C*-ALGEBRAS
VOLUME2: BANACH ALGEBRAS AND COMPACT OPERATORS
ELSEVIER SCIENCE B.V. Sara Burgerhart5traat 25 P.O. Box 2...
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C*-ALGEBRAS
VOLUME2: BANACH ALGEBRAS AND COMPACT OPERATORS
ELSEVIER SCIENCE B.V. Sara Burgerhart5traat 25 P.O. Box 211 , I 000 A E Amsterdam, The Netherlands
• 2001 Elsev1er Sc1cnce B.V All nghts reserved Th1< work ts pro1cclcd under copyr1ght by Elsev1er St1cncc, and the followmg terms and condlllons apply to 11s use: Photoeopy1ng S1ngle pholocop1es ofsmgle chapters may be made for personal use ns allo"'ed by nanonal ropynghtlaws. remuss1on of the Pubhsher or and pa}ment of a fee 1s requued for all o1her pho1oeopymg. mcludrng mult1ple or sySierrouc copy1ng, ropymg for advcms1ng promouonal purposes, resale, and all forms of document dell\ cry Spec1al rates arc avaJ!able for l and {x, y} is linearly independent,
~y + (~ + 1)x # 0. Thus /3--1,
O~2
_0,
a=0,
b=c=0. Hence 1, x, y are linearly independent.
Proposition 2.1.4.20
Let E
II
be a f i n i t e - d i m e n s i o n a l
real unital algebra,
which is a division algebra, and take z , y E E linearly i n d e p e n d e n t with X2 = y2 ___ _ I . T h e n there is an injective unital algebra h o m o m o r p h i s m u : IH -4 E such that
x=us,
ue u ( ~ ) .
By Proposition 2.1.4.19, there is an a E] - 2, 2[ such that xy + yx = a l.
Then yxy = y(al - yx) = ay + x,
( z y ) 2 = x ( a y + x) = a x y - 1.
Set a
2
a'= c,/4 ,~x+v'4-~,2u' a
2
b := x a = - ~/4"" - a 21 + v'4"':- a':: x y .
Then
~ . I Algebras
~
a2 x
s +
4 - as
4
2a _ as (xy + yx) +
2a s 4 4 - a 21 - -4- --- - a5 1
Oe2
- ~ 1 + 4 - - O~2
~s
b~ -
4
-
as -4- --' - ~a
ab =
4
4a 4 4 ~ a 2 x y + 4 - a 2 ( a x y - 1) =
4a
4
4 ~ a ~xy + 4 L a~ xy -
+ 4
a2 - 4 - ' --- - -a' ~ x
a2
-
a2- 4 = --'----~ 4 1 = -1,
4a
1 -
a2 - 4 " - '-- - 'a~ x
y~ -
a~
4a
c~s
4 - a~
4 4 ~ a 2
2a sx2y a
a2- 4
-4- '-- - -a~ 1 = 4 - a ~
2a 4 4 - a 2y + ~ y x y
2a
2a
4 - a 2y
4a
a ~(ay + x) =
4
4 -
a2
a u y + 4 ---Z---~x = 4 - a ~ x = x ,
ba = x a 2 -- - x
a x -- a2b -
bx - x a x -
=
4
4 - a 2y + 4-
4a
- 4 " - - ' - ~a x - -4- --' - 'a~ y + 4
--1,
-xb
,
-b ,
-- -x2a
~ a.
Define u . M ---+ E ,
~ l + T i + 6 j + e k ~--+ ~ l + T x + 6a + e b .
By the above, u is a unital algebra h o m o m o r p h i s m . By C o r o l l a r y 2.1.2.10 (and Corollary 2.1.4.17), u is injective. Hence
x=~i,
yeu(~).
U
46
~. Banach Algebras
T h e o r e m 2.1.4.21
(Frobenius, 1878) Every f i n i t e - d i m e n s i o n a l real unital
algebra, which is a divison algebra, is isomorphic to JR, to C,, or to ]H .
Let F be a finite-dimensional real unital algebra, which is a division algebra. Assume E # IR1. By Proposition 2.1.4.18, there is an a E E with a2 =
-1.
Hence, if E is two-dimensional, then E is isomorphic to C. Assume that E is not two--dimensional. By Proposition 2.1.4.18, there is a b E E , with a, b linearly independent and b2 - - - 1 .
By Proposition 2.1.4.20, there is an injective unital algebra homomorphism u : l-I ---~ E .
Assume that u is not surjective. Define x:=ui,
y := u j ,
By Proposition 2.1.4.18, there is
a c E
z := u k .
E\u(IH) with
c2 = - 1 . By Proposition 2.1.4.20, there is an injective unital algebra homomorphism v :I-I ~ E such that z = vi,
c E v(lI-l).
Y:=vj,
g := v k .
Define
Assume that Y E u(lH). Then z = vk = v ( i j ) = ( v O ( v j ) = x Y ~ u(ZH),
~ v(~) c u(~), which is a contradiction. Hence Y ~ u ( ~ ) . Similarly, Z ~ u ( ~ ) . By Proposition 2.1.4.19, there are a,/~ E R with yY + Yy = al ,
~. 1 Algebras
47
z Y + Y z = 31. Hence
az = ( y Y + Y y ) z = y Y z + Y y z = y(/~l - z Y ) + Y x = = 3Y- yzY - Z = 3y- xYz = ~~-
2ze
Z =/3y- Z-
Z,
u(]a)
which is a contradiction. Hence u is surjective and thus E is isomorphic to lH.
II
Proposition
2.1.4.22
Let S, T be set, 4i the set of bijective maps S ---} T ,
and bl the set of algebra isomorphisms I~176 ---} t~176 u~" g~(T) -----+e ~ ( S ) ,
x"
Given ~o 9 f~, define
" xo~o.
Then u~, 9 11 and the map
.
is bijective. We may replace t ~ by co or c in the above enunciation. It is easy to see t h a t u~ 9 b / f o r every ~a 9 4~ and t h a t the m a p
is injective. Take u 9 L/ and s 9 S . Then there is a t = 99(s) 9 T such t h a t
u-l(e,)et ~t O. Then ]Ket C u-l(lKes), i.e. u(]Ket) c lKe,. From e,u(et) = u(u-'(e,)et) # 0 it follows t h a t lKe, C u(lKet). Hence u(lKet) = lKe,. The m a p ~ is obviously injective. By symmetry, it is bijective. Take x 9 e ~ ( T ) and s 9 S . Then
x(~(~))~(,) = ~(,~, SO
(~)e,.
-(~)(u~co~)-
Thus u~x = ux and u~ = u . Hence the m a p ,,, } / g ,
is surjective.
~:
~u~
II
48
~. Banach Algebras
Definition 2.1.4.23 ( 0 )
For every vector space E and for every m, n E
1~I, we denote by Era,. the vector space of m • n-matrices with entries from E , the addition and the scalar multiplication being defined pointwise. If E is an algebra, m, n, p E ~ , a E Era,n, b E E.,p , then we denote by ab the matriz of Em,p defined by II
(ab),,k
"= ~
j=l
a,,bak
for all i E l q . and k E I~lp.
Proposition 2.1.4.24 ( 0 ) a) a E E~,n, b E E.,p,
b)
Let E be an algebra and m, n, p, q ~. IN.
c E Ep,q ~
(ab)c = a(bc) .
E . , . endowed with the bilinear map en,n • En,.
~ En,. ,
( a, b ) :
;ab
is an algebra.
c)
If E is unital then
Proposition 2.1.4.25
[6,,l],a~.
is a unit of E . , . .
II
( 0 ) Let E be a normed space and m, n E ~q. We
set ~,
171
II
I----I 1----1
for every a' E E~, m . If Era,. is endowed with a norm such that ~rl
It
sup Ila,jll _< Ilall _< ~ ~ Ila,,ll
sEINm jEINn
/or every a E Era,., then a' E
I--1 J--I
(Era,.)'
and
sup lla;,ll < Ila~ll < ~
*EINm 2Elan
Ila},ll
i=1 ) = 1
.for every a' E E~,m and the map EIn,m
,(Era,n) t,
is an isomorphism of vector spaces.
a t.
~ a'
P,.1 Algebras
~g
We have m
,.,.,
la'(a)l =
n
~ U,E ,
(x,y):
~. xy
is continuous and its restriction to E # • E # is uniformly continuous.
The assertion follows from Proposition 1.2.9.2 c =:, a & d.
I
Corollary 2.2.1.7
( 0 ) If E is a normed (umtal) algebra, then A c is a closed (unital) subalgebra of E for every A c E .
By Proposition 2.1.1.17 a), Ac is a (unital) subalgebra of E. Let x E A. Since the map E
>E,
y.~ ; x y - y x
is continuous (Proposition 2.2.1.6), the set
{y ~ E lxy = yx} is closed. Hence
gEA
is also closed.
I
75
~. Banach Algebras
Corollary 2.2.1.8
( 0 ) If E is a normed algebra and F i.s a subalgebra (left ideal, right ideal) of E , then F is also a subalgebra (left ideal, right ideal) oIE. is a vector subspace of E (Corollary 1.1.5.4). Define ~o " E x E
~E,
(x,y) ~
xy .
Since ~o is continuous (Proposition 2.2.1.6), ~ ( F ) is closed. Now F x F C ~ ( F ) C ~(F)
( E x F c -~~o(F) C ~ ( F ) ,
FxEc~(F)
C~(P)
),
so that _
_
fxr=Fxfc
~,(~x ~) c
,
~o(~)
~
( E x F -C _ ( F ) ,
-E x E C S ( F )
) ,
(~,(Ex ~)c ~, ~,(Tx E) c ~),
and F is thus a subaigebra (left ideal, right ideal) of E .
m
Corollary 2.2.1.9 ( 0 ) Let E be a normed (unital) algebra, A a subset of E , and F the (unital) subalgebra of E generated by A . Then F is the smallest closed (unital) subalgebra of E containing A . It is called the closed (unital) subalgebra ol E generated by A . I.f A is countable, then F is separable. The first assertion follows immediately from Corollary 2.2.1.8. Suppose that A is countable. If E is unital, we may assume that 1 E A. N
Let B be the set of the elements of E of the form l-I xk, where (xk)~e~ is k-'l
an arbitrary family in A. Then B is countable and F is the vector subspace of E generated by E. Hence F and F are separable. Let E be a normed (unital) algebra and take x, y E E. The closed (unital) subalgebra of E generated by {x}, (by {x,y}) is called the closed (unital) subalgebra of E generated by x {by x and y). If E is complete, then we often say "(unital) Banach subalgebra generated by" instead of "closed (unital) subalgebra generated by". Remark.
~.~ Normed Algebras
Corollary 2.2.1.10
73
( 0 ) Let E be a normed (unital)algebra and A a
commutative subset of E . The closed (unital) subalgebra of E generated by A is commutative.
Acc is a commutative closed (unital) subalgebra of E containing A (Proposition 2.1.1.17 e), Corollary 2.2.1.?). I Corollary 2.2.1.11
(
0 )
Let E be a Banach algebra, (xt)te, a summa-
ble family in E and take x E E . Then (xxt)tet and (xLx)tcn are summable families in E and
I.s
LGI
tel
tel
The assertion follows immediately from Proposition 2.2.1.6.
1 co
Corollary 2.2.1.12
Let E be a Banach algebra. Take x E E . Let ~ tnxn n'-O
be a power series in E . Assume there is an r > 0 smaller than the radius of convergence of this power series such that
/or every a E U~(O). Then
for every n E IN U {0).
Take a E Urn(0). Then f oo
O=x
oo
oo
oo
n=O
n=O
n=O
(Corollary 2.2.1.11). Thus XXn - X n X ~ 0
for every n E IN t.J {0} (Corollary 1.3.3.10).
1
74
~. Banach Algebras
C o r o l l a r y 2.2.1.13 Let E be a normed algebra and E the completion of the underlying normed space. Then there is a unique multiplication on E , rendering E a Banach algebra such that E is a subalgebra of E . It is called the completion o.f E . If E possesses a unit 1, then 1 is a unit of E . Being uniformly continuous (Proposition 2.2.1.6), the map E #xE #
;E,
(x,y):
;xy
can be extended continuously to a map
E x E-----} E ,
(x,y):= ~ xy.
All of the assertions follow from the h c t that E x E is dense in E x E.
I
P r o p o s i t i o n 2.2.1.14 ( 0 ) Let E be a normed (unital) algebra and F a closed (proper) ideal of E . Then E / F is a normed (unital) algebra (Theorem l.g.4.g a), Proposition 12.1.1.13 b), d)). If E is complete or commutative, then so is E / F . Take X, Y E E l F , x E X and y E Y. Then xy E X Y
and
IIXYII_ IIx~ll < Ilxll Ilull. Siace x and y are arbitrary, I]XYII ~ ]IXII JlYII. Hence E / F is a normed algebra. Now let E be a unital algebra and F a proper ideal of E . Let q" E -+ E l F be the quotient map. Then IlqliI_< IIIII- 1 and
Ilqlll = II(ql)2ll ~ IlqllI ~ (Proposition 2.1.1.13 d)). Thus
I[ql[I- 1 and so E / F is a normed unital algebra. The last assertion follows from Theorem 1.2.4.2e) and Proposition 2.1.1.13 e). I
~.~ Normed Algebras
75
Definition 2.2.1.15
( 0 ) (Peter-Weyl, 1927) Let E be a normed algebra. An a p p r o z i m a t e unit of E is a filter ~ on E such that E#\{0} E ~ and lim xy = lim yx = x
for every x E E . A normed (Banach) algebra possessing an approxsmate unit will be called quasiunita! normed (Banach) algebra.
If E is unital, then the neighbourhood filter of the unit of E is an approximate unit since the multiplication is continuous. If ~ and O are approximate units of E , then ~NO is also an approximate unit of E. For all p E [1, c~[, the Banach algebra obtained by endowing the Banach space ~ with the pointwise multiplication is not quasiunital. Proposition 2.2.1.16
( 0 ) Let E be a Banach algebra. If ~ is an appro-
ximate unit of E then
lim llxll = 1. x,~
If E has a unit, then ~ converges to this umt.
Take y E E. Then
Uyll
= lim z,#
Ilxull c~- ~ -
~
for every t E I and
L f3 (Lt) = 0 for every t E T \ K . Take t, tr I and t E T \ K . Then
f
Ixtlt'dA =
feL(s)lx~(st)l'da(s)=
L
=
.st-',x,c.s,PdA.s.f e L,~,,~,,~ f
Ix~(s)lPd,~(s)
~ - -~ a ~-- I ~
>
(z )' (! }x, lPd~
1
(-1)'~
IXt -- X~t
/
1
Ix~lPdA
(2);
1
lVdA
lx~lPd~X
-
1
,~-
_
Ix, - z,~ L
O t ,
1
(~)~,
1
x,~'11[, =
-
1
,
>
IIx,-
~
Ix,~lPd,\ < ~ ,
L 1
>
f
T\L
>
1
,~=
T\L
Ix, lPd),
7
=
.
II
108
I~. Banach Algebras
C o r o l l a r y 2.2.2.21
Let )~ be Haar measure on the unimodular locally com-
pact group T . Take p E [1, oo[, p E .Mb(T) , and x E I_)'(A) , such that t~*z#O (Proposition ~,.~.~.14 a) ). We set yt'T
~ ;~,
s:
~y(st)
.for every y E L~(A) and t E T . If T is not compact, then there is a sequence (t,,)net~ in T such that I1~ * x'" - ~ 9 x'" lip > I1~ * xll,,
for all distmct m, n E IN.
Take t E T. Then (,.
=
f
= f ~(~-'~t)d~,(~) = (~, 9 z)(st)
= (~ 9
x)t(s)
for every s E T . Thus * x' - (~, * x ) ' ,
I1~ * x'll,, -- I1(~ * *)'lip = I1~ * *lip.
We construct the sequence inductively, starting with t~ := 1. Let n E lN\{1 } and assume the sequence has been constructed for all m E INn-1 9 By the above considerations and by Lemma 2.2.2.20, there is a tn E T such that I1~, * z"- - u . ='" I1,, = II(u * *)'" - (u 9 *)'"lip > Ilu * *11,, 9
m
E x a m p l e 2.2.2.22 ( 0 ) Let (T, ~,tt) be a measure space and take p E [1, cr Gwen x E L~176 define
~. L"(~,) a)
Given x E L~176 ~ E s cation operator on LP(~).
b)
Themap
, L~(~,),
L~(#) --+ s
y:
, .y.
I1~11= Ilxll~. ~ in called a multipli-
x:
~
is a continuous unital algebra homomorphism (Examples 2.2.2.2 and
e.e.e.~).
~.~, Normed Algebras
109
c) For each x e L ~ 1 7 6 o(~) = (~ e ~: I~ > o ~
d)
~ ' ( u ~ ( ~ ) ) i~ not ,, t,-n,,u ~ a } .
Let tt be a a-finite measure or a Radon measure on a locally compact space and put
L~(tt) := { S i x e L~(]t)}. Then 9= Loo(l~ ) . e)
Let # be a positive Radon measure on the locally compact space T . Take p < oo and put
K:(T) := {x e C(T) I Supp x is compact}, K:(T) := { F i x e K:(T)}, L~(tt) := { ~ l x E L~176 Then =
o~
=L~lt.
Moreover if ~ denotes the upper section filter of
{x e K:(T) I 0 < x < eT}. Then
lim ~y = y x,~
.for every y e I ) ' ( # ) .
a) ~ is obviously linear and
II~yllp--(fl~l~lul~d~)~ 0.
A c {1~1 > ~ ) , We have
IJxll JleAJl~'_> II'~eAIIP
-
(L
IzlPd~) x"
_> olleAIIp
if p ~ oo. This inequality also holds for p - co and so
II~li _> ~. Since a is arbitrary, we get
il~il _> II~llo.. b) is easy to see. c) Take a e a(~) and ~ > O. If
~'(uy(~)) is a p-null set then in s
aeT--X
is invertible in L~176 By b), a l - -
is invertible
which is a contradicition.
Now take a E ~ \ a ( ~ )
and put
u := (al - ~ ) - l . Take e E ]0, JJuJJ[ and let A be a /~-integrable set contained in x l ( U ~ ( a ) ) . Then ]J(~l--X)eAJJp-
( L JaeT -- xjPd~) ~ ~_e]JeAJJ,
if p ~ oo and the same inequality holds for p = oo. Since
JJeAJJp -JJu(al - x)eAJlp _< IJuJJ JJ(al - ~')eAJJp _~ IlulJ:JJeAllp, we see that ea - 0. A being arbitrary, it follows t h a t
-xl(U~(a))
set. d) Take u G
. First suppose that T E ~ . Put x :=
UeT e Y ' ( ~ ) .
is a ~-null
~.~ N o r m e d A l g e b r a s
111
Then x y -~ y x -
for every y E L~176
y U e T -- u y e T - - u y
Take c~ > 0 such that A "=
{Ixl > ~}
is not a /z-null set. Define y" T ---4 ]K
t ~
{ ~_s
if
tEA
0
if
t ET\A.
