Topological Algebras with Involution

TOPOLOGICAL ALGEBRAS WITH INVOLUTION MARIA FRAGOULOPOULOU Department of Mathematics University of Athens Greece 2005 ...

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TOPOLOGICAL ALGEBRAS WITH INVOLUTION

MARIA FRAGOULOPOULOU Department of Mathematics University of Athens Greece

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A Pn(z) := max

\xm\,

l<m—> p[/(x) := inf{A > 0 : i £ AC/},

6

Chapter I. Background material

called gauge (or Minkowski functional) of U, is a seminorm; see, for instance, [198, pp. 94-95]. Concerning submultiplicativity of pu we have the following (cf. [262, p. 3, Proposition 1.4]). 1.5 Proposition. Let A be an algebra and p : A —> R a real function on A, The following are equivalent: (1) p is an m-seminorm; (2) p = pu with U an absorbing absolutely-convex and multiplicative subset of A. Proof. (1) => (2) Let U = Up(l) be as in (1.9). Then, U clearly has all the properties mentioned in (2). It remains to show p — pu- Notice that (1.11)

xeA

with

p(x) = 0 =4> pu(x) = Q.

Indeed, x G A with p(x) = 0 implies p(nx) = np(x) = 0, for all n G N; hence, x G — U, for all n G N, that yields (1.11). Still we have to prove n (1.12)

p(x)=pu(x),

Vx GA

with

p(x) > 0.

Let x G A with p(x) > 0. Then, —— G U, that is x G p(x)U, consep{x) quently pu{x) < p{x). Now, for each A G {A > 0 : x G AC/} one clearly has p(x) < A, therefore p(x) < pu(x) and (1.12) is proved. (2) => (1) According to the comments before Proposition 1.5, it suffices to show that pu is submultiplicative. Let x, y G A. Put pu{x) = £ and Pu{y) = rj. Then, for each e > 0 there exist A > 0 with x G XU and fi > 0 with y G /iC such that A < £ + £, respectively /i < n + £. Moreover, xy G (\U)((iU) = \n{UU) C A/i£7, where A/i > 0. Thus, Pu(xy) x + y : A[r] —> A[r],y G A) are homeomorphisms, an ^-neighborhood in A[T] is of the form x + V, with V £ 21. A closed, absorbing and absolutely convex subset of a topological algebra A[T] is called barrel. An m-barrel is a multiplicative barrel of A\r\. 1.7 Definition. A locally convex algebra is a topological algebra whose the underlying topological vector space is a locally convex space. In particular, a locally convex algebra whose the underlying locally convex space is a Frechet space (that is, metrizable and complete), or a barrelled space (in the sense that every barrel is a O-neighborhood), is called Frechet respectively barrelled locally convex algebra.

Every Frechet locally convex algebra is barrelled [198, p. 214, Corollary]. The topology T of a locally convex algebra A[T] is defined by a fundamental O-neighborhood system consisting of closed absolutely-convex sets. Equivalently, the topology r is determined by a family of (nonzero) seminorms. Such a family will be denoted by F — {p} (or eventually, for distinction, by FA — {p}) and always will be assumed saturated, without any harm of the generality. That is, for any finite subset F of F the seminorm (1.13)

PF{X) := maxp(x), x £ A, eF

belongs again to F. Stating throughout that F = {p} is a defining family of seminorms for a locally convex algebra A[T], we shall mean that F is a saturated family of seminorms denning the topology r of A. That is r = Tp, with Tp completely determined by a fundamental O-neighborhood system given by

8


the e-semiballs (1.14)

Up{e) := eUp(l) = {x G E : p{x) < e}, e > 0, p G F;

more precisely, for each 0-neighborhood V in A[T], there is an e-semiball Up(e), e > 0, p G F, such that Up(e) C V. The neighborhoods f/p(e), e > 0, p €. F, are called ftasic 0-neighborhoods. Taking the completion of a topological algebra A[r] (that is, taking the completion of the underlying topological vector space of A[r]), we may fail to get a topological algebra, unless the multiplication of A[r] is jointly continuous [262, p. 22, (4.1)]. If r = Tp, the respective completion of A[rr], when it exists, will be denoted by A[rj;], where F consists of the (unique) extensions of the elements of F to the corresponding completion of A[rr]Given a Frechet locally convex space E[r], every bilinear separately continuous map from E[T] X E[T] in E[T] (with the product topology on its domain) is jointly continuous (see, for example, [198, p. 357, Theorem 1]). Hence, every Frechet locally convex algebra has a jointly continuous multiplication. Now, let A[rr\ be a locally convex algebra and F' — {p1} a second family of seminorms on A.

We say that F and F' are equivalent and

1

we write F ~ F ', whenever the corresponding topologies rp and Tp> are equivalent. We respectively use the notation Tp ~ rpi, in the sense that the identity map id A

A[rp] —> ^4[T'], is a topological isomorphism (see

discussion after Definition 2.2). Further, suppose that

-A[TA], B[TB]

are two

locally convex algebras with TA = TpA and TB = TpB. We say that a linear map (f> : A[TA] —

-B[rs] is continuous, if and only if it is continuous at

zero; that is, for every q e FB there is p G FA and k > 0 such that (1.15)

q{4>{x))

|, V T G C{H) and £,r? G H,

Denote by TW the locally convex topology

corresponding to the family of seminorms F = {p^v,

£>?? G H}; TW is

called weak operator topology and C{H) endowed with rw is denoted by CW(H).

