finitely many, pairwiHe diHjoint, measurable subsets L of S with 0 < IL( L) < 00, and set SI --< S2 if each member of SI is the union of a subfamily of S2. The biorthogonal system corresponding to S= {L 1 , •.. ,L",} is ( (XL'! M(L 1)1/p, ... , XL,,/ M(L n )I/ P) . (XL'! M(Lt}l/q, ... , XL,! M(L]
)l/q)) ,
the corresponding homomorphism into F(LP) is denoted by 1>s, and the corresponding projection is denoted by Ps . For each L E s, PS(XL) = P~(xd = XL, and so it follows from AA.I0(ii) that 5.6.62(i) is satisfied.
Continuous and discontmuous derwation8
751
Consider S = {L 1 , .•. , Ln}. and let G s be the group of matrices of the form DtE,n where D t is the diagonal matrix specified by t = (tiDi,j), where we have tl, ... ,tn E {-1. 1}, and EO' is the matrix corresponding to a permutation 17 of Nn. Certainly Gs is an irreducible n x n matrix group. Note that
II~QiXL.!M(L;)'j,[ ~ tlad' It follows that, for each I 11$ (x 1111:
~
lit
t :;t
=
=
LP and x = DtE"
',p( L i )
p(Li)-p/q
J
III P = Li
E Gs, we have
j, (1. I) M( ['0))- ,j'XL.,.1
(Iii II) ii III P
IL(L i )-p/qj1,(L i )p/q
tj '1,=1
E
(a,,·.A EC).
P
by Holder's inequality
IIIII~ .
Thus IIs(x)11 :::; 1, and so 5.6.62(ii) is satisfied. Thus LP has property (A.), and the result follows in this case. Now suppose that p = 1 and that fL(S) < 00. Then the above argument, with small notational changel:l, l:lhows that L1(fL) has property (A.), and hence is an approximate diagonal of bound 1 for K( L1 (fL)). Finally, conl:lidcr the case where p = 1 and fL is a gcneral positive measure. not necessarily l7-finite. For each measurable subset T of S, we regard L1 (fL I T) a.q a closed linear subspace of U(Ji). The approximate diagonals for K(L1(fL IT)) of bound 1 conl:ltructed as above fit together in an obvious way to give another bounded net in L1 (fL) ®L1 (rL) such that this net is all approximate diagonal for K(£1(fL)). Thus K(L1(Ji)) is amenable. 0 Theorem 5.6.65 (Gr0nbrek, Johnson, and Willis) Let E be a Banach space. (i) Suppose that K(E') zs amenable and K(E) has a bounded approxzmate identity. Then K(E) is amenable.
(ii) Suppose that A(E') zs amenable. Then A(E) 28 amenable. (iii) Suppose that K(E) is amenable and K(E') has a bo'unded approximate identity. Then K( E') is amenable. Proof (i) The algebra K(E)a is a closed left ideal in K(E'), and it is antiisomorphic to K(E), and so has a bounded approximate identity. By 2.9.66, K(E) is amenable. (ii) Since A(E') is amenable, it has a bounded approximate identity, and so, by 2.9.37(i), E' has BAP. By 2.9.37, (iii) and (v), A(E) has a bounded approximate identity, and so, as in (i), A(E) is amenable. 0 (iii) This is similar to (i).
752
Automatic contznuity theory
Corollary 5.6.66 Let n be a compact space. Then K(C(n)) is amenable. Proof By A.3.60(v), C(O) has BAP, and so K(C(O)) = A(C(n». AI!-lo, by A.3.71(iii), C(n)' has the form L1(p,) for some positive measure p,. Since A(L1(p,» = K(£1(p,» is amenable, the result follows from 5.6.65(ii). 0
So far, we have not given an example of a 'natural' Banach space E- say. E is a space with BAP- such that K(E) is not amenable. The following theorem, whose proof we omit, gives such examples. Theorem 5.6.67 (Gnmbrek, Johnson, and Willis) (i) Let r, s E (1,2) U (2, x) with r =I- 8. Then K(fr EB fS) is not amenable.
(ii) Let T, s E (1, (0). Then the following conditions ar-e equwalent fOT the algebra 2( = qf r ®fS): (a) 2( is amenable; (b) r + s < TS; (c) 2( has a bounded [right] approximate zdentity. 0
In particular, K(f3/2 EBf 3 ) and K(f2f§f2) are not amenable. Theorem 5.6.68 (Dales, Ghahramani, and Gr(i'lnbrek) Let E be a reftel:ive Banach space wzth AP. Then N(E) zs (2n - I)-weakly amenable for each n E N. Proof By 2.6.24, N(E) is a closed ideal in N(E)". By a remark after 2.8.57, N(E) is weakly amenable, and so, by 2.8.77. N(E) is (2n -I)-weakly amenable for each n E No 0
We now turn to consideration of derivations from a C* -algebra A and of the higher cohomology groups 1t1t(A, E), where E is a Banach A-bimodule. Recall from 5.3.7 that all intertwining maps and, in particular, all derivations from A into E are automatically continuous. There are many deep and beautiful theorems in this area. Unfortunately. their proofs require a more sophisticated background in the theory of C* -algebra..,; than is available to us, and so we can only summarize some key theorems without proof. Fortunately other excellent accounts cover this material thoroughly. Further results and references are given in the notes; indeed, some terms that are to be used are only defined in the sources. The first result is an easy corollary of 5.ti.51 (ii). Proposition 5.6.69 Let A be a unital C*-algebra, let E be a umtal Banach Abzmodule, and suppose that the Banach space E' has the Radon-Nzkodym property. Then 1tl(A, E') = {O}. Proof Let U be the unitary group of A, and set B = f1(U). Then there is a continuous embedding () : B -+ A such that ()(8u ) = U (u E U). Since linU = A, we have B = A. By 5.6.51(ii), 1t1(B, E') = {O} because E' has the Radon-·Nikodym property, and so, as in 2.8.64(ii), 1t 1 (A, E') = {O}. 0 Theorem 5.6.70 (Sakai) Let A be a C*-algebra. Then 1t 1 (A, A) = {O} whenever A is either a von Neumann algebra or a simple, unital C* -algebra. 0
Contznuous and discontinuous derivations
753
Corollary 5.6.71 Let 2t be a C* -subalgebra oj B(H), where H is a Hilbert space, and let D be a derivatzon on the algebra 2t. Then there exists T E l2(wo c B(H) such that D(A) = AT - T A (A E 2t). 0 As we noted in §2.7, a derivation on a C'-algebra is not necessarily inner; it is more natural to ask if it is 'inner in a larger algebra'. In the following result, M(A) denotes the multiplier algebra of A.
Theorem 5.6.72 Let A be a separable C*-algebra. Then 1-{l(A, M(A» = {O} if and only zJ A = Al EB A 2, where Al is a C*-algebra with a continuous trace and A2 is the direct sum oj a Jamily oj simple C* -algebras. 0 The question of the amenability of C*-algebras involves very deep mathematics.
Theorem 5.6.73 (Connes-Haagerup) Let A be a CO-algebra. Then A is amenable zJ and only zJ A is nuclear. 0 Corollary 5.6.74 Let H be an injinzte-dimensional Hilbert space. Banach algebra B(H) is not amenable.
Then the 0
The only result on derivations from C* -algebras that we shall prove is that all C'-algebras are weakly amenable. We require two lemmas.
Lemma 5.6.75 Let R be a von Neumann algebra, and let DE Zl(R,R'). Then there exist Do E Zl (R, R') and a positive trace T E RbJ such that
I{b, Doa)1 ::; and D - Do
E
23/211Dllllall T(b*b)1/2
(a, bE R)
(5.6.37)
Nl(R, R').
Proof We may suppose that IIDII = l. The map (a, b) f---+ (b, Da), R x R --> C, is bilinear and continuous with norm 1, and so, by the generalized Grothendieck inequality 3.2.44, there exist states AI, A2, JLl, JL2 on R with
I{b, Da)1 ::; (Al(a*a) Set JL = (JLl
+ JL2)/2
E
+ A2(aa*»1/2(JL1(a*a) + JL2(aa*»1/2
(a, bE R).
SR' Then JLl ::; 2JL and JL2 ::; 2JL in (PA , ::;), and
I{b, Da)1 ::;
211all (p(b*b + bb*»1/2
(a, bE R).
(5.6.38)
By 3.2.41(iii), there is a positive trace T E RilJ and a net (Po,) in R' with POI in (R',a(R',R», and n"
POI = L
-->
T
=
1,
ta'JV~,j . JL . Va,j ,
j=l
where, for each a, we have na E N, ta,l,"" ta,n" E (0,1] with L;~l ta,j and Va,l, ... ,va,n" E R with v~,J Va,j = e R (j E Nna)' For each a, define n"
Wa
=
Lta,jD(v:;',j) . Va,j E Rill' j=1
A utomatzc contznuzty theory
754 By passing to a subnet, we may suppose that Let a, bE R. Then, by 1.10.23, we have
Wa -
win (R[lJ,a(R' ,R)).
no
(b, a .
W -
W •
a) = li~
L ta,j (b. a . D(v~,j) . vOI..j -
D(v~,j) . Va,Ja)
j=l no
= ~m L
ta,j
(b. V~,j . D(vQ,Jm'~) . v~.j - Da) .
j=l
Define Do = D + /jUll where /jw is the inner derivation implemented by w. Then D - Do E Nl(R,R' ). The following calculation gives the e::;timate (5.6.37) for a, b E R. First, I(b, v* . D(vav*) . 1')1
= I(vbv*, D(vav*))1 ::; 2110.11 (JL(vb*bv* + vbb*v*»1/2
whenever Ilvll = 1, and so n"
I(b, Doa) I ::; 2 110.11 lim ::;up Ct
L tOl..) (/L( v,...jb* bv;',j + vO!.jbb*v~.j))
By the usual Cauchy- Schwarz inequality I(b, Doa)1 ::; 2 II all lim sup a
1/2 .
j=l
(
OIl
JR.n", we have
t ta.j/L(vOl.,jb*bv~,j
+ va.jbb*v~,j)
1/2 )
j=l
=
21Ialllimsup(Pa(b*b + bb*»1/2
=
2110.11 r(b*b + bb*)1/2
a
= 23 / 2 11all r(b*b)1/2 ,
as required.
o
Since 0 is the only positive trace on a properly infinite von Neumann algebra, the above lemma already implies that each such algebra i::; weakly amenable.
Lemma 5.6.76 Let A be a unital C* -algebra, and let D E Zl (A, A'). Suppose that there is a posztzve tmce r in A' such that, for each a E A, there exzsts Ca > 0 with I(b, Da)1 ::; ca r(b*W/2 (b E A). Then D E Nl(A, A'). Proof Define IT = {a E A : r(a*a) = O}. Since r i::; a positive trace, IT i::; a closed ideal in A. As in §3.1, there i::; an inner product [" ·l on AI IT such that the completion of the corre::;ponding normed ::;pace i::; a. Hilbert space, defined to be H T. Further, as explained in §3.1, HT is a Banach A-bimodule. For each x E H T , define Ax E A' by (a, Ax) = [a + IT> xl (a E A). The map A: x f---+ Ax, HT - A', is a continuous, conjugate-linear injection, and so A(HT) is a linear ::;ubspace of A' and a.lso a Hilbert space with respect to the inner product defined by [Ax, Ay] = [y, xl (x, y E H T ). Let II· 112 be the associated norm on A(HT ). For a,b E A and x E HT> we have
(b, Ax . a) = [7l"T(a)(b + IT), x] = [b + IT, 7l"T(a*)(x)] = (b, A(7l"T(a*)(x») ,
Contwuo/J..
(b E A).
A(HT ) is a derivation. 0
Theorem 5.6.77 (Haagerup) Each C* -algebra is weakly amenable. Proof Let A be a C*-algebra. By 3.2.42(iii), B(A, A') = W(A. A'), and so, b~' 2.8.59(iii). it suffices to show that each von Neumann algC'bra it' weakly amenable. However. this is immediate from 5.6.75 and 5.6.76. D Corollary 5.6.78 Each C* -algebm is perrna:n.ently weakly amenable. Proof Let A be a C* -algebra, so that A is Arens regular. By 3.2.36, the iterated duals A{271} for 71. EN are von Neumann algebras. Let DE Zl(A, A"). By 2.7.I7(iv). there exists jj E Zl(AI,A") such that jj 1 A = D. By Sakai's theorem 5.6.70, jj E Nl(A",A"), and so D E Nl(A.A"). A Hmall C'xtension of this argument shows that A is 2n-weakly amenable for each 71. E N. We prove by induction on 71. that A is (271. - I)-weakly amenable for each C*-algebra A and for each 71. E N. By 5.6.77, this is true for n = 1. AHsnme that the result holds for n, and take a C*-algebra A and D E Zl(A,A{271+1}). By 3.2.42(iii), D is weakly compact, and so, by A.3.56(ii), D"(A") C A{2n+l}. Thus we may suppose that D" E Zl(A",A{2n+l}). Since A" is (2n -I)-weakly amenable, there exists A E A{2n+l} such that D"(iP) = iP . A - A . (iP E A"). and so D E Nl(A,A{2 n +l}). This shows that A is (2n + I)-weakly amenable, and continues the induction. D In the final part of this section, we consider when there are discontinuous derivations from commutative Banach algebras. Let A be a commutative Banach algebra such that A has no closed, prime ideal of infinite codimension. vVe showed in 5.3.22 that all intertwining map.'i from A into a Banach A-bimodule are continuous provided that A is separable and there are no discontinuous point derivations on A. We now consider the case where A does have a closed, prime ideal of infinite codimcnsion. It is possible that, in this case, there is always a Banach A-module E and a discontinuous derivation from A into E; however We can only prove this result in special cases. Clearly it is sufficient to consider algebras A such that all point derivations on A are continuous; by 4.3.13, the disc algebra A(Jij) falls into the class under consideration. In fact, our results will be established for Banach algebras of power series.
A utomatic con tinuity thr077J
756
Theorem 5.6.79 (Bade and Dales) Let A be a Banach algebra of power- 8("I"le8. Then ther-e is a Banach A -module E and a dzscontmuous den71atwn D fmm A mto E such that D I qXl = 0, D(expX) =1= 0, and D(A) C E t . Proof By 4.6.25, there is a weight w such that A is continuously embedded in Cl(w). Let E = co(Z-,C:;-l), in the notation of §4.6. Then E is a Banach A-module for the module product specified by a . ,\ = (a * ,\) I Z-" and the torsion module E t h; identified with the submodule X described in (1.6.8). Let D : .J ---+ X be the derivation constructed in 1.8.18, taking ,1'0 E X·. Then the restriction of D to A has the required propertieH. 0
Note that, for each a E A, neceHimrily D I qal iH a continuous derivation. We now Heek discontinuous derivations from Banach algebras of power series into torsion-free modules. We first note that, at least for ordinary Banach algebra...'l of power series. we must find X-divisible subrnodules of a Banach module. Proposition 5.6.80 Let A be an or-dzna1'y Banach algebm of power' ser'ies, and let E be a non-zero, torsion-free, unital Banach A-module. Then the follo'Unng ar-e eqmvalent:
(a) there zs a dzscontmuous denvatwn from A mto E; (b) there zs a denvatwn D : A
---+
E wlth D I qXl = 0 and D(expX) =1= 0;
(c) E contams a non-zero, X -dzVLsible submodule; (d) E contams a non-zem, closed submodule F such that X . F
=
F.
Proof (b)=}(a) This is trivial.
(c)=}(b) Let F be a non-zero. X-divisible submodllie of E. By 1.6.25, F is divisible. Choose Xo E F·. By 1.8.17, there is a derivation D : A ---+ FeE such that D I qXl = 0 and D(expX) = Xo· (d)=}(c) Set G = n{X n . F : n EN}. By the Mittag-LeIDer theorem A.1.25. G = F, and so G =1= O. Clearly G is an X-divisible submodule of E. (a)=}(d) Let D : A ---+ E be a diHcontinuous derivation. Then 6(D) =1= O. By 5.2.15(iii), there exists no EN such that
xn . 6(D)
= Xno .
6(D)
(n;:: no).
Set F = xno . 6(D), a closed submodule of E. Then F =1= 0 because E is torsion-free. Also X . F = F, and so F has the required properties. 0
Theorem 5.6.81 (Dales) Let A be a Banach algebra of power series. Then ther-e is a tor-sian-free, umtat Banach A-module E and a dzscontinuous der-ivatwn from A into E such that D I * sppeified aboV(' is unique). £ 1 (S) i5 amC'na ble if and only if S has only finitely many idempotents and ('very subgroup is amenable (ibid.). The followillg furth('r rC'sults are known: (i) if 8 i~ abelian. then £ I(S) is amellabl(' if and only if SO is a finit(' :-;('milattice of abelian groups (Gnmba>k 1990a); (ii) if Sis cauccllative ami unital. I (S) is amenable if and only if S is an amenable group (Gr0nba>k 1!)8~). For a survey. set' (Lau 1990). ThC' theorem that e'en) is amenablp was proved by .Johnson (1972a). Kadison and HiuglOsP (1971b), and bv Helemskii: S(,(' (Helemskii 1984. 1989b, ChaptC'r VII) for a fuller account. IndC'Pd, more gpneral results are proved in thesp sourc('s. The result also follows from theorems about bound question whether or not Il('cpssarily HI (21. B(H)) = {O}, A derivation from 21 to B(H) extpnds to a derivation from the von Neumann algebra 21" , and so it sufficcs to determine the von Neumann algebras 21 such that HI (21. B(H)) = {o}. The mnjecture is that this is true for all von Neumann algebras 21; it is the major open quer:>tion in the theory of derivations OIl C* -algebras, The conjl'cture has heen C'stablished in many cases; sec (Sindair and Smith 1995. Chapter 2). It is known that a derivation from 21 to B(H) is inner if and only if it is 'completely bounded' (Chrhitensen ]982), and this makes contact with the substantial modern theory of completply bounded operatorr:>. See also (Effros and Ruan 2000). We dbcuss TheorC'm 5 6, 7~. A major theme of C* -algebra theory ovcr many years has been the characterizat ion of amenable C* -algl'bras. It was proved in (Connes 1978) as a consequence of the seminal study of von Neumann algebras in (Connes 1976) that an ampnable C*-algebra is nuclear (spc Notes :3.2.45). The cOllverse was proved in (Haagerup 1983); the proof IIses the generalized form of Grothendieck's inequality which was stated &'l 3.2.44. Sub;;equent work, particularly that of (Effros and Kishimoto 1987) and (Effros 19S8), led to r:>implificatiolls of the original proof, and it is now possible to prove the result, and to pstahlish many other equivalent conditions to aml'nabiJity, without using the r:>tructure theory of (Connes 1976). Many additional characteri,mtions of amenable C* -algebras play an important role in the story. For example, two internwdiate characterizations are that 'A" is inJectwe' and 'A" is sem~di8crete'. Sl'e (Effr'os and Lance 1977). Let A be a von Neumann algebra, and let E be a Banach A-bimodule. Then B' is normal if the maps a ~ a ' A and a ~ A . a from A into E' are continuou;; when Aha;; the ultraweak topology and E' has the weak * topology Then A is amenable as a von Neumann algebra if pvery df'rivation into a normal Banach A-bimodule is inuf'r: r:>el' (Connes 1994, v.7) and (Helemskii HIR9b, Postscript). Also A is an AF algebra if it is the closure of an a;;cending chain of finite-dimensional C* -subalgebras: ;;pe (David;;on 1996, III). It i;; shown in (Johnson lOt al. 1972) that an AF algebra is amenahle as a von Neumann algebra; that an injective von Nl'umann algebra is an AF algebra is proved in (ConnE's 1976), with a simplified proof in (Popa 1986). Finally. a C* -algebra A is amenable if and only if the enveloping von Neumann algebra A" is amenable as a von Neumann algebra. For a discussion of these characterizations of amenable C* -algebras, see (Paterson 1988, 2.35) and the introduction to (Dixmier 1981) by E. C. Lance. Corollary 5.6.74 follows from a result of Wassermann (1976) that characterizes von Neumann algebras which are nuclear, and hence shows that the algebra 13(H) is not nuclear whenever H is an infinite-dimensional Hilbert space, as we noted in Notes
762
A-'ifomatic ('07lhn'U.ity
theory
~t2.'l5. For a morE' direct proof of tlH' chan,ctl'rb:ation of allwnable yun NE'lllllann algPi)ras and the fact that l3(H) i5 not amenable whenevpr H is an infillitt'-dimensiollal Hillwrt space. see (RundE' 2001). The dass of pxad C* -algE'bras is activelv studipd at presCo'nt: seE' (\Vassprmann 1994) for the d{'finition and a survey. It is proved that amenahle C* -algebras ar(' exact. that C* -subalgebras and quotiE'nts of exact C* -algebras are exact. and that t.here exist C* -algebras which are E'xact. hut not uuclear. and henc(~ not amenable For pach infinite-dimensional Hilbert space H. l3(JI) b not exact. III fact it is proved in (Kirchberg 19!)5) that a separable C* -algebra is exact if and only if it is a C* -subalgdml. of a uuclear C* -algebra. Sec alsu (Kirchberg and Phillips 2001. 2.8). In the paper (1983). Haagerup also establi"hed that all C* -algebras are weakly amenable: howevE'r, our proof of fl.6.77 copies (Haagerup and Laustsen 1998). The curollary fl.G.78 is from (Dale" ct al. 1998). It is shown in (Johnson 1996) that L 1 (G) is symmetrieally amenable for each amenable locally compact group, but that, for example. the 'Cunt:;: algPl)ra' ib an amenable C' -algebra which is not. symmetrically amenable. TIH're is now a vast literature un the cohumology theory of Banach algebras. and. in particular. of C*-algebras: we cannot discuss this seriously in this work. but we briefly reyiew some hi!?,hlightli herf'. SCf' (Kadison and Ringrose 1971a). (Helemskii 19R9/i, 2000). and (Sinclair and Smith 1995) for many more results. Clearly each amenable C' -algebra is simplicially trivial: this is also true for various other examples. including l3(H) for each Hilbert space H. See (Christensen and Sinclair 1989). However, it is not known whether or not every C'-algebra is simplicially trivial It is also Hot known whether 01 not dg A 2: 2 for each C* -algebra A: this is true for ('ach non-unitaL separable C*-algebra A (Arist.ov 1995). It is shown in (Ermert 1998) that, for each non-unital C* -algebra A the map (a. b) f--> a 09 b, A x A ---t A ~ A, is a cocyde which is not a coboundary. and so '}-e(A,A~A) =1= {O} and elbA 2: 2. A C* -algebra is biprojectiVE' if and only if it ili thE' co-sum of a family of full matrix algebras: see (Selivanov 1979) and (Helemskii 19~9b. IV.fl.15). Let Qt be a von Neumann al!?,ebra contained in l3(H). For each n 2: 2. it is an open question wllPtlter or not llecessarily 7-{" (Qt. Qt) = {O}. By (.Johnson and Parrot 1972). 7-{1 (Qt. K(H» = {O}: for n 2: 2. it b open whet.her or not necessarily 7-{n (Qt. K(H)) = {O}. See (Sinclair and Smith 1995) fur SOIIle strong partial rebult" These questions are studied by considering 'completely bounded cohomology'; se(~. for example, (Christensen et al. 19~7). (Effros and HUlln 20(0). and (Sinclair and Smith 1995). \Ve HlPntion onE' further 'automatic continuity result' result for C' -algebras. This is the deep proof of a conjPcture of Karollbi that the algebraic and topological K-theories of 'stable' C' -algebras are equal (Suslin and Wodzicki 1992) TIH'orclll 5.6.79 is from (Bade and Dale~ HJ89a), alld 5.6.R1, which is algebraically more elctllf'Iltary than 15.6.79, is from (Dales 197:3); fl.6.83 is from (Dales 1976). For cf'rtain spedal radical Banach algebras A = £1 (w). there exist derivations from A into a Banach A-himodule such that D I qX] is discontinuolls (Steiniger 1998): it is not known witethE'r or not a derivation from the disc algebra can he discontinuous on IC[X].