Ix(t)l
' Then y E L~176 so that
IxleA -- x y -- try. Thus Or
Hence
~ U IXleAIlp -- IluYlIP ~-- I1~11 IlYlIP --" II~IIIIleAIIP,
x E L~176
Since xy = xy = uy
for every y E L~176 and since L~176 is dense in LP(#), we get that u - 5 E L~176 If /Z is either a a-finite measure or a Radon measure on a locally compact space, then there is a set 2 of pairwise disjoint sets of "Z and such that for every A E 2
the set
A\UB BEtA
is a p-null set, where TA := {B E T I/Z(A n B) # 0}. Given A E fii, define XA :~ ~eA and
~. Banach Algebras
11~
if t E A foran AE~t if t E T \ I,J A .
XA(t) x'T---~K,
t~--~
0
AE~I
By the above considerations, x E f-2~
and
u = ~ E L~176
Hence
L=(.
c L~(~,).
The reverse inclusion is trivial since L~(/~) is commutative. e) Take u E
. We set ~" := {x E L~176 I ~u = uS}.
Let x be a lower semicontinuous function in s176176 G := {y E K:(T)+Iy _< x}, and ~r the upper section filter of ~ . Then x-uz = x u z = lim y u z = lim ~uz = lim u~z =
=
lim u ( y z ) = u ( x z ) = u'~z
for every z E LY(~u), where the limits are taken in LP(/z). Thus X~ ~
UX,
xE~.
Let (xn),,er~ be a monotone sequence in ~" converging pointwise to an x E L~176 Then ~uy = x u y = lim x,,uy = lim ~,,,uy = l i m u~,,y = n -~ oo
n --~ o o
n -4 oo
= lim U(Xny) = u ( x y ) = u~y n-+ oo
for every Y E / P ( p ) , where the limits are taken in / / ' ( p ) . Thus XU ~
UX,
~.~ Normed Algebras
113
xE~'. By the above considerations, .T = L~176
Hence, by d), uE
"T)
oo/~ =
=Loo~u ,
oo/~
=
L o o ~u "
The final assertion is easy to see.
I
E x a m p l e 2.2.2.23 Let E be a Banach space with topological cardinality R > No and R', R" cardinal numbers such that: I)
R' 0. There is a p (5 IN, such that
< e_. 3
I ,.t r + t l! n=p+l
Since P
lim ~
y--+z
P
Oln~ln ~ Z
n=O
Olnxn
n--O
(Proposition 2.2.1.6), there is a 5 (5 ]0, r-2--~ [ such that Otny n --
n=o
Otnxn
n=o
I1
< -
3
whenever y (5 U~(x). Then r-
IIYil < try - ~11 + It'll
o
there is an no E IN such that n
(~
~ O~n+l
-1) >
(1+~)o-1 1 n
for every n E IN, n >__no. Thus c~__..An
n + l ] ~'
> (--~,
,
O~n+l
for all n ~ IN, n >_ no, so that oo
I n--O
2. Banach Algebras
Lemma
( 0 ) (Binomial Theorem) If a E IR+, then the power
2.2.3.12
series
"(n) e
~
n=O
converges absolutely and uniformly on [-1,1] to (1 + t)" .
We may assume a > O. We have for every n E IN, n > a , ~ ( ~ - 1 ) . . . (,~ - ,, + 1)
-(I
n!
=n
) n+l-n+a 1 =n
n+l
(n + 1)! a(c~ - 1)... (a - n)
=(l+a)~
-1) -
n
SO
lni~moon
(I li~
-(n+~) - 1 )
=l+a>l.
By L e m m a 2.2.3.11
"(:)l n=O
Hence the power series
"(n) e
~
n'-O
converges absolutes and uniformly on [-1, 1] to an x E C([-1,1]). x is differentiable on ] - 1, 1[ and for every t e ] - 1,1[ we have
- n (:) t n - t = ~ k ( : )t ~~- l +
(1 + t ) x ' ( t ) = (1 + t ) ~
n=O
= ~-~ (n + 1) n=O
=~
k=l
( ) a
n + 1
~(
n--o
(n+l)a(a-1)"'(a-n) (n + 1)!
n
(:)
tn=
n=O
tn -I- ~-~ n
(:)
tn -
n--O
+ n a(c~- 1)...(a- n + 1)~ tn _
ni
/
177,3
~.~, Normed Algebras
-
_ ~
~(c~ - 1 ) . . . ( a - n + 1 ) ( ~ _ n + n ) t " = ~ ~
n-'O
n!
tn = ~x(t)
n=O
If we define y:]-1,1[
~ ,
t.
-
(I
~(t) +t)~'
then y'(t) = (1 + t)~x'(t) - a(1 + t ) ~ - l x ( t )
(1 + t) ~a
=
(1 + t)x'(t) - a x ( t ) (1 + t) '~+1 = O.
Hence ~(t)
(1 + t) a
= v(t) = v(o) = ~(o) = l ,
x(t) = (1 + t) a .
By continuity, the above relation holds for every t E [-1, 1]. P r o p o s i t i o n 2.2.3.13
( 0 ) Let E be a unital Banach algebra, c~ E IFt+,
and x E E such that Ilxll < 1 or
lim IIx"ll ~ < 1.
n---4oo
a)
The f a m d y ((~)xn)ner~u{o} ss absolutely summable; we put
[l+x]~':=~
x n. n=O
b)
II
The semes ~
xn
n--O
converges uniformly on E # , so that the map
E#
~E,
x:
"[l+x] ~
is continuous.
e) For every k E IN, ([1 + x]];) ~ -- 1 + x.
I~4
~. Banach Algebras
a & b. Since
ii(:) ll
E
be the resolvent of x . Then f is differenttable and ft
_. _ f2
.
~. Banach Algebras
I~8
Take a E lK\a(x). Then 1
~(f(/3)-
f(o~))= -f(ot)f(~)
for every ~ E 1K\a(x), ~ ~ a (Proposition 2.1.3.9). The assertion follows.
II
#
C o r o l l a r y 2.2.4.7 t 0 ) invertible element of E with
Let E be a unital Banach algebra and z an
I1~11 = I1~-~11 = x.
Then a ( ~ ) c { ~ e ~ 0. Since
(llz.ll).~ is summable in IR, the sequence (I
+
IIx.ll).~
is multipliable in IR. By Proposition 2.2.4.34 a), there is an A E ~I(IN) such that
H (1 + ]]x.I] ) - 1 < nEB
for e~e~y B ~ ~ / ( ~ \ A ) .
If B ~ ~ ( ~ \ A ) ,
IIT-[(1 +
x.)-
111= II~ ~cll_
E,
a" : ~, ( a l - x ) - '
is analytic (Corollary ~.~.4.5) and lim
(al
- x)-'
= 0 .
o-~oo
Take so E lK\a(x) and set r
:--
II(aol - x)-'ll "
Take ~ E U~ (c~o) and set X:=c~ol-x,
Y:=c~l-x.
Then 1
I I X - YII : I ~ - ~01
E ,
o~:
> (al
-
x) -1
.
Then f is analytic and lira f ( ~ ) = 0
~---4OO
(Theorem 2.2.5.1). It follows that
9 -~ = - f ( o ) = 0 (Corollary 1.3.10.7), which is a contradiction.
ll
Remark. a) The above theorem does not hold in the real case without the supplementary condition a~((x,O)) C l R . Indeed, C is a real unital Banach algebra and we have that z ~r
aCi) = O , . rCi) = o,
lira [linll~ -- 1. n--~oo
b) Example 2.1.4.4 shows that any nonempty compact set of IK can be the spectrum of an element of a unital Banach algebra. c) The above theorem was proved by Beurling in 1938, for E := LI(IR). d) r(x) depends on the algebraic structure of E only. The above theorem shows therefore that the number
lim I1~"11~
n--4oo
does not depend on the particular norm of E , provided it defines a Banach algebra on E . Corollary 2.2.5.5
( 0 ) (Gelfand 1939, Mazur 1938) Let E be a Banach
algebra. For lK -- lit, we assume in addition
a~((x,0)) c ~t for every x G E . If E is a dimswn algebra, then
E = IK1.
158
~. Banach Algebras
Take x E E and choose a E a(x) (Theorem 2.2.5.4). Since a l - x is not invertible and E is a division algebra, al-x=0,
x=al.
Hence E=~I.
I
Remark. By the above corollary, r is the only complex Banach algebra which is a division algebra. In particular, there is no norm on the algebra of rational functions with complex coefficients making it a complex Banach algebra. C o r o l l a r y 2.2.5.6 (Le Page, Hirschfeld-Zelazko) Banach algebra. If there is an a E IR+ with
Let E be a unital complex
I1~11~ 0. Thus
Ilxll
= 1} is compact, so
1
I1~il~ < ~11~11 for every x E E . By Corollary 2.2.5.6, E is commutative. C o r o l l a r y 2.2.5.8
I
(Wielandt, 1949) I f E is a complex unital normed algebra,
then xy - yx ~ 1 for ever,j x, y E E .
By Corollary 2.2.1.13, we may assume that E is complete. Assume that xy-
y x -- 1.
Take c~ e a ( y x ) \ l R + . Then E a ( x y ) -. a ( y x + 1) - a ( y x ) + 1
(Proposition 2.1.3.10 b)), so that a-
1 E a(yx)\lR+.
By complete induction,
and
I ~ - ~1 < ~(y~) < Ily~ll for every n E IN (Proposition 2.2.4.1), which is a contradiction. Take a E a ( y x ) N lit+. Then
12. Banach Algebras
160
oz e o ' ( x y - 1) -- a ( x y ) -
1,
a + 1 e o'(xy), a + 1 9 a(yx) Ci R+ (Proposition 2.1.3.10 b)). By complete induction, a + n e a(yx), and
a + n __. r(~x) _< Ilyxll for every n E IN (Proposition 2.2.4.1), which is a contradiction. By the above proof, a(yx) = 0, and this contradicts Theorem 2.2.5.4. C o r o l l a r y 2.2.5.9
II
Let E be a complex umtal Banach algebra and take x E E
such that
a(x) c { a e r Then there is an a > 0 with r(1 - ~x) = iim I1(1 - ax)~ll ~ < 1 I1--~oo
and for any such a and for any n E [q, (a-~ [1- (1-oex)]~ln = x . Since a(x) is compact (Corollary 2.2.4.5) and
U
c
a>0
there is an a > 0 with 1
c Thus
a ( a x ) C U~l(1), a(1 - ax) - 1 - a(ax)
C 1 - U~l(1) -- U~l ( 0 ) ,
lim [[(1 - ax)n[[~ - r(1 - ax) < 1 n--~oo
(Theorem 2.2.5.4). By Proposition 2.2.3.13 a),c),
( a - ~, [1 -
,).
(1 - az)]~
=
1( 1 -
(1 - am)) --- x.
II
~.~ Normed Algebras
161
2.2.6 P o l e s o f R e s o l v e n t s
P r o p o s i t i o n 2.2.6.1
( 6 ) Let E be a complex unital Banach algebra. Take
x E E , ao E r and r l , r2 E IR+, such that
Let oo
(t- ~o)-~. n--" --OO
be the Laurent series of the resolvent of x
in U
(Theorems ~,.2.5.1 and
i.s.io.8). a) {x. In e ~} c {z. In e ~}" = {xF. b)
For every n E IN, X_lXn_I
-- Xn_IX_
1 --- O ~
(x - a o l ) " x , - : = x_: - 1,
x - n = (x - aol)'*-lx_l. c)
x_l is idempotent (i.e.
X_I
-~ X _ 1 ) .
oo
d) a e V =~ x_: = (al-x) E (a-ao)-'~x-- = 1-(al-x) ~ (c~-ao)"X.. n=l
n=O
a) Take y E E and a E U. By Corollary 2.2.1.11,
B'----O0
11-'--OO
If y E {x} c, then
n----O0
I1"---OO
16~
~,. Banach Algebras
by Proposition 2.1.3.9. Hence OO
OO
n--'---O0
~=--00
and so zny = !lXn , by Theorem 1.3.10.8. Thus
e {x. In s 2zy,
(~)~ c (x. In s =}~,
x ~ { ~ . l n ~ = } ~, {:,:.l,~Tz}c{xy. If y E { x , l n E ~ } e ,
then
OO
oo
I1 ---'-- - 0 0
ft'- --00
and so ~ E {(al - x) -I }c by the above relations. By Proposition 2.1.3.9,
~ {x}':,
{x. In ~ ~}~ c {x} ~,
{x,, In ~ ~}': = {x}':.
b), c), and d). Now O0
1 -
(al
-
x) E
(a
-
a o ) n x . --
lPI--' - - 0 0
OO
OO
/1"----00
O0
~'---00
(30
-- E ( o - ~0)"~.-, + E (~- o0)"l~0;- ~)~.}1=--OO
•-'--OO
OO
=
E
(a - ao)n(:rn-i + (aol - x ) x . )
n=--O0
for every a E U (Corollary 2.2.1.11, Proposition 1.1.6.11). Hence x-, + (c~ol - Z)Zo = 1
~.~ Normed Algebras
163
and Xn_ 1 "-
for every n E ~ \ { 0 } that
(X
aol)x.
--
(Theorem 1.3.10.8). It follows, by complete induction,
x-n = (x - a o l ) n - l x _ l , x-1-
1 = ( x - a01)"z,,_l
for every n E IN. Then
n--O
n--1
oo
oo
= (. - ,~o)~
(- - ,~o)--._.
+ r
- ~)~
B--l
oo
= ~-~ + ~
I,~ - , ~ o ) - - . _ .
=
B-'I oo
(~ - , ~ o ) - " ~ - . - ,
+ ~
n=l
(,~ - ~o)-"(,~ol - ~:)~:_. =
n=l O0
---X--1 + Z (0~--~0)--n(x--n--1 n--1
"~" ( O r 0 1 -
X)X--n)"-- X--l,
oo
Z
(or - ao)nXnX_l = (cel
-
x)-lx_I
=
n-'--O0 O0
= Z
O0
r
O0
~0)-~. - Z ( - - ~o/-~. = Z r
n---oo
n=O
n=l
OO
E
( a - ao)"x-xxn = x - x ( a l - x ) - ' =
n--'--O0 co
oo
n=--O0
n"O
oo
n'l
for every a E U (Corollary 2.2.1.11, Proposition 1.1.6.11). Hence X2_l - - X - 1 ,
and XnX-1
--- X - l X n
for every n E IN U {0} (Theorem 1.3.10.8).
--" 0
ii
164
~. Banach Algebras
Corollary
2.2.6.2 (
6
) Let
E be a
complex unital Banach algebra. Take
x E E and let C~o be a p o l e of order p of the resolvent of x . Let y be the residue o.f x in ao.
a)
The principal part of the resolvent of x in (~o is P
(t - ~o)-"(~ - ~0x)~-~. n=l
b) (x- aol)ny #
0 .for all n E lN),_t and ( x - a o l ) J ' y = O.
The assertion follows immediately from Proposition 2.2.6.1 b).
Corollary 2.2.6.3 ( 6 )
i
Let E be a complex unital Banach algebra. Take
x, y E E , a,/~ E ~ , rl and r2 E IR+ such that
u := {~ e e l , ' , < I ~ - ~1 < r2} c r v := {, 9 e l f , < I , - ~1 < ~} c r Let oo
oo
( t - ,~)"x,,, tll--'--O0
~
( t - z)"y,,
tl------O0
be the Laurent series of the resolvents of x and y sn U and V , respectively.
Ii xy
-
-
yx ,
then Xmya = ynxm for every m, n E ~ .
By applying Proposition 2.2.6.1a) twice, we see that
e {=}~= {=~ In e 2z} ~ and
{x~ In e 7z} c {zA~= {z~. In ~ zz} ~.
m
~,.~ Normed Algebras
165
Corollary 2.2.6.4 ( 6 "} Let E be a complex unital Banach algebra. Take x E E such that q~\a(x) is connected. Let a, ~ be two distinct isolated points of a(x) and y, z the residues of the resolvent of x in a and ~, respectively. Then yz=O. Let (30
n-'--OO
be the Laurent series of x in a . By Proposition 2.2.6.1 d), oo
Y = (71 - x) ~
(7 -
a)-nx_n
n--I
for every 7 ~ a in a neighbourhood of a . Since the radius or convergence of
the power series oo
E
tnx_n
n'-'l
is infinite (Theorem 1.3.10.8), the functior. oo
f "qJ\a(x) = ; E ,
~/~
('yl - x ) - l y - ~_~ ('7- a)-nX-n n--I
is well-defined. Take x ~ E E t . The function x t o f is analytic (Theorem 2.2.5.1, Proposition 1.3.10.4) and vanishes on a nonempty open set. Hence it vanishes everywhere, it follows that co
((71 -
x)-ly, x ')
= (~--~ (7 - a ) - ~ x - , x') n--1
for every 7 E ~J\a(x) and so oo
lim
((71 -
x)-ly, x') = (~__, (1~ - a)-nxn, x')
-y~\~(z)
n=l
(Theorem 1.1.6.23). From Corollary 2.2.1.11, Proposition 1.2.1.16, and the classical theory of analytic functions, we see that
{zy, x') = O . Since x' is arbitrary,
zy=O (Corollary 1.3.3.8).
NN
166
12. Banach Algebras
T h e o r e m 2.2.6.5 ( 6 ) Let E be a complex umtal Banach algebra. Take x E E and a E a(x). a is a pole of the resolvent of x iff there is an idempotent element y of E satisfying:
I) x y = y x . 2) (x - a l ) y is nilpotent. 3)
There is a z E E with yz -- zy "- O,
z ( x - a l ) - (x - a l ) z = 1 - y . If these conditions are fulfilled, then the order of the pole ss the smallest number p E ~ l , such that ((x - a l ) y ) p = 0 , and --p
so
( t - ,~)" (~ - ,~l)-"-~u - ~ n=-
1
(t - ,~)"~"+~
n=O
is the Laurent series of the resolvent of x in a . It is defined in
Zer
0 Gp,
Gp
> Gp ,
X ~
"tvx
is the inverse of the map x .~ -~, ttx .