We show that the multiplication in CW(H) is separately but not

10

Chapter I. Background

material

jointly continuous. If T is a fixed element in C(H) and T* its adjoint (see Section 3, (3.3)), we easily get that for any £,rj G H Pt,v(TS) = PS,T*(T,)(S) resp.

Pi,v(ST)

= Pr^S),V

S e

£(H)-

Therefore, the linear maps CW(H) —> £W(H) :S^TS,

CW(H) —> £ „ ( # ) : 5

> ST,

are continuous (see (1.15)). To show that the multiplication (1.19)

CW{H) x CW{H) —> £„,(#) : (T, 5) —> TS,

is not jointly continuous suppose, for instance, that H is separable (cf. [283, p. 450]). Then, we can choose an orthonormal basis {ek}kefi

m

H

and define S G C(H) such that S(ek) := Cfe-i, for A; = 2 , 3 , . . .

and 5(ei) = 0;

put T n := 5 " , n G N. Clearly, Tn e £(iJ), n G N and (1.20)

Tn

> 0.

It is easy to see that the adjoint S* of 5 is given by S*(ek) = efc+ii k £ N, so that T* = (Sn)* = (5*) n , n G N. Moreover, TnT* = I,

(1.21)

neN,

where / is the identity operator in C{H). Now (1.20), (1.21) show that the map (1.19) cannot be jointly continuous. Thus, CW(H) is a Hausdorff locally convex algebra with separately (not jointly) continuous multiplication; in particular, CW{H) is neither metrizable nor complete, (cf., for instance, [191, 2; p. 154]). R e m a r k . The weak operator topology rw on C{H) is, in particular, a topology induced by a dual system. More precisely, the functions (1.22)

u€iJ, :CW{H) - ^ C : T ^ « f (T) :=< T£,V >, V £,v e H,

1,

Topological algebras. Definitions, examples

11

belong to CW(H)', for all £,77 e H. Then, if F{H) denotes the linear subspace of £W(H)' generated by the functions «£iT?, £,,rj G H, the pair {C{H),!F{H)) is a dual system [198, p. 183, §2.] and the corresponding a(£(H),J:'(H)) topology on C(H) coicides with TW; that is, (1.23)

TW =

q(x). Then, we find A > 0 such that p(x) > A > q(x) and since p, q are absolutely homogeneous, one gets — ) > 1 > q ( —), which contradicts (2.5). Hence, p(x) < q(x), for every A/ VA/ x <E A .

16

Chapter I. Background material Now, for any p,q G F with p < q, one has Nq C Np so that the

connecting maps (2.6)

gpq : A[q]/Nq —> A\p]/Np : xq = x + Nq i—> a;p EE a; + i\Tp,

are well defined continuous surjective morphisms, such that gpq o gq = gp. Hence, gpq, p < q, are uniquely extended to continuous morphisms between the Banach algebras Aq, Ap; we retain the symbol gpq for the extensions too. The families (A[p]/Np, gpq), {Ap, gpq), p < q in F, form inverse systems of normed, respectively Banach, algebras. Their corresponding inverse limits denoted by lim A\p]/Np and lim Ap give an m-convex respectively an ArensMichael algebra to which our initial algebra A[rr] is related as follows (2.7)

A[rr] ^ lim A\p}/Np -» limAp, p e r, p

p

where "^->" means topological injective morphism. In the case when the initial algebra A[rf] is moreover complete, that is an Arens-Michael algebra, one has equalities in (2.7); namely, (2.8)

A\Tr\ = lim A\p]/Np = lim Ap, p € F,

up to topological isomorphisms (see, for instance, [272, Theorem 5.1], [262, p. 88, Theorem 3.1] and [191, p. 259, §2]). The expression of A[rr] in (2.8), as an inverse limit of Banach algebras, is called by A. Mallios Arens-Michael decomposition of

J4[TT]

[262, p. 91]. For distinction, in the non-complete

case, we shall name the expression of the m-convex algebra A[r^] in (2.7) Arens-Michael analysis of

A[TJ^].

Before we proceed to examples we present a result of R.M. Brooks [84, Theorem 3.2] (also see [174, p. 64, (3.1.7)]), analogous to the Gel'fand theorem for Banach algebras, which asserts that the norm of a unital Banach algebra A may always be taken to preserve identity. 2.3 Theorem. Let A[rr] be a unital m-convex algebra. Then, there is a family F' = {p1} of m-seminorms

on A, defining the same topology as F

and such that p'(e) — 1, for every p' £ F'.

2.