fl.7
EMBEDDING ALGEBRAS IN BANACH ALGEBRAS
We now approach the climax of our theory, the construction of embeddings from a variety of algebras into Banach algebras; we shall show that many ah~ebras arc normable and t.hat there are discontinuous homomorphisms from many (mainly commutative) Banach algebras into other Banach algebras. Most of the algebraic preliminaries for our constructions have already been given in Chapter 1, and especially in §L7. The reader should review the theory of valuation algebras,
Ernbeddmg algebras in Banach algebras
763
Mittag-Leffler algebras, and Henselian algebra..:;. For example. for each totally ordered, divisible RrouP G, the (complex) valuation algebras ~(G+) and ~(1)( G+) are algebraically closed (by 1.7.19), Mittag-Leffler (by 1.7.7(ii)), and Henselian (by 1.7.11); a key role will be played by the algebra M# = ~(l)(G+) (Hee 1.3.65 and 1.7.38(i)). Again, let P be a non-maximal, prime ideal in poc = C((3N). and set Al' = Cum)/ P. Then AI' is an algebraically closed valuation algf'bra (by 4.8.24(ii)), and it is a l\'Iittag-Leffler algebra. Our embeddings will be constructed by using the algebrau: cxtenszon theorem of 1.7.42 and the main exten8wn theorem 1.7.44 and its sequel 1.7.45. In these results, it is required that the codomain algebra A be local and Hensclian; in the case where A = R# for a commutative, radical Banach algebra R, the algebra A is indeed local and Henselian (by 2.4.39). The main task of the present section is to construct suitable framework maps 1/J, as defined in 1.7.27. First we shall present in 5.7.1 the seminal result of Allan on the embedding of the algebra ~ = q[X]] of formal power series into Banach algebras. In this case the framework map 'IjJ is essentially trivial; this result implies that the algebra ~ is normable. We shall then present in 5.7.7 a similar result for the algebras ~n of formal power series in n indeterminates. The framework map in more general cases will he constructed by using T/l-concs over QH (see §1.2). We shall then be able to exhibit some universal algebras; these are algebras containing M, and hence, with CH, a copy of every non-unital integral domain of cardinality c. The first universal radical Banach algebra will be R o, a quotient of HO'(IT) (see 5.7.16): with this algebra we shall prove that (with CH) there are discontinuous homomorphisms from the algebras C(O) for every infinite, compact space n. Perhaps the most interesting universal algebra is the convolution algebra A of Definition 4.7.29: using this algebra. we shall exhibit several further universal radical Banach algebras in 5.7.25. We shall then give two striking characterization results of Esterle: in 5.7.28, we shall characterize those commutative, radical Banach algebras which are universal. and, in 5.7.30. we shall characterize those commutative, unital Banach algebras A for which there is a discontinuous homomorphism from C(O) into A for each infinite, compact space n (working in the theory ZFC + CH). We believe that there is a discontinuous homomorphism from each infinitedimensional, commutative Banach algebra A; this is proved for many, perhaps all, such algebras A in 5.7.32. Let A be a commutative Banach algebra, and let a E A. vVe shall next seek homomorphisms from A that are discontinuous on the "ubalgebras Cola] or even on Coral: we shall obtain some positive results in 5.7.33, 5.7.37. and 5.7.38. Finally, we shall extend some of our results to give discontinuous homomorphisms from commutative normed algebras. The proofs of our results have evolved over a number of years, and are now significantly shorter than the original ones, which involved some detailed computation,,; Home history of the evolution is given in the notes. Recall that we are careful to note which of our results depend on the continuum hypothesis, CR. The continuum hypothesis indeed plays an important role in our story. To explain this, let UH consider the sentence NDH ('no discontinuous homomorphisms'), formulated by Solovay. This is the sentence: for each compact space n, each ho-
A 'Utomatzc contm'Uity theory
764
momorphzsrn from C (fl) into a Banach algebra is contin'Uo'Us. Theorem 5.7 .20( iii) asserts that the theory ZFC + CH proves the sentence ,NDH, and so ,NDll is relatively consistent with ZFC. Could it be that CH is redundant. and that -,NDH can be proved in ZFC itself? In fact, this is not the case: there is an extension (obtained by 'forcing') of a model of ZFC which is a model of the theory ZFC + MA + NDH. (Here MA is Mart~n's axiom, a well-known axiom of set theory.) Thus the sentence NDH is independent of the theory ZFC. The next question that naturally occurs is whether or not -,NDH is relatively consistent with ZFC + ,CH: to prove that this is the case. one must start with a model of ZFC, and show that there is a model of ZFC + ,CH in which there is a discontinuous homomorphism from C(D) into a Banach algebra for some compact space fl. Now CH is used at just one key point in the main proof: the result follows from 5.7.20(ii), a theorem of ZFC itself, provided that there exists p E ,6N \ N such that Ap is a BI-valuation algebra. That there is a model of ZFC + ,CH in which some Ap is a ,61-valuation algebra is another theorem of Woodin; it was to open the door to such a result that we expended a considerable amount of effort to obtain results, involving ,61-valuation algebras, that hold in ZFC itself. The re~;ult shows that CH is independent of the theory ZFC + NDH. \Ve first consider embeddings of ~ in Banach algebras. Recall from 4.6.2 that ~ is not a Banach algebra with respect to any norm. Let r E Z+. As in §1.6. we write nr : L QsXs ~ Q r for the corresponding projection on ~, and AIr
= X,·~ = {a
E ~:
o(a) 2: r}
for the corresponding ideal; by 1.6.20, {Mr : r E Z+} is the family of non-zero ideals in~. By 1.7.26, ~ is Henselian; certainly ~ is an aI-valuation algebra. The notation la = n{ an A : n E N} was introduced in (1.3.20). Theorem 5.7.1 (Allan) Let A be a comm'Utatwe, 'Unital Banach algebra, and let a E A. Then there zs a unital embeddmg () : ~ -+ A wzth ()(X) = a zf and only if a E rad A and a has finite closed descent m A. Moreover! if a satisfies this condttion, if fo E ~ zs transcendentalvnth respect to qX]. and if bEla! then there are umtal ernbeddmgs ()I,(h : ~ -+ A with ()l(X) = ()2(X) = a and ()2(fO) = ()I(fO) + b. Proof Suppose that there is a unital embedding () : ~ -+ A with (}(X) = a. By 1.5.28, (T(a) C a(X). and so a E D(A) = rad A. Set IIfll = 11(}(f)II (f E ~), and let S denote the closure in (~, 11·11) of a subset S of~. By 2.3.5, .!'vh = .!'vh. For each r E N, .!'vIr is an ideal in ~, and so there exists SEN" with Mr = .Ms. Assume towards a contradiction that Mr = .!'vI" (r EN). By A.3.8(i), each projection nr is continuous on (~, 11·11). Set f = L~l r IInrll xr. Then I E ~ and n,,(f) = r lin" II (r EN), and so 11/11 2: r (r EN), a contradiction. Thus there exists kEN such that M I , ... , Ah are closed in (~, 11·11) and lth+l = M k . We have a k E ak+I(}(~) C ak+l A . Since a k =1= 0 and a E md A, necessarily a k rt ak+l A, and so b'A(a) = k. Conversely, suppose that a E radA is such that b'A(a) = k. By 2.2.12(ii), b'radA(a) = k. By replacing A by (radA)#, we may suppose that A is local and Henselian. Let ~o and ~ be as in 2.2.20, so that ~ is a Mittag-Leffler set in A.
Embedding algebms in Banach algebms
765
First suppose that {an}-L = 0 (11 EN). Then't/J: n t---t an, N ~ A. is a freely acting framework map in the sense of 1.7.27, and 'lj;(N) . Inv A C ~, so that ¢(N) . lnv A is a Mittag-Leffler set in A. The Illap Bo : p t---t p(a), qXl --> A. is a homoIllorphism which is compatible with 1/). By the main extension theorem 1.7.44, there is a unital embedding fh : J --> A with Bl (X) = a. Now let fo and b be as specified. and set U = Alg;Jqxl. so that fo rt u. Since is Henselian, U is a Henseliall valuation algebra. ChoosE' (11k) in N such that nk+l > nl,; and 7rnk (fo) f= 0 for kEN, and set iJk = L7~~l 7rj (fo)xj E u, so that the sequence (o(fo - Yk» is cofinal in Z+. Then W(Yk+l - Yk) = an. (k E N) (where W = 'ljJ 0 0), and HO b E n{W(Yk+l - y!,;}A : kEN}. By the remark following 1.7.43, Bl(f) + b is also Bl-compatible with fa. By 1.7.31, there is a homomorphism B2 : U[fol --> A with B2 I U = Bl I U and 82 (fo) = 81 (fo) + b. Again, by 1.7.44. B2 extends to a homomorphism 82 : J --> A. Thus the result holds in the present special case; we do not giYe the proof of the general case here because it is an easier version of an argument to be given in 5.7.7. 0
:s
Certainly there are many commutative, radical Banach algebras which contain elements a of finite closed descent (and such that {an}.L = 0 (n E N». For example, this iH true of the Volterra algebra V and of the algebras U(w), where w is a radical weight on jR+ (see §4.7). Thus we' have the following concluHion. Corollary 5.7.2 The algebm
J is normable.
o
Corollary 5.7.3 Let A be a commutative. unital Banach algebm, and let a E rad A have fimte closed descent. Then a has a discontmuous functional calculus map. Proof Since 0-(0) = {OJ, we have (Ja(a) = C{X}. Let e a : C{X} --> A be the unique continuous functional calculus map for a. Choose fo E C {X} Huch that fo is transcendental with respect to qXl. Since a has finite cloHed deHcent, Ia f= {O}, and so, by the theorem, there is a unital homomorphism 8 : C{X} --> A with B(X) = a and 8Uo) f= ea(Jo). The map 8 is a discontinuous functional calculus map for a. 0 Let A and B be Banach algebras. A major theme of the earlier sections of this chapter was that it is often the CaHe that each homomorphism from A into B is automatically continuous on a large subalgebra of A. \Ve now give an example which shows in a strong way that there is no general result of thiH nature. In 5.3.3, we showed that certain bilinear maps related to intertwining mapl> are automatieally continuous; the following result shows there is no analogous result for homomorphisms. We use the notation and terminology of §4.6. Theorem 5.7.4 (Esterle) Let w be a radical, baszs weight sequence un Z+, and let R be a non-zero, commutative, radzcal Banach algebra wzth a bounded approximate zdentity. Then there is a homomorphism (j : Ml (w) --> R such that, for each r E N and each non-zero subalgebra B of Ml(W), the r-linear map (II, ... ,fr) t---t (j(II··· fr) is discontinuous on B(r). In partzcular, (j is very discontinuous.
766
A utomatic continuity theory
Proof Set Ai = lVh(w). By 4.6.26, for each I E Me there is a constant k sllch that (IIrllw /w(n»)l/n ~ k (n EN). By 2.9.17. we may suppose that R is separable, and so, by 2.9.30(iv). there exists c E n(R). By 2.9.42, there exist a, b E A with c E aA. a E h, and (IIan ll Iw(n))l / n ~ 00 as n ~ 00; clearly, a, bE n(R). By 2.2.11(iv), 8R (b) = 1. By 5.7.1, there is a unital embedding 0 : ~ ~ R# with O(X) = b. Let B be a non-zero subalgebra of M, let r E N. and take I E Be with o(f) = m, say. Then I E xm . Inv~, and so O(f) E bm . Inv R#, whence (L E bmr R c O(r)R, say a = O(r)c, where c E B. For each n E N, we have
( 1IO(frn)ll)l/n> 1l/n . (1Ianll)l/n. ( w(n) )l/n > _1_. (1Ianll)l/n IIrll w Ilcnll w(n) IIrll", - k IIr.II w(n) and so (IIo(frn) II I IIrlU l/ n ~ 00 as n ~ 00. Assume towards a contradiction that there exists K
> 0 such that
II II 1Iw··· IIlrllw (iI,···, Ir E B). Then (II0(frn) II I IIrlU l/ n ~ Kiln IIrllcS'"-l)/n ~ 0 as n ~ 00, a contradiction. II 0 (h ... Ir)II ~ K
The result follows.
0
We now give a result related to 5.7.1; it characterizes elements of finite closed descent. The preliminary proposition follows easily from the main extension theorem 1.7.44, but we give the elementary direct proof, working in the cont('xt of (F)-algebras.
Proposition 5.7.5 Let A be a commutatwe, umtal (F)-algebm, and let a have finite closed descent m A. Then there is a umque unital embedding 0 : ~ ~ AlIa with O(X) = a + Ia. Proof Suppose that 8A(a) = Tn, and set B = amA, so that B is an (F)-algebra and a E n(B). By 2.2.19, n(B) is a Mittag-Leffler set. For p E qX], set O(p) = p(a) + la, so that, by 2.2.11(ii), 0 : qX] ~ AlIa is a unital embedding with O(X) = a + I a • Take I E ~\ qXJ, say I = E~l arkxr k , where (rk) is strictly increasing in Z+ and (ark) c B be a homomorphism, and suppose that there exists m E N such that am E am+lA. Then there exists 71 E N such that O(a)n E B(a)n+l B. This result is immediate in the case where am E am+! A, and so we may suppose that 8A (a) < 00. Set b = (}(a). By 2.2.l2(ii), we may suppose that A, B, and 0 are unital. By 5.7.6, there is a unital embedding", : ~ -> A/Ia with ",(X) = a + Ia. Clearly B(Ia) C h, and so there is a unital homomorphism "if: A/Ia -> B/h with O(a + Ia) = b + h. Now 0 0 '" : ~ -> B/Ib is a unital homomorphism with (00 T/)(X) = b+h. Suppose first that Bo '" is an embedding. Then 8B (b) < 00 by 5.7.6. Suppose next that 0 0 '" is not an embedding. Then there exists n E N with (0 0 ",)(xn) = 0, and then bn E h c bn +! B. The result holds in both cases. We now extend the main part of 5.7.1 to show that the algebra ~n is normable for each n E Nj the result is also an immediate corollary of 5.7.18(ii). below. Theorem 5.7.7 (Haghany) Let R be a commutatzve. radzcal Banach algebra con taming an element a of finite closed descent, and let n EN. Then there zs a unital embedding () : ~n -> R# such that O(Xn) = a. Proof Set A = R#, so that MA = R. By 2.4.39, A is Henselian. Let W = Wn be the valuation algebra defined in1.7.14j W.is an Ql-valuation algebra containing ~n as a subalgebra, and Pw = (zn, :;)+-. First, we consider the special case in which {an}l. = 0 (n EN); again let .1. be the Mittag-LefHer set specified in 2.2.20. Set an = a. By 2.2.20(ii), we can successively choose an-l, ... , al E .1. with ai E n{ai+l . .1. : n E N} for i E Nn - 1. Then (5.7.2) whenever i E Nn - 1 and ri E N. We now define a framework map 'I/J : Pw i E N n with r =
(0, ... ,0, ri, ri+l> ... ,rn )
-
->
R-. For r E Pw, there exists
(0, ... ,0, Si+l, .•. ,sn)
(5.7.3)
A utomatic continuity theory
768
for some Ti EN and Ti+l, ... ,Tn ,Si+l .... ,8 n E Z+. By (5.7.2), there exists x E b. such that a~i = a~+"\1 ... a;n :r, and, by 2.2.20(iv). x is uniquely specified. We set 'I/;(r) = a;"+"\1 ... a~n:r. Then '1/'(1') is independent of the representation OfT in (5.7.3), and 1/): (Pw ,+) ~ (R-,') is amorphism. Clearly 7/J is afreply acting framework map, and 'I/;(P~v) . Inv A c b. is a l\,littag-Leffier set. The map eo : p t-+ p(al, ... , an), C[XI, ... , Xnl .--+ A, is a homomorphism which is compatible with 1/). By the main extemiion theorem 1.7.44, there is an embedding e : W ~ A which extends eo, and then e I In is the required map. Second, we consider the general case. Set 1= {ak}.l, where k = 6R(a), and write 7[" : A ~ AI I for the quotient map. Let n E N. We claim that {7["(a n )}.l = 0 in AI I. For suppose that b E A and n 7["(a b) = O. Then anb E I, and so, rmccessivcly, ak+"b = O. akb = 0, bEl. and 7["(b) = 0, as required. Since 6A/I(7["(a)) < :xJ, the first part of the proof applies to give an embedding e : J" ~ AI I with e(X,,) = 7["(a). Set B = {J E In : vU) C:: (0 .... ,0, k)}. where v is the valuation on J". so that B is an ideal in In. Take bEak A n I. Then there exists c E A with b = OkC; we have a2k c = 0, so that c E {a 2k }.l = {ok}.l and b = 0. Thus a k A n I = 0 and 7r I akA is an injection, with inverse 7["-1 : 7["( a k A) ~ akA. Clearly e(B) c 7["(a k A). Set 8 = 7["-1 0 (e I B). Then 8: B ~ A is a well-defined embedding with 8(X;:') = am (m ~ k). Each element f of In has the form
f = 0;0 + Cl'lX" + ... + ak_1X~-1 + g, where
0;0, ... , O;k-1 E
eu)
C and 9
E
B; set
= rYOeA + (tla + ... + O;k-1a k-1 + e(g).