From this and from wv=O, we get that w =
~ o j-1
e) follows from Theorem 2.2.6.5.
o (1 -
v)o. II
174
~. Banach Algebras
2.2.7 Modules The definitions and results of this subsection are special cases of the theory of Banach categories (Section 1.5).
2.2.7.1 ( 0 ) Let E be a (unital) Banach algebra. A (unitaO E - m o d u l e is a Banach space F together with bilinear maps
Definition
E x F
t F,
(x,a).~ ~. x a ,
F x E
~F,
(a,x) .~ ~, a z ,
such that
(~,)a = x(~,a),
Ilxall _< Ilxll Ilall,
(xa)y = x(,W),
,,(~:~) = (a~)y
Ilaxll _< Ilall Ilxll (la- al --a)
for every x, y E E and a E F . An E - s u b m o d u l e o] F is a Banach subspace G of F such that (x, a) E E • G ==~ xa, ax E G .
A (tmitaO E-algebra ts a (unital) E-module F endowed with the structure of a (unitaO Banach algebra such that x(ab) -- (xa)b,
(ax)b--- a(xb),
(ab)x - a(bx)
for all x E E and a, b E F . Let F, G be E-modules (E-algebras}. u E s
G) is called a homomor-
phism of E - m o d u l e s (of E - a l g e b r a s ) i f
uC~a) = x ( u a ) ,
uC,=) = C,,a)~
.for all (x, a) E E • F (and u is an algebra homomorphism).
Every ideal of E is an E-algebra in a natural way. Proposition 2.2.7.2
( 0 ) Let E be a (unital) Banach algebra and F a
(unital) E-module. Given (x, a') E E • F', define xa' : F ~
lK,
a~
(ax,s
~.~ Normed Algebras
a' x : F
~ IK ,
a:
175
" {xa, a') .
F' endowed wsth the bilinear maps
ExF'--+F', F'xE
;F',
(x,a')'
~xa',
(a',x):
~a'x
is a (unitaO E-module.
It is obvious that xa', a'x belong to F ' , that
II~a'll < II~ll lla'll,
Ila'~ll < Ila'll I1~11
for all (x, a') E E x F ' , and that the corresponding maps are bilinear. Take x, y E E and a' ~ F ' . Then, given a E F , {a, (xy)a') = {aCxy), a') = {(ax)y, a') = {ax, ya') = {a, x(ya')),
{a,a'(zy}) = {(xy)a,a') = {x(ya),a'} = {ya, a'z) = {a, ( a ' z ) y ) ,
((a,la') = (al,a'>-- (a,a'>, (a,a'l)= (la,a')=
(a,a')),
so that (xy)a' = x(ya'),
(xa')y = x(a'y),
a'(xy) = (a'x)y
(la' = a'l = a').
II
E x a m p l e 2.2.7.3 Co(T)', xx'
If T is locally compact, then, given (x,x') E Co(T) x is equal to x.x' in the sense of integration theory (Example
1.e.e.lo). For y E Co(T),
and so XX
P r o p o s i t i o n 2.2.7.4
I _--- X . X
t "
( 0 ) Let E be a Banach algebra and x' E E ' .
m
176
2. Banach Algebras
a) x E E c =~ xx' = x ' x . b) If ~d is a approximate umt of E then x' =
lim xx' = lim x'x z,~
zE~
in the topology of pointwise convergence.
a) For every y E E,
(y, x~') = (vx, ~') = (xy, ~:') = (v, x'x) so that XX I _--- X t X .
b) For every y E E , (y, x') = lim(yx, x') = lim(y, xx') x,# x,~
'
(y, x') = lim(xy, x') - lim(y, x'x) x,;}
z,~
so that x' = lim xx' = lim x'x z,~ z,~ in the topology of pointwise convergence. P r o p o s i t i o n 2.2.7.5
( 0 )
I
If E is a Banach algebra and F ts an E -
module, then the evaluation J F : F ~ F" is a homomorphtsm of E-modules.
By Proposition 2.2.7.2, F' and F" are E-modules. Take (x, a) E E x F . Then, given a' E F ' , (jF(xa), a') = (xa, a') = (a, a'x) = (jea, a'x) = ( x ( j f a ) , a') ,
( j f ( a x ) , a') = (ax, a') -- (a, xa') = (iRa, xa') "- ( ( j f a ) x , a')
so that jF(xa) -- x ( j f a ) ,
P r o p o s i t i o n 2.2.7.6
jF(aX) = (jFa)X.
I
( 0 ) Let F and G be modules over the Banach algebra E . If u: F --r G is a homomorphism of E-modules, then so is u'.
~,.~ Normed Algebras
177
Take (x, b') E E x G'. Given a e F , (a, u'(xb')) --- (ua, xb I) = ((ua)x, b') =
= (u(ax), b') = (ax, u'b') = (a,x(u'b')),
(~, ~'(b'~)) = (~a, b'x) = (x(u~), b') =
= (u(xa), b')= (xa, u'b ~) = (a, (u'b')x). Thus
u'(xb') - x(u'b'),
u'(b%) = (u'g)x.
m
/
P r o p o s i t i o n 2.2.7.7 ( 0 ) Let E, F be Banach algebras and u" E -+ F a continuous algebra homomorphism. Then
~(~'v') = ~ , ' ( ( ~ ) v ' ) ,
(u'v')~ = ~ ' ( v ' ( ~ ) )
for every (x, y') e E x F'. Take y E E . Then
(v, ~(,,'u'))= (ux,,~'u')= (u(v~),u')= = ((,~v)(u~), v')= (~v, (u~)v') = {y, u'(C,~x)u')), (v, (~'u')~) = {~v, ~'v') = {~(~v), v') = = ((~)(~y), Definition
2.2.7.8 ( 0 )
u') = (~y, v ' ( ~ ) )
= {u, J ( v ' ( ~ ) ) )
9
II
Let E be a Banach algebra and F an E-module.
Given (a, a') E F x F ' , define aa" E
~ ; IK,
x:
" (xa, a'),
a'a" E
; li,
x:
; (ax, a')
If E = F then the above definitions coincide with those introduced in Proposition 2.2.7.2.
178
~,. Banach Algebras
P r o p o s i t i o n 2.2.7.9 ( 0 ) Let E be a Banach algebra and F an E-module. Then, given (a, d) E F x F', ~a',a'a e E',
Ilaa'll E"
is bilinear. P r o p o s i t i o n 2.2.7.14 ( 0 ) Let E be a Banach algebra and F an E module. Then, given x", y" E E" and a' E F', a'(u" "~ x") = (a'u")x",
(u" ~" x")a' = y"(~"a') .
By Proposition 2.2.7.12 b), given a E F (~,~'(y" -~ ~')) = ( ~ ' , u " -~ ~") = (~", (~a')y") = (~",~(~'y")) = (~, (a'y')~") (~, (u" ~- ~")~') = (~'~,u" ~- ~") = (u",~"(~'~)) = (u", (~"~')~) = (~,u"(~"~')) .
Thus
a'(u" -~ ~") = (a'u")~",
Theorem
2.2.7.15
(~" ~- ~")a' = ~"(~"a').
m
( 0 ) Let E be a (unital) Banach algebra.
a) E" is a (unita O E-algebra with respect to -q (with respect to ~-). We denote it by E~ (by E~ ). b)
The map E - - - ~ E'~ (reap. E~),
is a homomorphism of (unita 0 E-algebras.
x,:: ~ jEx
~,.~ Normed Algebras
e)
181
(~, X u ) E E • E" =~ ~ x"x = x"~- ( j s x ) = x" -q (j~x), 9 ~ " = (js~) e ~ " = (j~x) ~ x"
t
d) The map E~,
~ E'b ,
x" :
" x" e u" (resp. y" -~ z")
is continuous for every y" E E " . e) If ~- and -~ coincide, then F' is an E~_-module whenever F is an E module.
f) If F is an E-module and a predual of E such that (ax, y) = (a, xy) = (ya, x) for all a E F , and x , y E E , then:
t"1) jF(aX) = (iRa)x, j f ( x a ) = x ( j f a ) for every (a,x) e F • E". f~) (3fa)x" = jf(a(3'FX")), X"(jfa) = jf((j'FX")a) for every (a,x") e F x E".
f3) j'F(X " ~- Y") -- 3'F(x" -t y") -- " "t3FX""It3FY" "') for every x", y" e E " . t"4) If ~- and A coinczde, then J'F and the canonical projection of the tmdual of F (Proposztzon 1.3. 6.19 b)) are algebra homomorphisms.
g) If E is commutative, then
gl)
XX' -- XtX f o r every ( x , z ' ) E E x E'.
g2) x'x" = x"x' for every (x', x") E E' x E " . g3) xU& y" = y" -t x" /or every x", y" E E" . In particular, if m ad&tzon ~- and -q coincide, then E" is commutative.
a) By Proposition 2.2.7.2, E" is a (unital) E-module. We first show that Eg and E~ are (unital) Banach algebras. Take x", y", z" E E". By Proposition 2.2.7.14 (and Proposition 2.2.7.2), ((~" ~ r
~ ~", i ) = (z", ~'(~" ~ y")) =
--- re', (~'x")~") ---- (~" ~ z", i ~ " ) - -
(~" -~ ( r
z"), x'),
18~
~. Banach Algebras
((x" ~ y") ~- z",~') = (:~"~- ~/",~"~') = (:,I/"(z"~')) =
= (~", (~"~- z")~')= (~" ~-(:~- z"),~') for every x' E E ' , and so (x" -~ y") -t z" = x" -~ (y" -~ z " ) ,
(x" ~- ~") ~- z" = x" ~- (y" ~- z"). Hence -t and F- are associative. Now suppose that E is unital and take x" E E ' . Then
((jEI)-'IX",X')= (X",X'(jEI))=(Xo,X'I)= (X",X'),
(x""l(jEI),x')-(jE1,X'X")-'-(I,X'X")-- (X",lX'>: (X",X'>, (0~I) ~- ~", ~'> = (3~I, ~"~'> = (I,:~') = (~",~'I> = (~",~'), (x" i- ( j E l ) , x ' ) = (x", (jEl)x') - (x", I x ' ) = (x",x')
for every x' E E' (Proposition 2.2.7.10, Propositior, 2.2.7.2), so (jE1) -~ X # -- X" -~ (jE1) -- X", (j~l) I- x" -- x"k- (3El) -- x". Hence jE1 is the unit of E~ and E~. We now show that E~ and E~ are E-algebras. Let x E E and x", y " E E " . By Proposition 2.2.7.9 (and Proposition 2.2.7.2)
((~') ~ y", ~ ' ) = (~", ~ ' ( ~ " ) ) = (y", (~'~)~"> = = (~" ~ ~", ~ ' ~ ) = r
~ ~"), ~'>,
((~") e ~", ~') = ( ~ " , ~"~') = (~", (:~')~> = = (~", ~"(~'~)) = (~"e ~", ~'~) = (~(~" F y"),~'),
~.~ Normed Algebras
((~" -~ ~")~,~')
183
(~" -~ u",~x') = (~", (~')~") =
= (~",x(~'~")) = ~ 9 "~ , ~'~") = (~" ~(~"~),~'), ((~" ~ ~")~, ~') = (~,,e y", ~ ' ) = (~,,, y"(~')) = = (~", (~"x)~')= (~"~ (y"~), ~'),
((~"~) ~ y", ~') = (~", ~'(~,,~)) = (~-, (~'~")~) = = (~,,, ~,~,,)= (~,,~ (~r
~'),
((~"~) F u", ~') = (~"~, y"~') = (~", ~(y"~')) = = (~", (~y")~')= (~"F (~u"), ~') for every x' E E ' . Thus
( ~ " ) ~ ~"= x(x" ~ ~"),
( ~ " ) e ~ " = x(x" e u"),
(~" ~ y")~ = ~,, ~ (u"~),
(~"~ y,,)~ = ~,,e (y"~),
(~"~) ~ y,' = ~,, ~ (zy"),
(~"~) ~ y" = ~,,e (~y,,)
and E~, E~ are therefore E-algebras. b) By Proposition 2.2.7.5, jE is a homomorphism of E - modules. Take x, y E E . By Proposition 2.2.7.10,
(j~(~), ~ ' ) = (~y, x ' ) = (x, ~ ' ) = 9
Xt
= (jEX, ( 3 E Y ) ) =
(~, ( j ~ ) ~ ' ) =
((jEx)I" (jEy),x'),
(jE(xy),~') = ( ~ , ~ ' ) = (~,~'~)= (u, ~ ' ( j ~ ) ) = = (jEy, x'(jEx)) "- ((jEx) -'t (jEy),x') for every x' E E ' , so that
184
2. Banach Algebras
jE(Xy) = (jEX) F- (j~y) = (jEx) "d (jEY).
Hence jE is an algebra homomorphism. From the proof of a), we see that jE is unital whenever E is unital. c) By Proposition 2.2.7.10,
(X"F (jEx),x') = (X",(3EX)Xt) = (X",XX')= (X"X,X'), (: ~ (j:), ~') = (j:, x':) = (~, x'~") = (:~, ~') ,
((jEx) "~ x " , x ' ) = (X",X'OEX)) = (X",X'X) = (XX",X'),
( ( j : ) e ~",~') = ( j : , : ~ ' ) = ( ~ , : ~ ' ) = (xx",~')
for every x' 6 E'. Hence X" t" (3EX) = X"X = X" H (jEX),
(jEx) H X" -- XX" ---- (3EX) I- X u .
d) Given x' 6 E', lira (x" }-- y" x') = !,!m (x" V"X') = 0
z"--cO
~
Z
--4'0
~
lim (y" -~ x", x') = lim (x", x'y") = 0
z " --r
y".-r
where the limits are taken in E~,. e) follows from Proposition 2.2.7.14. fl) Given y 6 E , (jr(a~), V ) = (ax, V ) = (~,~V) = ( J ~ , ~ V ) = ((Jr~)~, V) ,
Or(*a), V) = ( ~ , V) = (", W ) = 0,~", V') = (*(J~"), V) . Thus jf(ax) = (jfa)x,
j f ( x a ) -- x ( j f a ) .
~.~ Normed Algebras
185
f2) By fl), if x e E then (~. ( j ~ a ) z " ) = ( ~ ( j F a ) . z " ) = ( j ~ ( ~ a ) . z " ) = (za. j k x " ) =
= (x,a(j'Fx"))=
(X,X"(jFa))
= ((jFa)x,x")
(x, j p ( a ( j ' F x " ) ) ) ,
= (jF(aX),X")
= (ax,3Fx" " ' ~ =
= (~, ( j k ~ " ) a ) = ( ~ , j F ( ( j k z " ) a ) ) .
Hence (jFa)X" -- jF(a(?'FX")) ,
x " ( j F a ) -- j F ( ( S F X " ) a ) .
t"3) By f2), if a E F , then (a, I 9f' ( :
I'- y"))
--
=
( j f a , ~,, F- ~ . ) = ( ~ , , ( j f a ) , ~") = ( j f ( ( j ' F y " ) a ) , ~,,) =
: '~ x ""~ = (a. (:~' ~" ) ( ~ '
((Jkr
= (jF(a(j'FX")),y")
= ((jFa)X",y")
" ))=
(a(Jk:).:Ft, ' ""~ =
-- (jFa, X"--I y") -- ( a , j ' F ( X " - I
Hence J~-(~" ~- v") = :,~:""" -~ r
= (:F~" ")(JF~" ") 9
f4) follows from b), f3), and Proposition 1.3.6.19 c). gl) Since
(v. ~ ' ) = ( ~ . . ' ) = (*~. x') = (u..'~) for every y E E , it follows that XX I .-
Xt X .
g2) By gl), if x E E , then
(x,x'x") = (xx', x")= (x'x, x " ) = (x,~"x').
y")).
2. Banach Algebras
186
Hence XtXtt--.XtlXt
"
g3) By g2),
(~'.~" F ~")= (r
~") = (~'~". ~")= (~,. y. d ~-)
for every x' E E'. Hence x" }" y" = y" "1 x".
I
Proposition 2.2.7.16
Let E, F be Banach algebras, u 9 E ~ F a continuous algebra homomorphism, and x 6 E , x" 6 E " , y~ 6 F ' , y" 6 E " .
a) u"(xz") = (ux)(u"x"),
b)
u"(x"x) = (u"x")(ux).
x"(u'y') = u'((u"x")y'),
(u'y')x" = u'(y'(u"x")).
c) ~"(~" -~ ~") = (~"~") -~ (~"~").
~-(~,,~- y.) = (~-~.) ~- ~-~-.
a) By Proposition 2.2.7.7,
(="(~"). r
{~". ='~')= (~,'.(='r = (.."~". r
(~".='(r
= ((..~)(..~"). r
(="(~"~). r = (~"~. ='r = (~".~(='r
= (~".='(=~)y')) =
= (="~". (=~)y')= ((="~")(=~).r for every y' E F'.
b) By a), (*.*"(='r
= (**". ='y')= (="(**"). r = (,... (u'~")r
(~, (~'u')~")= (x"x, ~'r = (~, r
=
(..,.
I
((~.)(="z"). r
((u 11~ I t )~,I )).
(~"(~"~), r (~, ~'(r
((~"~")(~), r
~.~ Normed Algebras
187
for every x E E . c) By b),
(u"(~"-~ ~"), r = (r u'(r
(~"-~ r u'~')= (r (,,'r (,~"r r
(~"(x" e r
((,,"~") -~ (u"~,"), r
r = (~"F r ~'r = (~", r162
= (~", ~'((~"r162
=
= (~"~", (~"r162 = ((~"~") ~ (~"y"), r
for every y' E E'.
II
P r o p o s i t i o n 2.2.7.17
Let E be a Banach algebra, E an associated unital
Banach algebra, ~ an approximate unit of E , and F an E - m o d u l e such that
lim ]lxall = lim Ilaxll = IlaII x,~ x,~ for every a E F . Then F admits a unique E - m o d u l e structure extending its structure as E-module.
We may assume that the unital algebra IK • E defined in Proposition 2.1.1.7 is the underlying unital algebra of E . The uniqueness being trivial, we prove the existence. We endow F with the bilinear maps E • F - ~ F,
((a,x),a) :
'~ aa + x a ,
FxE----+F,
(a,(c~,x)):
"aa+ax,
Take (a, x), (~, y) E E and a E F . Then
((., ~)(~, u))a = ( ~ , ~
+ ~
+ xy)a = ~ .