Arens—Michael algebras. Basic examples

17

Proof. Since p ^ 0, for every p £ F, it follows that p(e) ^ 0, for all p £ F. The function (2.9)

p'{x) := sup{p(xy) : p(y) = 1}, x £ A,

is a well denned seminorm on A with the properties (2.10)

p'{x) < p{x), V i e i

a n d p'(e) = 1.

In particular, (2.11)

p(xy) < p'(x)p(y),

V x, y £ A and p £ F.

Indeed, if y £ A with p(y) = 0, then p(xy) = 0 = p'(x)p(y),

for every

y

x £ A. If y £ A with p(y) ^ 0, then, p(-r-r) = 1, therefore from (2.9) PU/) p(xy) = p{y)p X—— < p(y)p'(x), V x £ A.

\ p(y)J Let F' = {p1}. For any p£ F, p' £ F1 one has from (2.10) and (2.11) that p'(x) < p{x) < p(e)p'(x), V x £ A. Hence F ~ F' (equivalently Tp ~ Tp> on A) with p'(e) — 1, for every p' £ F'. It remains to show that each p' is submultiplicative. Fix y in A with p(y) = 1 and let x,z be arbitrary in A. Then, from (2.11) P((xz)y) o Km

an

d for each K C Rn compact, there is Km such

that K C Km. Furthermore, let p = {pi,

be a multi-index; that

is an ordered n-tuple of nonnegative integers pi, i — 1 , . . . , n with length I P I : = YH=iPi-

L e tx

G R n and dj = — , j = l , . . . , n be

= (xi,...,xn)

the symbol of the j-partial derivative. We set Q\P\

d if\p\

= O,dpf

-d1

= f,fe

multi-index q = (q\,..., (2.16)

cr2

C°°{Rn).

dn

-

dxv^..dxvn^

Now for a n y f,g e C ° ° ( E n ) a n d a second

qn), we have (Leibnitz formula)

dp(fg)(x) = Y^(P)dpf(x)d*'-ig(x),

x e Rn,

where q < p means q{ < pt, for every i = 1 , . . . , n and (£) = (^) (^) . For any m G N U 0 and p = (pi,... ,pn), we define (2.17)

Pm(f):=snp{\dpf(x)\:xeKm,\p\<m},

f G C°°(K").

(^).

20


Each Pm, m <E N U 0, is a seminorm on C°°(Kn). For simplicity's sake we employ the same symbol Jô as in Example 2.4(1) to denote the family of seminorms {Pm}, m G N U 0. Let Trx be the corresponding topology on C°°(Rn). Then, C°°(R n )[T r J is a Hausdorff Frechet locally convex space [232, p. 73, Theoreme 26]. Using (2.16) one also obtains that (2.18)

Pm(fg) < aPm(f)Pm(g),

V f,g e C°°(Rn),

where a is a positive constant depending on m; thus C°° (M.n)[Trooj becomes a Frechet locally convex algebra with jointly continuous multiplication. On the other hand, see [335, p. 53, 5.4] (2.19)

C°°(R n )[TrJ = lim(C°°(Rn)[Pm}/Nmr,

m e N U 0,

up to a topological isomorphism (of locally convex spaces), where Nm = ker(P m ), m e N U O , and (C°°(R")[Pm]/Nm)~ the completion of the normed space C°°(Rn)[Pm]/iVm, under the norm || || m induced by Pm (see (1.6)). Now, let C(m>(Km), m g N U O , denote all functions on the compact subset Km of R n , which are continuously m-differentiable on the (non-empty) interior Km of Km.

That is, an element / €E C^m'(Km) is a function / :

Km —> C such that dpf exists on Km and is continuous for every multiindex p (as above) with | p | < m. Under algebraic operations defined pointwise, C^m'(Km) is a unital commutative algebra. Moreover, C^m'(Km) is a Banach space endowed with the norm (see [232, p. 73, 3]), (2.20)

| | / | | ( m ) := sW{\dPf(x)\

:xGKm,\p\<m},

/ €

C^(Km),

that moreover fulfils the inequality (cf. (2.18)) (2.21)

||/m{f) := \\f\Km \\'m, f e C°°(Rn),

m e N U 0.

Then, T^ = {P^}, m G N U 0, is a family of m-seminorms on C°°(lRn), equivalent to the original one 7 ^ = {Pm}, m £ N U 0, (see (2.17) and (2.20)). In this way (also see (2.19) and (2.23)), C°°(Rn) viewed retopologized (that is, as Coo(Mn)[rr^j]) turns into a Hausdorff Frichet algebra, whose the Arens-Michael decomposition consists of the Banach algebras C^\Km){\\

.||(m)], meNUO.