Then 8: In ~ A is a linear map. Suppos(' that 8U) = O. Since 8(y) c akA, it follows from 2.2.1l(ii) that 0;0 = ... = O;k-l = 0, and so 8(y) = o. 9 = O. and f = 0. Thus 8 is an injection. Clearly (7r 0 8) (g) = e(y) (y E B). Also e(Xn) = 7["(0) = (7[" 0 8)(X,,) and e(l) = fAil = (7[" 0 8)(1). Hence 7r 0 8 = e 011 In. Now take h, 12 E In. and set b = 8Ul12) - 8(h)8(12). Then 7["(b) = 0, and so bEl. But also bE a," A from the definition of 8 on In, and so bE akAnI; since a k A n I = {O}. we have b = 0. This shows that 8: J" -> A is a homomorphism. 0 Hence 8: In -> R# is a unital embedding with 8(Xn) = a. We now move towards embedding theorems for which the construction of the framework map 'l/J is not trivial. The key algebra that we shall work with is M. and we shall use M in the definition of a 'universal algebra'. Recall from 1.1.19(i) that Q is universal in the class of totally ordered Ih-sets. from 1.2.29(i) that G is universal in the class of totally ordered fh -groups, and from 1.7.37 that M# is universal in the class of (31-valuation algebras. Definition 5.7.S An algebra A is universal if theTe is an embedding of M zn A.
Let A be a unital universal algebra. Then there is a unital embedding of M# in A. Let A be a universal Banach algebra, and let e : M ~ A be an embedding. Then e(M) c Q(A) = rad A, and so rad A is a universal algebra.
Embedding algebras zn Banach algebras
769
Proposition 5.7.9 Let A be a umversal algebra. Then A contazns a non-zero, real semigroup. Proof By 1.3.66(iii), M contains a non-zero, real semigroup.
o
Theorem 5.7.10 Let B be an zntegral domazn wzth a transcendence degr-ee at most ~1' and let A be a unzversal algebra. (i) Suppose that B zs non-umtal. Then there is an embedding of B in A. (ii) Suppose that A and B are unital and that B has a character. Then there is a unital embedding of B in A. Proof This is immediate from 1. 7.38.
o
We now come to a point at which we must invoke the continuum hypothesis; it is used to discern integral domains of cardinality ~1. The following result is immediate from 5.7.10 because IBI = ~1 with CH. Theorem 5.7.11 (CH) Let B be an integral domazn with IBI = c, and let A be a unzversal algebra. (i) Suppose that B zs non-unital. Then there zs an embedding of B in A. (ii) Suppose that A and B are unital and that B has a character. Then there is a unital embedding of B in A. 0 We now seek examples of universal algebras; we shall use the embedding theorem 1. 7.45. Clearly the main extra ingredient that we require is the existence of a morphism '1/) : P ---+ 1\(4.; we shall construct such morphisms by finding a suitable 7]l-cone over Q+. in MA and applying 1.2.32. Theorem 5.7.12 Let 0 be a compact space, and let P be a non-maximal, prime ideal zn C(n). Then C(O)/ P is a unzversal algebra. Proof Set Ap = C(O)/ P, as in 4.8.10. We apply 1.7.45. By 4.8.11(i), Ap is Henselian, and aJ. = 0 (a E Ap) because Ap is an integral domain. By 4.8.20, (Mp, .) contains an 1]l-cone over Q+., and so, by 1.2.32, there is a morphism 'I/J : P ---+ Mp. By 4.8.11(ii), AI' is a Mittag-Leffler algebra. Thus the conditions in 1. 7.45 are satisfied, and so there is an embedding of M# into Ap. 0 The importance of universal algebras for the existence of discontinuous homofrom the algebras C(O) is shown by the following theorem.
morphisIIL.~
Theorem 5.7.13 Let 0 be a compact space, and let A be a commutative, unital Banach algebra. (i) Assume that there is a discontinuous homomorphism from C(O) into A. Then there is a closed ideal K in rad A such that (rad A) / K is a universal, radical Banach algebra. (ii) Suppose that 0 is infinite and that A is a universal algebra. Assume further that there exists p E /3N \ N such that Ap is a /31 -valuation algebra. Then there is a discontinuous unital homomorphism from C(O) into A.
770
Automatic contznuity theory
Proof (i) This follows from 5.4.32 and 5.7.12. (ii) Take p E I:IN \ N such that AI' is a ,8I-valuation algebra. By 1.7.37, there is an embedding of AI' into M#, and hence there is a unital embedding, say fJ, of AI' into A. The map () 0 7l"P : too ...... A is a discontinuous homomorphism. The result now follows from 5.4.37(ii). 0 The following application of 1. 7.45 will be used to identify some universal radical Banach algebras.
Proposition 5.7.14 Let R be a commutatwe, mdical Banach algebm. Suppose that there exist an 'r/l -cone (S, 58) over Q+. and a morphism X: (S. +) ...... (O(R), . ). Then (X(S) . Inv R#) U {O} contains a universal algebm.
Proof By 2.4.39. R# is a Henselian algebra. By 2.2.19, n(R) is a Mittag-Leffler set in R. By 1.2.32, S is a universal cone, and so there is a morphism
7ro(F), (S, +) ...... (RO, .), is a morphism. Since FE O(HO'), we have 'IjJ(S) c O(Ro). By 5.7.14, Ro is a universal algebra. 0
771
Bmbeddwg algebms m Banach algebms
There is a closely rel~ted, perhapH even simpler, univerHal Banach algebra: fie replace the function F of the above proof by the function Z
f->
F( z + 1),
II
-?
.7.18.
0
Theorem 5.7.20 (Dales. Est£'rlc) Let 0 be an mjimte, compact space. (i) (CH) Let P be a przme ideal in C(O) such that IC(O)/ PI = c. Then 0(0)/ P zs norrnable. (ii) Assume that there exists" E im \ N such that Ap is a (it -valuation algebra. Then there is a discontmu01Ls unital homomorphism from C(O) mto some Banach algebra. (iii) (CH) There i..~ a dzscontinuous umtal homomorphism from C(O) into some Banach algebra, and there is an algebra norm on C(O) which is not equivalent to the uniform norm. Proof (i) This is a special case of 5.7.19(ii). (ii) This follows from 5.7.13(ii); the Banach algebra is (iii) With CH, Ap is a i31-valuation algebra for each " follows from (ii).
R't. E
(3N \ N, and so this 0
772
A'utomat?,c contmw,ty theory
Corollary 5.7.21 (ClI) Let rp E /3IR \ R an: eq7Jivalent:
Then the following condztwns on :p
(a) 'P is a remote pomt of ;3IR:
(b) J", is a prime ldeal zn C(;3IR); (c) CC/3IR)/Jy? is normable. Proof By 4.2.27, (a){:}(b); by 4.2.25, (c):::}(a); by 5.7.20(i), (b):::}(c).
0
In §5.4, we discussed the continuity of intertwining maps from the algebras = ((,(n) (ll). !!·!I n ), where n E N. We now resolve two points that were left open in §5.4; we use the terminology given in that section.
c(n)
Theorem 5.7.22 (Dales) (ClI) Let n E N.
(i) There is a permanently discontinuous homomorphism from c(n) (ll) mto 80me Banach algebra. (ii) There is a unital homomorphi.sm e fmm c(n) (lI) znto some umtal Banach algebra such that e I C(2n+l)(lI) is 1I·1I2n+l-continuous, but e I C(2n)(1l) ZS riot " . "2n -contznuous. Proof (i) By 1.3.44(ii), there is a prime ideal Pin C(lI) with Zk ~ P (k EN). Let () : C(ll) --. B be a homomorphism into a Banach algebra B such that kcr 0 = P. For each k 2: n, P n C(k) is not closed in (C(k), II· Ilk) because Zk+l ~ P, and so (j I C(l.;) is not ". Ilk-continuous. Thmi (j is permanently discontinuous.
(ii) Set AI = {f E C(ll) : f(O) = O}, and define L = {j E C(ll) :
IJ(t)1
= OW) as
t
---+ ()
+ for each
E
> o}.
Then L is an ideal in C(ll) with L c AI. Take 90 = 1/10g(1/Z) (with go(O) = 0). Then go E At, but g~ ~ L (k EN). and so. by 1.3.44(ii). there is a prime ideal Pin C(ll) with L c Pc AI and 90 ~ P. Let (j : C(lI) ---+ B be a unital homomorphism into a unital Banach algebra B for which ker (j = P. Let '13 = c n x B(n+l) as a linear space, and write 7rj : '13 ---+ B for the projection onto the (j + 1)th coordinate for j E Zt", where we are idelltifying C with CCH. For G, bE '13, define ab E '13 by the formulae j
7rj(ab)
= L 7ri(a)7rj-i(b) (j
E
ztJ·
i=O
Then '13 is a commutative algebra with identity (1,0, ... ,0), and '13 is a Banach algebra with respect to the norm " . " : a f--> L~:;;'o "7rj (a) " . We define linear maps Sk : Mn,n-l ---+ C(lI) for k E z;t inductively as follows. First, So(h) = h/Zn (h E Jl,fn,n-d. Now take k E Nn , and assume that So,···, Sk-l have been defined. Then Sk is any linear map such that
Sk(h) = Sk-l(h/Z)
(h
there is such a linear map because zn
E E
ZMn,n-l),
Sk(zn) = 0;
Mn,n-l \ ZMn,n-l'
Embeddzng algebr'as in Banach algebms
For
J E e(n),
p(f)
=
define n-1 J8j (f)ZJ i=O
2:
773
= 8n(f)zn + RnJ
E
zn(8n(f)1
in the notation of (4.4.2). For J E M n ,n-1, we have p(f)
Tk = (}
0
Sk
0
P
+ Mn,o)
C
M".n-1
= f. DefinE'
(k E z~) .
We make some calculations. Take h E 1\[n,n-1 and j, k E z;t. If J < k, then S",(Zi h) = Sk-J(h), so that. in particular, Sk(Zj+n) = O. If j = k, then Sk(Zjh) = So(h) = hjzn, so that, in particular. Sk(Z)+ll) = L If j > k, then
Sk(Zj h)
= So(Zj-kh) = Zj-k . (hjzn)
E
L.
Now suppose that h1,h2 E M n .n- 1. Then h1h2 E znIl"In,n_1 by 4.4.5(v), and so Sk(h1h2) = 0 (k E Z~_l)' whereas
Sn(h 1h2 ) = So(Z- n h]h 2) = Z- 2n h 1h 2 = So(hdSo(h2)' Take
J, 9 E e(1I).
(5.7.5)
ThE'n
p(fg) - p(f)p(g) n-1 = (8j (f)p(g) j=O
2:
11-1 n - ]
+ 8j (g)p(f»zj + 2: 2: 8i (f)oll+j_i_1(g)Zll+ j -1. j=1 .=j
It follows that, for k E Z~_1' we ha\'e
n-1
k
Tk(fg) = 2:(6j (f)Tk-.i (g) j=O
+ OJ(9)Tk-j(f)) +
2:
6i (f)6 n +k- i(g):
i=k+1
here we are using the facts that S",(p(f)p(g» = 0 and Tk(ZTr!) = 0 (m EN). Also, it follows from (5.7.5) that n-1 Tn(fg) = To(f)To(g) + 2:(8J (f)Tn - j (g) + 6j (g)Tn - j (f» . j=O Now define e : e(n)
e:
J
----> ~
f->
by the formula
(60 (f), .... 6n - 1 (f), To (f) , ... , Tn (f» .
Clearlye is a linear map with 8(1) = e'}3, and the above calculations show that
e is a homomorphism.
Let f E e(2n+1). Then p(J) = L;'=o 6J+1l (f)Zj+n + z2ng for some gEL, and so Tk(f) = 8k+n(f)eB (k E z;t). Thus 8 agrees on c(2n+1) with the map f f-> (60 (f), ... , 62n (f», and so e I C(2n+1) is II· II 2n+ I-continuous. The function go belongs to A2n (where we are using the notation of 4.4.3), and so Z211g0 E M 2n ,2n' Also 8(Z2ngo) = (0,0, ... ,0, 8(go» i= o. But z2n gO is the 1I·1I 2n -limit of a sequence (gk) in M 2n+ 1 ,2n, and 8(gk) = 0 (k EN). Thus e I C(2n) is not II . 112n -continuous. 0
774
A utomaizc ('ontmuitll
thCOT1j
\Ve now Heek Home further universal Banach algebras, including some which arc integral domains. In the following results, /" denotes the Hahn valuation on M#. \Ye fin,t extend our main extension theorem 1.7.44.
Theorem 5.7.23 Let 1.'1 be a commutative. non-v:nztal Banar:h algebra. Suppose that there 1.8 a morphism 1/J : P --> Af with 'lp(P) c OeM). and suppose, fur-ther. that fo E M-. that r u {fo} 7S a transcendence bas'is for M, and that 'Y : r --> lR+ o is a map. Then there 78 a hom.omorph1,sm () : qr U {fo}] ---> l-v[ such that (}(M-) C 4J(P) . (Af# \ 1\/), such that (}(fo) = (V' 0 1')(fO). and such that 11(}(f)11 = -y(f) (f E 1'). Proof Set
\]I
=
I/;
0
'lJ
and ao
= \f!(fo).
(}o : p(fo)
H
The map
Co[Jo]
p(ao).
--->
AI.
is a homomorphism with (}o(fo) = a() and range contained in 1~'(P) . (Af# \ AI). Take TO < W1 with fo E M~. and s('1 l' = (1' n M~) U {fo} (TO sa < ...:t). \Ve shall construct by transfinite recursion a family {(}O' : TO sa < wd, where each (}O' : qr0'] ---> AI is a homomorphism with (j".(M-) C 'l,i(P) . (11,{# \ M), with (}a(fO) = ao, with II(}U)II = -yU) U E [0'), and with (}r I qr".] = (}rr whenewl' (T
TO
sa < T < Wl.
Indeed, take T > TO, and assume that the homolllorphism 0". has been defilwd whenever TO S a < T. Define UT = u{qr".] : TO S (j < T}, and define a homomorphism ()~ : UT Af by requiring that O~ I qr".] = ()". for TO S (j < T. \Ve shall extend ()~ to a homomorphism Or : qr r] ---> A[. There is a cofinal seqllcnce (8 n ) ill P r' Set -40
I = nrl/;(81) ... 1i'(8 n )Af : n E fil} .
Since t,0(P) c n(Af), it follows from 2.2.4(ii) that 7 = exists s E P r with 0(8) E I. Then there exists n so I c ?p(s)JU and '1/;(8) = I/;(.~)a for some a E 1\1. so b = ba (b E A1), a contradiction of the fact that
1,0(P r
)
C
AI \ I.
AI. Assume that thcle E fil with 8 n > s, and But '0(8) E n(Af), and AI is Iloll-unital. Tllll~
Set A = lIf# I I. Since [ is a primary ideal in /11#, A is a local algebra with Af/1 = Mil, and. by 2.4.39. A it; Henselian. Let n : M# ---> A be the quotient map. and sd, ;;: = (n 0 1/)) I P r and (f = n 0 ()~. Then;j;: P r ---> AlA is a morphism. Let 8. t E Pr, and take tL E AI with ~'(s)tL E I. For each n E fil with Sn > 8, Wc' have 1/'(s)'/1, E l!{5d" ·1,0(sn}.'I, and so there exists an E '/1,-'ljJ(st)··· l/;(sn)M with w(s)a n = O. Since ljJ(s) E n(M). we have 1,0(t)an = 0, and this implies that 1,0(t)u E 'Ij;(st) .. . 1,0(sn)l'vi. Thus ?j;(t)u E I. We have shown that 0(8)1. = ;j;(t)l.. in 1\IA' and so J; is a framework map. Clearly if is cornpatiblp with ;;. We now apply the main cxtension theorem 1.7.44. By 2.2.19, n(M) is it :t\Iittag-Leffler set in M. and so n(rl(M)) is a Mittag-Leffler set in Mil; we have J;(P T ) • Inv (AI1) c n(n(M)). By 1.7.44, there is a homomorphism JL : Mf --> A which extends '0 and which is compatible with :;p. Let s E T n Up Then s E U{ru : TO S (j < T} because the set r is algebraically independent over C. Set Or(s) = O~(s). Now let s E r T \ UT , and
r
Ernbeddmg algebras zn Banach algebras
775
choose O.,.(s) E ]I.{ with 7r(O.,.(il» = Jl(.9) and 1I0T(8)1I = ,"y(. Af. extends to a unique hOIllomorphism 8.,. : qr T 1 ---> AI. Clearly 0.,. I UT = O~, and 7r 0 0.,. = J.L hecause these two homomorphisms agree on r.,.. Finally. take f E qr T 1. Since J.L is compatible with;;;. there exists 0' E C- with ().,.(f) E w(f) . (neA +A1)+I. But I c \I!(f)M. and so O.,.(f) E w(f) . (M# \ M). This completes the construction of 0.,.. The construction of ().,.o is similar. and SO we have the required family {e" : ro ~ (T < WI }. Finally, we define a map () : qr u Uo} 1 -+ AI by the requirement that I C[r,,1 = ()" whenever ro ~ (J' < WI· The map 0 is well-defined and has the required properties. 0
o
We next show that the algebra A (defined in 4.7.29) is a universal algebra: recall that A is an integral domain which is the union of a chain of semisimple Banach algebras. We make a small modification to tIl(' cone S used ahove. Indeed, define go(t) = It IIj'l. (t E JR), and define
T =
n{
nuo
+S
:n
E
N} ;
the function gl : t 1-+ (It I /2)1/2 belongs to T. By 1.2.21(ii), (T. ~T), is an 1fl-cone over Q+ •. Let F and P be associated with f E T. as above. By A.2.33, (v) and (vi).
Znp
E HOC
(T/
E
N) and :r-1\og IF(x)1
-+
0 as
x -+
00.
In particular, the
functionH associated with the special function g1 are Zl/2 and cxp( _Zl/2), respectively. Define
as in (4.7.16). ho is the inverse Laplace transform of exp( _ZI/2).
Theorem 5.7.24 (Dales) Let fo E M- and (0 E C-. Then there it; an embedding () : M - 7 A such that ()(fo) = (oh o and such that meh II E ()(M-) has the following properties:
(i) It is injimtely dzffer'entzable, and h("')(O) = 0 (k E Z+): (ii) a(h) = 0.(iii) II. E U{O(Ll(c.:,,» : (J' > O}. In partzcula''', A is u umversal algebra.
Proof Fix (J' > 0, and write (M, .) for (Ll (c.:.,.), *). For JET, define r(J) = .c.-I(F) IlR+. Since znp E H'X! (n EN), necessarily P E HI(II), and so, by 4.7.16, r(J) E M, r(f) is infinitely differentiable on JR+. and r(f)(k)(O) = 0 (k E Z+). Since X-I log Ip(x)l-+ 0 as x -+ 00, r(J) does not satisfy 4.7.19(b) for any a > 0, and so, by 4.7.19, a(r(J» = O. Hence, by 4.7.65, r(J) E OeM). In particular, r((09d = (oh o .
A utomatzc continuity theory
776
The map T : (7, +) ----> (M, . ) is a morphism. By 1.2.32. there is a morphism ----> 7 with 'T/(v(fo)) = (Ogl. Definp IjJ = T 0 'T/, so that 'I/; : P ----> M is a morphism with 'I/;(P) c n(M). Take r such that rU Uo} is a transcendence basis for M. By 5.7.23, there is a homomorphism e: qr u {.fo}]----> M with e(M-) c nov!) and (J(fo) = (011,0. "Ve finally apply the algebraic extension theorem 1.7.42. By 4.7.30, the algebra A# is local and Henselian. We regard 'If; : P ----> A-as a framework map and e : qru{.fo}] ----> A# as a homomorphism which is compatible with '1/;. By 1.7.42, (J extends to an embedding e : M ----> A which is also compatible with '1/;. Since O(M-) C T(T) . lnv A#, each h E e(M-) has properties (i) (iii). 0 'T/ : P
Of courst'. it call be arranged that many functions in A other than (011,0 are the image of a specified element of M- . It is immediate from 5.7.24 that we obtain the collection of universal Banach algebras specified in the next theorem; for clause (iii), we use 4.7.67. Theorem 5.7.25 (i) Let w be a radzcal weight fnnction on IR+ snch that w is bounded near the orzgzn. Then LI(w) 1S a universal. radzcal Banach algebm. (ii) The Volter-m algebm (V. 11·111) and the subalgebra (Co.*. sal, radical Banach algebras.