+ ~ +
Zxa + xya =
= ~(Za + ya) + ~(~a + ~a) = (a, z)CCZ, U)a), ((a,x)a)(/~, y) -- (aa -4- xa)(~, y) = a ~ a + ~ x a + a a y + x a y
- ~(Za + ~ ) + ~(Z~ + ~v) = (a, x)(~(Z, y)),
-
-
~. Banach Algebras
188
a((a, z)(Z, ~)) = a(aZ, ay + Z~ + zv) = ~Za + aav + Z a z + ~ = ~(~
=
+ a~) + (~a + ~ ) ~ = (a(~, ~))(,, u),
(1, O)a = a = a(1, 0). Thus the algebraic axioms are fulfilled. Now
I1(,:,(~, ~))~:11 = II(aa + a~)zll = I1~ + axr
= Ila(-z + ~)11 _
_
II~,jll ~ .
3EINn I=l
For all ((, x) E E n x E ,
3--1
j----1
for all i E ~ m , so
and ~ e Y ( E n, Em). c) The first assertion follows from b). Assume now t h a t E is unital and let
u E :F(E n, Era). For every z E iNto and 3 e lNn, put al~ :-" (U((~3k)ke~n)) i 9 Then for every ~ E E n and i E INto,
i =1
=
I
j=l
~((~)~,) j=l
SO a--u
and the m a p
)
s
= (~(~)),, I
~. Banach Algebras
E~,.
~ ~ ( E " , Era),
is surjective. d) For every ~ E E " ,
= ~ ( ~ ) = (~)~. sEIN~
e) follows from Example 2.2.7.22. f) is easy to see. g) By b),
i.~(.)i-< E
io',,_ Ila'll I1~11-> a'(+)=
$=1
a',,(+,,.,) i=1
)+
=
ll,+~,llai,(b,.,) >_ o
3----1
t=l
J=l
lla~ll _> o
'~+ II~.~,II' i--1
lla.~,ll:+, t--1
3"1
9
$=1
Since 6 is arbitrary, it follows that
Ila~ll ___
~ ;--1
Ila'+,ll2
.
j--1
By Proposition 2.1.4.25, the map e'.,,.
> (E,.,.)',
a':
~ a'
is linear and bijective.
h) By c),f), and Example 2.2.7.22, E.,. is an E-algebra. Take b E E.,.. Then
(b, aa') = (ba, a') = t
( (ba)+k, a'k, ) =
I,k.=l
---
n
yl
z,k=l
3=1
tt
(b., ajka~,) =
t , 3 = l k--1 n
n
= E (b,,,E o,~o~.>, s,3=l
k=l
SO n
k=l
for all z, j E INn. Similary n
k--1
for all z, k E ]5/,.
II
196
~. Banach Algebras
P r o p o s i t i o n 2.2.7.25 ( 0 ) Let E be a Banach algebra, Ij an approximate unit of E , and F an E-submodule of E' (Proposition 2.~. 7.2} such that EF is Haussdorff. a) If ~ converges m EF, then its hmit ss a unit of E and ~ converges to this unit in norm. b) If E#F is compact, then E is unital. a) Let x be the limit of ~ in EF and let y E E . Then for every x' E F , (xy, x') = (x, yx') = lim(z yx') =
= lim(zy, x') = (y, x'). z,~
Since EF is Hausdorff, it follows xy=y. Similarly, yx=y. Hence x is a unit of E . By Proposition 2.2.1.16, ~ converges to x in norm. b) Let .5 be an ultrafilter on E finer than ~. Since E # is compact and E # E ,5, .5 converges in EF. By a), E is unital.
~.~ l~ormed Algebras
197
Exercises E 2.2.1
Let p be a norm on IK. Show t h a t lK is a normed algebra with
respect to p i f f t h e r e is an a E [1, c~[ such t h a t p = c~].I. It is a unital normed algebra if a = 1. E 2.2.2
Let E be a Banach algebra and take x E E . Show t h a t the following
are equivalent: a)
(xn)ne~ is summable.
b)
(xn)ne~ is absolutely summable.
c)
[Ix"I[ < 1 for some n e IN.
E 2.2.3 xEE
Let E be a Banach algebra and a unital algebra such t h a t for each
the maps E
~E,
y.~
;xy,
E .
~ E,
y:
" yx
are continuous. Show that there is a norm on E , which is equivalent to the initial norm, such t h a t E is a unital Banach algebra with respect to this norm. Deduce t h a t every finite-dimensional unital algebra F admits a norm, which renders F a unital Banach algebra. E 2.2.4
Let E be a unital Banach algebra, U a subset of IK, and c~ a n o n -
isolated point of U. Let f, g : U --+ E be maps which are differentiable at c~. Show t h a t f g is differentiable at c~ and t h a t (Sg)'(,~) = f'(,~)g(,~)+
f(,~)g'(,~) 9
Moreover, if f(~)f'(~)-
f'(~)f(~),
then (fn)~(o~) = n f ( o O n - l f ' ( a ) ,
for every n E IN.
198
2. Banach Algebras
E 2.2.5
Let Ea be the unital algebra defined in E 2.1.3. Show:
a) Ea is a unital Banach algebra with respect to the norm
E,~-
~ ~.+,
(~,,7):
-" I~1 + I~vl.
b) Define E,
~ lq.+,
(~, r/) ~
sup{l~ - r/l, I~ + r/I).
El is a unital Banach algebra with respect to this norm and E l , as such, is isometric to r ({0,1 }). E2.2.6
Take a, b E l ~ with a < b ,
nEINU{0},and
E := {x E C([a, b]) Ix is n-times continuously differentiable} Show: a) E is a unital subalgebra of C([a, b]).
b) The map E
>Ft+,
x"
1 Ig~)(t)l
~ sup
tE[a,b] k=O
is a norm. c) E is a unital Banach algebra with respect to the norm defined in b). E 2.2.7 map
Let E be a normed space and a unital algebra. Suppose that the
E x E -----~ E ,
(x,y):
~ xy
is continuous. For each x E E , define ux : E ~
E,
y:
~xy.
Show: a) u= E L:(E) for every x E E . b) The map E
;s
z:
~u:
is an injective, continuous unital algebra homomorphism.
~.~ Normed Algebras
199
c) The map
p.E
>n~+,
~'.
;llu=ll
is a norm equivalent to the initial norm and E is a unital normed algebra with respect to p.
E 2.2.8
Let T be a group. Take p e [1, oo] O { 0 } , x E ~ ( T ) , and define ~" T x T
~ IK,
(s,t) ~ *. x ( t s - ~ ) .
Show that 5 E ~~176 T) and
i1~11 = II~ll~. E 2.2.9
Generalize Theorem 2.2.2.10 to Hausdorff topological groups, without
assuming local compactness. E 2.2.10
Let E be a unital Banach algebra. Take x E E with [[x[[ < 1.
Prove that 1 1 1 + [Jx[[ - [[(1 - x)-l[[ _< 1 -[[x[[ " E 2.2.11
Let E be a unital Banach algebra. Take x E E and f'lK,
3, E ,
(x:
;e "~.
Prove that f is differentiable and that
f ' = xS = I z . E 2.2.12
Let E be a Banach space. Take u E s
and x E E . Show that
there is a unique differentiable map f " lK -+ E such that
I'=uoI.
S(0) = ~ ,
E 2.2.13
Let E be a unital Banach algebra. Take x E E , and
ote lK\(a(x)U
{0}). Prove: if x is invertible, then
(1), 1 - x -1
E 2.2.14
-
- c ~ x ( a l - x) -] .
Let U be the set of invertible elements of the unital Banach algebra
E . Take x e U and Y ' = lim
-
U \ { x } . Prove t h a t
IIy-1 - x - l + x - l ( Y - x ) x - 1 + x - l ( Y - x ) x - l ( Y - x)x-ll[ = O.
~00
~. Banach Algebras
E 2.2.15
Let E be a complex unital Banach algebra. Suppose that for some
aElR+
IIxll II~ll _ ~llxyll, for every x, y E E . Show that E is isometric to ~ . E 2.2.16
Let E be a unital Banach algebra, F a closed unital subalgebra of
E and U (resp. V ) the set of invertible elements of E (resp. F ) . Show that the boundary of V in F is contained in the boundary of U in E . E 2.2.17
Let F be a closed unital subalgebra of the unital Banach algebra
E . Take x E F . Show that if ~ \ a E ( x ) is connected, then aE(x) = aF(x). E 2.2.18
Let E be a unital Banach algebra. Show that {x' e E I IIx'll = ~'(1)}
is a sharp convex cone of E ' . E 2.2.19 Let E, F be Banach algebras and u : E -+ F an algebra homomorphism. We assume that F is unital and finite-dimensional, that
IIx211 = Ilxll 2 for every x E F , and that the ideals of the subalgebras of F are trivial. Show that u is continuous with norm at most 1. If E is quasiunital, then u -- 0 or
IiulI--
1 (gencralization of Proposition 2.2.4.19).
~.3 lnvolutive Banach Algebras
~01
2.3 Involutive Banach Algebras All of Chapter IV is devoted to C*-algebras, which are Banach algebras endowed with an involution, satisfying a very strong condition. The theory of involutive Banach algebras without this condition is of mathematical interest in its own right with applications in harmonic analysis for example. 2.3.1 I n v o l u t i v e A l g e b r a s D e f i n i t i o n 2.3.1.1
( 0 )
Let E be a set. A n i n v o l u t i o n on E is a map E
~E,
x. ~ ;x*
such that X**
--" X
.for every x E E . A n i n v o l u t i v e space is a set endowed with an involution. Let E be an involutwe space. For x E E , we call x* the a d j o i n t o f x . The elements of the set R e E := { x e E i x = x*} are called selfadjoint. For A c E , A*
define
:={x*lxeA}.
A zs called an involutive set o f E if A* C A (in which case A* = A ) . Let E, F be involutive spaces. Given u E F E , define
u* : E----+ F ,
x~--~(ux*)*.
The map FE
~F E ,
u,
~ u*
zs an involution on F E ; it is called the (canonical) i n v o l u t i o n o l F E 9 The selfadjoint elements on F E are called i n v o l u t i v e m a p s . u E F E is an involutive
map iff
~ ' = (~)* for all x E E .
202
~,. Banach Algebras
Let E be an involutive space. Then Re E is an involutive set of E and every involutive set of E is an involutive space with respect to the induced structure. P r o p o s i t i o n 2.3.1.2
( 0 ) Let E , F be involutive spaces, A (resp. B } a n
involutive set of E (resp. F ) and u : E ~ F an involutive map. Then u(A) (resp. -ul(B) ) ,s an involutive set of F (resp. E ) and u(Re E ) C Re F .
If x E A, then (ux)" = ux" ~ u ( A ) .
If x E u 1(B), then uz" = (ux)" e B .
Finally, if x E Re E , then (ux)* = ux* = UX.
Definition 2.3.1.3 ( 0 )
I
An involutive v e c t o r space (resp. an involutive
algebra) is a vector space (resp. an algebra) endowed wtth a conjugate linear involution, which, in the case of an algebra, satisfies (xy)" = y*x ~ for every x, y E E . An i s o m o r p h i s m oj' involutive vector spaces (ol involutive algebras) is an tsomorphism of vector spaces (of algebras), whwh is involutive. Let E be an involutive vector space. For x E E , define 1
rex := ~(x + x*). It is called the real part of x and ts selfadjoint. Let E be an tnvolutive algebra. The elements of the set
NoE := {x E E I x ' x = xx*} ,
Sn E := I x E E
xENoE
if ~ =r
xEReE
if IK = IR
S
are called n o r m a l (0. Toephtz, 1918) and self-normal, respectwely. 1] E is unital, then the elements o.f the set
UnE := {x E E Ix*x = xx* = 1} are called unitary (L. A utonne, 1902}. A conjugate involution on E is an isomorphism of involutwe real algebras u : E --+ E whwh is conjugate linear and for whwh u ~ is the identity map.
~.3 lnvolutive Banach Algebras
~03
If E is an involutive vector space, then from O* = 0 + O* it follows that 0 is selfadjoint. IF: endwoed with the involution ]K
,']K,
x *--+ 9
is an involutive algebra.
Example 2.3.1.4
( 0 ) Let (El)leT be a family of involutive (unital) alge-
bras. The algebra I-I Et (Example ~.1.4.1) endowed with pointwise involution lET
is an involutive (unitaO algebra. If Et = IK for every t E T then the algebra IKr endowed with the involution ]K T
~ ]K T ,
x ~
"x
is an involutive commutative umtal algebra. An element of IKr is selfad3oint iff it ss real. The element x of IKT is umtary iff Ixl- 1. e~ is an involutwe unital subalgebra of IKT . If T is a topological space, then C(T) is an involutive unital subalgebra of e~176 If T is a locally compact space, then Co(T) is an involutive ideal of C(T). II
E x a m p l e 2.3.1.5
The algebra lK[s, t] (of polynomials in the variables s and t with coe~cients in ]K ) endowed wzth the involution
tO
t,3
is an mvolutwe commutative umtal algebra.
E x a m p l e 2.3.1.6
( 0 ) Let E , F be mvolutive vector spaces and L ( E , F )
the vector space of linear maps of E into F . L(E, F) is an involutive set of F s . L ( E , F )
is an involutive vector space with respect to the canonical
involutzon.
Take u e L(E, F ) , x, y e E , and a,/3 E lK. Then u * ( ~ + ~ ) = ( ~ ( ~ * + ~y')}" = C ~ *
+ ~')*
= a(u~*)* + ~(uv*)* = au*x + ~ * v
=
~04
~. Banach Algebras
and so u* E L ( E , F ) . Take u, v E L(E, F) and c~,/~ E IK. Then
(,~u + av)'x = ((,~u + av)x')" = (,~,x" + #vx')" = a
~
m
= n(ux')" + a(vx')" = nu'x + av'x = (nu" + # v ' ) x for every x E E , so that
(au + #v)" = nu" + #v'. P r o p o s i t i o n 2.3.1.7
( 0 )
I
If E, F are involutive vector spaces and u "
E ~ F an involutive hnear map, then u(rex) = re(ux) for every x E E . If, in addition, u is injective, then -1
u (Re F) C Re E .
Now
~,(~) = u(~1 (x
+ x" )) = ~1 ( ~ + ~ ' ) = ~1 (ux + (ux)') = re(ux)
Now suppose that u is injective and x E ul(B.e F ) . Then u(rex) = re(ux) = u x , so that x=rex P r o p o s i t i o n 2.3.1.8
I
I
E ReE.
0 ) Let E be an involutive unital algebra. Then X -1
=X*
for every unitary element x of E and the set of unitary elements oi' E is a subgroup of the multiplicatwe group of invertible elements of E . If x, y are unitary elements of E , then (xy)* xy = y *x" x y = y * y =
1,
(xy)(xy)* = xyy*x* = xx* - 1. Hence xy is also unitary.
I
~.3 lnvolutive Banach Algebras
Proposition 2.3.1.9 IK
x
~05
( 0 ) Let E be a non-un,tal involut,ve algebra and
E the unital algebra associated to E . Then IK
x
E endowed with the
involution
IKxE-~.IKxE,
(a,x)~: ~.(~,x*).
is an involutive unital algebra. It is called the involutive unital algebra associated to E . If E is an involutive unital algebra, then the involutive unital algebra associated to E is E .
It is obvious that the above involution is conjugate linear. Given (a, x), (~, y) e lK x E, ((~.~)(a, u))" = (~a, ~y + a~ + ~u)" = ( ~ , nu" + ~ " + y'~') = = (L ~')(n, ~') = (a, y)*(~, ~)'.
Proposition 2.3.1.10
m
( 0 ) If E is an mvolutive vector space, then Re E
is a real vector subspace of E .
Take x, y E Re E and a, ~ E lR. Then (ax + #y)* = ax + #y. Hence c~x +/~y E Re E.
I
Proposition 2.3.1.11
If E is an involutwe vector space and F a v e c t o r subspace of E , then F* ss also a vector subspace of E .
Take x, y E F* and a, ~ E lK. Then x*, y* E F and a z + ay = (~x* +/3y*)* e F ' .
Proposition 2.3.1.12
I
( 0 ) If E is an involutive algebra, then
for every x E E and n E IN.
Use proof by complete induction. Proposition 2.3.1.13
( 0 ) Let E be an involutive algebra.
a) x*x is selfadjoint for any x E E .
i
~06
~. Banach Algebras
b) If E is unital, then 1 is selfadjoint. c)
If x e E and xy=y for every y E E , then x is a unit of E .
a) We have that X ' X ) * --- X ' X * *
-- Z* Z .
b) I* = I ' I , so by a), 1 = 1". c) We have that XX* -~- X*
and so X* ---~ X
by a). It follows that for y E E ,
Thus x is a unit of E .
II
P r o p o s i t i o n 2.3.1.14
( 0 ) Let E be an mvolutive unital algebra and take
x E E . If x is invertible, then (x-,) 9 = (~.)-~.
In particular,
~(~') = { ~ l a e o(x)} for every x E E . If x is normal (selfadjoint) and invertible, then x -1 is also normal (selfadjoinO.
It follows from x x -1 = x - I x = 1
that (x-1)*x* = x * ( x - l ) * = 1 " = 1
~.3 lnvolutive Banach Algebras
~07
(Proposition 2.3.1.13 b)), so (x-')"
= (x*) -~ .
Hence, given x ~ E , a ~_ a(x*) r
r
(al - x* not invertible) r
(~1 - x not invertible) ~
~ E a(x).
If x is normal and invertible, then ( x - ~ ) * x - ' = ( x * ) - ' x -~ = ( x x * ) - '
= (x'x)-'
= x-~(x*) -' = x-'(x-1)"
,
so that x -l is normal. The corresponding assertion for selfadjoint is obvious. m C o r o l l a r y 2.3.1.15 algebra is normal.
The resolvent of a normal element of an involutive unital
Let x be a normal element of an involutive unital algebra and take a E I K \ a ( x ) . Then a l - x is normal and by Proposition 2.3.1.14, (al - x) -1 is also normal. P r o p o s i t i o n 2.3.1.16 ( 0 ) Let E, F an involutive algebra homomorphism. If then u(G) is an involutive subalgebra of then ux is a normal element of F . If involutwe unital algebra homomorphism, element x of E .
be involutwe algebras and u" E --+ F G is an involutive subalgebra of E , F . If x is a normal element of E , E and F are unital and u is an then ux is unitary for every unitary m
P r o p o s i t i o n 2.3.1.17 ( 0 ) If E is an involutive algebra (resp. involutive unital algebra), F a subalgebra (resp. a unital subalgebra) of E , and G an ideal (resp. a proper ideal) of E , then the same is true for F* and G* . II P r o p o s i t i o n 2.3.1.18 ( 0 ) Let E be an mvolutive (unitaO algebra. If (F,),e~ is a nonempty family of involutwe (unital ) subalgebras of E , then F, is an involutive (unitaO subalgebra of E . If A (resp. x ) zs a subset (resp. tel
an element of E ), then there is a smallest (unita 0 subalgebra of E containing A (resp. x ) , called the involutive (unitaO subalgebra of E generated by A (resp. x ). The same holds - mutatzs mutandis - if we replace "(unital) algebra" and "(unitaO subalgebra" by "vector space" and "vector subspace", respectively. The result is called the involutive vector subspace of E generated by A (resp. x ). m
208
~. Banach Algebras
P r o p o s i t i o n 2.3.1.19
( 0 ) Let E be an involutive (unital)algebra, A a subset of E , and F the involutive (unital} subalgebra generated by A . Then F is the (unital) subalgebra of E generated by A U A* . If A U A* is commutative, then so is F .