Remark. If in place of Mn we take an open subset U of R n , we can find again a sequence of compacts Sm, m € N U 0, exhausting U and such that every other compact subset of U to sit inside a member of this sequence. Thus, C°°(U) becomes as in the case of C°°(Rn) a Prechet algebra, whose an Arens-Michael decomposition is given by the Banach algebras

C^m'(Sm),

m e N U O (see (2.23) and the comments following it); for more details, see, for example, [232, p. 73], [330, p. 32, 1.46] and [371, p. 87]. (3) The algebra 2)(R"). Let X be a topological space, E a (topological) vector space and / a map from X in E. We call support of / (and

22


denote by supp (/)) the smallest closed subset of X in the complement of which / vanishes; namely, (2.25)

supp(/):={seX:/(a;)Ô},

where "—" means (topological) closure. Let now (2.26)

2)(Mn) : = {/ <E C°°(Rn) : supp (/) is compact}

Clearly, 1 ^ 2)(K n ), where 1 stands for the constant function 1. Endowed with pointwise defined algebraic operations, £>(]Rn) becomes a nonunital commutative algebra. Moreover, the sequence of the compacts Km := {x e Rn : | x | < m + 1}, m e N U O , with | | the standard norm on R n , cover R n . Set (2.27)

VKm(Rn)

:= {f € C°°(Rn) : supp(f) C Km}, m € N U 0.

Equipped with pointwise defined algebraic operations and the relative topology, say r™, from C°°(IRn)[rr^] (see discussion after (2.24)) lDKm(Wl) becomes a Frechet algebra (for the completeness of 25icm(Kn)[r™] cf. [344, p. 64]). Since Kn is a-compact, the algebras (2.27) form, in particular, an increasing sequence of ideals in S)(Mn), such that 2)(Kn) = |j3)/f m (R n ), m > 0.

(2.28)

Consider the natural injections, (2.29)

j

m:Dxm(I")

—> S)(R"), m G N U 0.

The usual topology on 2)(R n ) called "inductive limit topology" and denoted by rj), is defined to be the finest locally convex topology making the injections (2.29) continuous. In other words, a subset V of 23(Kn) is a TX>neighborhood of 0, if VnDKm(M.n) is a -^-neighborhood of 0 in ®Km(Rn), for each m € N U 0. On the other hand, if r® is the relative topology on 2)(Kn) induced by the standard topology r roo of C°°(Mn) (cf. Example 2.4(2)), we readily have that (2.30)

T £ -< r 3 .

2.

Arens-Michael

algebras. Basic examples

23

Later on we shall discuss some specific interesting properties of the preceding topologies 735, r ^ (see, for instance, Example 6.12(2) and Proposition 28.7). Now, note that S)(Kn)[r2)] is acomplete Hausdorff locally convex space (see, e.g., [198, p. 164, Corollary and p. 165, Example 6] and/or [232, p. 74, 2]). The topology T£> is described by a family of seminorms given in the following way: Take a sequence a = (km), m G N U 0, of nonnegative integers and define oo

(2.31)

k p Pa(f) : = J2 ™ snp{\d f(x)\

: x € Km+1 \Km,\p\
] is a non-unital commutative Arens-Michael algebra, called algebra of test functions and/or algebra of the compactly supported C°°-functions on Rn. Note that 2)(Kn)[T®] is dense in C°°{M.n)[Troo], therefore it is not complete [232, p. 77, Theoreme 30]; but S)(Rn)[r®] attains a weaker concept of completeness, the so-called advertible completeness (Warner), which in several cases makes a topological algebra to behave as being complete; see Section 6 and, in particular, Example 6.12(2). For the construction of a function in £>(Rn), see [198, p. 166] and/or [232, p. 78, Lemma 2]. For the general case of the algebra 1)(X), with X a finite dimensional 2nd countable C^-manifold, see [262, p. 133] and Proposition 28.7. (4) T h e algebra 5(M n ). Let p = (pi, n) be a multi-index with n length [ p \— XîLiPi (i this regard, also see Example 2.4(2)). Let / £

24


C°°(R™). We say that / is rapidly decreasing (at infinity), if (2.32)

pm,k(f)

:= sup{(l+ | x \)k\dpf{x)\

: x G R n , | p |< m} < oo,

m, A; = 0,1, 2 , . . . , where | | is the usual norm on R"; cf. for instance, [198, p. 91], [232, p. 75, 3)], [330, p. 168], [371, p. 92]. Define (2.33)

5(R") := {/ G C°°(Rn) : / is rapidly decreasing}

and consider on it pointwise defined algebraic operations. Let Fs = {pm,k}, m,k G NUO. Each pm^ G fg is a seminorm and Fs is clearly countable; thus if TS = rps is the locally convex topology induced by Fs, one has that 5(R n )[rs] is a (Hausdorff) Frechet locally convex space (cf.

[371, p. 92,

Example IV], and/or [379, p. 404, (1)]). If q = (qi,... ,qn) is a second multi-index with [ q \ < 2fc, k > 0, the topology TS is equivalently defined by the following (countable) family of seminorms (2.34)

p'mik(f)

:= SvLP{\x"dPf(x)\

m > 0, where xq := xfxf...xln,

: x G R " , | p \ < m}, / G x = {xi,...,xn)

S(Rn),

G M n . From (2.32)

and/or (2.34) it follows that each / G S(Rn) is a bounded function on Rn. Furthermore, using Leibnitz formula (cf. (2.16)), we conclude that (2.35)