I· lIT)
are Uni1)Cr-
(iii) Let R be a commutative, radzcal Banach algebra contazning a non-zem, ('ontznuou8, bounded real semzgroup. Then R zs a unzversal algebra. 0 Corollary 5.7.26 (CH) Let n be an infinite, compact spare. Then there zs a dtsrontinuous homomorphism from C(n) into each of the followzng local Banach algebms: (i) Ll (w)# for each contimtO'u,s. radzcal wezght flLnctw71 w on ~+; (ii) V#; (iii) 0
ct*.
Note that Ll(u)) is an integral domain. but that V and C o.* have dense sets of nilpotents; L 1 (:..;) and Veach have a bounded approximate identity, hut Co,* doeb not. By 2,9,43(i). each commutative, radical Banach algebra with a bounded approximate identity satisfies the condition ill 5,7,25(iii). and so is univen;al. It is notationally amusing that there is a discontinuous homomorphism from (Co (II) , . ) into (Co (II). * ). Our next objective is to characterize universal, commutative, radical Banach algebras. Recall from 5.7.1 that a commutative, radical Banach algebra R COIlt.ains a copy of the maximal ideal Ah of ~ if and only if R contains an element of finite closed descent. We shall prove that R iI:; universal if and only if it satisfies exactly thiH conditioIl, Theorem 5.1.27 Let R be a commutatzvc, radical Banach algebra containzng an element a of finite closed descent. Then there 1,S a closed zdeal RI in R such that Ia n neRd contain:; a universal algebra. Proof Suppose that 8R (a) = k, and set RI = akR, so that RI is a closed ideal in Rand a k E n(RI ) by 2.2.11(i). Clearly n{anRl : n E N} = Ia. The result follows from 5.7,14 and 4.9.16, 0
Embeddmg algebras in Banach algebras
777
Theorem 5.7.28 (Esterle) Let R be a commutative, rad,teal Banach algebra. Then the following condztwns on Rare equzvalent: (a) R contains an element of finite closed descent; (b) R contazns a nO'Tt-zem, rational semigmup;
(c) R contams a non-zero, real semigroup; (d) there zs a sequence (an) in R- such that an E a;+lR (n EN);
(e) R zs a unzversal algebra. \ Proof It was proved in 4.9.14 that (a)¢:}(b)¢:}(d). By 5.7,9, (e):::}(c), and, 0 trivially, (c)=Hb). By 5.7.27, (a):::} (e). Corollary 5.7.29 Let....; be a radical wezght on Q+-. Then f 1(Q+-, w) is a universal algebra, and it contazns a non-zero, real semigroup. Proof Certainly fl(Q+_, w) satisfies condition (b) in 5.7.28.
o
We note that there is no obvious, non-zero, real semi group in the algebra £l(Q+_, w). Theorem 5.7.30 (CH) (Esterle) Let A be a commutative, unital Banach algebra. Then the followzng conditwns on A are equzvalent:
(a) A zs a universal algebra; (b) rad A r.ontains an element of finzte closed descent; (c) there 'ts a 'unital embedding of'J = q[XJ] into A; (d) there lS a discontinuous homomorphism from C(O) into A for some compact space 0; (e) there is a dzscontinuo1Ls homomorphism from C(n) into A for each infinite, compact space n; (f) there zs a unital embedding of B in A for each unital zntegral domam B with a character and with iBI = c. Proof Condition (a) holds if and only if radA is universal, and so (a)¢:}(b) by 5.7.28. By Allan's theorem 5.7.1, (b)¢:}(c). The implications (f):::}(e):::}(d) are trivial. Finally, (a):::}(f) by 5.7,11(ii), and (d):::}(b) by 5.4.32. 0
Let A be a commutative, unital Banach algebra, and let P be a prime ideal with lA/Pi = c. Then, by 5.7.19(ii), the algebra A/P is normable (with CH); we now show that there is a discontinuous homomorphism from A with kernel P, even in the case where P is closed. Theorem 5.7.31 (Esterle) (CH) Let A be a commutative, unital Banach algebra, let P be a non-maximal, prime zdeal such that lA/PI = c, let 111·111 be any norm on A, and let B be a unital, universal Banach algebra. Then there is a homomorphism e from A into B such that ker e = P and e is discontinuous on (A, III· III)·
778
A'ittornahc confinU1fy th('()'~1J
Proof Let 7r : A ----> AlP be the quotient map. By 5.7.1O(ii). we may regard AI P as a :mbalgebra of M#. By 4.8.5. AlPhas an infinite tnwscendeuGc ba. . i~, and so there is a Sl't {an : n EN} in A such that {7f( an) : n EN} iH algebraically independent in M; take r tu be a transccndence basis of M containing this set. Since B is a universal algebra. rad B is It universal algehra. and so. by 5.7.27. there is a closed subalgehra Rl of rad B such that O( R 1 ) contains a univl'rsal algebra. and so there is a framework map II' : P ----> R J with i)(P) c n(R J ). By 5.7.23. there is a homomorphism JL : qr] ----> HI which is compatiblE' with lj) and such that 11(1-£ 0 7r)(a n )11 = n Illanlll (n EN). By the algebraic extension theorem 1.7.42. the map It extends to an l'mbedding II,: M# ----> Rf. Set () = It 0 7r. Then () : A ----> B is a homomorphism with ker () = P; abo. 1I(}(a,,)11 = n Illanlll (n EN), and so () is discontinuous on (A. 111·111). 0 For example. let A and B each be one of the following algebra.... : Ll(w), where is a radical weight on 1R+ and w is hounded near the origin: the Volterra algebra V; the algebra Co.*. Let 111·111 be any norm on A. Then the1e is a discontinuous homomorphism from (A III . III) into B. We now seek to show that there iH a discontinuouH homomorphism from each infinite-dimensional, comnmtativl' Banach algebra.
u)
Theorem 5.7.32 (Esterle) (CR) Let A be a cornm'ILtative Banach algebm. and let III· III be any norm on A. Suppose that A satzsjies at least one of the foliounng conditions:
(i) there zs a non-rnaxzmal. przrne 1.deal P m A with IAI PI = c: (ii) A ~8 mjinite; (iii) A is algebraic and injinite-dimensional; (iv) A 1S radzcal and A2 has injinite cod'tTnens~on zn A. Then there is a d~scontznuous hornomorphzsmfrom (A, 111·111) znto .~ome Banach algebra. Proof Suppose that A satisfies (i). By 1.3.54, we may suppose that A is uuital. and then the required discontinuous homomorphism existH by 5.7.31. Suppose that A satisfies (ii). Then, by 4.8.7, A already satisfies (i). Suppose that A satisfies (iii) or (iv). Then the required homomorphism ('xistb by 4.8.9 or 2.7.12, respectively. 0 Assume that there is a commutative Banach algebra A not satisfying any of the conditions in the above theorem. Then there is a commutative, radical, Banach algebra R such that R2 has finite codimension in Rand IRI PI > c for each prime ideal P in R. No such algebra is known. Question 5.7.A Let A be an injimte-dimensional, commutative Banach algebra. Is there necessarily a dzscontinuous homomorphism from A into some Banach algebra? Let A be a commutative Banach algebra, and let a E A. We now consider whether there is necessarily a discontinuous homomorphism () from A into some
Embeddm.q algebras zn Banach algebras
779
Banach algebra such that 0 I Coral is discontinuous, or even such that 0 I coral is discontinuous; clearly a necessary condition for this is that a be non-algebraic.
Theorem 5.7.33 (CH) Let A be a commutatzve, unital Banach algebra, and let a E A be non-algebrazc. Suppose that either IAI = c or that O'(a) is mjinite. Then there is a umtal homomorphism 0 from A mto some Banach algebra such that O(exp a) i= exp O(a). In partzcular, 0 I Cla] zs discontinuous. Proof First note that A contains a prime ideal P such that a ~ CeA + P and IAI PI = c: this follows from 1.6.9 in the case where IAI = c and from 4.8.6 in the case where 0'( a) is infinite. Let rr : A --+ AlP be the quotient map. The map f f--+ rr(J(a» , O(C) --+ AlP, is a homomorphism; by 4.8.4. it is an embedding. By 5.7.11(ii), there is a unital embedding J.L1 : AlP --+ M#. The pair {(JL1 0 rr)(a), (JLl 0 rr)(exp a)} is algebraically independent in M# because {Z. exp Z} is algebraically independent in O(C). By 1.7.39, there is an automorphism J.L2 on M# with (J.L2
0 J.Ll 0
rr)(a) = (JL1
Let JL3 : M# and set
--+
0
rr)(a)
and
(JL2
0
JL1
0
rr)(exp a)
i= (JLl
0
rr)(exp a).
B be an embedding into a unital, universal Banach algebra B,
(h
= JL3
0
/l2
0
J.L1
0
rr,
O2 = JL3
0
JLl
0
rr.
Then 01 and O2 are homomorphisms from A to B such that 01(a) = 02(a) and (h (exp a) i= O2 (exp a). At least one of 01 and O2 satisfies the required condition on 0. 0
Theorem 5.7.34 (Dales) (CH) Let A be a p,.echet function algebra. Then the lollowzng conditions on A are equivalent: (a) there zs a discontznuous homomorphism from A into some Banach algebra;
(b) there zs a discontmuO?LS homomorphism from A znto some LMC algebra;
(c) there is an injimte subset S of EA such that f(S) lEA.
Z8
bounded for each
Proof (a)*(b) is trivial, and (b)*(c) is 4.10.19. (c)*(a) The result is trivial if A is not functionally continuous. and so we may suppose that EA = R# be a non-zero homomorphism such that e I J = 0, where R is a commutative. radical Banach algebra, and define e to be the restriction of ~z ® e to ~, so that e : ~ -> M2 (R#) is a discontinuous homomorphism into a Banach algebra. Clearly e I M2(J) = 0 and I(e) = Mz(M). The space F of 5.4.40(iii) is equal to lin{Eu,Ezz} in this case, and so the singular part p, of e coincides with on Mz(J) and is such that p,(Ell ) = p,(E22) = O. As~;ume that there are linear maps Ji'l,tlz : Q{ ----> M 2(R#) such that p, = 11'1 + JLz and {11 19J11 and Jl2 I 9J1 2 are homomorphisms with Jl1(P(9J1 1 )) = JL2(P(9J1 2)) = {a}. Since P(9J1 1 ) = 9J1 1 and P(9J1 2) = 9J1 2 , necessarily JlI (9J1 1 n 9J1 2 ) = {I I M 2(M) = 0, a 0 contradiction. Thus there is no decomposition of {I.
e
Let A be a commutative Banach algebra, and let a E A. \Ve now seek to construct a homomorphism () from A into a Banach algebra such that I Cola] is discontinuous; for this, it is sufficient to arrange that {"()(a n ),, / "anll : n E N} be unbounded, and we now attempt this. \Ve fix ho to be as in 5.7.24. and take 'Yo E jR+- such that 'Yo Jo1 ho = 1.
e
Theorem 5.7.37 (CH) Let A be a commutative, umtal Banach algebra, and let a E A. Suppo8e that there ar'e a prime ideal P in A and 'P E A 8uch that P c M L1(w)# such that IIO(an)11w / "an" -> 00 as n -> 00. Proof First we consider the case where vA(a) > 0; we may suppose that \D(a) = vA(a) = l. Take (En) in jR+- with En -> 0 and nEn -> 00 as n -> 00 and such that !Ian" ::.; (1 + En)n (n EN): for each kEN, there exists Nk 2: k with
(n) k
k
En
2:
~ ~
(n).
j=O
J
(5.7.6)
j En
Let w be a continuous, decreasing, radical weight function on jR+ such that wen) 2: 2nE~ (n EN). By 5.7.11(i) and 5.7.24, there is a homomorphism e : M.p ----> L1(W) with ker e = P and e(a - eA) = 'Yoho; extend 0 to be a unital homomorphism, also called e, from A to £l(w)#. Take kEN. Then, for n 2: N k , we have
II()(an)IIw = 11(150
2: 2k
+ 'Yoho)*nllw 2: ~ C)W(j)
t (r~)E~ t (r~)E~ 2: k
J=k
J
j=O
by 4.7.31(ii) by (5.7.6)
J
= k(l + En)n 2: k Ilanll . Thus lIe(an)lIw / II an II The case in which
-> 00
as n -> 00. 0 is an easier version of this calculation.
VA (a) =
0
782
Automatic continuzty theory
Let A be a commutative, unital Banach algebra with IAI = c, and let a E A. Suppose first that a E rad A. Then the condition on a in 5.7.37 is satisfied if and only if a rt- IJl(A), and cle&rly this is a necessary condition for the conclusion to hold. If A is semisimple, then it follows from 4.8.6 that the condition on a is satisfied if {z E a(a) : Izl = vA(a)} contains a non-isolated point of a(a); however, a slllall elaboration of the argument in 2.3.36 shows that, if this condition on a fails, then {IIB(an)11 / Ilanll : n E N} is hounded for each homomorphism () from A into a Banach algebra, and so we must work a little harder to produce homomorphisms from A whose restriction to Coral is discontinuous.
Theorem 5.1.38 (CH) Let (A, 11·11) be a commutative, umtal Banach algebra, let a E A be non-algebrazc, and let III· III be any norm on C( a). Suppose that either IAI = c or a(a) zs mjinite. Then there are a continuou.~, rad~cal wezght function w on 1R.+ and a unital homomorphzsm () : A ----> £l(w)# such that: (i) ()(a) E lin{80 , ho};
(ii) B I C(a) is discontmuous with respect to 111·111; (iii) () I Cola] is discontmuous with reHpect to 11·11. Proof We may suppose that Illxlll 2: IIxll (x E A). Set C = Cla]. By 2.3.21(iii). 80'c(a) C aA(a) C ac(a). We clazm that there are a prime ideal P in A and
giving the result.
n--+oo
L
dm(xm, Bm(xm+d) = 8,
m=l
o
793
Topological and metTzc spaces
Corollary A.1.25 Let {Xn; ()n} be a projective sequence. Suppose that, for each n E N, (Xn,J n ) is a complete metrzc space and ()n : X n +l ~ Xn zs a continuous map such that ()n(Xn+I) is dense in Xn' Then 7rk(X) = X", for each kEN. Proof Set d l = 151 , amI inductively define (d n : n
d,,(:r:, y) = 6n (:1:, y)
~
2) by the formulae
+ dn- l (()n-l (x), ()n-l (Y))
(x, Y E Xn) .
Each dn is a complete metric defining the topology of Xn, and (A.1.2) holds. Let kEN, and take Xk E X k and c > O. Then we may inductively choose Xn E Xn for n ~ k + 1 sHch that dn(:r: n , ()n(Xn+l)) < c:/2" (n ~ k). By A.1.24, for each n ~ k there exists Yn E Xn such that ()n(Yn+d = Yn (n ~ k) and dk(xk, Yk) ::; c. Set Yj = 7r),k(yd (j E Nk-d. Then (Yj) E X, and so dk(Xk, 7rk(X)) ::; c. Thus 7rk(X) = Xk. 0
Proposition A.1.26 Let (w n ) be a sequence in IR+·.
Wn _
Wn·
(1.) l'nn SUPn_.oc lin < l'1111 SUPn--->oc Wn + 1 / (ii) Suppose that L:~=1 Wk/Wk+l < 00. Then limk-+oc(k!/Wk)l/k (iii) Suppose that Wm+n ::; WmWn (m, n EN), and set p · -+ Wnlin = p. Th en IIm n oo
= O.
= inf{w!/n : n EN}.
Proof (i) Set p = lim sUPn-+oc wn+dwn. We may suppose that p < 00. Take T > p. There exists N E N such that Wn+l < TW n (71 ~ N), and then we have Wn ::; WNT n - N (71 > N), so that limsuP n-+ oo w~/n ::; T. (ii) Take c > O. There exists N E N with L:':=N Wk/Wk+1 < c/4. For each n EN, we have lin (
W::n
and so, for each 71
)
~
N, we have
(N + n)! < _1_ . (N + n)! . (~)n ::; N! . (~)n WN+n - WN (2n)n 2 WN 2 Thus, for n sufficiently large, ((N
+ n)!/WN+n)l/(N+n) < c.
The result follows.
(iii) Take c: > O. Then there exists mEN such that Wm < (p + c:)m. Set M = max{wl, ... ,wm }. For n > Tn, take l' E N with n E {rm+ 1, ... , (1'+ l)m}. Then Wn ::; Wn-TmW~, < ]vI (p + c: m , and so lim sup w~/n ::; p + c:. Sinct' p::; w~/n (n EN), it follows that p = lirnw~/n. 0
r
Notes A.1.27 Standard texts on topology include (Bourbaki 1960), (Engelking 1977), (Kelley 1955), (Kuratowski 1966), and (Munkres 1975). The results on disconnected spaces and on the order topology are also in (Gillman and Jerison 1960). The name 'Mittag-Leffler theorem' for A.1.25 (Bourbaki 1960, Chapter 2, §3) arises because the classical Mittag-Leffler theorem on meromorphic functions (Rudin 1974, 15.13) is an easy consequence of A.1.25. For further versions of the Mittag-Leffler theorem, see (Esterle 1984a).
794 A.2
Appendix' COMPLEX ANALYSIS
We shall recall here some results in the theory of analytic and suhharmonie functions: basically. we shall consider functions of oIle complpx variable. but we shall also make a few remarks about analytic functions of several complex variables. The set of complex numbers forms the comple:r plane e; til(' plane is taken to have the usual topology. The coordmate functzonal on e is Z : Z f--> z. For z E e, we set: x = ~z, the real pa.rt of Z; y = ;.rz. the zmagmary part of Z; r = Izl, the modulus of Z; and (if z =I- 0), () = argz. the argument of z, taking () in the interval (-7f. 7fl. Thus z = x + iy = re i8 . The complex conjugate of z is Z = x - iy. In the case where ( E e, we set ~ = IR( and Tf = ;.r(. There are certain subsets of e for which we have a fixed notation. Let Zo E e and r E ~+ •. Then
][))( Zo; r) = {z
E
e:Z1
Zo 1 < r}
1r( Zo; r)
and
= {z
E
e:z1
Zo 1 = r}
are the open disc and circle, with centre Zo and radius r, respectively. \Ve spt
][)) = ][))(O; 1) and 1r = 1r(0; 1). For lTl, a2 E ~ with al < a2. define TIO'l,0'2 = {z E e : a1 < x < 0'2}, TIO'l.O'J = {z E e : lT1 ::;.r ::; a2}, so that TI"'1,0'2 is an open vertical strip and ITO'l,0'2 is its closure. For a E ~. spt
TI",
= {z : x > a}. IT", = {z : x 2 a},
= {z : x < a}. uTI = {z : .r ::; a},
uTI
so that TIO' is an open right-hand half-plane, with closure nO', and uTI is a lefthand half-plane. We write TI for TIo and IT for ITo. Finally. set V 0' = {z : x = a}. so that V 0' is a vertzcal line; V = Vo is the imaginary axis. Let () E (-7f, 7fl. The ray Ro is the half-line R8 = {z = re iO : r 2 o}. A sector is a set of the form U{R.(I : () E I} or U{Re : () E I} \ {O}, where [ is an interval in (-7f,7fl (so that e itself is a sector); the angle of a sector is the length of [. For 1j; E (0, 7f), we write S'Ij; for the sector S(z, t) is an analytic function on U for each t E J; (ii) t H 1'>(z, t) is a measurable function on J for each Z E U; (iii) for each compact subset K of U, there exists PK E L1(1J-li) such that 11'>(z, t)1 ::; PK(t) (z E K, t E J). Define w(z) = fJ 1'>(z, t) dt (z E U). Then W E O(U), and w'(z)
=
1~~
(z, t) dJ-l(t)
(z E U).
o
The gamma function f is defined on II by the formula
r(z) =
1=
(z E II).
tZ-1e-tdt
(A.2.2)
By A.2.14, f E O(ll). We have f(z + 1) = zf(z) (z E II), and this implies that f( n + 1) = n! (n E Z+); also, f(1/2) = Ti I / 2 . The reciprocal of the gamma function ha.'5 an extension to an entire fUllction on C; the extension has zero set Z-. Fix ;3 > O. Then we have the asymptotic estimate: (x - .B)L3r(x - !3)/f(x)
->
1
as
x
->
00.
(A.2.3)
Definition A.2.15 Let u : U -> [-00, (0) be a function on a non-empty, open subset U of C. Then: (i) u satisfies the mean inequality on U if u is bounded above on non-empty, compact snbsets of U and if, for each z E U and r > 0 with Jl))( Z; r) C U, the function () H u(z + re iO ) is measurable on (-Ti, Til and
u(z) ::; - 1
1'"
2Ti _'"
n(z + re iO ) d();
(A.2.4)
(ii) u is subharmonic if it is u.s. c. and satisfies the mean inequality on U; (iii) 1£ : U -> IR is harmonic on U if both 'u and -1£ are subharmonic on U. A function u : U -> C is harmonic if ~u and <Ju are harmonic in the above sense. Thus 1£ is harmonic if and only if 1£ E C(2)(U) and u satisfies Laplace's equation: fPu fPu &x2 + &y2 == 0 on U.