Let G be the (unital) subalgebra of E generated by A U A*. Since A*CF*CF,
it follows that G c F . G* is a (unital) subalgebra of E (Proposition 2.3.1.17) containing A U A*. Hence G c G*, i.e. G is an involutive (unital) subalgebra o f E . Thus F C G and so F = G . The final assertion follows from Proposition 2.1.1.17 f). [] P r o p o s i t i o n 2.3.1.20
( 0 ) Let E be an involutive (umtal) algebra. Take x 6 E and let F be the involutive (unital} subalgebra of E generated by x . F is commutative iff x is normal, in which case the map u:G
~ :~ E ,
P"
~ P(x,x*)
ss an involuhve (unital) algebra homomorphism and F = u(C), where
G := {P 6 ~[s, tllP(O, (G :=
o)= o},
1K[,, t]).
By Proposition 2.3.1.19, if x is normal, then F is commutative. The con-
verse is trivial. Assume x to be normal. Then
1,1
s.l
,
10
s,))
for every ~ c~us'tJ 6 ]K[s,t] (Proposition 2.3.1.12 and 2.3.1.13 b)). Hence t,$
u is an involutive (unital) algebra homomorphism (Proposition 2.1.1.19). By Proposition 2.3.1.16, u(G) is an involutive (unital) subalgebra of E and so F c ~(G).
The reverse inclusion is trivial.
II
~.3 Involutive Banach Algebras
P.09
( 0 ) Let E be an involutive vector space, F an involutive vector subspace of E , and q" E --+ E / F the quotient map.
P r o p o s i t i o n 2.3.1.21
a)
X ~_ E / F =~ X ~ e E l F .
b)
E/F
endowed with the involution ElF
"~E l F ,
X'.
~ X*
is an involutive vector space and q is involutive. c)
If E is an involutive (unitaO algebra and F is an involutive (proper) ideal of E , then E l F is an involutive (unita 0 algebra (Propositzon e.1.1.13). a) Take x, y E X*. Then x*, y* E X , so that x* - y* E F . We get
x-y
= ( x - y ) ' * = (x" - y * ) * ~. F * = F .
Now take z E E with x - z E F . Then
x*-z* EF*'-F, z* E X , zEX*. Hence X* E E / F . b) Choose X , Y
E E / F and c~,13 E lK. Further, take x E X , y E Y.
Then a x +/3y E c~X + BY, x* E X * , y* E Y* , so m
~ x * + ZY" ~ ~x* + Zy* = (~x + Oy)* e ( ~ x + Z Y ) ' . Hence ( a X + flY)" = 6 X " + flY*. c) Let X , Y ~. E l F and x E X , y E Y. Then
xy ~. X Y
x* ~. X* y* ~. Y*
so that
Y ' X * ~ y'x* = (xy)* ~. ( X Y ) * (xY)" - Y ' X ' .
a
1~I0
~. Banach Algebras
( 0 ) Let E be an involutive complex vector space and take x E E . Then there are y, z E Re E uniquely such that
P r o p o s i t i o n 2.3.1.22
x := y + iz. Define i m x := z
and call it the imaginary part ol x . Then y=rex=~
1 (x + x* ),
rex = im (ix),
1 ( x - x*),
imx=
i m x = - r e (ix) ,
and the map E----4ReE,
x:
(resp. imx)
;rex
is R-linear. Put 1 ~:=~l ( x + x , ), ~:=~(x-x'). Then y.
z*=
1 ~(~'+~)= ~l ( x
=
1 -~(~"
-
~)
=
+x.
)=~ ,
1 (x-x*)=z,
1 l(x ~+i~=~(x+~')+~
-
x* )=~
which proves the existence. Now take y, z E Re E with
x=y+iz. Then
x* -- y - iz, 1 ~(~" + ~) =
and 1
proving the uniqueness. The other assertions are trivial.
,
X)
Z
1
~.3 lnvolutive Banach Algebras
~orollary 2.3.1.23
( 0 )
Let u " E ---, F
~II
be a linear map between the
involutive complex vector spaces E and F . Then the following are equivalent:
a) u is involutive. b) u(ReE) C R e F . c) u(rex) = re ( u x ) ,
u(imx) = im(ux) f o r every x e E .
d) u(rex) -- re(ux) for every x ~. E . a =} b follows from Proposition 2.3.1.2. b = } c . Now u x = u(rex + i imx) = u(rex) + iu(im x).
3y b), u(rex), u(imx) e Re F so that re(ux) = u(rex),
im(ux) = u ( i m x ) .
is trivial. d =~ c. By Propositioa 2.3.1.22,
c =} d
u(imx) = uC-re(ix)) = - r e ( u ( i x ) ) = - r e ( i u x ) = imCux). c =} a. Given x E E , ux* = u ( r e x - i i m x ) -- u(rex) - i u ( i m x ) --
= re(ux) - i i m ( u x ) = (ux)*. Corollary 2.3.1.24
m
( 0 ) L~t E be an ~,~,ot~,t~v~ ~o,~pt~, at~b,'a. T,,k~
x E E and put
y := r e x ,
z := i m x .
a) x E R e E c : } z = O .
b)
~'~ = y~ + z ~ + i ( y z - ~u),
xx* = y2 + z 2 _ i ( y z -
c) x E N o E v } y z = z y v } x * x = y 2 + z
zy).
2.
d) If E is unital, then x umtary
r
and y 2 + z ~ = 1 ) .
2. Banach Algebras
~,I~
a) The assertion follows from 1
z = y~(~ -
~').
b) x ' x = (y - i z ) ( y + iz) = y2 + z 2 + i ( y z -
zy),
++" = (u + i + ) ( u - ++) = y2 + z 2 _ i(u+ - + y ) .
II
c) and d) follow from b). Remark.
The following calculation would be false: x ' x = ( x ' x ) " = ~2 + z 2 _ i ( y z -
zu) = x x " ,
i.e. all elements are normal! The error arises from the fact that yz - zy is not selfadjoint. Corollary 2.3.1.25
If all the elements of the involutive complex algebra E
are normal, then E is commutative.
By Corollary 2.3.1.24 c), Re E is commutative. Hence ~ is commutative (Proposition 2.3.1.22). 1 Definition 2.3.1.26
( 0 ) An involutive algebra E is called symmetric
(strongly symmetric) if
a(z'x) c lrt+ .for every x E No E (x E E).
If E is commutative, then the notions "symmetric" and "strongly symmetric" coincide. Example 2.3.1.27
Wsth respect to the involution ~
>~,
(~,a)
;(~,n),
the commutatwe unital algebra IK ~ (pointwise multiplication) is an involutive algebra, whzch is not symmetric.
$.3 Involutive Banach Algebras
$13
Given (c~, 13) e IK2 ,
o((c,, ~)*(a, ~)) = o((~f~,a/~)) = {af~,a~} Remark.
I
Another example of a non-symmetric involutive algebra is provided
in Example 2.4.3.7 e).
2.3.1.28 ( 0 ) Let E be a (strongly) symmetric involutive algebra. Then every involutive ideal of E is (strongly) symmetric.
Proposition
The assertion follows immediately from Corollary 2.1.3.15. P r o p o s i t i o n 2.3.1.29
I
( 0 ) If E is a symmetric involutive algebra then
o(~)
~t,
c
.for every x E Re E . Take a E a(x) and let F be the involutive unital algebra associated to E . Then a l - x is not invertible in F . Since
~Zl
-
x 2
= (al
-
x)(al + x),
it follows that a21 - x 2 is not invertible in F (Corollary 2.1.2.6). Thus
,~ e oF(~ ') = oE(~*~) C ~ + . Hence c~ E lR and
o(~) c u t .
mm
( 0 ) Let E be an involutive vector space and take rn, n E IN. We denote for every m x n-matrix a with entries from E (i.e. for every a E Sm,n ) by a* the n x m-matrix with entries from E (i.e. a* E En,m) defined by D e f i n i t i o n 2.3.1.30
(~*),, = (~j,)* for all i E INn and j E INto. P r o p o s i t i o n 2.3.1.31
take m, n, p E IN.
( 0 ) Let E be an involutive (unital) algebra and
Ig. Banach Algebras
214
a)
The map Em,n
~ En,m ,
a :
~ a*
is conjugate hnear.
b)
a E Em,n =~ a** = a.
c) a E Era,n, d)
b E En~ =~ (ab)* = b'a*.
The algebra E,,,, (Proposition 2.1.4.24 b)) endowed with the involution En,n
~ En,n ,
a .~ ~ a*
is an involutive algebra. e)
I f (Et)tET iS a family of involutive (unital) algebras then for every n E IN
the map
I-I(E,),,,. tET
)
1-I E,
~.tET
, n,n
( [z,a,kb,l, er~,, )ter " ; l(X,j,I,)teT],,,e~,, is an isomorphism of mvolutwe (unital) algebras.
I
Example 2.3.1.32
( 0 ) Let n E IN. IKn,n is a strongly symmetric involutive umtal algebra (Proposition 2.1.4.2.t b),c) and 2.3.1.31 d)). In particular, .for every a E ]Kn,n with w
az3 ~ ajs
for all i, j E INn the zeros of the polynomial
Det[a,j - 6,,t] are real.
By Example 1.2.2.7a) (and Example 2.1.4.10 a)), Kn,n is strongly symmetric. The last assertion follows from Proposition 2.3.1.29. 1
~.3 Involutive Banach Algebras
E x a m p l e 2.3.1.33
~15
Take n E )N and define I1
id=l
]or a := [a,,] ~ lKn,n (Example e.3.1.3~). Then "5 ~ (IKn,,,)', whenever a IKn,n and the map IK.,.
~,( ~,~),
a:
~a
is an isomorphism of involutive vector spaces (Example e.3.1.6), where (lKn,~)' denotes the algebraic dual of lKn,n.
Let c~ := [c~,j] E lK~,n. Then ([~,,1,~*) = ([~,,]*,~ = ~
~j,a,__ = ([~,,1, h-')
I,$=1
for every [c~,j] E ~n,n. Hence ,.,.. -- a*
a*
and the map IK~,.
~(IK.,.)',
a"
;
II
is an isomorphism of involutive vector spaces. E x a m p l e 2.3.1.34
The matrix
7
~
is a normal element of the involutive algebra ]K2,2 (Example ~.3.1.3~) iff lal = 171,
~ ( ~ - z) = a ( ~ -
~).
x ,s an idempotent selfadjomt element of ]K2,2 (Example 2.3.1.30) zff either
~=7=0, Or
a,~ E JR+,
c~,6E{0,1}
~,. Banach Algebras
216
Note that
6
~ ~
~?+~6
171~+161 ~
~
? 6
a~
I~12 + 1612
+ ?~
'
The first assertion now follows. For the second assertion assume first that the matrix is idempotent and selfadjoint. Then a,6 ~ R,
~=~;.
Moreover,
[ ][ ] [ ~6
~6
=
~2 + i~12 ~(a + 6)
~(~ + 6) ] I#12+ 6 2
Thus [ ~ = ~ + I#1 ~
= 62 + I~12 .
If ~ = O , then a,66.{O, 1 } . I f ~ # O , c~,6 6. ~ + ,
then c ~ + 6 = 1,
I~I 2 = a - a 2 = a ( 1 - ~ ) = ~ 6 .
The converse is easy to see. L e m m a 2.3.1.35
1
Take rt 6. IN" and define
i=1
for x, y 6. IK". If E is a vector subspace of IK", then there ts a finite subset A of E such that
~.3 lnvolutive Banach Algebras
z917
for every x, y E A and
yEA
for every x E E . A is an algebraic basis of E . Let 2t be the set of subsets A of E such that
whenever x, y E A and let A be a maximal element of ~1 with respect to inclusion. Since A is linearly independent, it is finite. Take x E E and let z := ~-
2~ly~u. yEA
If z # O, then
(zlz) > 0 and
A U
{1 } .iZlz) 89Z
E
which contradicts the maximality of A. Hence z = 0,
yEA
I
and A is an algebraic basis of E . E x a m p l e 2.3.1.36
Take n E IN" and define 11
I-----1
.for x, y E IKn . A matrix u is an idempotent and selfad3oint element of lKn,,, (Example ~.3.1.32) iff there is a finite subset A of IK" such that
(xly) = ~,v whenever x, y E A , and
=E[xlx] xEA
~18
2. Banach Algebras
Assume u is idempontent and selfadjoint. P u t E := I m u . By Lemma 2.3.1.35, there is a finite subset A of E such that (~1~) = 6,,, for every x, y E A and
xEA
for every y E E . Hence
zEA
zEA
for every y E E . Take y E ~ " . Then uy E E and
s=l
s=l
3=1
,/=1
s=l
,=1
j=l
.7=1
for every x E E , so that
zEA
zEA
xGA
zEA
Hence
xEA
Now assume that there is a finite subset A of I ( n such that (xly) = a,,~ for every x, y E A and
= ~[~1~]. xEA
Since [xlx ] is selfadjoint for every x E E:", u is also selfadjoint. Since
[~I~] [~I~]
=
a,,~[x I x]
for every x, y E A, it follows U 2 - - U.
I
~.3 Involutive Banach Algebras
~19
Proposition 2.3.1.37
Let E be a strongly symmetric involutwe algebra and u a surjective involutive algebra homomorphism of E into an involutive algebra F . Then F is also strongly symmetric.
Take y E F and x E ul(y). Then u(x'~) = (~,x')(~=) = ~'~, so that
OF(U'U) C
aE(X'~)U{0} C rt+
(Corollary 2.1.3.12). Hence F is strongly symmetric.
II
Proposition 2.3.1.38 ( 0 ) Let E be an involutive vector space and E the complexification of the underlying vector space of E (Lemrna 2.1.5.1~)). Then endowed with the involution
;k,
(~,y), ,(~',-u')
is an involutive complex vector space called the complexification ol E .
Let (a, b), (c, d) e k and a,/~ e ]R. We have ((a,b) + (c, d))* = (a + c,b+ d)* = (a* + c*,-b* - d*) =
- (a*,-b') + (C*,-d*) = (a, b)* + (c, d)*, ((c~ + i~)(a, b))* = (c~a - t~b, l~a + v&)* -- (c~a* - l~b*, -t3a* - ~b*) = (~ - il~)(a',-b*) = ~ + i~(a, b)*.
II Proposition 2.3.1.39
( 0 ) Let E , F be involutive real vector spaces, L(E, F) the involutive real space of the linear maps of E into F (Example o E ~.3.1.6), E, "~, and L( , F) the complexifications of E, F and L(E, F) respectively, and L(E, F) the involutive complex vector space of the linear maps of k into b (Example e.3.1.6). For (u, v ) E L(E, F) define
(~, v'-'~ k
--~ b ,
(~, y):
- ( ~ - ~y, ~ + w ) .
~0
2. Banach Algebras
Then the linear bijectwe map L(E, F)
, L(E, F) ,
(u, v) ~
(u, v)
(Lemma e.1.5.3) is involutive. (We may thus identify L(E,F) and L(E,E) ). If (u, v) E L(E, F), then (u, v) is selfadjoint iff u = u*, v = - v * . If E and F are endowed with norms such that the canonical bijections 0
o
o
~ExE,
F
~FxF
are isomorphisms of real normed spaces, then we may replace L by s in the above enunciation. Choose (u, v) E L(E,F). Then, for every (x, y ) E E ,
((~, v~l ((~,~))- ((~)((~, ~).))"= ((~, v~((~.,-r
l=
= (uz" + v v ' , - u v " + w ' ) " = ((,,z')" + ( , v ' ) ' , (uv')" - ( w ' ) ' ) =
= (,,'x + v'y, ='V - v ' x ) = ( , , * , - v ' ) ( ( - ,
y)) = (=, v)*((~, y)) 9
Thus
((~))" (-~)" -"
U~V
.
Hence the map o
L(E, F)
~ ~)
~ L(E,
,
(u, v) : ~ (u,v)
is involutive. In particular, (u, v) is selfadjoint iff
(~',-r
= (~, v)
i.e. iff u = u*, v = - v * . The last assertion follows from Proposition 2.1.5.6. P r o p o s i t i o n 2.3.1.40
I
( 0 ) q E is an involutive real algebra, then the
complex algebra E (Proposition 2.1.5. 7) endowed wtth the revolution
k--~,
(~,v):
~(~',-v')
~.3 Involutive Banach Algebras
~$1
is an mvolutwe complex algebra called the eomple=illcation of E . If E is (strongly) symmetric, then E is also (strongly) symmetric. If F is an involutive subalgebra of E , then F • F is an involutive subalgebra o
of E and is therefore a complexification of F . For every n E IN the map ,
" E
,
(x,y)~
~x+iy
lltn
is an isomorphism of involutive complex algebras.
If (a, b), (c, d) E/~, then ((a,b)(c,d))* = (ac - bd, ad + bc)" = (c'a* - d*b',-d'a* - c'b*) = ---(c*,-d*)(a*,-b*) = (c,d)*(a,b)*.
This proves the first assertion (Proposition 2.3.1.38). By Corolla'ry 2.1.5.14, if /~ is (strongly) symmetric, then E is also (strongly) symmetric. The last assertions follow easily from Propositions 2.1.5.10 and 2.3.1.31 d). II Proposition 2.3.1.41
( 0 ) Let E , F be involutive real algebras, E, F thezr
complexifications, and u" E --+ F an mvolutive algebra homomorphism. Then ~ .~
~~ ,
(~, ~) ~ , ( ~ , ~ )
zs an involutive algebra homomorphism, surjective iff u is surjective.