PmMfd) < apm,k(f)Pm,k(9), V / , g e S{Rn)

and m.feeNU 0,

where a is a positive constant depending on m, k. It follows from (2.35) that the ring multiplication in S"(Mn) is jointly continuous, and particularly that 5(Mn) verifies Michael's theorem [272, p. 18, Proposition 4.3] (also see [262, p. 31, 5 (1) and p. 37, Corollary 5.12]); thus, TS can equivalently be described by a fundamental 0-neighborhood system consisting of mbarrels. Hence, S(IRn) turns to be an m-convex algebra. More precisely, the algebra 5(K") of all rapidly decreasing C°°-functions on W1, is a non~unital commutative Hausdorff Frechet algebra. Compared to the algebras C°°(IRn) and 2)(E n ) (see Examples 2.4(2) and 2.4(3)), 5(E") always sits between them; that is, if 2)(R n ), 5(Mn) carry respectively the relative topologies T ^ , T^ from C00(Mn)[Tr00], one has that (2.36)

S)(Kn)[rS] ^ S(Rn)[r^} A : x \—> x*, with the following properties: (1) (x + y)* = x* + y*, V x, y G A,

3.

Topological *-algebras

(2) (Ax)* = Ax*, V A e C ,

27 x£A,

(3) (xy)* = y*x*, V x,y E A, (4) (a:*)* = z, V i e A . An algebra A equipped with an involution is called an involutive algebra, or an algebra with involution. If E is a vector space, a map * : £ ' — > £ ' : I H I * ,

with the properties (1), (2) and (4) is called a linear involution.

It is clear that a linear involution, respectively an involution, is always a bijection (because of (4)), which in the first case is anti-linear (because of (1) and (2)) and in the second case an anti-morphism (because of (l)-(3)). A subalgebra B of an involutive algebra A is called a self-adjoint subalgebra, whenever it is invariant under *; that is, x* E B, for every x G B. 3.2 Examples of involutive algebras. (1) The algebra C(X), X locally compact, of all continuous functions on X, becomes an involutive algebra under the involution defined by the complex conjugate. That is, (3.1)

* : C{X) — > C{X)

* w i t h f * { x ) := f ( x ) ,

V i e l

For X = N we have the particular case of the algebra C N (see Examples 1.4(1), 2.4(6) and (3.24)). Thus, one has x* = (5^) n 6 N , V x = ( z n ) n e N G C N .

(3.2)

(2) The function algebras C°°[0,1], C°°(R"), £>(Rn), S(Rn) (see Examples 2.4(l)-(4)) become all involutive by defining involution as in (3.1). (3) The algebras CW{H), CS{H) of the Examples 1.8(2), 1.8(3) become involutive under the involution defined by the adjoint of an operator, given as follows

(3.3)

* : C(H) —> C{H) : T — » f with < T*(£), JJ >=< f, Tfa) >,

for all £, r? e H. (4) The n-disc algebra A{Jbn) (CI- (2-40)) becomes involutive under the involution

(3.4)

* : A(Bn) -^

A(Bn) : / — > / * : f*(z) := f(z), V z e D n ,

28


n G N. That / * G A(B>n) follows from (2.43). In the same way, one defines an involution on the algebra O{C) (see Example 2.4(5)). Indeed, because of (2.44) each / G O(C) is of the form (ZbJneN with f\Dn E A(Bn) such that Qnm{f\nm) — / | o n , V n < m in N, where gnm, n < m, are the connecting morphisms between the Banach algebras of the inverse system (ACDn))neN (cf. (2.6) and the comments after it). Thus, one may define (3.5)

* : O(C) - ^ O(C) : / ^

/ * := ((/| D J*)

and check easily that * is compatible with the inverse system; that is Qnm((f\BmT) = ( / | D J * , Vn<m,

therefore / * G O(C).

3.3 Definition. Let A be an involutive algebra. An element x E A is called self-adjoint respectively normal, if a;* = a; respectively £*£ = xa;*. If moreover A is unital an element x G A is called unitary, if x*a; = e = ra*. We fix the following notation for a given involutive algebra A. (3.6)

H(A) = {x G A : x* = x},

will stand for the self-adjoint elements of A. Clearly 0 G H(A) and, when A is unital, e G H{A) too. H(A) is a real vector space and (3.7)

A = H{A)®iH(A),

where "©" means direct sum. Indeed, for each x G A there are unique elements xi,X2 S H(A) such that (3.8)

x = x\ + 1x2 with x\ = — - — and X2 = ———

The normal elements of A will be denoted by (3.9)

N(A) = {x G A : x*x = xx*},

where clearly H(A) C N(A) and N(A) = A when A is commutative. Furthermore, if A is unital we shall use the symbol (3.10)

U(A) = {xeA:x*x

= e = xx*},

3.


29

to declare the unitary elements of A, where readily U(A) C N(A).

On

the other hand, every x G U(A) is invertible with inverse x~l = x* and x~l

e U(A) with (ar 1 )* = x= (a;*)"1.

3.4 Definition. Let A be an involutive algebra. A seminorm (respectively m-seminorm) p on A is called *-seminorm , or *-preserving seminorm (respectively m*-seminorm , or ^-preserving m-seminorm) if (3.11)

p{x*) =p{x),

V i e i

The terms *-norm and m*~norm are similarly denned.