Let f E O(U). Then f, ~f, and ':Sf are harmonic on U. Suppose that ~ is a disc in C and u : ~ -> IR is harmonic. Then there exists f E O(U) such that 1£ = ~f; the function f is unique up to an additive constant. Thus each harmonic function is infinitely differentiable.
799
Complex analyszs
To verify that a u.s.c. function u : U -+ [-00,(0) is subharmonic. it suffices to show that, for each z E U, there exists rz > 0 such that (A.2.4) holds for each r E (0, rz). It follows that log If I is subharmonic on U for each f E O(U).
Proposition A.2.16 Let U be a non-empty, open set
me
(i) Let u and v be subharmonic functions on U, and let u V v and (Xu + j3v are subharmonic on U.
(x,
j3 E IR+. Then
(ii) Let (un) be a seq1lence of subharmonzc functions on U such that (un(z) is decreasing for each Z E U. Set u(z) = infnEN un(z) (z E U). Then u is subharmonic on U. (iii) Let u be subharmonic on U, and let
0 such that 'll'(z; r) n U =I 0, and then J::'", u(z + rei/}) dO = -00. Since u satisfies the mean inequality, u(z) = -00. (ii) Choose c, dE IR with a < c < d < b, and let Zo be the centre of a square S in C, one of whose sides is the interval [e, d] in lR. Set
v(z)
= u(z) + u(zo + i(z -
zo))
+ u(zo -
(z - zo»
+ u(zo -
i(z - zo»
(z
E
q.
Then v is subharmonic on C, and v(z) = -00 (z E DeS). It follows from A.2.18 that v(z) = -00 (z E S). But v(zo) = 4u(zo), and so u(zo) = -00. This proves that u == -00 on a non-empty, open set of C. By (i), 7.l == -00 on ce. 0
800
Appendzx
The following two results are, respectively, Liouville's theorem and a form of the Phragmerz ~ Lzndelof theorem for subharmonic functions. Theorem A.2.21 (Liouville) Let 11 be a subharmonic function on C such that u is bounded above on C \ !DI(O; r) for some r > O. Then u I,S a constant. In partic71.lar, a bounded ent'tre f71.nctwn is constant. 0 Theorem A.2.22 (Phragmen-LindeI6f) Let S be an open sector in C of angle 7r ja, where a::" 1, and let u be a s71.bharmonic function on 8 wzth :t(z) = 0(lzI 6 ) as Izl --+ x in S for some (3 < a and such that limsupz-+zo u(z) ::; 0 for each Zo E 88. Then u(z) ::; 0 (z E 8). 0 Corollary A.2.23 Let u be a subharmonic function on II s71.ch that: (i) limsupz~iyO u(z) ::; 0 for each Yo E lR; (ii) 71.(z)
= O(lzl 13 ) as Izl
--+
(iii) limsuPr->oou(reie)/r'
l. Fix n > /3, take () with lei::; 7r(a - 1)/2a, and let Uo be the open sector {z E C·: largz - el < 7rj2a}. It follows from (iii) and the fact that u = 0 on V that there exist constants Go and Do such that u( z) ::; Go + Dox (z E 8Uo). By A.2.22 applied to the subharmonic function z f-7 u(z) - Ge - Dex on the sector U(), we sec that u(z) ::; Ge + Dox (z E Ue). Since II can be covered by finitely many such seeton, Ue. there exist constant" C and D such that
'l1(z)::;G+Dx
(z=:r·+iYEII).
We now repeat the above argument, with a = 2. on thE' first and fourth quadrants. Take N E lR with N > limsnPr-+oo 11,(r)/r. By A.2.22, there is a constant G > 0 such that u(z) ::; C + Nx (z E II). By again applying A.2.22. now to the function z f-7 u(z) - Nx, which is bounded above on II, we see that u(z) ::; Nx (z = x + iy E II). The rer:mlt follows. 0 Let X be a subset of C. For r > 0 such that {z EX: Izl FE C(X), define
MF(r) = sup{IF(z)1 : z E X, Izl
= r},
= r} =f.
0 and for
(A.2.5)
so that 0 ::; lvh(r) ::; x. Definition A.2.24 Let 8 be an open sector in C, and let F E O(S). Take p, T E lR+. Then F has order p in 8 if P = limsuPr-+oo log+ log+ MF(r)j logr, and F has exponential type T in 8 if T = limsuPr-+oo log+ MF(r)jr. Further, F has exponential type if F has exponential type T for some T E lR+, and F has minimal exponential type if F has exponential type O.
Complex analysis
801
Thus F has exponential type at most T on S if, for each E > 0, there exists C c > 0 such that W(z)1 :S Co expeCT + E) Izl) (z E S), and F has exponential type on S if there are constants a and b in lR+ such that IF(z)1 :S aexp(b Izl)
(A.2.6)
(z E S).
Theorem A.2.25 (Phragmen-Lindel6f) Let S be an open sector of angle n/n, where n ~ 1, and let F E O(S) have order at most /3, where /3 < n. (i) Suppose that limsupz~zo W(z)1 :S M for each Zo E
as.
Then
Wls
:S M.
(ii) Suppose that there exzsts ro > 0 such that limsupz--->zo IF(z)1 :S M for each Zo E as with Izol 2: roo Then limsuPlzl->oo W(z)1 :S M.
Proof (i) This follows from A.2.22, applied with u = log+(WI - M). (ii) Set G(z)
= z(z + a)-l F(z)
limsupz~zo IG(z)1
:S M (zo
E
as).
(z E S), where a > 0 is chosen so that By (i), IGls :S M. The result follows. 0
In a standard form of the theorem, S = IT, and it is then required that /3 < 1; a bounded function satisfies this condition. By using a conformal map, we obtain a similar result for a strip.
Corollary A.2.26 Let F E A b (ITa! ,(2)' where lTl < lT2 in R. Suppose that + iy) --+£ as Iyl--+oo for j = 1,2. Then F(z) --+£ as z --+ 00 zn ITO"!,0"2' 0
F(lTj
There is a modification of A.2.25 that we use; the proof is similar to part of that in A.2.23.
Theorem A.2.27 Let F E O(IT). Suppose that F is of exponential type zn IT, that limsuPz->iyO W(z)1 :S M (Yo E lR), and that there exist r.p E (-n/2,n/2) and c E lR such that IF(rei'P)1 = O(e(c+c)rCOS'P) as r' --+ 00 for each E > O. Then IF(z)1 :S Me cx (z E IT). 0 Corollary A.2.28 Let F be an entire function of minzmal exponential type. Suppose that there exist k, £ EN such that IF(x)1 I Ixl i is bounded on lR and liminf W(z)1 I Izlk Izl--->oo
= O.
Then F is a polynomial of degree at most k - 1.
Proof Set G(z) = F(iz)/(z + 1)£ (z E II). Then G satisfies the conditions on Fin A.2.27 (with r.p = c = 0), and so G is bounded on IT. It follows from this and a similar etltimate on oil that there exists M > 0 with
W(z)1 :S M(lz + 11£ + Iz - 11£)
(z
E
q.
Since IF(n)(o)1 :S n!MF(r)lr n (n EN), we have F(n)(o) = 0 (n> f), and so F is a polynomial of degree at most £, say aF = m. Assume that m 2: k. Then there exists n > 0 such that W(z)1 > n Izl m for Izi sufficiently large, a contradiction of the fact that liminflzl~oo W(z)1 I Izlk = O. Thus aF :s: k - 1. 0
802
Appendix
A related result is slightly stronger for entire functions than the above Liouville theorem. For an entire function F and r > 0, set
= sup{(3W)+(z) : Izl = r}.
AF(r)
Proposition A.2.29 Let F =
2::'=1 anZn
lanl rn :S 4AF(r)
be an entire lunction. Then
(n E N, r
> 0).
Suppose that Ap(r) = O(rk) as r ---7 Xl. Then F is a polynomial most k. In particular, F = 0 whenever AF(r) is bounded.
01 degree
at 0
Definition A.2.30 For ( = pei'P E lDl and () E (-7r, 7rJ, set P,
k. Then lim sup IF(z)1 :::; em Izl-+oo (v) Suppose that inf J >
k/ 2 .
and limltl-+oo J(t)/ It!,,' = Then, for each n EN, we have zn F E H OO (II). (vi)
X-I
log IF(x)l-- 0
-00
00,
where a > O.
as x -- 00.
Proof (i) For z E II, we have (~F)(z) 2 -m, and so lexp(-F(z»1 ~ em. (ii) Apply (i) to the function - {
Appendix
804 (iii) For z E II, we have (~F)(z) ~ -m
k
+ -1f
1
xdt
00
to
X
2
+
(
y- t
)2
=
-m
k
+ -1f
jy-to _00
xdu 2 X +u 2
.
Thus (~F)(z) ~ -m + k/2 whenever y ~ to. Similarly this inequality holds whenever y :S -to· (iv) Since F E H''''(II), this follows from (iii) and A.2.25(ii). (v) Take to E JR such that f(t) > Iii'-" (It I > to). By (iii), there is a constant
~ to.
C > Osuch that iF(z)i:s Cexp(-IYIl> /2) for z E II with IYI
exp(Zl»F is bounded on II, and so zn F is bounded for each (vi) For x > 0, we have
~ log iF(x)i = -~ j:xJ X
1f
and so x-llog iF(x)i---+ 0 as x
---+
_00
71
By A.2.25(ii),
EN.
( 1 ++t22) f(t) dJL(t) , t X
2
o
x by A.4.6.
The following set of functions will be used in §5.7: we write 'lim f = an abbreviation for 'limiti--+oo f(t) = 00'.
00'
as
Definition A.2.34 Let S be the set of locally bounded functions f in Li (11.) such that f(t) = f( -t) (t E JR) and lim f = 00. Let f E Sand n E N. We temporarily set fn = f 1\ n, and write Fn and Fn for the corresponding functions given in (A.2.9) and (A.2.1O), respectively. Clearly fn ---+ f in Li(JL) a.'l n ---+ 00. By A.2.33, (i) and (ii), Fn , 1/ Fn E HOO(II). We denote by Ho(II) the set of functions F E HOG(II) such that IF(z)1 ---+ 0 as z ---+ 00 in II. Proposition A.2.35 Let f E S. Then: (i) FE Ho(II);
(ii)
!F(z)i :S iFn(z)i :S IFm(z)1 for z
E II and m:S n m N;
(iii) for each bounded subset B of II, F / Fn (iv) Fn
---+
F m H=(II) as n ---+
---+
1 uniformly on B as n ---+
00;
00;
(v) for each G E Ho(II), GF / Fn ---+ G as n ---+
00.
Proof (i) By A.2.33(i), FE Hoo(II): by A.2.33(iv), IF(z)l---+ 0 as z
---+ 00.
(ii) For m :::; n, we have fm :::; fn :::; f on JR, and so RFm :::; RFn :::; ~F on II. (iii) Take k > 1 such that B c ][)leO; k), and then take no E N such that f(t) = fno(t) (t:::; 2k). For z E Band t E JR with It I ~ 2k, we calculate that I(tz + i)/(t + iz)1 :S 2k + 1, and so W(Z) - Fn(z)1 :::;
Hence
F/ Fn =
2k+
11
-1f-
exp(Fn - F)
---+
00
-00
IJ(t) - fn(t)1 dJL(t)
1 uniformly on B as n ---+
(z
E
00.
B, n ~ no).
Complex analyszs
805
> O. Then there exist no EN and a bounded subset B of II such that IFno(z)1 < I:: (z E II \ B). Using (ii), we see that (iv) Take
I::
IF(Z) - Fn(z)/ :::; IFno(z)IIF(z)/Fn(z) and so IF - Fnl
n\B
-11:::; 2IFno(z)1
(z E II, n 2: no),
< 21::. The result follows from (iii).
(v) This follows similarly.
0
For n EN, define
En(z) = exp( -zln)
(z
E
IT).
(A.2.11)
Then En E A b (II) and E;: = E 1 . Proposition A.2.36 Let A be H(f(II) 01' Au (II). (i) The map LEI: G r--+ El G, A -* A, zs a linear zsometry, and
LEI (A)
is a
proper, closed subspace of A.
(ii) For each G E A, EnG
-*
G zn A as
71 -* CXl.
Proof (i) Far each G E A, IE1Gln = IGln. and so LEI is a linear isometry. Thus LEI (A) is a closed subspace of A. Set Go = (Z + 1)-1. Then Go E A, but Go fj. LEI (A) because e T I(x + 1) -* 00 as x -* 00, and so LEI (A) I- A. (ii) We have lEnin = 1 (n EN), and En -* 1 uniformly on bounded subsets of II. The result follows. 0
Let V be as in (A.2.9). By calculation, we have V(x+iy)
= _If"" ( 7r
We set Fx(Y)
-00
x
2
Y(-~t)2 +-1 tt )f(t)dt. y + '2
+
(A.2.12)
= F(x + iy) for a function F on II.
Theorem A.2.37 Let f E L1(f.L) have Poisson zntegml U.
(i) FOT almost all y E JR, lim:r--+o+ U(x + iy) = fey). (ii) Suppose that f is bounded and continuous on JR.. Then limx--+o+ Ux = f uniformly on each compact subset of JR, and U has an extension U in C b (IT) such that U(iy) = fey) (y E JR). (iii) Suppose that f is real-valued and continuously diffc'rentiable 011 JR, and that f and f' are bounded. Then V has a contznuous extension to II. Proof (i) This is immediate from A.2.31(i). (ii) Since fIR P:r = 1 (x> 0), U is bounded on II by
(U" - f)(y) =
I:
Px(t)(f(y - t) - f(y» dt
Ifl lR , and (x
+ iy E II).
Let K be a compact subset of JR.. For each 8 > 0, we have
lUx - 11K::;
(r
J 1tl:5,O
+
r)
J1tl>O
Px(t) sup yEK
I/(y - t) - l(y)1 dt (x> 0).
806
Appendlx
Now ~fl>t5 PAt) dt -+ 0 as x -+ 0+, and sUPYE:K If(y - t) - f(y)1 -+ 0 as t --+ 0 bccause f is uniformly continuous on a neighbourhood of K. The result follows. (iii) For lJ E lR. decompose the integral for V(x+iy) in (A.2.12) into the parts where It - yl :::; 1 and It - yl > I, respectively. and let x -+ 0+ in the integrals so formed; by A.4.6, -71"V(J: + iy) converges to
J.1
fey - s) - fey
o
+ s)
db _
,1
yt). f(t\ dt. l+t
y-t
This limit is called -71"V(iy). It is straightforward to chcck that
W(x
+ iy) -
V(iy)1 :::; -1 (21f'IR 71"
+ Ifla)
1
00
2x
-:)0
2
+ s 2 ds = O(.c)
X
uniformly in y. and so there is a continuous extension of V to
as x
-+
IT.
0+
o
\Ve shall require some basic facts about the Hardy spaces HI on half-planes. Definition A.2.38 Let F E o (llT ), where
17 ::::
O. Then F E HI (llO") if
~~~l: IF(x + iy)1 dy < 00.
IIFIII =
Theorem A.2.39 Let F E HI (llO") , where
17 ::::
O.
(i) The functwns F,r, converge zn LI(JR) as x
and 1
F(x+iy)=11'
(ii)
IlFxliI
;.00 ( -00
x-
x -
17
)2
(iv) Set f(()
-+ 00
-+ 17+
to a functwn, say FO",
17
+ ( y - t )2 FO"(t)dt
zs a decreasing function of x for.r:
(iii) !F(z) I -+ 0 as z
(A.2.13)
in llT for each
(x+iyEllO").
> 17.
7 > 17.
= F((j + (1 + 0/(1 - ()) (( E j[))). Then sup O m. Set G(z) = F(z + (]') (z Ell). Then G E N(II), and so there exists M > 0 with IG(z)1 = 0 (e M1z1 ) for z E S",. We have IF(z)1 = 0 (e M1z1 ) on S"', and we may suppose that the same estimate holds on -S",. Set S = {z E C : 1/) < arg z < 7r - 'Ij;}, and define
H(z) = F(z)eiMzCosec",
(z
E
S) .
Then S is a sector of angle 7r - 2'1j;, H is bounded on as, and H has order at most m. Since m < 7r / (7r - 2'1j;), it follows from the Phragmen- Lindel6f theorem A.2.25(i) that H is bounded on S. Thus W(z)1 = 0 (eMlzlcosec",) on B. Again We may suppose that the same estimate holds on -B. It is now clear that F is of exponential type on C. 0
Theorem A.2.49 (KreIn) Let F be an entire function which is a quotient of bounded analytzc functions on each of oIl and II. Then F is of exponential type on Co Proof By A.2.41, F satisfies (A.2.25) with m from A.2.4B.
= 4,
and so the result follows 0
810
Appendi:r
Notes A.2.50 The elementary complex function theOIY that we a.:;SUIll(' to he knowlI can be found in (Rudin 1974) and many other texts For the theory of analytic functioll1, of several complex variableH. Hee (Gunning and ROHHi 1965) and (Hiirrnander 1973). for pxample; A.2.29 b taken from (Browder 1969). For an attractive introduction to the theory of harmonic and subharmonic functions. see (Ransford 1995b): proofs of most of the results that we haw statp([ are given in thib sourCt', but A.2.2:3 iR from (Ransford 1997). A substant.ial trC'atrnent of subharmonic functions is given in (Hayman 19S9). For the theory of entire funct.ions (If exponential type'. see (Boas 1951). The relationships between a function and its extension calculatpd by lIsing a Poisson integral are explored in many texts, including (Duren 1970), (Garnett ]081). and (Hoffman 1962): the theory of Hardy spacps is also given in these sources. Nevanlinna's theorem A.2.46 and the Ahlfors-Heins theorem A.2.47 are proved in (Boa~ 1954,6.5.4 and 7.2.9): somewhat different proofR are given in (Sinclair 1982, Appendix 1). A stronger form of the Ahlford HE'ins theorem A.2.47 (for subharmonic functioll;') is given in (Hayman 1989. Theorem 7.39). A formula for the constant c is 2
c= lim -r--'= 7fr
j7'"/2 logIF(reili)lcos()d(). -,,/2
Krein's theorem A.2.41 is takpll from (Krein 1947).
A.3
FUNCTIONAL ANALYSIS
In this section. we shall gather together some results of functional analysis for reference. Some theory of linear spaces and linear maps to which we appeal is given in 31.3. Definition A.3.l Let (E. T) be a linear space (over q 'which is a topolog1,cal space for the topology T. Then E is a topological linear space if T 'tS a Hausdorff topology and 1,f the maps
(.1', y)
f---7
X
+ y.
Ex E
-7
E,
and
(a. x)
f---7
ax', ex E
-7
E.
are continuous.
Let E and F be topological linear spaces. A bijection T E L(E, F) is a linear homeomorph'ism if both T and T- 1 arc continuous; in this case E and Fare linearly homeomorphzc, written E ~ F. Let E be a topological linear space. We write NB for the family of open neighbourhoods of 0 in E. Let (xv) be a net in E. Then (xv) is a Cauchy net if, for each U ENE, there exists A such that xp - Xv E U whenever /1, ~ A and 1/ ~ A; a Cauchy sequence is a Cauchy net which is indexed by N. The space E is complete [sequentially complete] if each Cauchy net [Cauchy sequence] in E is convergent. Definition A.3.2 A topological linear space (E, T) is an (F)-space if there is a complete metnc on E which defines the topology T. A metric d on a linear space space E is invariant if d(x+z, y+z) = d(x, y) for all x, y, z E E. The topology of an (F)-space is induced by a complete, invariant metric; an (F)-space is a complete topological linear space.
F'unctwnal analysis
811
A subset S of a linear space E is: convex if tS + (1 - t)S C S (t E II); balanced if as c S (a E ~): absolutely convex if it is both convex and balanced; absorbing if U{ n5 : n E N} = E. The convex hull (S) of 5 i:s the smallcHt convex set containing S. Let S be a convex set. Then an element x E 5 is an extreme point of 5 if S \ {x} is convex; the set of extreme pointH of S is denoted by ex S. To show that x iH an extreme point of 5, it suffices to show that a; = y = z in the caHe where y, z E Sand 2x = y + z.
Definition A.3.3 A topological linear space is a locally convex :space if there is a base of neighbourhoods of 0 conszsting of convex sets. A Frechet space zs a locally convex (F) -space. Each topological linear space has a base V of neighbourhoods of 0 consiHting of balanced, absorbing sets, and a locally convex space has a base V of neighbourhoods of 0 consisting of absolutely convex, absorbing sets; in both cases, for each V E V, there exists WE V with W + We V.