We have ~,((~, ~)') = ~((~', -~*)) = ( ~ ' , -~y') = =
((~)',-(~)')
= (~x, ~ ) "
= (;,((x, y)))"
for every (x, y) e /~. Thus ~ is involutive. The other assertions follow from Proposition 2.1.5.11. II Proposition 2.3.1.42
( 0 ) Let E be an involutive algebra and F an
involutwe ideal of E . Then the map
E/F
"~~ / b
defined sn Proposition 2.1.5.10 c) is an ssomorphism of involutive algebras. In o
~2f2
~. Banach Algebras
The proof is by direct verification.
Proposition 2.3.1.43
II
( 0 ) Let E be an mvolutive complex algebra and U
the set of conjugate involutions on E . Given u E II , define
F,, := {z e Eluz = z } . Further, let .~ be the set of mvolutive real subalgebras F of E for which there o
is an isomorphism of involutive complex algebras v : F
~. E such that
,((~,o))=x .for every x E F .
a) If u E hi, then Fu E .~ and the map F,,
~. E ,
(x,y)-
~x+iy
is the isomorphism appearing sn the definition of ~'.
b) Themap U
;Y,
u'-~F=
is bljective.
c) If G is an involutive real algebra such that G is isomorph to E , then G is lsomorph to an F E Jr.
d) For every u E hi, the structure of Fu may be extended to that of an involutive complex algebra iff there is a linear map v : E ~ E , such that V 2 ~--1~
ltV -- VU
and v(~u) = (v~)u = xCvy),
v~" = - ( v x ) "
for all x, y E E ; in this case the complex structure on Fu is defined by the map r • F~ --~ F~,
(a + i~, x)'.
~ ax + f3vx,.
~.3 lnvolutive Banach Algebras
e)
For all u, v E hi, the involutive real algebras Fu and F, are isomorphic iff there is an isomorphism of mvolutive complex algebras w 9 E -~ E such that W O U - - - - 1) o ~ 1 .
a) It is easy to see that Fu is an involutive real subalgebra of E . Define ,
O
v'F~
~ E,
(x,y):
~x+iy,
o
and take (x, y) E Fu. Then , ( i ( . . y)) = , ( ( - y . ~)) = -z~ + i . = i ( . + iy) = i.,((., y ) ) .
, ( ( . , y)') = , ( ( . ' , - y ' ) )
= ." - iu* = (~ + iv)" = , ( ( . , y))*.
Hence v is linear and involutive. If v((~, u ) ) = 0 ,
then x+iy=O
and x-
iy = ux - iuy = u ( x + iy) = 0,
so that (x, y) = 0, whence v is injective. o
If (xl, Yl), (x2, Y2) E Fu, then ,((~,, ~)(~,
= (~
y~)) = , ( ( ~
- y~,
~,y~ + u , ~ ) ) =
- ~,y~) + i ( ~ , ~ + ~,x~) = (~, + i y ~ ) ( ~ + iy~) = ,((x~, u , ) ) v ( ( ~ , ~ ) ) .
Hence v is an injective homomorphism of involutive complex algebras. Finally, we show that v is surjective. Let z E E . Put X:--~
l(z + uz)
1 y := ~,(z - uz).
Then x , y EFt, and
2~4
~. B a n a c h
Algebras
,,((~, u)) = x + iu = z.
b) Take u, v E/4 such that Fu = Fv. Take z E E and put
{ Then x, y E Fu and
~ := 89 + ~ )
z = x + iy.
uz = ux -
y := ~(~ - u ~ )
We have
iuy = x -
zy = vx-
ivy
= vz.
Hence u = v and the map
U
;Y,
u:
~,F~
is injective. Take F E ~'. We identify /~' and E using the isomorphism F --~ E described in the definition of ~'. Define u'E
~E,
(x,y):
.~(x,-y).
Then for any (x, y) E E
,,(,(x, u)) = u ( ( - u , x ) ) = ( - u , - ~ ) = - i ( ~ , -,.,) = - i u ( ( ~ , u)) , u((x, u)') = ~ ( ( ~ ' , - u ' ) )
= (~', u') = ( ~ , - y ) " = ~((~, u ) ) ' ,
so that u is involutive and conjugate linear. If (xl, yl)(x2, y~) E E , then ~ ( ( ~ , ~ ) ( ~ , , u,)) = u ( ( ~ , ~ , - u~u,, ~ y , + ~ ) )
= (~,
-
u,u,,-~,u~
-
u,~) = (~,,-u~)(~,-u2)
=
= u((~, ~))u((~,
~,)).
Since u is bijective it is an isomorphism of real algebras. Moreover, u 2 = 1. Hence u E H. Since F = Fu the map It----}.,~ ,
u ~--} F~
is surjective. c) Let v 9 G -+ E be an isomorphism of involutive algebras and put
r .= ~ ( a • {o}).
gg5
g.3 lnvolutive Banach Algebras
Then F E ~" and the map G
>F,
x : : ;v((x,0))
is an isomorphism of involutive algebras. o
d) We identify F~ with E via the isomorphism o
F~
}E,
(x,y):
}x+iy.
First assume that the structure of Fu may be extended to that of an involutive complex algebra. Consider O
O
v. r,
~ F=,
(x,U) ~
(,~,iy).
o
Take (x, y) E F~. Then v(i(x,y)) = v ( ( - y , z ) ) = ( - i y , iz) = i(ix, iy) = i v ( ( x , y ) ) , v((x, y)') = , , ( ( . ' , - y ' ) )
= (i~',-iy')
= -((i.)*,-(iy)')
=
= - ( i x , iy)* = -v((x, y))*, v ' ( ( . , u)) = v ( ( i . , iy)) = (i=~. i'.j) = - ( ~ . ~). v~.((~. ~)) = v ( ( . . - y ) )
= (i..-iy)
= ~.((i~. iu)) = ,.,,(., ~)).
o
Take (xl, yl), (x2, Y2) E Fu. Then ( , ( ( ~ , , u~))(~, u~) = ( , ~ , iu~)(z~, y~) = ( m ~ = ,((~,~
- ~y~, ~,y~ + y , ~ ) )
- iy, u~, i~,y~ + i u , ~ ) =
= , ( ( ~ , , y~)(x~, y~)),
( ~ . ~ ) v ( ( . ~ . y,)) = (~.. y.)(i.=, i~,) = ( m * = - i~.y,, i..~= + i~.~=) =
= v((~,, y , ) ( ~ , y~)).
Assume next that there is a v with the above properties. Take x E Fu. Then ttVX
"-- V t t X
"-
VX
~
~26
~. B a n a c h Algebras
so that v x ~. F , , . We endow F~ with the map r215
, F,,,
(c,+z#,z):
~.ax+#vz.
Take a, ~, 7, 6 E ~ and x E Fu. Then ((,~ + ia)('r + i6))x = ((a'r - ~6) + i(a6 + a.r))x =
= (a3, - # 6 ) z + (a6 + # 7 ) v x = a'yz + #6v2x + a 6 v x + #'rvx =
= a('rx + 6vx) + #v('rz + 6vx) = (a + s#)(('r + i6)x),
(iz)" = (vz)" = - v x " = - ~ x ' .
Thus F~ is an involutive complex vector space. If x, y E Fu then
(i.)u = (,,x)y = v ( x y ) = i(xu), x(iu) = , v u = v ( ~ ) = i ( , ~ ) .
hence Fu is an involutive complex algebra. e) Let wo" F~ --, F~ be an isomorphism of involutive real algebras. Then o
o
w" Fu
~ Fv,
(x,y):
; (wox, w0y)
is an isomorphism of involutive complex algebras (Proposition 2.3.1.41). We o
o
o
canonically identify E with Fu and F , . Then for all x, y E Fu, w o u(z + iy) = w(uz
- iu~) = w ( x - i~1) = WoZ - i w o y =
= VWoX - iVWoy = V(WoX + i w o y ) = v o w ( x + ~y)
so that w o u = v o w . Conversely, let w 9 E ~ E be an isomorphism of involutive algebras such that w o u = v o w . Then for each x E Fu, WX " - W ~ X ~ VWX~
so that w x E F v . Hence the map fu
~==; F v ,
x:
; fox
~.3 lnvolutive Banach Algebras
~7
I
is an isomorphism of involutive real algebras.
Remark. Let E be the involutive complex algebra C,,n (Example 2.3.1.31 d)) and u the map
Then u is an isomorphism of involutive real algebras, which is conjugate linear and satisfies u s = 1. We have ]r~,. = {x e r
= x}
and Cn,n is the complexification of lKn,n (Example 2.1.5.9 and Example 2.3.1.31
d)). E x a m p l e 2.3.1.44 We take IK =ffJ. Let T be a set, 9 the set of involutions of T , and yc the set of involutive real subalgebras F of t~176 such that the map ~
s176176 (x,y) ~
x + iy
is an isomorphism of involutive complex algebras. For each ~p E 4i, put F~ := {x e l~176 x = x--6~}. a) F~ e jr for every ~o e O. b)
The map 4~
~ Y:,
~o;
" F~
is bijective. c)
Take ~p E O, put S := {t e Tic(t) = t}, and let R be a subset of T \ S such that r
= T \ ( R O S).
Then F~ is isomorphic to e_~(R) • e ~ ( S ) , where ~ ( R ) denotes the underlying involutive real algebra of e~(R) and s the corresponding mvolutive real algebra e~176 constructed over IR.
2~8
d)
~. Banach Algebras
For each ~ E 4, the real involutive algebra structure of F~ may be extended to that of a complex mvolutive algebra iff ~ has no fired-point. (This implies that Card T is not an odd natural number).
e)
Given ~o, tp E 4~, the following are equivalent: el)
F~ and F,p are isomorphic.
e~)
There is a bijective map f : T --, T such that
~,of = f o ~ . e3) Card {t e Tl~(t) = t} = Card {t e TI~p(t) = t}. f) Let s be the class of involutive real algebras the complexifications of whwh are isomorphzc to e ~ ( T ) . Given F, G E F. , define F ,~ G :r
F and G are isomorphic.
Then
1+
Card E / ~ =
Card T--[ ain(|Card T)[
2
,
sup{ Ro, Card ~}
if T ss fimte
/f Card T = R~
(where ~ is an ordinal number). We may replace s above.
by co or c in the
Let /4 be the set of conjugate involutions on s176176 Given u E / 4 , put E, := {x E g~176
= x}.
Then by Proposition 2.3.1.43 a),b), Eu E .~" and the map U~
~Y,
u:
;E~
is bijective. For ~ E ~ , define u~ : g~176 ---, g~176
It is easy to see that u~ E/4 and F~ - Eu~. a) follows from the above considerations. b) It is easy to see that the map
x:
~,~ o ~ .
9.3 I n v o l u t i v e B a n a c h Algebras
~9
is injective. Take F e ~'. By the above considerations, there is a u E / 4 such that F = F u . Define v" t~176
:
~ t~176
x::
~ u'g.
Then v is an isomorphism of complex algebras. By Proposition 2.1.4.22, there is a bijective map ~" T -+ T such that v x - x o~o
for every x e t~176
Thus ux - xo~p
for every x E t~176
Take t e T . Then u e t ---- et o ~p -- e ~ - l ( t ) ,
et = u2et = e~-~(~-l(t)),
so t h a t ~,-'(v-'(t))
= t.
Hence ~ is an involution, i.e. qo E 4i. We get u = u~, Eu = F~, whence the map
-y,
~ ~--~ F~
is surjective. c) For (x, y) e ~ ( R )
x ~(S),
define x(t)
(x, y)' " T
"r
t:
~
x(~o(t)) y(t)
if t e R if t e T \ ( R u S) if t e S
and consider
u" ~~
x ~(S)
~ tc'~(T),
(x, y ) ' - ; (x, y)'.
Then u is an injective homomorphism of involutive real algebras. Take (x, y ) e ~ ( R )
x t ~ ( S ) . Then
~30
1?,. Banach Algebras
(~, y), o ~,(t) = =(~,~(t)) = =(t) = (=, ~)'(t) for t q R ,
(=, y),
o
~,(t) = ~(~,(t)) = (=, z,)'(t)
for t E T \ ( R O S), and
(=, y),
o
~(t) = (=, y),(t) = ~(t) = y(t) = (=, y)'(t)
for t E S. Hence
(=, y),
o ~
= (=, ~)',
i.e. (x, y)' E F~. It follows that
~(e~(R) x ~ ( S ) ) c F~,. Take z E F~. Then ~ o ~o = z. Define
x:R
~r
y:S--,~,
t:
~ z(t),
t:
~ z(t).
Then (=,u)'(t) = =(~,(t)) = ~(~,(t))= ,(t) for all t E T\(R U S). Hence z = (=,y)' = ~((=, ~)) e u(t~(n) x i t ( S ) ) , .(~(R)
x tt(S))=
F~.
Hence F~ is isomorphic to ~ ( R ) x t ~ ( S ) . d) follows from c). el =~ e~. By Proposition 2.3.1.43 e), there is an isomorphism of involutive complex algebras v : e~176 --+ e~176 such that
By Proposition 2.1.4.22, there is a bijective map f " T --~ T such that UX--XO f
~.3 Involutive Banach Algebras
P,31
br every x E s176176Hence xocpof =xofor for every x E s176176It follows that ~oof=for e2 =~e3. Take t E T such that r
~0(f(t)) = f ( r
= t. Then
f(t),
so that f ( { t E TiC(t) = t}) c {t E Tl~(t) = t}. The reverse inclusion follows by symmetry, and so Card {t E Tic(t) = t} = Card {t E T I r
= t}.
e3 =~ el follows from c). f) follows from b) and el =* e3 (and Proposition 2.3.1.43 c)). Remark.
I
a) and b) will be generalized in Proposition 4.3.5.6.
P r o p o s i t i o n 2.3.1.45 Let E be an involutive complex algebra E . Then the following are equivalent: a) E is the complexification of an involutive complex algebra ( i.e. there is an involutive complex algebra F such that if F denotes the complexification of zts underlying involutive real algebra, then E and F are isomorphic). b)
There is a homomorphism of involutive real algebras p : E --+ E such that p~ = p ,
p(E) fq ip(E) = {O} ,
and p(ip(i~)) = -p~
for all x E E .
p(E) + ip(E) = E
1232
2. Banach Algebras
a =~ b. We identify E and i~" and consider
~ .~,
p ~
(~, ~) ~
(~ + i~, o).
Then
p~ = p,
v(~) n iv(#) = 1(o, o)}
are obvious. Take (x, y) E F . Then p((.. ~)') = p((x'.-~'))
= (x" - i ~ ' . 0) =
= ((~ + iu)', o) = (x + iy. o)" = p((~, ~))', (x,y) = (x,O) + i ( - y , O ) = p((x,O)) + ip((O, iy)) E p(F) + ,p(F) , p(ip(i(x, y))) = p(zp((--y,x))) = p ( i ( - y -t- ix, 0)) =
= p((o,-y
+ i.)) = (-iy
- x, o) = - p ( ( ~ , u ) ) .
Hence p is involutive,
p(g + ip(~) = ~ , and the last condition is fulfilled. Take (xl, Yl), (x~, y~) E ~'. Then p((x~, u , ) ( x 2 , y~)) = p ( ( x , x 2 - u~y~, x~y~ + ~ 2 ) )
=
"-- ((XlX2 -- YlY2) "~" $(XlY2 + ~IX2), O) --" ((X 1 ~" ~yl)(X2 "~" ~Y2), O) -~
-- (Xl + $Yl, O)(X2 -I- ~Y2, O) -- p((Xl, Yl))P((X2, Y2)).
Thus p is a homomorphism of involutive real algebras. b =~ a. We put F "= p ( E ) . F is an involutive real subalgebra of E . We extend the scalars of F to C by defining a o x = p(ax)
for all (a, x) E 9 x F .
~.3 Involutive Banach Algebras
~33
Take a,/~, 7, (f E lR and x E F . Then (.~ + i a ) o ((7 + i.~) o . ) = (.~ + i a ) o ('r* + .~p(i.)) = + Z.~p(.r.(ix)) =
= .~.y,~ + . ~ a p ( i . ) + / ~ T p ( i . )
= (aT - ~(i)x + (a5 + ~ 7 ) p ( i x ) = = ((,~7 - a.~) + i(.~.~ + aT)) o .
= ((.~ + iO)(.r + i,~)) o ~ .
((.~ + iZ) o x)* = ( . . x + ~ p ( i x ) ) * =
= ,~x* + Z p ( - i x * )
= ,~x* - apCix*)) = (~ - i/~) o , ' .
Take tr E q~ and x, y E F . Then o (.y) = p(~(~u)) = p((~.)y)
= p(~.)~
= (~ o . ) ~ .
Hence F endowed with the map q~xF
~F,
(a,x):
}aox
is an involutive complex algebra. Define u'/~
rE,
(x,y):
;x+iy. o
By the assumption on p , u is bijective. For every (x, y) EF, ,,(i(~, u)) = , , ( ( - y , ~)) = - u + ix = ,(~ + iu) = i,~((~, y ) ) , ,~((~, y)') = , , ( ( ~ ' , - y ' ) )
= ." - i~" = ( , + iu)" = ,~((~, y ) ) ' .
Hence u is linear and involutive. Take (xl, yl), (x2, Y2) E F . Then " ( ( X l , Yl)(X2, Y2)) -- ~((XlX2
-
-
YlY2, XlY2 + ylX2)) --
= ( . , ~ , - y,y,) + i ( ~ , u , + ~,x=) = (~, + i y , ) ( ~ , + w , ) = , , ( ( ~ , y , ) ) , , ( ( * , , y , ) ) .
Thus u is an isomorphism of involutive complex algebras.
I
~34
~. Banach Algebras
E x a m p l e 2.3.1.46 ( 0 ) We identify the algebra ]H of quaternions with the unital real subalgebra of ~,2 generated by
o]
I :=
,
J :=
[o 1]
0 -i
-1
,
K :=
0
[o i
0
(Proposition 2.1.~.16, Corollary 2.1.~.17). ~-I is an involutwe real subalgebra
of 1~2,2 (Proposition ~.3.1.31 d)) such that (al +/31 + 7 J + 6K)*(al +/31 + 7 J + 6K) = = ( a l +/31 + 7 J + 6 K ) ( a l + BI + 7 J + 6K)* = (a 2 + B2 + 72 + 62)1
for all a l + b3I + 7J + 6K E IH and such that the map "r
(x,y) ~
x + ~.v
is an isomorphism of mvolutive complex algebras. In particular, for every n e IN the map o
l'In,n--'} r
,
(x, y) : ~ x + iy
is an isomorphssm of involutive complex algebras. The last assertion follows from the preceding one and from the final assertion of Proposition 2.3.1.40. 1
Remark. If E is a unital involutive algebra and a an element of E such that I12-- --1,
a* ~ - - a
then there is a unique unital algebra homomorphism u" ~ --r E~,~ such that
u is injeetive and involutive (Proposition 2.3.1.31 P r o p o s i t i o n 2.3.1.47
d)).