An involutive

algebra ^4 endowed with an m*-norm, respectively an m.*-seminorm, is called a normed *-algebra respectively m*-seminormed algebra. A complete normed *-algebra is said Banach *-algebra. In general, the term topological *-algebra refers to a topological algebra A[r] equipped with a continuous involution (this is, for instance, the case whenever (3.11) is true); in the opposite case we use the term involutive topological algebra. In the case of an involutive topological algebra A[T], we shall use the (algebraic) term self-adjoint subalgebra, in the sense of the discussion after Definition 3.1. But, when involution is continuous, that is A[T] is a topological *-algebra, then accordingly the term "self-adjoint subalgebra" will be replaced by the term *~subalgebra. A continuous involution on a topological algebra is readily a homeomorphism or more precisely a topological anti-isomorphism. Because of (3.11) a Banach *-algebra often is called a Banach algebra with isometric involution. Conditions under which an involution defined on a topological algebra is continuous, are given in Chapter III, Section 16. There are various sources of getting examples of normed topological *-algebras (see, for instance, [191, Chapter III]). One of them comes from function algebras, from where we mention: The Banach algebra C(X) of continuous functions on a compact space X; the Banach algebra Cb(X) of continuous bounded functions on a locally compact space X and its closed subalgebra CQ(X) of continuous functions on X vanishing at infinity; all of them are endowed with the uniform norm denoted by || ||oo and the (isometric) involution denned by the complex conjugate (cf., e.g., (3.1)).

30


A second source is that of operator algebras, with leading one, the Banach *-algebra C(H) of bounded linear operators on a Hilbert space H, equipped with the operator norm and the (isometric) involution induced by the adjoint T* of an operator T G C(H) (see (3.2)). Calkin and Fermion algebras are also of a particular importance in this category; see [127, p. 127], [279, pp. 30 and 123], respectively [191, p. 147] and [225, p. 759]. A third source is that of group algebras. From this category we particularly use the group algebra LX{G) of a locally compact group G. If dx denotes a left Haar measure on G, i 1 (G) consists of equivalence classes of Borel measurable functions f on G such that JG \f(x)\dx exists, define on L1 (G) vector space operations pointwise, multiplication by convolution, that is

(3.12)

( / * g)(x) := [ f(y)g(y-1x)dy}

f , g e LX(G),

x e G

JG

and norm by

(3.13)

H/ll := f \f(x)\dx,

fEL\G).

JG

In this way, L1 (G) becomes a Banach algebra. Further, it becomes a Banach *-algebra with respect to the (isometric) involution given by

(3.14)

f*(x) := Aix-^Jix^),

f € L\G), X e G,

where A is the modular function of G. I/1(G) is commutative if and only if G is commutative (equivalently abelian) and it has a left (or right) identity if and only if G is discrete (cf., for instance, [126, p. 81, Section 13]). In the general case of a locally compact group G, Ll(G) has a bounded approximate identity (abbreviated to bai), bounded by 1 (ibid., p. 85, Theorem (13.4)). Particular classes of topological *-algebras, to which a considerable attention is given in this book, are the locally convex *-algebras and m*convex algebras. 3.5 Definition. A locally convex *-algebra respectively m*-convex algebra is a topological algebra whose topology is denned by a (saturated)

3.

Topological ^-algebras

31

family of (nonzero) *-seminorms respectively m*-seminorms. A complete m*-convex algebra is called Arens-Michael *-algebra. A Frechet locally convex, respectively metrizable Arens-Michael, *-algebra is said Frechet locally convex, respectively Frechet, *-algebra. When no continuity of the involution is involved, we use (according to Definition 3.4) the terms involutive locally convex, involutive m-convex, involutive Arens-Michael algebra. Similarly, for Frechet. Examples of non-normed topological algebras with continuous or not involution are given in Subsection 3.(2). 3.(1) Basic properties Let A be an involutive algebra and U C A. Set (3.15)

U* := {x* : x G U}.

U is called a *-set if U* = U. In this regard, an ideal / of A for which /* = / , is said *-ideal. 3.6 Definition. A map <j> : A —> B between two involutive algebras A, B is called -^-preserving, when {x)*, for every x G A. Thus, a *-morphism is a *-preserving morphism between two involutive algebras. In the category of involutive topological algebras (respectively topological *-algebras) a topological * -isomorphism is a *-preserving topological isomorphism. The term topological injective *-morphism applies to a * preserving topological injective morphism. Let A be an involutive algebra and / a *-ideal of A. Considering the corresponding quotient algebra A/1 we may define (3.16)

(x + I)* :=x*+I, V

xeA.

Then, the natural quotient map g : A —> A/I with g(x) :— x + I, x e A, becomes a *-morphism and g(H(A)) = H(A/I); (3.17)

V x +1 e H(A/I)

3 zeH(A)

namely, with x + I = z + I.