Definition A.3.4 Let E be a lmear space. p : E --t lR such that: (i) p(x) ~ 0 (x
E
A seminorm on E zs a function
E);
(ii) p(ax) = lal p(x) (a E C, x E E); (iii) p(x + y) ::::; p(x) + p(y) (x, y E E). A norm on E is a seminom~ p such that: (iv) p(x) > 0 (x E Ee). We shall usually denote a norm on a linear space by 11·11 (or 111·111, or some variant). A normed space (E, 11·11) is of course a metric: space, and hence a topological space, with respect to the metric d defined by d(x,y)=llx-YII
(x,yEE);
we shall occasionally use the fact that a seminormed space (E,p) is also a topological space. A normed :space is a Banach space if it is complete as a metric space. The closed ball of radius r ~ 0 in a :seminormed :space (E,p) is denoted by E[r]
=
(E,P)[r']
= {x
E E:
(A.3.1)
p(x) ::::; r}.
We extend the notion of linear homeomorphism to seminormed spaces. For example, standard Banach spaces are: f',
{a ~ (an) ,lIall, ~ (~I"nl') ,/, < oo}
(p E [1,00));
foo = {a = (an) : lah'l = Iialloo = sup lanl < oo}; Co = {a = (an) E foo : an --t 0 as n --t oo}. For a non-empty, locally compact space 0, (Co(O), I . In) is a Banach space. For p E [1,00], the fP-norm on <e k is denoted by II· lip; the Euclidean no~ is 11.11 2 ,
Appendix
812
Proposition A.3.5 Let kEN, and let 11·11 be a norm on Ck. Then there zs a 0 linear homeomorphism £: (Ck, 11·11) -> (C k , 11.11 2 ) with 1J£1111£-111 Jk.
s
Proposition A.3.6 Let E be a Banach space, and let {x(n) : n E Z} be a set in E such that, for a constant K > 0,
Ilx(m+n)+x(-m)+x(-n)1J SK Then there exzsts Y
E
(rn,nEZ).
(A.3.2)
E such that Ilx(n) - nY11 S 4K (n E Z).
Proof Let n E Z. It follows from (A.3.2) that Ilx(4n)+2x(-2n)1J S K and that Ilx( -2n) + 2x(n)1J S K. Hence
+ 2x( -2n)11 + 2IJx(-2n) + 2x(n)11
Ilx(4n) - 4x(n)11 S Ilx(4n) Set Yk,n
S 3K.
(A.3.3)
= x(4 kn)/4 k (k E Z+). It follows from (A.3.3) that IIYk+1,n - Yk,nll S 3K/4k+1 (k E Z+),
and so (Yk.n : k E Z+) is a Cauchy sequence in E. Since E is complete, the sequence converges, say to y(n). Let k E Z+ and n E Z. By (A.3.2) with -4 kn for rn and 4k n for n, we have
IIx(0)/4 k + Yk,n
+ Yk,-nll S
K/4k, and so y(n) = -y( -n). By (A.3.2) with 4k for rn and 4kn for n, we have IIYk,n+1
+ Yk.-1 + Yk,-nll S
K/4 k ,
and so y(n + 1) + y( -1) + y( -n) = O. Thus y(n + 1) = y(n) + y(l), and so y(n) = ny, where y = y(l). We have lJy(n) - x(n)11 S 3KL:~=o 4- k = 4K (n E Z), as required. D
Definition A.3.7 Let H be a (complex) Imea.,. space. A map
[', .] : (x, 'Y)
J---+
[x, y],
H x H
->
C,
is an inner product on H if: (i) x
J---+
[x, y], H
(ii) [y,x]
->
C, is a linear functwnal for each y E H;
= [x,y] (x,y
E
H);
(iii) [x, x] 2:: 0 (x E H) and [x, x]
=0
only if x
= O.
Let [., .] be an inner product on H. Then y J---+ [x, y], H -> C, is a conjugatelinear functional for each x E H. Set Ilxll = [x, X]1/2 (x E H). Then IJ-II is a norm on H. In the case where (H, 11·11) is complete as a normed space, (H, [-, .]) is a Hilbert space. For example, (2 is a Hilbert space with respect to the inner product given by [a,,8] = L:~=l a n73n (a = (an), ,8 = (,8n) E (2). The CauchySchwarz inequality asserts that l[x,y]1 ~ Ilxlillyli (x,y E H). A family P of seminorms on a linear space E is sepamting if, for each x there exists pEP with p(x) =f. o.
E
Ee,
Functional analysis
813
Let E be a linear space, and let K be an absolutely convex, absorbing set in
E. Define PK(X) = inf{t
>0:x
E
tK}
(x E E).
Then PK is the Minkowski junctional of K; PK is a semi norm on E. Suppose that E is a locally convex space, and that V is a base of absolutely convex, absorbing sets in E. Then {Pu : U E V} is a separating family of continuous seminorms on E. Conversely, each separating family P of seminorms on a linear space E defines a topology T on E with respect to which E is a locally convex space; one takes as a subbase of neighbourhoods of 0 the sets {x : p( x) < E} for pEP and E > O. Let P be a family of seminorms defining the topology of a locally convex space E. Then P is saturated if the seminorm
PV q:x
f-7
rnax{p(x), q(x)} ,
E
-+ ~,
belongs to P whenever p, q E P. Note that, if p, q are algebra serninorrns on an algebra, then lOO is p V q. vVithout further mention, we shall suppose, as we may, that a family of lOeminorms defining the topology of a locally convex space is both separating and saturated. A net (xv) converges to x in (E, P) if and only if p(xv - x) -+ 0 for each pEP. A locally convex lOpace is metrizable if and only if itlO topology can be defined by a countable family of seminormlO. Let (E, P) and (F, Q) be locally convex spacelO. Then a linear map T : E -+ F is continuous if and only if, for each q E Q, there exist pEP and C > 0 such that q(Tx) ~ Cp(x) (x E E). A seminorm p on a normed lOpace (E, [I'[[) 11O continuous if and only if there exists C > 0 with p(x) ~ C [[x[[ (x E E).
Proposition A.3.8 Let E be a topological linear' space. (i) A Imear j7mctzonal A on E is continuous zj and only if ker A is closed. (ii) Each Jinite-dzmenszonal subspace of E is closed.
o
Let E be a topological linear lOpace, and let F be a linear sublOpace of E. The quotzent topology on the qnotient space E / F is the strongest topology for which the quotient map 7r : E -+ E / F ilO continuous. Suppose that p is a seminorm on E. Then the corresponding quotzent semmorrn is given by
p(x + F)
= inf{p(x + y)
: y E F}
(x E E).
Proposition A.3.9 Let F be a closed ImeaT subspace oj a topo[ogzcal linearspace E. (i) The space E / F is a topoZogzcal Imear space wzth respect to the quotient topology. (ii) The quotient map 7r : E -+ E / F zs continuous and open; if G is a linear subspace oj E with F c G, then 7r (G) = 7r(G).
(iii) Suppose that E is locally convex or metrizable or an (F)-space or a Jilrechet space or a norrned space or a Banach space. Then E / F has the corresponding property. 0
814
Appendix
Proposition A.3.I0 Let E and F be topological linear spaces, ---+ F be a contmuous lmear map.
and let
T: E
(i) Let G be a topological linear space, and let S : E surjec.lion wzth ker S wzth T 0 S = T. (ii) The map x
c
G be an open lmear ker T. Then there is a continuous linear map T : G ---+ F
+ kerT
f-->
Tx, ElkerT
---+
---+
F, zs contmuous.
o
Corollary A.3.1l Let F and G be closed lmear subspaces of a topologzcallznear space E. Then the canonical bzjection FI(F n G) ---+ (F + G)/G is continuous. 0 Suppose that the inverse zs contznuous. Then F + G zs closed m E. Proposition A.3.I2 Let F be a proper, dense lznear subspace of a Banach space E, and take a E ElF and r > O. Then there exzsts x E E with Ilxll = rand a=x+F. 0 Let E be a linear space. As in §1.3, we often set (x, A) A E EX.
= A(X) for x E E and
Definition A.3.I3 Let E be a topological linear space. Then E', the space of continuous linear fttnctionals on E. zs the dual of E. Suppose that E is a normed space, and define IIAII = sup{I(X,A)I: x E E[l]} for A E E'. Then (E', 11·11) is a Banach space. We continue to define the higher duals E", EIII, ... ; for n EN, the nth dual space is denoted by E{ n}. The natural embedding of E in E" is denoted by I. so that
t(X)(A)
= (t(x). A) = (x, A)
(x
E
E, A E E');
we shall often regard E as a linear subspace of E". The space E is reJlexzve if t(E) = E". For example. the spaces fP for p E (l,ao) are reflexive. A Banach bpace E is reflexive if and only if E' is reflexive; if E is reflexive and F is a closed subspace, then F and ElF are hoth also reflexive. Proposition A.3.I4 (Riesz) Let (H, [" .]) be a Hilbert space. Then, for each A E H', there exists a ttnique y E H sttch that (x, A) = [x,y] (x E H). 0 Definition A.3.I5 Let E and F be lznear spaces. and let
(x, y)
f-->
(x, y),
E x F
---+
C,
be a bilznear functional sttch that, for each x E E-, there exists y E F with (x, y) =I- 0, and. for each y E F-, there exists x E E 'Unth (x, y) =I- O. Then (E, F) zs a dual pair.
Let (E, F) be a dual pair. For each y E F, define py(x) = l{x,y)1
(x
E
E).
Then Py is a seminorm on E. The weak topology on E determined by F is the weakest topology on E with respect to which each of the seminorms Py is
Functional analysis
815
continuous; it is denoted by a(E,F). The sets {x E E: l(x,y)1 < I} for y E F form a subbase of open neighbourhoods of 0 in (E,a(E,F)), and (E,a(E,F)) is a locally convex space whose dual space is F. Let E be a linear space. Then (E, EX) ancl (EX, E) are both dual pairs for the corresponding bilinear functionals; a(EX, E) is the weak* topology on EX. We have All ----) A in (Ex. a(EX, E)) if and only if (x, All) ----) (x, A) (x E E). A subhnmr f1Lnctwnal on a linear space E is a map p: E ----) JR such that p(x + y) ::::: p(x) + p(y)
(x, y E E),
p(ax) = apex)
(x E E, a E JR+).
The following three theorems are versions of the H ahn- Banach theorem.
Theorem A.3.16 Let F be a lmear subspace of a rwl-lmear [hnear] space E.
(i) Let p be a s1J,blinear' functional [seminorm] on E, and let A be a reallinear [linear] functwnal on F S'uch that (x, >..) ::::: p(x) [I(x, >")1 ::::: p(x)] (x E F). Then there is a real-linear [linear] functional A on E such that A I F = A and (x,A) ::::: p(x) [J(x,A)1 ::::: p(x)] (x E E). (ii) Let C be a convex subset of E such that {x - y : x E C} zs an absorbmg set for some y E C n F, and let A be a real-lmear functwnal on F such that (x, A) 2 0 (x E C n F). Then ther-e 2S a real-linear functional A on E such that A I F = A and (x,A) 20 (x E C). 0
Theorem A.3.17 Let A and B be dzsjoint, non-empty, convex sets in a real topological linear space E. (i) Suppose that A is open. Then there exist A E E' and a E !R such that
(x, A) < a ::::: (y, >..)
(x E A, y E B) .
(ii) Suppose that A lS compact, B zs closed, and E is locally convex. Then ther-e eX2st A E E' and a, j3 E !R such that (x, A)
"11 = 1. The above extension A is a norm-preserving extension of >... Let E be a topological linear space. Then E has a total set of functionals if, for each x E E·, there exists>.. E E' with (x, >..) t= O. In this case, both (E, E') and (E', E) are dual pairs; the aCE, E')-topology is the weak topology on E, and the a(E', E)-topology is the weak* topology on E', By A,3.18(i), each locally
816
Appendix
convex space has a total set of functionals. We !lhall use terms such as 'Weakly convergent for a net (xv) in E to indicate that (xv) is convergent in the weak topology; this occurs if and only if (xv, A) ~ (x, A) for each A E E'. The space E is 'Weakly sequentzally complete if every weakly Cauchy sequence is weakly convergent. Let F be a closed linear subspace of E. Then a(F, F') = a(E, E') I P; if E is weakly sequentially complete, then F is also weakly sequentially complete. Theorem A.3.20 (Banach-Alaoglu) Let E be a normed space. Then Efl] ts a(E', E)-compact, and each net in Efl] has a a(E', E)-accumulation point and a a(E', E)-convergent subnet. The set (Efll' a(E', E)) 1,S metrizable 1,f and only if E is separable. 0 Let E be a topological linear space. A subset B of E is bounded if, for each U ENE, there exists 0: > 0 such that B c /3U for each {3 > 0:. Clearly finite unions of bounded sets. the closures of bounded sets, and compact sets are bounded. A subset B of E is weakly bounded if and only if sup{l(x, A)I : x E B} < oc
(A E E').
Let (E, P) be a locally convex space. Then a subset B of E is bounded if and only if {p(x) : x E B} is bounded for each pEP. A topological linear space E is locally bounded if N E contains a bounded !let; for example, each normed space is locally bounded. The space E is countably boundedly generated if it is a countable union of bounded subsets. An (F)-space is countably boundedly generated if and only if it is locally bounded, and so the standard examples are the spaces LP(p,). where p > 0 and p, is a positive measure (see Appendix 4). The space E has a fundamental sequence of bounded sets if there is a sequence (Bn) of bounded sets in E such that E = U~l Bn and such that. for each bounded set B in E, there exists n E N with B C Bn. Clearly each locally bounded space has a fundamental sequence of bouuded sets. Definition A.3.21 Let E and F be topological linear spaces. A lmear map T: E ~ F is bounded 1,fT(B) is bounded in F for each bounded subset B of E. The set of bounded linear maps from E to F 1,S denoted by 8(E, F). We write 8(E) for 8(E, E). Certainly 8(E, F) is a linear subspace of £(E, F). For example, IE E 8(E), where Ie is the identity operator on E. Theorem A.3.22 Let E and F be topologicallmear spaces, and let T E £(E. F). (i) Suppose that E
1,S
metrizable and T is bounded. Then T is continuous.
(ii) Suppose that T is continuous. Then T is bounded.
o
Suppose that E and F are locally convex spaces. Then 8(E, F) is a locally convex space for the topology T of uniform convergence on the bounded subsets of E. Suppose that E and F are normed spaces. Then (8(E, F), T) is a normed space with respect to the operator norm II . II, which is defined by the formula IITII = sup{IITxll : x
E
Ell]}
(T
E
8(E, F));
Functwnal analysis
817
the space B(E, F) is complete whenever F is complete. An element T E B(E, F) is an isometry if IITxl1 = Ilxll (x E E).
Theorem A.3.23 (Open mapping theorem) Let E and F be (F)-spaces, and let T : E - F be a continuous linear surjectwn. Then T is open. Suppose that T is also an injection. Then T zs a lmear homeomorphism. 0 In particular, in the case where E and F are Banach spaces and T : E - F is a continuous linear surjection, there exists m > 0 such that, for each y E F, there exists x E E with Ilxll :::; mIlyll and Tx = y. The next result follows from the open mapping theorem ..
Proposition A.3.24 Let E and F be (F)-spaces, and let"'£;E B(E, F). Suppose that T(E) has finite codzmension in F. Then T(E) ,zs closed m F. 0 Theorem A.3.25 (Closed graph theorem) Let E and F be (F)-spaces, and let T : E - F be a linear map. Then the Jollowmg condztwns on T are equivalent: (a) T is continuous;
(b) T has a closed groph;
o
(c) zJx n -0 inE andTx n - y inF, theny=O.
Corollary A.3.26 Let E and F be (F)-spaces, and let T : E - F be a linear bijectwn which has a closed groph. Then T is a linear homeomorphzsm. 0 We shall use the following consequences of the Mittag-Leffler theorem many times. In the result, lim sn(E) +-
= {x
E E : there exists (xn ) in E such that} .
x
= Xl
and Xn
= SXn+1
(A 3 4)
(n E N)
..
Proposition A.3.27 Let E be an (F)-space. (i) Let F be a countable subset oj B(E) such that See) = E (S E F) and ST = TS (S, T E F). Then there is a dense lmear subspace F oj E such that S(F) = F (S E F). (ii) Let S E B(E) be such that See)
= E. Then limSn(E) and +-
n:=l sn(E)
are dense in E. Proof (i) Let En = E (n EN), and let (On) be a sequence composed of elements of th0 set F, with each element occurring infinitely often. Define X = lim proj{En;On}, and set F = 7rl(X), in the notation of Appendix 1. By A.1.25, F is a dense linear subspace of E, and clearly S(F) = F (S E F). (ii) We apply the argument of (i) with F
= {S}, so that F = ~ sn(E);
note
o Proposition A.3.28 Let E be an (F)-space, let (un) be a sequence in E, and
818
Appendix
= E (n
let (Tn) be a sequence in B(E) with Tn(E) x E E such that
EN). Then there eXlsts
n J; -
I)TI ... Tk)(Uk)
(n E N).
E (TI ... Tn+I)(E)
k=l
Proof For n E N, set (}n(x) = Tnx + U n- l (x E E) (where 110 = 0). Then (}n : E -+ E is continuous, and (}n(E) = E. By A.1.25. there exists (Xn) in E with (}n(X n+l) = Xn (n EN). Then the equation «(}l 0··· 0 (}n+I)(X n +2) = Xl becomes (Tl ... T n +1)(xn +2)+ EZ=-l (Tl .. ·1k)( Uk) = Xl, giving the result. 0
Let E be a normed space. Then For T E B(E'), set
L :
(E, 11·11)
(YX)(A) = (TA)(X)
(x
E
-+
(E", 11·11) is a linear isometry.
E, A E E').
Then T f--+ Y, B(E') -+ B(E, E"), is an isometric linear bijection: we shall often identify B(E') and B(E, E"). Theorem A.3.29 Let E be a Banach space.
(i) (Goldstine) For each E E", there lS a net (xv) zn E such that -+ in (EI,a(E",E'». (ii) (Mazur) For each convex set S m E, the closures of S m (E. II· II) and (E, aCe, E'» are equal. (iii) (KreIn Smulian) Let K be a weakly compact sub8et of E. Then (K) lS relatively weakly compact. (iv) (Eberlein -Smulian) A subset of E is weakly compact if and only lJ it is weakly sequentially compact If and only if it is weakly countably compact. (v) (Rainwater) Let (xn) be a bounded sequence in E, and let x E E. Then Xn -+ x weakly ~f and only if (x, A) = limn-+oo(x n , A) for each A E exE[I]'
Ilxvll ::; 1111 for all v and t(xv)
(vi) (Principle of local reflexivity) For each fin·ite-dlmensional subspace X of E", each fimte subset F oj E', and each £ > 0, ther'e is a linear map S : X ~ E with S I X n E = IxnE and IISIlIIS-l I S(X)II < 1 + £ such that (S(.(T) = I(Tx, A)I
(T E B(E,F)).
The weak* operator topology on B(E, F'), denoted by wo*, is the topology defined by the family of seminorms {Px,y : x E E, Y E F}, where
Px,y(T) = I(y, Tx)1
(T
E
B(E, P')).
820
Appendz:c
Thus (8(E, F), wo) and (8(E, F'), wo*) are locally convex spaces; we write
T1 ~ T
or
wo - lim T1 = T -y
if (T-y) converges to T in (8(E, F). wo). Theorem A.3.35 Let E and F be Banach spaces.
(i) The umt ball 8(E)[1] zs wo-compart iJ and only iJ E is reftexwe. (ii) The unit ball 8(E. F')[l] is wo*-compact.
0
Let EI, ... , En. F be normed spaces. An n-linear map T E Cn(E1, ... , En; F) is bounded if there exists AI > 0 such that IIT(xl,"" xn)11 :<s: M Ilxlll·· '11£nll
(Xj E
E j • j E N n );
the map T is bounded if and only if it is continuous. We write 8 n (E 1 •.•. , En; F) for the set of all bounded maps in Cn(E 1 , .•• , En: F); 8 n (El."" En: F) b a linear subspace of Cn(El"'" En; F). and it is a normed space with respect to the norm 11·11 given by IITII
= sup{IIT(xl, .. "
xn)11 : Xj E (Ej)[l]
(j
E N n )};
this space is complete whenever F is complete. We write 8 n (E, F) in the case where El = ... = En = E. We now list some standard consequences of the Baire category theorem A.1.21. A family F of maps from a topologicallillear space E into a topological linear space F is equzcontmuous if. for each V E N F , there exists U E NB such that
T(U) c V (T
E
F).