( 0 ) Let E be an involutwe real algebra such that
Re E is one-&mensional and such that x E E\{O} ==~ x ' x # O.
Further let ~ll be the set o/]imte subsets A of E\{O} such that x E A ==~ x + x * = 0 , x, y E A, x r y ===}xy + yx = O.
~35
~.3 lnvolutive Banach Algebras
a)
There is an a e Re E\{O} with a 2 =a.
b)
For every x E E \ { 0 } , there is an ~z E IR\{0} such that X* X ---- O t x a .
c)
Take A e ~ and let F be the subalgebra o.f E generated by A E ~ F , then there is a z E E \ F
d)
U
{a}.//"
such that A U {z} E fll.
If E is unital, then for every A E ~ and for distinct x, y E A , A and {x, y} generate the same subalgebra of E .
e)
If E is unital and if.for every x E E , there is an a E lit+ with x*x = a l , then E is isomorphic to IR,C, or IH. a) Take b E R e E k { 0 } . Then b2 = b*b ~_ Re E , so that there is an a E
lR\{0} with 52 "- orb.
Put a :-- l b E ~ .~e ~~ \tl,n, t j Then aS = -~b2 = l b = a -
-
~
b follows from x*x E Re E . c) Take y E E \ F . Since y+y* E ReE, there is an ~ E IR such that y+y* --aa. For x E A , ( ~ + y~ - ~ ( ~
O~
,
+ ~a))" = y*~* + ~'~" - ~ ( ~ ~ + a~*) =
236
2. Banach Algebras
= -(~
-
= -aax
Y)~
~(~"
-
a (ax + xa) = ~)+ 7
-
Ol
+ yx - axa + xy + -~(ax + xa) = Z-
a (ax + = x y + y x - -~
xa)
so that there is a /~x E P~ with
Put a 1~ -~ a + ~
z := y -
~---~x e E \ F . otz
Then
1~ a'-'~ &(x+x') =o
z + z* = y + y* - ota + -~
and for every x E A, OL
~ x X2
x z + z x = x y + y x - -Zx-( x a + a x ) + - Olx
= &a - ~x'x
=
= & a - & a = o.
O~x
Hence A tJ {z} E ill. d) Let z ~. A \ { x , y } .
Then ( x y z ) * = z*y*x* = - z y x
= xyz,
so that there is an a E P~ with xyz = al.
It follows that
~
= ~y~
.~ 9
-~y~
=
-- ( X * X
)(~ * y)~ = - ~ ~ ,
z = --~xy. O~xOLy
Hence A and {x,y} generate the same subalgebra of E .
~37
~.3 lnvolutwe hranach Algebras
e) If 0 is a maximal element of ~t, then by c), E is generated by a, so that it is isomorphic to IR. Otherwise, there is an x E E such that {x} E ft. By the supplementary hypothesis of e), we may assume that x2 = -x*x = -1.
If {x} is a maximal element of ill, then by c), E is generated by x, so that it is isomorphic to q~. Otherwise, there is a y E E \ { x } such that {x, y} E fil and we may assume y2 = --1.
Let F be the subalgebra of E generated by { x , y } . Suppose E ~ F . Then by c), there is a z ~. E \ F such that {x, y, z} e ill. This contradicts d). Hence E = F and E is isomorphic to IH. II P r o p o s i t i o n 2.3.1.48
( 7' )
Let E be an involutive complex algebra,
the complexification of the underlying mvolutive real algebra of E , E x E the involutive complex algebra defined m Example 2.3.1.4, and u : E ~ E a homomorphism of involutwe real algebras.
a)
The m a p v'F,
(x, y) " "~ux + iuy
= ~.E,
is a homomorphism of involutive complex algebras.
b)
I1 u
zs a
conjugate involutzon, then the map w'E
(x,y):= ~ ( x + i y , u x + z u y )
;ExE,
is an isomorphism of involutive complex algebras.
c) If E is the complexification of an involutive real algebra, then E and E x E are isomorphic.
a) For (x, y), (xl, Yl), (x2, Y2) E/~, v(i(~,y)) =
v((-u, ~)) =
.((~. ~)') = . ( ( x ' . - y ' ) )
v(Cx,, ~ ) ( ~ ,
-~
= ux"
y~)) = v ( ( ~
+ iu~ = i ( ~
-
+ ~)
= iv((x,u)),
i~.y" = ( u . + iuy)* = v((~. y ) ) ' .
- y~2,x,~
+ ~))
=
~38
~. Banach Algebras
= u(xlx2 - YlY2) + iu(xly2 + ylx~) =
= (u~)(u~,) - (uu~)(uu,) + i((ux,)Cuu~) + (~,u~)Cux,)) =
= ( - ~ + i ~ u , ) ( ~ , + i-u,) = ,((~,, u~))v((~,, u~)). b) By a), w is a homomorphism of involutive complex algebras. For (x, y) E ExE,
w(( (x + uu), ~i(z - uu)) =
1 (~ + ~,y) + i uu), 1 (~ + uu)) + ,u( 1 ,,~))) = (~ ~,(~u(~ ~(~= 1 = (~, ~
1 + ~-
1 ~
1 + ~ ) = (~, u),
so that w is surjeetive. From (x, y) E/~ and
w(~, u) = 0 it follows successively that
0 = ux + iuy = u(x - i y ) ,
x-iy=O,
(x + iy, x - iy) = O,
(x,y) = 0 , i.e. w is injective. Hence w is an isomorphism of involutive complex algebras. c) By Proposition 2.3.1.43 b), there is a conjugate involutive on E. By b), /~ and E x E are isomorphic. 1 P r o p o s i t i o n 2.3.1.49 ( 0 ) Let E be an involutive algebra such that the real vector space Re E is finite-&mensional. Then the involutive subalgebra of E generated by any finite subset o] E zs finite-dimensional.
~.3 lnvolutive Banach Algebras
~,39
By Proposition 2.3.1.22, we may assume lK = IR. Let A be a finite subset of E . We show that the involutive subalgebra of E generated by A t9 Re E is finite-dimensional. Since y+y* E ReE
for every y E A, we may assume y* _- - y for each y E A. Let ( x , ) i ~ be an algebraic basis of the real vector space Re E and let (yk)~e~, be a family with A = {y~lk e l N , } .
For i, j E lNm and k, s e INn, x, xj + xjx, , x~y~ - ykx, , Y~Yt + YtYk E Re E .
It follows that the vector subspace of E generated by the elements of the form Xzl X,2 9 9 9 Xsa Ykl Yk2 9 9 9 Ykb ,
where il < i2 < . . . i a ,
k l < k2 < ... < kb,
is equal to the involutive subalgebra of E generated by A U Re E , which is therefore finie-dimensional. I P r o p o s i t i o n 2.3.1.50
( 0 ) Let E be an involutive algebra and E the invo-
lutwe algebra obtained from E by reversing the multiplication, i.e. by endowing E with the multiplicatwn E x E
~ E,
(x,y) ~
yx =: x S y.
Let bl the be set of isomorphisms of involutive real algebras E --+ E , which are conjugate linear and )) the set of isomorphisms of involutive algebras E --+ E . For every u E LI , put u " E---r E ,
x:
; ux*.
Then ~ e V for every u e bl and the map bl
;V,
u:
; u
is bijective. In particular, if IK = IR, then the involutive real algebras E and 4-4
E are zsomorphic.
~. Banach Algebras
e40
Let x, y E E and a,/~ E IK. Then
~ ( a z + #u) = u(ax +/3v)" = u(nz" + -~v') = aux" + auv" = a u x + #~u, ( ~ ) " = (,=')* = u~'" = ~ ' ,
~(~v) = -(~v)" = . ( v ' ~ ' ) = ( ~ v ' ) ( . ~ ' ) = ( - ~ ' ) $ (~v') = ( ~ )
$ (~v).
so that ~ E V. Take v E V and put u'E~E,
x'.
~vx*.
Let x, y E E and a,/3 E IK. Then m
m
o
u(ax + Zy) = v ( a x + ~v)" = v(~x' + f~v') = ~vx" + f~vv" = ~ u x + / ~ u y .
(~z)" = (,~')" = . x " = ~ ' ,
. ( ~ u ) = ,(~u)" = v ( u ' ~ ' ) = ( , u ' ) $ ( , ~ ' ) = ( , ~ ' ) ( . v ' ) = ( ~ ) ( u u ) ,
UX
--
UX*
-"
VX**
--
YX,
so that u E/,4 and ~ = v. Hence the map bl
~ r,
u~----~ u
is surjective. Since it is obviously injective, it is bijective. If lK = R , then /4 contains the identity map, so that V is nonempty.
m
2.3 Involutive Banach Algebras
e41
2.3.2 Involutive Banach Algebras D e f i n i t i o n 2.3.2.1
( 0 )
A n involutive n o r m e d space is an involutive
vector space endowed with a norm such that
IIx*ll = Ilxll for every x E E . If, in addition, E ss a normed (unital} algebra, then it is called an involutive n o r m e d (unitaO algebra. If, furthermore, E is complete, we speak of an involutive B a n a e h space and an involutive B a n a e h (unitaO algebra. Given an involutive normed space E , put Re E # := (Re E) # . If E, F are involutive normed spaces, then E(E, F) is an involutive normed space with respect to the canonical involution E x a m p l e 2.3.2.2
]H is an involutwe real unital Banach algebra wzth respect
to the involution IH
~, IH,
~ + i~ + j T + kS :
~, a - i ~ - j T - k5
II
and to the Euclidean norm (Corollary 2.1.4.17). E x a m p l e 2,3.2.3 ( 0 ) If (T, ~ , p ) is a measure space and p E then IY(#) is an invoutive Banach space with respect to the involution LP(p)
~. LP(#),
x:
[1, cx~[,
;5
and Leo(#) is an involutive commutative unital Banach algebra wzth respect to the involution Leo(l~) ' ~ Le~
x:
;5
(Example 2.2.2.2). If q is the conjugate exponent of p (and if Leo(#) is order complete when p = 1 ), then the canonical map Lq(#)
; LP(t~) ' ,
ss an isomorphism of involutive Banach spaces.
II
~4~
2. Banach Algebras
Example 2.3.2.4
( 0 ) Let (Et)teT be a family of involutive normed (unital)
algebras and
E := {x e I I E, I (llx(t)ll),eT e e~(T)}. tET
14qth
E ~
R+,
z ~
sup fix(011 tET
as a norm and with E
, E,
~, (z(t)')ter
(x(t))ter:
as involution, E is an involutive normed (unital) algebra and
{z e E l(llz(t)ll)t~v e co(T)} is an involutive closed ideal of it. If all Et (t E T) are complete, then E is complete too (Ezample ~.2.E3). In particular, P~ is an involuttve commutative unital Banach algebra and co(T) is an mvolutivc closed ideal of it. If T ts a topological space and all Et (t E T) qre equal, then
{ x E E I z is continuous} is a closed invuluttve (unttal) subalgebra of E . If in ad&tion JEt = ~f, for all t E T , then C(T) is a closed involutive unital subalgebra of s176176and if T is locally compact, then C0(T) ts a closed involutwe ideal of C(T). In this case, every closed mvolutive subalgebra of Co(T) is quasiunttal.
Only the final assertion requires proof. Let ~ be a closed involutive subalgebra of Co(T). Put
s := f'l sE~"
Let K be a compact set of T \ S and let e be a strictly positive real number. By Theorem 1.3.5.14, there is an x E ~" with
[x-lll
Ic~l if [ a [ < l
~.3 lnvolutwe Banach Algebras
~S
By Corollary 1.3.5.17, f o x E ~'. Given a compact set K of T \ S and e > 0, put ~'(K, e) := (x E Re~" I eK - e 0}. It is easy to see that ~ is an approximate unit of ~'. E x a m p l e 2.3.2.5
( 1 ) Let T be a set. For k 6 [~,~(T,T) define k*'TxT
~IK,
;k(t,s).
(s,t):
Then k* E ~2'2(T, T) for every k E/2'2(T, T) and the Banach algebra t2,2(T, T) (Example 2.2.2.5 a)) is an involutive Banach algebra with respect to the involution ~'~(T, T) ", e~'~(T, T) , k: ~ k* m E x a m p l e 2.3.2.6 ( 0 ) For n E IN, cons:der IKn with the euchdean norm and the unztal Banach algebra s n) (Example 2.2.2.4) w:th the involution obtained by the identification with the involutwe algebra ]Kn,n of n x n-matrices (Example 2.3.1.32). Then s n) is a strongly symmetric involutive unital Banach algebra. Take u E s By Example 1.3.4.12, the transpose of the matrix associated to u is the matrix associated to u *, and by Theorem 1.3.4.2 b).
I1~11 = I1~'11. By Example 2.3.1.32, s algebra, P r o p o s i t i o n 2.3.2.7
n) is a strongly symmetric involutive unital Banach i
( 0 ) If E is an involutive normed space, then
Ilre ~11-< I1~11 for every x E E and the map E
~ReE,
x'
.'rex
is both IR-linear and continuous. If IK --qJ, then
2. Banach Algebras
e44
Ilim~ll < II~lJ :for every x 6_ E and the map
E
~ReE,
x:
"imx
is both lR-hnear and continuous.
Given x E E, 1
1
Ilrexll = I1~(~ + x*)ll < ~(llxll + IIx'li) = Ilxll, 1
1
Ilimxll = II~(x - ~')11 /o~ ~,~,~ ~ e s
and x,y e L2(p)).
c) The involution s
~/:(L2(p)),
u:
; u"
renders ./:(L2(//)) a unital involutive Banach algebra such that Ilu" o ull = I1~,11~
for every u E s
d) Let L:~(/_t,IR), L2(/.t,r
denote the real and complex vector space L2(p),
o
respectively. Let ~'~'(#, IR) and ~(L~(#, IR)) denote the complexification of L2(p, lit) and s IR)), respectively. Given (u, v) E ~(L2(#, IR)), define 0
0
(~, v~-3. " ~ ( . , n~)
~ " ~ ( . , n~),
(~, ~)-
; ( ~ - v~, ~y + w)
o
and identify " ~ ( # , IR) wiih L'(U,q3) via the map o
~"(#,lR):
; L~(#,r
(x,y):
; x+zy
(Example 2.1.5.2). Then the map N
s
IR)) ----+ s162
(~, v):
"
(~, .)
(Lemma 2.1.5.3) is a isomorphism of involutive unital algebras.
a) is easy to check (Theorem 1.3.4.2). b) We have
c) By b),
for all x, y E s
so that
256
2. Banach Algebras
Hence the map s
--r s
u:
;u"
is an involution. This map is obviously conjugate linear and 1" = 1. Take u, v E s
By Corollary 1.3.4.5 (uv)*x = (uv)"e = v'u"e = v'~"~ = v*u*z
for every x E L~(/~), so that
(~V) Hence •(L2(Iz)) Take u E s
* -"
IJ*U*.
is an involutive Banach algebra. By a),
Ill-it ~ = < ~ 1 ~ ) =
O. Take x E E with Ilxll < 1. Then IIx'~ll _< II~'ll Ilzll = II~ll ~ < 1.
Hence, by Propositions 2.3.2.19 b) and 2.2.3.13 c), there is a selfadjoint element y of E such that y2 = 1 -- x*x.
Then x'(1) - x'(x*x) = x'(y 2) >_ O,
~.3 lnvolutive 1J'anach Algebras
f291
Ix'(:~)l ~ _< x'(l*l)x'(x*x) _< x'(1) 2 (Proposition 2.3.4.6 c)). Thus x' is continuous and
IIx'll ]K,
z:
~, X t~(Y * Z Y) ~
.
I~91~
~. Banach Algebras
Then
~'(z'~) = x ' ( y ' z ' z ~ ) = ~'((~)'(z~)) e rt+. Hence y' is positive. Then
W(y'xy)l
=
ly'(x)l ~< Ilxll Ily'll
=
Ilxlly'(1)= Ilxllx'(y*y),
by Corollary 2.3.4.7. P r o p o s i t i o n 2.3.4.9
Ill ( 0 ) Let E be an involutive Banach algebra and
an approximate unit of E . Define
f:E
~E,
x~-~x*,
g:E
~E,
x~---~rex,
h:E
~ E,
x~-+x*x.
Then f ( ~ ) , g ( ~ ) , and h(~) are approximate units of E and
iim x* yx = y for every y E E . In particular, E possesses an approximate untt r
such that
Re E # E ~ . Since 0 = lim [ly*x - y*[[ = lim [Ix*y - y][ = lim i l x y - Yll =,~
=,8
x,l(8)
'
0 = lim ]]xy* - y*[[ = lim ]lyx* - y[] = iim ]lyx - Y]I x,~
x,8
x,/(8)
'
for every y E F , f(~) is an approximate unit of E . Take y E E . Then 1 1 1 , I I ( r e x ) y - Yll = ll~( ~ + x')Y - Yll -< 511xy- Yll + 5119 Y - YlI, 1
1
1
IlY(rex) - Yll = lilY( x + x*) - y[] < ~llyx - Y[I + -~]IYx* - YlI ,
I1~'~ - ~11 _< II~'~y - ~'~11 + I1~'~ - yll _ [Ix'JJ. b) y' is positive. Take (/~,x) 6 E: x E. Then (D1 + x)*(f?l + x) -[~J2l + "~x + l~x* + x ' x , y'((f~l + x)'(~l +
x)) =
a[/~l2 + ~x'(x) + flx'(x*) + x'(x*x).
a =~ b. We may assume that x' # 0. By Proposition 2.3.4.10 b), IIx'Hy'((~l + x)*(~l + x)) > IIx'll21DI 2 - 2]DJ IIx'll Ix'(x)l + Ix'(x)l 2 =
~.3 lnvolutive Banach Algebras
~95
= (llx'll Iz~l- Ix'(x)l) 2 >_ 0 for every (f~, x) ~ IK x E . Hence y' is positive. b ~ a. It is easy to see that x' is positive and that a E IR+. Now, since 0 < y'((f~l 4- x)*(f~l + x)) < a ~ 2 + 2/~x'(x) + IIz'll for every (f~,x) e IR x Re E # ,
9 '(x) ~ _< ~llx'll for every x E Re E # . By Proposition 2.3.2.22 j),
IIx'll 2 = sup{x'(z)21 x ~ Re E ~ } _< allx'll 9
IIx'll _< ~.