32


Indeed, given x + I G H(A/I)

there is y G A with g(y) = x + I. Then, the y -\- y* required z G H(A) is served by . Let now A[rr] be an m*-convex algebra. Since, each p G F is an m*seminorm, the ideal Np (see Example 1.2(2)) is a *—ideal, for every p G F. Thus, for every p G F, the correspondence (3.18)

A\p]/Np

—> A\p]/Np :xv = x + Np^^x*p

= x* +NP,

is a well defined isometric involution on A\p\jNp. In this way A[p]/Np becomes a normed *-algebra and its completion Ap a Banach *-algebra. At the same time, the quotient maps gp, p G F, as well as the connecting maps gpq, p < q in F (see (2.2) and (2.6)), turn to be continuous *-morphisms. Thus, the families (A\p]/Np, gpq), (Ap, gpq) with p < q in F, form inverse systems of normed, respectively Banach, *-algebras such that (cf. (2.7)) (3.19)

A[rr] ^-> \im A\p]/Np ^> \imAp,

p G F,

up to topological injective *-morphisms. In the case when A[rr] is moreover complete, that is an Arens-Michael *-algebra, then (see (2.8)) (3.20)

A[Tr} = \im A\p]/Np = \imAp,

p G F,

with respect to topological *-isomorphisms. The topology of a locally convex algebra A[rr] with continuous involution can equivalently be defined by a family of *-seminorms. More precisely one has the following. 3.7 Theorem (Brooks). Let A[rr] be an involutive locally convex algebra. The following statements are equivalent: (1) The involution of A[rp] is continuous. (2) The topology Tp is defined by a family F' = {p1} of *-seminorms. Proof. The implication (2) => (1) is evident. We prove (1) =^ (2). For each p G F define the function (3.21)

p'(x) := ma,x{p(x),p(x*)},

x G A.

3.


33

It is easily checked that each p' is a *-seminorm on A. If F' = {p1}, using the statement (1) and (3.21) it follows that rp ~ rpi. Furthermore, it is an easy observation that F' is saturated, provided that F has the same property. 3.8 Remarks. (1) If A[rr] in Theorem 3.7 is an involutive m-convex algebra, each p' G F' (see (3.21)) is clearly an m*~seminorm. (2) Let A[rp] be as in Theorem 3.7. Consider the e-semiballs Up(e), Upi(e), e > 0, corresponding to p G F, respectively p' G F1. Then,

Up,(e) = Up(e)nU;(e)

and Up, (e) = Up (e),

so that when the involution of A is continuous, the topology rp can be denned equivalently by a fundamental 0-neighborhood system consisting of ^-barrels (that is barrels, which are *-sets; cf. discussion after (3.15)). In this language, the two equivalent statements (1) and (2) of Theorem 3.7, can be reformulated as follows [84, Theorem 3.1]: (1') The involution of A[rp] is continuous. (2') The topology of A[rp] is defined by a fundamental ^-neighborhood system consisting of * -barrels. If A[rp] is an involutive m-convex algebra, then ^-barrels are replaced by m*-barrels (namely, m-barrels that are *-sets). Every complete locally convex algebra is embedded as a closed ideal in a locally convex *-algebra, in the way Proposition 3.9 describes. 3.9 Proposition. Every locally convex algebra A[TA] is topologically embedded, through a topological injective morphism, in a locally convex *-algebra B[TB\- A[TA] becomes an ideal in B[TB], which is closed whenever A[TA] is complete. If A[TA] has an identity, so does B[TB). Proof. Take B := A[TA] X ^[TVI] and define algebraic operations on B in the following way {x, y) + {z, w) := (x + z, y + w), \(x, y) := (Ax, Ay) (x,y)(z,w) := (xz,wy),

34


for any (x,y), {z,w) G B and A G C B becomes an involutive algebra under the involution: (x,y)* :— (y,x), (x,y) G B. The functions q(x,y) := max{p(cc),p(y)}, V (x,y) G B and p G F, are *-seminorms and define a (Hausdorff) locally convex topology 773 on £?. Thus, B[rj3] is a locally convex *-algebra and the correspondence A[TA] —> B[TB] x 1—> (x,0) is a topological injective morphism. If A[TA] is unital, the pair (e,e) is an identity for B[TB]3.(2) Examples of topological algebras with continuous or noncontinuous involution Several of the examples of locally convex and/or Arens-Michael algebras we have already met are topological algebras with continuous or not continuous involution. We refer briefly to a number of them and we elaborate some new ones. Among them is the field algebra and the algebra C(H), of bounded linear operators on a Hilbert space, endowed with some of the known locally convex topologies it accepts, that we shall use in later Chapters. 3.10 Examples. (1) The algebras C°°[0,l], C°°(Rn), £>(R"), S{Rn), O(C) endowed with the involutions described in Examples 3.2(1), 3.2(2) and 3.2(4) are all Arens-Michael *-algebras. (2) The Arens algebra LM[Q, 1] (see (i) before Definition 2.2) endowed with the involution defined by the complex conjugate is a Frechet locally convex *-algebra (also see [338, p. 56, Example 2.5.5]). (3) The algebras £W(H), Caw(H), H a Hilbert space (see Examples 1.8(2) and 1-8(4)) are locally convex *~algebras, while LS(H), H a Hilbert space (Example 1.8(3)) is an involutive locally convex algebra. It is easily seen that the involution (3.3) is continuous in the case of CW(H) and Caw(H), but this is not true for LS(H). To see the latter, take the bounded linear operators T n , n G N, defined in Example 1-8(2) on a separable Hilbert space H (also see [283, p. 450]). If (ek)ken is an orthonormal basis in H,

3.