Theorem A.3.36 (Uniform boundedness theorem) Let E be an (F)-space, let F be a topological Imear space, and let {T1 : 'Y E r} be a Jamzly oj continuous linear' maps from E into F. Suppose that {T-yx : 'Y E r} is bounded in F Jor each x E E. Then {T-y : l' E r} is equicontznuous. 0 Corollary A.3.37 Let E be a Banach space, and let F be a normed space.
(i) Let {T-y : 'Y E r} c 8(E, F). Suppose that, Jor each x E E, there ensts Mx > 0 such that IIT-yxil :<s: AIx ('Y E r). Then there exists AI E JR.+ such that IIT-yxil :<s: AI Ilxll (x E E, 'Y E r). (ii) Let (Tn) be a s('-Quence in 8(E, F). Suppose that limn Tnx = Tx (x E E). Then (Tn) is bounded in 8(E, F), T E 8(E, F), and IITII :<s: liminfn IITnll. 0 Corollary A.3.38 A subset oj a locally convex space is weakly bounded iJ and only zJ it is bounded. 0 Corollary A.3.39 Let E l , ... , E n - 1 be (F)-spar-cs, let En be a metrizable topological linear space, and let F be a topological linear space. Then each separately continuous, n-linear map from n~l Ei into F is continuous. 0
Functzonal analyszs
821
Corollary A.3.40 Let B be a closed, absolutely convex, ab80Tbing set zn a Frechet space E. Then B is a nezghbourhood of 0 in E. 0 Corollary A.3.41 Let (E, T) be a Prechet space. Suppose that there is a norm 11·11 on E such that the zdentity map (E, 11·11) --; (E, T) zs continuous. Then (E, T) is a Banach space. Proof Let P be a family of seminorms defining the topology T on E. For each pEP, there exists kp > 0 such that p(x) :::; kp Ilxll (x E E). Define Illxlll = sup{p(x)jkp : pEP} (x E E). Then 111·111 is a norm on E, and the map (E, 111·111) --; (E,T) is continuous. Let B = {x E E : Illxlll :::; I}. Then B is absolutely convex and absorhing, and it is closed in (E. T). By A.3.40, BENE, and so the map (E. T) --; (E, III· lID is continuous. Thus (E, T) is linearly homeomorphic to the Banach space (E, 111·111). 0 Let E be a topological linear space, and let F and G be linear subspaces such that E is (algebraically) the direct sum of F and G, so that E = F 8 G. We write E = F GJ G if both F and G are closed in E; further, we write E = F ffip G if Iiy + zilP = IlyliP + Iizll P (y E F, z E G), where p E [1,(0), and E = F ffioc G if Iiy + zll = max{llyll ,llzll} (y E F, z E G). The direct sum E Co) F is topologzcal if the projections of E onto F and G are both continuous; this implies that E = F ffi G. A closed ::mbspace F of E is complemented if there is a closed subspace G of E such that E = F ffi G. For example, let F be a dosed subspace of a Hilbert space H, and set FJ..
= {y
E H : [x,
y] = 0 (x
E F)} .
Then H = F ffi FJ.., and F is complemented in H; FJ.. is the orthogonal complement of F. and (FJ..)J.. = F. The projection in B(H) with range F and kernel FJ.. is the orthogonal proJectzon onto F.
Theorem A.3.42 Let E be a topologicallznear space. (i) Let F be a closed linear' subspace of finite codimenszon in E. Then F is complemented in E. Suppose. further, that G is a topological lznear space and that T : E --; G is a Imear map such that T I F is contmuous. Then T is continuous.
(ii) Suppose that E is locally convex. subspace of E is complemented.
Then each finite-dimenswnal Imear
(iii) Suppose that E zs an (F)-space and that E has closed lmear .mbspaces F and G such that E = F ffi G. Then the direct sum is topological. 0 Theorem A.3.43 Let F and G be closed linear subspaces of a Banach space E such that E = F + G. Then there exists a> 0 such that, for each x E E, there 0 exist y E F and z E G with x = y + z and Ilull + Ilzll :::; a Ilxll. Definition A.3.44 Let E, F be topological lmear spaces, and let T E B(E, F). Then T is admissible if kerT is complemented in E and T(E) is closed and complemented in F.
822
II
Appc'fl(i1:r
Let (En: n E ;E) be a sequence of Banach spaces, and suppose that, for each thc'I'{' E'xists Tn E 8( En tl. Ell) snch that
E ;E,
is a complex of linear spacE'S and linear mappings, as in §1.3. Then we have a comple;r; of Banach 8pacp-s lLnd contimw1Ls linear' mapping8: E h; admisslble if it is ('xaet and each Tn is admissibk. Thus E is admissible if it 'splits as a complex of Banach spacE's': E is admissible if and only if. for each n E ;E, there exbts Q'/1 E 8(En' E n+1) with Q,,-1 0 T Il - 1 + 1'r, 0 Qn = II<J n • A short exact sequen("(' L : 0 --+ E !i." F !: a --+ 0 is admissihle if and only if S(E) is cOlllpkmented. Definition A.3.45 Let E be a locally convex space, let F be a linear subspace of E. and let a be a lmear- subspace of E'. The annihilators FO of F and °a of G are defined by:
= {A E
FO
E' : (x. A)
=0
°a = {x E E: (x. A) = 0 °a
(,I' E F)}: (A E
an,
Thus FO and are closed linear subspaces of E' and E, respectively. In this case, O(FO) is the closure of Fin E, and (Oa) is the a(E', E)-closure of a in E', We set FOO = (FO)O c E".
°
Theorem A.3.46 Let E and F be normed spaces, and let T E 8(E. F). Then th(,re f.L'18ts a 1Lnique T' E 8(F', E') such that (Tx. A) Moreover. IIT'II = contznuous.
II TIl ,
= (x, T' A) (x
E E. A E F') .
and the map T' : (F',O"(F',F))
--+
(E',a(E',E)) is 0
\Ve call T' the dual of T, reserving the term 'adjoint' for the 'Hilbert space adjoint' of 3.1.4(iii). For a closed subspace 2( of 8(E, F), define 2(a
Then T
I->
T'.
2( --+ 2(a,
= {T'
E
8(P'. E') : T E
2(}.
(A.3.5)
is an isometric linear bijection.
Theorem A.3.47 Let P be a closed linear' subspace of a nO'f7ned space E. (i) For each A E F', take A E E' wzth IIAII = IIAII and A I P = A. Then the A I-> A + pO, P' --+ E' / pO, Z.'I an isometric linear' bZ)ectzon.
map
(ii) Let 7r: E ts
(J
fI
--+ E/F be the quotient map. Then the map 7r': (E/F)' ....... tsornctric linear b~1ec:tzon.
(iii) P"
= FOO = p(O).
whem o (j\') (E/F)" ~ E"/po .
0"
=
pO
O"(E", E'). 0
Theorem A.3.48 Let E and P be Banach spnces, and let T E B(E, F). Then: (i) kerT = O(T'(P')); (ii) kerT' = T(E)O;
Functwnal
analysz.~
823
(iii) T' zs znjectzvr 7j and only if T(E) is dense zn F: (iv) T is m]ecti7l e 2f and only 7JT'(F') is u(E',E)-derl,!;e in E':
(v) the following condztions on T ar'e equivalent: (a) T(E) is closed m F; (b) T'(F') lS u(E'. E)-closed mE'; (c) T'(F') is closed mE': (vi) T"
lS
surjectzve zJ and only zf T is suryective. a.nd, m this case, (ker T)"
= ker T" =
---(cr)
(ker T)
•
where u = u(F".F'); (vii) zJ T zs injective and has closed mnge, then T" zs a.lso m]ective with closed mnge. 0 Corollary A.3.49 Let E and F be Banach spa.ces, and let T E B(E, F). 8-0.1)pose that there are constants II < 1 and m > 0 such that, Jor each y E Fhl- the're exists x E E[ml with IITx - yll < k. Then T is a surjection.
yll < k. Then A)I :::; k II All + m IIT'(A)II .
Proof Let A E F'. For y E F[ll' take x E E[m] with IITx -
and so
I(y. A)I :::; I(y - Tx, A)I + I(Tx, IIAII :::; k IIAII + m IIT'(A)II. Thus IIT'(A)II ~ (1 - k) IIAII 1m.
(A.3.6)
It follows that T'(F') is closed in E'. By A.3.48(v). T(E) is closed in F. Now take A E F' with A I T(E) = O. Then T'(A) = O. and so A = 0 by (A.3.6). Thus T(E) is dense in F. 0 Definition A.3.50 Let F be a closed lmear subspace oj a normed space E. Then F is weakly complemented in E iJ F 0 zs complemented m E'. For example, the space Co is weakly complemented in tOO, but it is not complemented in fOC. Suppose that F is complemented in E, so that there is a projection P in B(E) with range F. Then (I - P)' is a projection in B(E') with range FO, and so F is weakly complemented in E. Let E, F, and G be normed spaces, and let B : E x F ---> G be a continuous bilinear map. We successively define maps Bl : G' x E ----> F', B2 : F" x G' ---> E'. and jj : E" x F" ---> G" as follows. Here. for x E E, y E F. z E G, A E E', JL E F' _ 1/ E G'. A E E", and 1Vl E F", we define:
(y, B 1 (I/,x» (x,;.. B2(M,I/» (B(A, M), 1/)
= =
=
(B(x,y), 1/); } (M, B 1 (I/,x»; (A, B2(M, 1/» .
(A.3.7)
Clearly each of the maps Bb B 2 , and jj is continuoub and bilinear, and we have B(x, y) = B(x, y) (x E E, y E F), where we are identifying E, F, and G with their canonical images in E", F", and G", respectively. Further IIBII = IIBII· Definition A.3.51 Let E, F, and G_be normed spaces, and let B : Ex F ---> G be a continuous bilinear map. Then B : E" x F" ---> G" is the extension oj B.
824
Appendzx
The following result is clear from the definition. In each case, the second dual spaces have their weak* topologies.
Proposition A.3.52 Let E, F, and G be normed spaces, and let B : Ex F be a contmuous bzlznear map. (i) For each x E E, the map M f---> B(x,M), F" --> G", is continuous.
(ii) For each M E F", the map A f---> D(A, 1.1), E" --> G", is continuous.
-->
G
0
Let A E E" and 1\1 E F". By A.3.29(i), there are nets (xa,) in E and (ye) ill F such that Xn --> A and Y3 --> 1\1. It follows from A.3.52 that
B(A.M)
= limlimB(xo r A is n-linear. Hence there is a unique linear map £ : F ----> E' with £(A1 ® ... ® An) = r A • and £ is again an injection. We identify F with the linear subspace £(F) of E'.
Definition A.3.61 Let E 1 , • .. ,En be normed spaces, and set E = ®7=1 E i . The injective tensor norm on E is the restriction oj the norm on the Banach space Bn(E1, .... En; q to E. The closure oj E in thzs space is the injective ---n
....
....
tensor product oj E 1, . .. , En, denoted by ®i=l Ei = E1 ® ... ®En. We denote the injective tensor norm by
II· lie:; explicitly,
Ilxlle: = sup { f)X1,j, At} ... (Xn,j, An) } ,
3=1
the supremum being taken over all representations x = where n E N and all Ai E (ED[lj (i E N n ). We have
IIXI @ ... @ xnlle: = IIxd" 'lIxnll
(Xi E
E:;:l X1.J®· . '@Xn,j E i , i E N n ).
E
E,
828
Appendix
In the next re::mlt, we write
11·11: for
the norm on the dual space (®Ei,
II·ILJ/.
Proposition A.3.62 Let E I , . .. , En be normed spaces, and take Ai E E~ for each i E N n . Then AI 0 ... 0 An is the unique element in (® E i , II . ILoY with
(AI 0···0 An)(XI 0···0 X.,) = (Xl, AI)··· (x." An)
(Xi E E i , i EN,.).
Further, IIAl 0···0 Anile = IIAI II·· . II An II· For each Z E ®Ei W'lth exist Ai E E~ (i EN.,) such that (AI 0···0 A.,)(z) 1:- O. I
-
Z
1:- 0,
there 0
Let E and F be normed spaces. For T E B(E, F), there is a linear functional AT: E0F' ---- C such that (x0A, AT) = (Tx, Al (x E E, A E F'). The operator T is integral if AT is continuous on (E 0 F', 11·11); the set of integral operators is a linear space which is denoted by I(E,F). We set IITIIT = IIATII. It is clear that N(E, F) c I(E, F) and that IITIIT ::::: IITllv (T E N(E, F)). Now take 7 E (E0F)'. Then there exists ST E B(F, E') such that
(x, STY) = (x 0 Y, 7)
(x E E, Y E F) .
Proposition A.3.63 Let E and F be Banach spaces. Then (I(E, F), 11·11x) is a Banach space, zdentified with a closed subspace of (E0F', 11·11)'. Further, I(E, F) c WeE, F), but I(E, F) O. Suppose that Z E E ® F has a r-epresentation z = 2:;=1 Xj ® Y1' where
Then IIzll71' ::; C. Proof Set
where (
= e211"i/n. Then in fact w = z. Also
o
as required.
Proposition A.3.69 Let El .... , En be normed spaces. let F be a Banach space, and let S E B(El, .... En;F). Then there is a unique continuous linear map Ts : E 1 ®··· 0En --+ F such that TS(XI ® ... ® xn) Jilurther, the map S >-> Ts, isometric linear bijection.
= S(Xl, ... , xn) (Xj sn(El, ... , En; F)
--+
E E j , j E Nn ) .
B(El®'" ®En, F), is an 0
830
Appendix
The following remark follows from A.3.69. Let E 1 , ..• , En, F I , ... ,Fn be Banach spaces, and let Ti E B(Ei . Fi ) (i E N n ). Then there is a unique element
TI®·· .®Tn in B (®Ei.®Fi) with (TI®··· ®Tn)(XI ® ... ® Xn) = TlXl ® ... ® Tnxn
(Xi E E i , i E Nn ).
Proposition A.3.70 Let n E N, and let E l , ... ,En and F be Banach spaces. For A E (El ®... ®En®F)' , define
(Xj E E j , y E F).
T>..(Xl, ... ,Xn)(Y) = (Xl ® ... ®xn ®y,A) Then A I----> T>..o (El~'" 0En®F)' bijectwn.
--+
Bn(El"'" En; F'), is an isometric linear
Proof For A E (E 1 0··· ®En0F)', x J E Ej (j E Nn ), and Y E F, we have
IT>..(xl, ... , xn)(Y)1 ~
IIAllllxl ® ... ® Xn ® YII71"
and so T>..(Xl, ... , Xn) E F' and T>.. .9 E Bn(El , ... , En; F'). The map (Xl .... ,
Xn, y)
I---->
IIAllllxlll·· 'lIxnllllyll, E Bn(El , ... , En; F') with liT>.. II ~ IIAII. Take
S(Xl .... , Xn)(Y),
El
=
X ...
x En
X
F
--+
C,
is continuous and (n + I)-linear, and so, by A.3.69, there existH an element AE (E1 0'" 0En®F)' such that T>.. = Sand IIAII ~ IISII. 0 Let E and F be Banach Hpaces. The bilinear map
(y,A)
I---->
Y ® A,
F
X
E'
--+
B(E,F) ,
is continuous, and so there is a continuous linear map R : F0E' --+ B(E, F) extending the identification of F ® E' with F(E, F). The range of R is exactly N(E, F). and the nuclear norm 11·lIv is the corresponding quotient norm.
Proposition A.3.71 Let E be a Banach space. (i) For each Banach space F, the map R: (F0E', 11·1171") --+ (N(E. F). II· IIv) an epimorphzsm wzth IIRII = 1. (ii) The following conditions on E are equivalent: (a) E has AP; (b) the map R : (E0E', 11·1171") --+ (N(E), II·IIJ is znjective: (c) the natural embeddmg (E0F.II·II71") --+ (E0 F, II· lie) zs injective for every Banach space F. (iii) Suppose that F zs a Banach space such that F' has AP and the RadonNzkodym proper-ty. Then we have the following zdentzficatwn.~:
zs
K(E, F) = A(E, F) = F0E' ; K(E,F)' =N(E',F') = E"0F'; K(E, F)" = B(E", F"). The map that identifies K(E. F) wzth its image in B(E", F") is T
I---->
T".
0
Let I = [a, bj be a compact interval of JR, and let n E N. A function f on I is n-times continuously differentiable if f', ... , fen) exist on I (taking one-sided derivatives at a and b) and fen) is continuous on I. The space of such functions is denoted by c(n)(I); we define C(oo)(I) = n~=l c(n)(I).
Punctzonal analysis
831
Theorem A.3.72 For' each n E N, the space c(n) (1) zs a Banach space wzth respect to the norm II· lin ' where
Ilflln =
t ~!
If(k)l!
(f
E
c(n)(1)).
k=O
The space C(oo)(1) zs a Frechet space wzth respect to the famzly of sernmorms.
{II, lin
: n E N} D
Similarly, let K = [a, b] x [c, d] be a compact rectangle in ]R2. A function f on K is n-times continuously dzJJerentiable if the partial derivatives {)j+k f /()xj{)yk exist and are continuous on K for j + k :-:; n. The set of these functions is c(n)(K): it is a Banach space with respect to the norm II· lin' where
IIflin = .
L
J+k~n
j/k!
I::j;~ IK .
n:=l
We define C(oo)(K) = c(n)(K), so that C(oo)(K) is a Frechet space. We sometimes write Dj,kf for {)j+kf/DxjDyk. Definition A.3.73 Let (K, d) be a non-empty, compact metric space, and take a E (0,1]. Then LiPaK is the space of complex-valued functions f on K such
that Pa(f) < 00. where Pa(f) = sup {
If(x) - f(y)1 } d(x, y)a : x, y E K, x =I- y ,
and liPnK zs the subspace of LiPaK consistmg of functzons f such that If(x) - f(y)1 _ 0 d(x,y)a For f
E
LiPa K , define IIfli a
ru>
d(x. y) _ O.
= IflK + Per(f).
It is clear that (LiPaK, II ·ll a ) is a Banach space and that liPerK is a closed subspace thereof: the functions in LiPaK are Lipschztz of ord('.r a. Note that the spaces LiPaK depend on the metric on K; even if d 1 and d 2 are equivalent metrics on K, LiPa(K, d 1 ) and LiPa(K. d2) may be very different. For compact subsets K of]Rn, d is assumed to be the Euclidean metric, unless stated otherwise. Note also that, if a < {3 :-:; 1, then Lip 13K C liPaK. For each Xo E K, the function x f-t d(x. xo)l~ belongs to LiPaK if {3 ~ a and to liPaK if 8 > Q. We shall also require some definitions and results from the tht."Ory of Banachspace valued functions. Definition A.3.74 Let {(E,,!, ta.ke p E [1,00). Then:
(p(r, E'Y) = {(x'Y) f
00
E
II· II"!) :, E r}
be a family of Banach spaces, and
I1'YErE'Y : II(x'Y)II =
(E'YEr II:r'YII~f/P
.. 0 I E O(U) lor e-ach >.. E E'. The set of analytzc E-valued Innctwns on U is denoted by O(U, E). Suppose that U is a non-empty, open set in C, that E and Fare Banaeh spaces, and that T E B(E, F). Then To IE O(U, F) (f E O(U, E»).
Proposition A.3.76 Let U be an OIJen set in C, and let E be a Banach space. Then I : U -4 E is analytic il and only if it is weakly analytic. 0 The space O(U, E) is a closed linear subspace of C(U, E). l\:Iany properties of O(U, E) follow from A.3.76 and the corresponding result about O(U).
Proposition A.3.77 Let E be a Banach space. (i) Let U be a non-empty, open set in C, and let IE O(U, E). Then analogues 01 Canchy'8 theorem. Canchy'8 integrallorrnnla, and the rnaximnm rnodnlns prmciple all hold for I.
(ii) Suppos(' that I is a bonnded Innction in O(C, E). Then I is constant. (iii) Let I E O(IDl, E). Then there is a seqnence (an: n E Z+) in E such that 00
n=O
Fnrlher, lIanll S; sup{lI/(z)1I :
Izl
S; r}/rn (n E Z+) lor each r E (0,1).
0
Functional analysis
833
Proposition A.3.78 Let E be a Banach space wzth closed linear subspaces F and G such that F + G = E. For each f E O(JI)), E), there eX'tsts g E O(JI)), F) and h E O(JI)), G) such that f = g + h. Proof Set f(z) = L::'=o anz n , as in A.3.77(iii). By A.3.43, there exist O! > 0, (b n) C F, and (en) C G with an = bn + en and Ilbnll + Ilenll ~ Q Ilanll for each n E Z+. Set g(z) = L:;:'=obnz n and h(z) = L:;:'=oCnz n for z E JI)). 0 Proposition A.3.79 Let E be a Banach space, and let F be a closed linear subspace with quotient map 7r: E ----> ElF. Suppose that!k ----> 0 in O(JI)),EIF). Then there exists (h) in O(JI)), E) such that 1,. ----> 0 and 7r 01,. =!k (k EN). Proof For kEN, set fk{Z) = L:;:'=oa... kZn (z E JI))), where an.k E ElF. Choose bn.k E E with 7r(bn.k ) = an,k and IIb n .k II ~ lIan.k II + 2- n - k for kEN and n E Z+, and set 1,.{z) = L:;:'=o bn.kZn (k E N, z E JI))). 0 Notes A.a.SO The seminal work on what are now called Banach spaces is of course (Banach 1932). The theory of topological linear and Banach spaces is expounded in (Diestel 1984), (Dunford and Schwartz 1958), (Edwards 1965), (Jarchow 1981), (Kothe 1969, 1979), (Lindenstrauss and Tzafriri 1977), (Megginson 1998), (Meise and Vogt 1997), (Rudin 1973), (Schaefer 1966), and (Wojta.'3zczyk 1991), for example. In particular. many of the early results in this section can be found in (Rudin 1973). Properties of (F)-spaces are in (Kothe 1969, 1.5.11) and (Megginson 1998, §2.3); for A.3.5, see (Jameson 1987, Theorem 5.6). Most of A.3.29-A.3.31 is proved in (Diestel 1984), (Dunford and Schwartz 1958), (Megginson 1998), and (Wojtaszczyk 1991); our form of A.3.29(iv) is (DiesteI1984, p. 18) and (Megginson 1998, 2.8.6); a short proof of A.3.29(v) is in (Phelps 1966, Chapter 5); clause (d) of A.3.31 is Grothendieck's iterated limit criterion, from (Grothendieck 1952, 1955). The Krein-Mil'man and RadonNikodym properties arE' discussed in (Diestel and Uh11977, Chapter VII), where a long list of propE'rties equivalent to the Radon-Nikodym property is given. The uniform boundedness theorem A.3.36 and its consequences are in (Rudin 1973. Chapter 2); for A.:J.39. see (ibid., 2.17); for A.3A2 and A.3A3, see (ibid., 4.21, 5.16, and 5.20); for A.3A7 and A.3A8, see (ibid., Chapter 4). ThE' results are also given in (Megginson 1998, Chapter 3). The two E'xtensions of a bilinear map were first given in (Arens 1951b); for a full account, see (Grosser 1979). For the remarks on compact and weakly compact operators, see (Dunford and Schwartz 1958, VIA and VI.5), (Palmer 1994, §1.7), and (Rudin 1973, Chapter 4); for strictly singular operators and A.a.58, see (Lindenstrauss and Tzafriri 1977, §§2.a,c). Details about bases and approximation properties in Banach spaces are given in (Diestel and Uhl 1977), (Lindenstrauss and Tzafriri 1977), and (Megginson 1998); CAP and BCAP are specifically dE'fined in (Dixon 1986). The only naturally occurring Banach space known to lack AP is B(H) for an infinite-dimensional Hilbert space H (Szankowski 1981). An example of a reflexive Banach space that has BCAP, but does not have AP, is given in (Willis 1992c); a separable Banach space without CAP is discussed in (Lindenstrauss and Tzafriri 1977, 1.g). Theorem A.3.60(iii) shows that A(E) = K.(E) whenever E has AP; the converse is a well-known opE'n problem. The theory of tensor products of Banach spaces is given in (DE'fant and Floret 1993), (Diestel and Uhl 1977, Chapter VIII), (Helemskii 1993), and (JamE'Son 1987); the seminal work is (Grothendieck 1955). A stronger form of A.3.64 is given in ('Ulger 1991, §4). For A.3.68 and other estimates on projective norms, see (Helemskii 1993, §2.6); for A.3.7l, see (Defant and Floret 1993, §§5.6, 16.7). For a recent account of Lipschitz spaces, see (Weaver 1999). For more sophisticated versions of the results about O(U, E), see (Laursen and Neumann 2000, §2.1).
834 A.4
Appendtx MEASURES AND INTEGRALS
Let 8 be a non-empty set. A a-algebra on 8 is a family 9J1 of subsets of 8 such that: (i) 8 E 9J1; (ii) if T E 9J1, then 8 \ T E 9J1; (iii) if (Tn) C 931, then U:=l Tn E 9J1. For each subset F of P(8), there is a smallest a-algebra containing F: it is the a-algebra generated by F. For example. let X be a topological space. Then the a-algebra generated by Ux is the Borel algebra of X, denoted by Bx; Bx consists of the Borel sets. Let Y be a topological space. A mapping f : 8 - 7 Y is 9J1-measurable if f-l(V) E 9J1 (V E Uy): a Bx-measurable mapping is a Borel mappmg. The spaces of Borel and bounded Borel functions on a topological space X are denoted by B(X) and Bb(X), respecti vely. Let 9J1 be a a-algebra on a non-empty set S. A measure on 9J1 is a complexvalued function f..l with domain 9J1 such that
f..l(V) =
L
f..l(Vi)
(V
E
9J1)
(A.4.1)
i=l
for each partition {Vi : tEN} of V in 9J1. The set of all measures on 9J1 is denoted by M(9J1). Let 11, // E M(9J1) and a,/3 E C. We define:
(al1'
+ /1v)(V)
=
ap,(V) + i3v(V)
(V E 9J1).
Then (M(9J1), + ) is a linear space, and the triple (S, 9J1, f..l) is a measure space. A positwe measure on 8 is a function f..l defined on 9J1 and taking values in [0,00] such that (A.4.1) holds; a positive measure f..l is finite if f..l(S) < 00, and a probability mea8ure if peS) = 1. Let It, v be finite measures. Then f..l ~ v if f..l(V) ~ v(V) (V E 9J1).
Theorem A.4.1 (Jordan decomposition) Let 11 be a measure on a set S. Then there exzst finite measures f..l1, Jl2, f..l3, f..l4 on S with f..l = f..ll - 112 + i(/l3 - 114), and such that Vj ~ /lj (j = 1,2,3,4) whenever f..l = Vl - V2 + i(v3 - V4) for fimte measures VI, V2, Va, V4 on 8. 0 Let (8, 9J1, f..l) be a measure space. We define the total varwtion I/ll of 11 by setting 1111 (V)
= sup {~I/l(Vi)1 : {Vi : lEN} is a
partition of V in 931}
(V E 9J1).
Then l/ll (V) < 00, and 111,1 is a finite, positive measure on 9J1. The linear space M(9J1) is a Banach space with respect to the total variation norm 11·11, given by II/lil
=
Illl (S)
(/l
E
1\1(001».
Let {1 be a positive measure on 9J1. A set V E 9J1 is a-finite if there exists a family {Vn : n E N} in 001 with /l(Vn ) < 00 (n E N) and U~=I Vn = V, and /l is a a-finite measure if S is a-finite. For example, Lebesgue measure on IR is a-finite. A subset N of S is /l-nllll if N E 001 and /leN) = 0; a property that holds on the complement of a /l-null set in S holds /l-almost everywhere. A subset N of S is It-locally nllll if V n N is /l-null for each V E 001 with /leV) < 00.
Measures and tntcgrals
835
Lpt 11 be a pOfolitive measure on a set S. An 9Jl-measurable function I : S ---+ C is simple if 1(8) is finite and p({s E S: 1(8) =I- OJ) < 00. Let f = "Lj'=l Cl!jXVj be a simple function. Then we define
1 :t I
dJi =
V
Let
I :S
---+
n Yj)
O!jJl(V
(V E 9Jl).
j=l
[0. xj be 9Jl-measurable. and let V E 9Jl; we d('finc !,IdP=sup{lvgdP':9 simple,
Let
I :S
Iv I
---+
dp
C be 9Jl-measurable and such that
=
Iv
(SRJ)+ dJL
-Iv
(SRJ)- dJL
+i
O~9~f}.
Iv III dp < 00: we define
(Iv
(CSJ)+ dp
-Iv
(s.JJ)- dP) .
Let JL be a positive mea..oojv
The set of these integrable functions is denoted by Ll (/l. E): Ll (fl., E) Banach space with respect to the norm defined by
Ilfll =
is Ilf(s)11
dJL(s)
IS it
(J E L 1 (11,E))
(where we again identify fund ions which are equal almost everywhere). The map f f--> f el/l. L 1 (11. E) -> E, is linear anel
Is
(A.4.5) Proposition A.4.16 Let /l be a a-finite, positive, regular Borel measure on a non-empty, locally compact space n. and let E be a Banach space. Then each f E C(n, E) with '/;l IIfll d/l < x is Bochner- integmble. 0 Theorem A.4.17 Let p. be a a-finite mea8ure on a set S, let E and F be Banach spaces, and let T E 8(E, F). Then, for each f E L 1 (11. E), To f E Ll(p. F) and (T 0 f) dJI = T f dI1). In partzcular, A f d/l) = (A 0 f) el/l for each
Is
Us
Us
A EE'.
Is
0
Let n be a non-empty, locally compact space, and let E be a Banach space. Then we can also define i;l f dJ1 for each bounded. continuous f : 0 -> E and each p. E 1\[(0): again A f d/l) = J~l(A 0 1) dfl. (A EE').
Un
Proposition A.4.1S Let U be a non-empty, open set in (z .. ). U -> Ll(I/lI), zs analytic. Proof The map z f--> fJ (z. t)g(t) dJ1(t) is analytic for each g E L'X>(IJLI) by A.2.14. and so the given map is weakly analytic. By A.3.76, the given map is analytic. 0 Notes A.4.19 1: 0 such that, for each y E F. there exzst Xi,l, ... ,Xi.m E Ei (i E N n ) with m
m
y= LT(X1,j, ... ,x n ,j)
and
L
Ilx1.j II" ·llxn.j I
::;
M
Ilyll
j=l
j=l
Proof For p, q E N, define Fp,q to be the set of elements y E F such that p
p
y =
LT(X1,j'" "Xnj) )=1
and
L
IIX1.j II ' , . IIJ;n,j II ::; q
j=l
for some XU, . .. , Xi,p E E. (i E N n ), Then Fp,q is an analytic subset of F, being the continuous image of a closed subset of the analytic space (Il~=l Ei)(p). Since lin T(Il~l E i ) = F. we have F = U Fp.q. Since F is a Banach space, Fr,s is non-meagre in F for 801I1e r, sEN. By A.5.17, Fr,s - Fr,s contain8 an open ball, say with centre 0 and radius 5. Then each y E F belongs to the set (211YII /5)(Fr,s - Fr,s)' The re8ult follows with Tn = 2r and M = 4s/5. 0 We conclude this appendix with an amusing remark that is referred to in §1.1.
Lemma A.5.22 Let X be a complete metric space, and let f : X ---> IR be a function such that f-l(B) E BPx for each B E BJR. Then there is a meagre subset Y of X such that f I (X \ Y) is continuous.
846
Appendzx
Proof For 11. E Nand k E Z, define Yn,k = 1- 1 ([kin, (k+l)ln)). By hypothesis, Yn,k E BPx. and so there exist Un.k E Ux such that Yn.k.6. Un,k is meagre. Since countable unions of meagre 83ts are meagre. there exhits a meagre set Y such that Yn.k n (X \ Y) = Un.k n (X \ Y) (11. E N, k E Z+). Take Xn,k to be the characteristic function of Yn.J.., and set h n = E~_')()(kln)Xn.k (11. EN). Then hn I (X \ Y) is continuous. Since (h n ) converges uniformly to f, f I (X \ Y) is continuous. 0 Lemma A.5.23 Let E be a separable (F)-space, let p be a semmorm Or! a lmear space F, and let T : E -> F be a lmear map. Suppose that, for each B E l3 1R , (p 0 T)-l(B) E BPE' Then T : E -> (F,p) zs continuous.
f = poT. By A.5.22. there is a meagre set Y Y) is continuous. Take (xn) E co(N, E), and define
Proof Set
f I (E \
c
E such tha.t
Z = U{k(Y - xn) : k, n E N} U U{kY: kEN}.
Then Z is a meagre subst't of E, and so Z -I- E: choose :r E E \ Z. We sec that Xn + xlk E E \ Y (k,n E N) and :t:lk E E \ Y (k EN), and so f(xn + xlk) -> f(xlk) as n -> oc for each kEN. Thus
li~~~pp(Txn) :::; (~) p(Tx) and so TX n
->
0 in (F,p).
(k E N), 0
Let DC be the axiom of dependent choice, and let BP be the Baire property axiom: every subset of every complete, separable metric space has the Bazre property. It is known that, in SoloV'Cty's model of ZF + DC, which is mentioned in §l.l. BP becomes a true statement. Thus we have the following conclusion. Theorem A.5.24 There is a model of the theory ZF + DC in which every linear map from an (F)-space mto a locally convex space is continuous. 0 Notes A.5.25 Analytic spaces are discussed in (Kuratowski 1966), in (Bourbaki 1960). where they are termed 'espaces sousliniens', in (J. P. R. Christensen 1974, Chapter 1), and in (Rogers and Jayne 1980). The latter two source P(S). For such a schem~. set A" = n{A(o- r k) : kEN} (0- EN). and set SeA) = U{A". : 0- E Af"}. ThE'll SeA) is the Souslin set of A. L(·t X be a topological SpaCE). We define Sx to be the family of Souslin sets of a Sou..'!lin seherne A with AVJ;) c F x. For each X, WE:' have Ax C Sx C BPx (Kuratowbki 1966, §:3H, II). (Rog~rf> and JaynE' 1980. 2.9.4). and Ax = Sx in the eahe where X if> an analytic space. These r~~marks lead to a plOof of A.5.14 (Kuratowski 1966. §:~!), II). Thrort'ms A.5.15 and A.5.W are due to Pettis (Hl50). generalizing earlier results of Banach for the metriza.hl(' ea, to solve problems in the throry of separahle Banach spaces was not.iced by (.J. P. R. Chri:.:;tensen 197{:i), where a form of A.5.20(i) was obtained by using results contained in (J. P. R. Christmsen 1974). The cOllnection was developed by (Loy 1976). where results dose to A.5.20 and A.5.21 are given. The argument of A.5.21 shows tha.t. if £1 ..... E" are separable Banach spaces. if T : f1 E; --> F is a continuous n-linear surjpetion onto a Banach space F, and if U is the open unit hall in E" then T(U) + T(U) is a neighbourhood of 0 in F. An example (Horowit7. 1975) shows that T(U) Jwcd not be a neighbourhood of 0, even if El, ... , ETI are finite-dimensional.
n
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Index of symbols
Z = {O, ±1, ±2, ... }, the integers N = {I, 2, ... }, the natural numbers Z+ = {O, 1, 2, ... }; Z- = {O, -1, -2, ... }
Ord, 7; Card, 8 w, WI, No, N j , 8; 2~o s O} Q- = {x E Q: x::::; O} JR, the real line JR+ = {x E JR : x ;::: O} = [0,(0) JR+. = {x E JR: x > O} = (0, (0) JR- = {x E JR: x::::; O} = (-00,0] c Gaussian, 541 ]{-. 13, 2tl. 1-14. 157 mull iplicat ive. 27 non-unital. 12 opposite. II ordered. 1G OV(,I ]{. 17 ratiollal, 14.43. 1U6. 543, 573-4. 579. 777 real, 14, 50, 543 4, 552. 576 -7, 769,7767
Index topological, 168 torsion-free, 13 totally ordered, 16 unital, 12 well-ordered. 16, 197 Set' also: algebra. scmigroup scmigrollp stability. 615. 688 seIuinorm. 811 algebra. 152. 156,416,433.693 uniformly convex. 744. 819 seminormable. 152·-3, 160. 184, 192. 420, 590, 783 sentence. :3 relatively consistent. 3. 7G4 independent. 3, 76·1 scparatcs the points, -.107 strongly. 407 separating. 169. 812 ideal. 620-1. 635, 639, 643 modnle, 617, G39, 641 space. 5!J7. 609. 617. 626. 6356 sequence. 5 analytic. 464 convex, 511 desccnding. 2 differentiable, 464. -.171 dual. 285 rlyadic, 10 increasing. decreasing. () logarithmically convex, 464. 471 long exact of cohomology. 134. 28:3 non-allal.nic, 464 radicaL 501 short exact. 2G l:>tar-shaped. 51L 513 set. (\1-. 10.21 ih-. 10. 768 111 -711-, 10 directed. 5 Ditkin, 411. 419, 428, 475, 478 rl1-, 10.21. 56G, 681 f"'u-, 785 G(j-. 168, 202, 785 Heboll, 412, 491, 742 of non-synthesis, 411, 620, 670· 1
Index of synthesi~, 179,411,415,476, 491, 495. 498, 517 open, 785 power, 2 Semi-1'/l-, 10, 566 Sierpin~ki. 11 transitive. 7 set theory, axiomatic. 2 Zermelo- Framkd. 2 simplicially trivial See: Banach algebra, ~implicially trivial: C~ -algebra. simplicially trivial Sierpinski set, 11 a-algebra. 834 SIN group, 402, 405, 746 Hingularity point, 647-8 singularity set, 647, 669, 672, 674, 687,713 Souslin scheme, set. 847 space. analytic, 841, 844 compact, 787 connected, 789 count ably compact, 787 disconnected, extremely, 437, 444, 789 totally, 214, 408, 663, 665, 789 discrete. 786 F-, 422, 439-40, 446, 665, 679, 789 (F)-, 810, 844 Frechet, 811 hemicompact, 582, 584. 588, 788 k-, 588, 788 Lindel6f, 584, 589-90, 787 locally compact, 787 locally connected, 789 locally convex. 811 metric, 785, 790 complete, 790 metrizable, 785 Polish,841 pseudocompact, 160, 788 relatively compact, 787
905 relatively sequentially compact, 787 replete. 446, 588 Heparable. 786 sequentially compact, 787 a-compact, 788 ::;tate, 146. 346. 356 Stone, 446 structure. 66. 178, 410 topological. 785 topological linear. 810 Hpectral analysis, 179, 404, 415, 498, 547 spectral radius. 78, 183. 203 continuous. 198, 258 formula, 193. 200. 206 9, 352 spectral seminorm. 190 spectral space. algebraic, 58-9, 87. 652. 675 analytic, 651, 659 spectral synthesi~, 411, 417, 432. 458, 475, 478 spectrum, 78, 166, 183 4, 193, 213 analytic locaL 650 continuous, 198, 214 joint, 80, 204, 229 Taylor, 229 splits See: complex, splits; extension, splits state, 146, 343, 348, 369 pure, 350. 448 tracial. 146, 348 strong boundary point, 447- 8 support, 20, 407, 786 supremum, 4 surjection, 2 Swiss cheese, 455 symmetric, 12, 27, 148, 186 See also: algebra, symmetric; module, symmetric symmetric difference, 2 symmetrizing map, 27 syntax, 3 system, m-, 43
906 tail. 5 Taylor expansion. 1.5tl. 459 tensor produet. 26, 32, 54. 164 injective. 185. 230. 827 projective, 165, 185. 193. 230. 381. 493. 829 Src also: algebra. tensor theory, consistent. 3 topological *-algebra, 342 topological algbera, 169, 1tl5 complete. 169 essentiaL 172 functionally continuous, 581. 58·1595 pliable, 175-6, 637 topological direct sum. 821 topological homology. 287, 304 topological linear space. 810 complete. 810 sequentially complete, 810 topologizable, ] 69. 1tl9, 236, 593 topology, 785 hase,785 compact-open, 186. 5R8, 590, 789 inductive, 795 completely regular, 186, 407, 786 coordinatewise convergence, 185, 499 Gcl'fand. 37, 178. 191. 407. 411 Hausdorff, 786 hull-kernel, 66. 68, 178, 411, 431, 659 Jacobson, 68 normal, 7tl6 order, 589, 789 p-adic,628 product. 792 quotient. 813 reg ular, 786 relative. 785 strong operator, 327, 363. 392, 539, 736, 819 stronger. 785 subbase, 786 T o-, 66, 786 T}-, 411. 786 weak, 814-15
Index weak opC'rator, 36:t 819 wmk*, 366. 815 ·]6 weak* operator. 819 weaker, 785 torsion module, 53, 1;6-7 torsion submodule. 53. 57, 654 total set of functionals, 815 totally bounded, 790 totally ordered, 4 tran" 29. 31,37,147,177,243,299 trace duality, 246, 351 trace extension prop