35

an element £ € H is uniquely written in the form oo

oo

with ^ | a f e | 2 < o o .

£ = ^2akek, ak= fc=i fc=i

Thus (cf. (1.24) and definition of T n before (1.20)), 2 2

2

n

p€(Tn) = ||rn(OH = âkS (ek)

=

fc=l fc=n+l oo

oo

2

2

Yl

a

kek-n

n

- £ iî = EK! -ENi 2 n -^ o fc=n+l

2

fc=l

k=l

But, dealing with T^ (for its definition see Example 1.8(2)), we have 2

2

2

a

e

p 5 (7;*) = i|T n *(0l! = E * * + «

= ll£U 2 >o,

v

^ ° -

fc=i

(4) The Arens—Michael *-algebra CC(X). Let X be a completely regular fc-space; a topological space X is called k-space if a subset B of X is closed (respectively open) whenever Bf)K is closed (respectively open), for every compact subset K of X [230, p. 230]. Consider the involutive algebra C(X) of the Example 3.2(2). Let K, denote the family of all compact subsets of X. For each K G K. define the function \\f\\K := sup \f(x)\ =

(3.22)

||/|K||OO,

/ G C(X),

which is clearly an m,*-seminorm. The Hausdorff topology induced by the m*-seminorms || \\K, K G /C, is denoted by "c" and called topology of compact convergence; under "c", C(X) becomes an m*-convex algebra denoted by CC(X). Since X is a fc-space, CC(X) is also complete [230, p. 31], hence an Arens-Michael *-algebra. In particular, (3.23)

CC(X) = limC(K)[\\ [U], K £ K,

with respect to a topological *-isomorphism. Indeed, if ./Vjf = ker(|| K e /C, the map (3.24)

C(X)[|| [k]/iV K —> C(/f), :f +

NK*-+f\K

\\K)>

36


is a well defined surjective *-morphism. Surjectivity follows from Urysohn's extension theorem for completely regular spaces and compact subsets [172, p. 43, (c)]. Moreover, (3.24) is an isometry, therefore the normed *-algebra £(^0[|]' \\K]/NK is, in fact, a Banach *-algebra. Thus, taking into account (3.20), we get (3.23). For extra information on the algebras CC(X) and C(X)[\\ \\K]/NK, K € /C, see Example 7.6(3) and Subsection 10.(2). Note that if X is just a completely regular space, then CC(X) is only an m*convex algebra imbedded in lixaC(K), K 6 K., with respect to a topological injective *-morphism (see (3.19)). Remark, (i) If X = N, the Arens-Michael *-algebra CC(N) is the Frechet *-algebra C N of all complex sequences (see Examples 2.4(6) and 3.2(1)); namely, one has the equality (3.25)

CN = CC(N),

up to a topological *-isomorphism, when CN carries the cartesian product topology (ibid.). In this case, compare the Arens-Michael decompositions of the two members of (3.25) according to (2.45) and the corresponding one of (3.23). (ii) A topological space X is called hemicompact , if there is a countable increasing family Kn, n £ N, of compact subsets of X, such that X = (J Kn, n € N, and each compact subset K of X sits inside of a Kn, n G N. The real numbers R and the complex numbers C are clearly hemicompact spaces. In particular, every locally compact a-compact space is hemicompact (see, for example, [130, p. 241]), therefore the same is also true for every 2nd countable locally compact space [130, p. 238, 6.3]. If X is a completely regular space, then the m*-convex algebra CC(X) is Frechet if and only if X is hemicompact [174, p. 69 Theorem]. Our last example, is the so called "field algebra", which is of particular importance for Mathematical Physics (see, for instance, [73, 74, 75] and [397, 398]). (5) The field algebra or Borchers algebra. Consider the Euclidean space K4n and the Frechet *-algebra Sn = S(R4n) of rapidly decreasing

3. Topological *—algebras

37

C°°-functions on R 4n (see Example 2.4(4)). Set S0 = C and define &

^ J 7 "Til n=0

where " 0 " means topological direct sum (or locally convex direct sum) of locally convex spaces (cf., e.g., [77, p. 75, Definition 5] and/or [235, p. 211, 5]). Endow S with the direct sum topology, that is the finest locally convex topology with respect to which the natural injections in-Sn

—> S, n = 0,1,2,...

are continuous. S is, in particular, the strict inductive limit of the finite partial sums ®^ = o Sk, where the latter is topologically isomorphic to rifc=o &k [235, p. 222 and p. 212, (2)]. The elements of S are terminating sequences of elements from £ n 's. Thus, for any / = (/o, / i , . . . ) , g = (go,

Topological Algebras with Involution

Topological algebras with involution

Topological algebras