B
C o r o l l a r y 2.3.4.12 ( 0 ) Let E be an involutive quasiunital Banach algebra and F an associated involutive unital Banach algebra. Then every continuous positive linear form x' on E has a unique posdive linear extension y' to F such that (Corollary 2.3.4.7)
I1~'11 = IIx'll. Then
y ' ( 1 ) - I1~'11. We may assume E to be non-unital. By Proposition 2.3.4.11 a =~ b, the linear extension y' of x' to F defined by
y'(1) = IIx'll is positive and, by Corollary 2.3.4.7, y' E F~ and
Ily'll
=
y'(1)
=
II~'ll.
Now take a positive linear extension z' of x' to F such that
IIz'll = llx'll. By Corollary 2.3.4.7,
z'(1) Hence z' = y'.
=
IIz'll- IIx'll. 1
296
~. Banach Algebras
Proposition
2.3.4.13 ( 0 )
Let E be an involutive quasiunital Banach
algebra and let C' be the set of contmuous positive linear forms on E .
a) C' is a sharp convex cone of E ' , closed in E'E. b)
There is a unique order on E' which renders E' an ordered Banach space such that
El@ .~.CIo c) E'+ # is a compact set of E'E , metrizable whenever E is separable. d)
l f lK = r , then E'+ = {x' e E' I x e E ==~ x ' ( x ' z ) e 1%}.
a) It is easy to see that C' is a convex cone of E'. Take
='
e
c' n ( - c ' ) .
Then Iix'lI = limx'(x*x) = o where ~ is an approximate unit of E (Proposition 2.3.4.10 a)). Thus x' = 0 and C' is sharp. By Proposition 2.3.2.22 c), Re E' is closed in E~. Since
c ' = r'l {~' e R~ E' I ~'(~'~) > 0}, zEE
it follows that C', too, is closed in E~. b) follows from a) and Proposition 1.7.1.3. c) follows from a), b) and the Aiaoglu-Bourbaki Theorem. d) Choose x' E E', such that
z'(z'z) e R+ for every x E E. Define f : Ex
E
~,
(x,y):
~ x'(y*x).
By Proposition 2.3.4.6 a), f is sesquilinear. Since
~.3 lnvolutive Banach Algebras
~,97
f ( x ) = xt(x*x) E. ]R for every x E E , it follows from Proposition 2.3.3.8 b =~ a that f is Hermitian. Take x E E and let ~ be an approximate unit of E . Then
x'(x*) = limx'(x*y) = lim f ( y , x ) = lim f * ( y , x ) ---y,~ ~,~ y,~ = lim f ( x , y) -- limx'(y*x) = x'(x) by Proposition 2.3.4.9. Hence x' is involutive and therefore positive.
Remark.
I
The result does not hold without E's being quasiunital. Let E be
the unital Banach algebra defined in E 2.2.6 and F the closed subalgebra
{x e E Ix(a) - 0 } of E . Take the involution F
"~F,
x:
"~
on F . Then F is an involutive Banach algebra and the map F
~IK,
x'
; x (1)(a)
belongs to C' r3 ( - C ' ) . Hence C' is not sharp in this case. C o r o l l a r y 2.3.4.14
( 0 ) If E is an mvolutive quasiunttal Banach algebra,
then
I1~' + ~'11 = I!~'11 + I1r for every x', y' E E'+. By Proposition 2.3.4.10 a),
IIx'll + Ily'll-- limx'(x*x) + limy'(x*x) = lim(x*x,x' + y') = Ilx' + y'll x,~ x,~i x,i~ for any approximate unit ~ of E .
I
Let E be an involutive quasiunital Banach algebra. Take z', y', z' ~ E'+ such that C o r o l l a r y 2.3.4.15
IIx'- y'll- IIx'll + Ily'll, z' < x',
z' < y'.
Then z' = 0 (t.e. the infimum of x' and y' m E'+ is 0).
2. Banach Algebras
298
By Corollary 2.3.4.14, IIx'll § Ily'll---IIx'- ~'11 = II(x' - z') - ( y ' - z')ll _
_
>_ ~ ( ~ + a)lu'(~)l ~ + ~ ( ~ + a)lz'(~)l ~ - ~ l y , ( x ) l ~_
-~21z'(x)l~
- 2 a ~ r e y ' ( x ) z ' ( x ) = c~ly'(x ) - z'(x)l 2 , y'(~) = ~'(x).
Hence y' = z' and x' E ro(E).
1
~. Banach Algebras
310
Proposition 2.3.5.7 a)
(4 )
Let T be a locally compact group.
The map A4,(T) --+ K ,
la:
~ ~(T)
is a pure state (Proposition ~.3.~,.28 a)).
b) If A is a left and right invariant Haar measure on T , then the map LI(A)
dA
is a pure state (Proposition e.s.e.eg
a)).
By Propositions 2.3.2.29 a) and 2.3.2.28 f), the given maps are involutive algebra homomorphisms. Hence they are positive (Proposition 2.3.4.5). Since they have norm 1, the assertions follow from Proposition 2.3.5.6. I Proposition 2.3.5.8
( 0 )
Let E
be an involutive quasiunital Banach
algebra. Then ~ E C(r(E)) whenever x E E and the map
E~C(r(E)),
V
x:
;x
is linear, involutive, and continuous with norm at most 1. Moreover, for every V
x E E , if
Ilxll = I1~11, then IJz_ ]Ixll =~ ]]cd - xl] _ c~.
gs) ::lc~e IR+,
I]c~l-xll < ~.
h) If x" is a linear.form on E' such that nuo?28.
x" ( E' +)
C lit+ , then x" is cont,-
2. Banach Algebras
314
a =~ b. By Proposition 2.3.2.22 1), there is an x ~ E Re E' such that
IIx'll- 1,
x'(x) - Ilxll.
By a), there are y',z' E E'+ such that
x'= u'- z',
Ilu'll + IIz'll
=
1.
We get
I1~11 = ~'(~) = u'(~) - z'(~) _< ly'(x)l + I~'(~)1 _< -< Ilu'll Ilxll + IIz'll I1~11- I1~11 9 We may assume z' # 0. Then I]TH z l ' e r ( E ) and 1
Ilxll- ii-~lllZ'(x)l _< I1~11,
I1~11--I1~11 9 b :~ c. Let K ' be the closure in E~ of the convex hull of ~'(E)U ( - r ( E ) ) . Assume there is an x ~ E Re E ~ # \ K R e E such that
~.
By Proposition 2.3.2.25, there is an x E
~up ~'(~) < ~'(~).
lfEK'
By b), we deduce that
Ilxll- I1~11-- sup lu'(~)l _< sup y'(x) < x'(x) < I1~11 v'e~'(E)
~'eK'
which is a contradiction. Hence Re E '# C K ' . The converse inclusion is trivial (Proposition 2.3.2.22 k)). c =~ a. We may assume that IIx'll = 1. We define f " IR x E' • E'
~ E',
(a,y',z') :
: ay'-
(1- a)z'.
It is readily verified that f([0,1] x v(E) x r ( E ) ) is convex. Hence it contains the convex hull of v(E) O (~-(E)). Let 11 be the neighbourhood filter of x' in E ~ . By hypothesis, for every U E 11, there is an element
~.3 lnvolutive Banach Algebras
(OtU, v v , t
z'v) e
815
[0, l] x r ( E ) x r ( E )
such that ~uY'u - (1 - cxv)z~ e U. Then
IIx'll -- 1 - cru 4- (1 - cru) - II~uybll + I1(1 - au)z[,ll 9 Order 11 by inclusion and let ~ be an ultrafilter on ~ ( E ' ) finer than the lower section filter of 11. E~ # is a compact set of E~ (Proposition 2.3.4.13 c)). Set y' := limauY'v 6 E'+ # V,~
'
z' := lim(1 - otv)z'v 6 E+ # V,~
'
where the limits are taken in E ~ . Then x ' = lim(v,~ vVu
Iir
+ IIz'll >~ Ily'- z'll -- IIx'll- n~(ll~uy~ll + 1](1 - CZu)Z[r[I) :
=
by
(1 -Cru)Z'v) = limc~vvbe,~ liv,~(1- a u ) Z ' v = V'-, z'
lim II uvuII + v,~
lim v,~
II(1 - ~u)4,11 >- I1r + I1~'11,
Proposition 1.2.6.6. Thus
I1~'11 = lien + IIz'll. b =~ d follows from Proposition 2.3.5.8. a =~ e follows from Proposition 2.3.4.17. f) We may assume I1~11 < 1. By e),
0 = (b,a) = ~ (bj,,a,,)sj=l
= ~ (%,,jE(~)a,~)= (b,(jE(p)a,j),je~.), s,J=l SO
P,.3 lnvolutive Banach Algebras
Jw(P) a -- ( j E ( p ) a l $ ) l d E ~ . . L.inln
327
.
~(p)
Assume now a E E ~p+3)n,n. Then, by the above result, for all b E ~n,n , n
(b,"3~(z) a ) =
(jE(?) b, a) =
((jEg,,,b)j,, ai.~) = td=l
n
n
"' o,,) = Z (b,,,,3E(,)av) __ (b, (3E(,)a,j),,j~n) ,
- E _
_
t.3=l
i,j=l
SO
j E ( n, t n) a ---- ( 3 E ( , ) a t j
2.3.6.8 Let E be an involutwe complex Banach algebra and F an involutive E-module. Let E and F denote the underlying involutive real Banach algebra of E and the underlying involutive E-module of F , respectively. For every a' E F' put
Proposition
a'~'F----~IR,
a:
~re(a,a').
Then a' E (F)' and the map F'
~ (F)',
a' : ~ a'
is an isometry of mvolutive E-modules. Take x E E . For every a E F ,
(a, xa"-') - re (a, xa') --- re (ax, a') - (ax, ~) - (a, za;),
(a, a'x) = re (a,a'x) = re (xa, a') = (xa, a') = (a,a'x),
(a, ~*) = (a*, ~;) = re (a*, a') - re (a, a'*) -- (a, a'~),
so that xa I _
xat~
alx
-- atx~
aI
-- al,
and the assertion follows from Proposition 1.3.6.26 b).
1
328
~. Banach Algebras
Exercises
E 2.3.1 Let f : S --~ T be a map between the sets S and T . Show that the map ]KT
~ K S,
fr~
27:
~ fr~
,~ 2 7 o f ,
x:
~xof
are involutive unital algebra homomorphisms.
E 2.3.2 Let E be an involutive algebra. Show that the algebra E x E (Example 2.1.4.1) is an involutive algebra with respect to E x E
~ E x E,
(x,y):
"(y*,x*)
as involution. If, in addition, E is an involutive normed algebra, then the above involutive algebra E x E is an involutive normed algebra with respect to the norm
E•
;R+,
(x,~):
;sup{llxll, ll~ll}.
E 2.3.3 Let 1 and j be the standard unit vectors of ~:2. Take a E { - 1 , 0, 1}, /3 E lK such that 1/31= 1 and /3 E { - 1 , 1 }, whenever a ~ 0. Let Ea~ be the involutive algebra lK 2, such that 1 is the unit, 3 2 = a l , and j* = / 3 3 . Show the following. a)
If lK = IR, then we have: al)
The involutive unital algebras Ea# are pairwise not isomorphic.
a2) Every two-dimensional involutive real unital algebra is isomorphic to one of the above involutive unital algebras. a3) Every unital algebra homomorphism Ea,~ --+ Ea,# is involutive. a4) The set of unital algebra homomorphism Ea,~ --~ ~
is empty if
a = - 1 or if a = 1 and /3 = - 1 . It contains exactly one element if a = 0 and exactly two elements if a =/3 = 1.
E3 lnvolutive Banach Algebras
3~9
as) The involutive algebras E1,1, E-I,t and E-l,-1 with the norms Et,l
~ IR+,
(~,r/):
; suP{IS-- 711,I~ + T/l},
E-l,1
~ lR+,
(~, ~/) ~
E_,_,
~ n~+,
(~,,):
~/'~ + 7/~ ,
- v/r
r
are involutive unital Banach algebras such that
IIx*xll = Ilxll2
(P)
for every element x. Let I := {(1,1), ( 1 , - 1 ) , ( - 1 , - 1 ) } . For every tow-dimensional unital involutive real Banach algebra E possessing the above property (P), there are an (a,f~) E I and a unital involutive algebra isomorphism E -+ Ea,~. Every unital algebra homomorphism Ea,~ -+ Ea,~ is involutive and an isometry for every (a, f~) E I . E-l,-1 is isometric to the involutive real Banach algebra q~. b) If ]K =q~, then: bl) The involutive unital algebras E~,, E76 are isomorphic iff one of the following conditions is fulfilled: a=7=O, a = 7 =/=0
and
f~ = (f,
= -7 # 0
and
f~ = - ( f .
b2) Every two-dimensional involutive complex unital algebra is isomorphic to one of the above involutive unital algebras. b3) Every unital algebra homomorphism El,, -~ El,, is involutive. b4) The group of isomorphisms E0,~ -+ E0,~ is isomorphic to the multiplicative group IR\{0}. bs) The set of involutive unital algebra homomorphisms of E~,, in q~ is empty if lal = 1 and /~ = 1. It contains exactly one element if = 0 and exactly two elements if la[ = 1 and /~ = - 1 .
~. Banach Algebras
330
be) E1,1 endowed with the norm E1,1
.~ IR+,
(~,7/):
~ sup{l~ - 7/I, I~ + 771}
is an involutive unital Banach algebra, such that I1~'~11 = I1~11~
for every x E E1,1. Every two--dimensional involutive unital Banach algebra with the above property is isometric to E~,l. E 2.3.4 Let E be a symmetric involutive unital Banach algebra and define C' := {x' e R e E ' [ [[s
C := {x E R e E I x '
= x'(1)},
E C' ==~ x'(x) E 1~+}.
Prove the following: a) C' is a closed sharp convex cone of E'. b) For every x E Re E and every a E a(x), there is an x' E C' such that
x'(~)
=~.
c) C is a closed convex cone of E and a(x) C •+
for every x E C .
d) If x = 0 whenever x E R e E and a(x) = {0}, then C is sharp. E 2.3.5 Let E be an involutive complex vector space (involutive commutative o
complex algebra), F the underlying real structure, and F its complexification. Show that the map
F ---~ E • E ,
(x,y) ~-~ (x + iy, x* + ~y*)
is an isomorphism of involutive complex vector spaces (of involutive complex algebras). E 2.3.6 Let E be an involutive commutative (normed) algebra. Show that if we endow the underlying (normed) real algebra of E with the involution
E---~ E,
x"
~x,
then the result is an involutive commutative (normed) real algebra.
~,.~ Gel]and Algebras
2.4 Gelfand
331
Algebras
By the Banach-Mazur Theorem the Gelfand transform can be defined on Gelland Banach algebras. In the special case of commutative locally compact groups this transform generalizes the Fourier transform, many properties of which are best understood in light of the Gelfand transform. But the most important role played by the Gelfand transform does not arise until we come to Gelfand C*-algebras. For then it is an isometry and hence a functional calculus can be developed for C*-algebras, providing the main tool for their study. 2.4.1 T h e Gelfand Transform Definition 2.4.1.1 ( 0 ) A (unital) Geifand algebra is a commutatwe (unital) Banach algebra such that if ]K = JR, then
,~((x. o)) c for every x E E . By Corollary 2.1.5.14, the above relation is equzvalent to
o~((~.0)) = ~ ( ~ ) . Let E be a Gelfand algebra. We denote by ao(E) the set of algebra homomorphisms of E into IK endowed with the topology of pomtwise convergence a,d b~ , ( E ) th~ ~ub~pa~ oo(E)\,{0} of oo(E). V~,~,, 9 e E . d~l~n~ E
9= ~ ' a ( E ) -
~lK,
x'~x'(x).
a(E) is called the spectrum of E and its elements are called characters of E.
By Proposition 2.2.4.19, every character is continuous. ~ is obviously continuous for every x e E . a(E) may be empty (see Remark of Corollary 2.4.1.14), but if E is unital, then a(E) ~ q) (Proposition 2.4.1.5). Let E be a real Gelfand algebra and let E be a c0mplexification of E . If q~ x / ~ is a unital Banach algebra associated to E (Proposition 2.2.1.3), then q~ x / ~ is a complexification of ]R x E . Hence every real Gelfand algebra possesses an associated unital Gelfand algebra. Proposition 2.4.1.2
( 0 ) IrE E
is a Gel/and algebra, then the map
} C(a(E)),
x~
33e
2. Banach Algebras
is a continuous algebra homomorphism of norm at most 1 ; it is called the Gelyand transform o.f E (Gelfand, 19~1). If E is unital, then sts Gelfand transform is a unital algebra homomorphism.
Let x, y E E and e,/~ E]K. Then
~'~'~"~'~(~'=) x'(~,x + ~,) = ,~'(~) + ~ ' ( ~ )
=
= a~(x') + X~(x') = (a~ + X~)(x'), @ ( ~ ' ) = ~'(x~) = x'(~)~'(~) = ~(x')~(~') = ( @ ) ( x ' ) ,
I~(~')1 = Ix'(~)l O, define U,,(A) .= U u~(t3). OeA
Gtven A, B C qJ , put d(A, B ) : = inf{a > OIA c U s ( B ) , B C U,~(A)}. Let E be a complex Banach algebra and take x, y E E such that xy = y x , then
d(a(x),a(y)) < IIx - ull.
336
2. Banach Algebras
We may assume E to be unital. Let F be the unital Banach subalgebra of E generated by
I~
{ x , y } U {(al - x ) - '
er
u {(al - y)-' l a e r
By Proposition 2.1.2.8 and Corollary 2.2.1.10, F is commutative. Since obviously
OE(~) OF(~), OE(V) OF(V), =
=
we may assume that E is commutative. By Corollary 2.4.1.7 a) and Proposition 2.4.1.2, a(x)
=
B(a(E)) C U U~(~) ~eA
for every a > I l x - yll, where A := ~(a(E)) = a ( y ) .
It follows that
d(a(~),o(~)) < Remark.
IIx- ~11.
I
The above corollary improves Corollary 2.2.5.2, when xy = xy and
lK=C. Corollary 2.4.1.9
0o tn Let y] an be a power series w:th radms of convergence n--0
r . Take
n=O
Let E be a un:tal complex Banach algebra and take x E E with
lim Hx"}}~ < r ,
n--4~
and co
f(~) = ~ ~.~" n-'O
(Propos:tzon 2. 2. 3. 2). Then
o'(f(x)) = f(o'(X)) (Proposition 2.2.4.1).
~.4 Gel]and Algebras
337
We may assume that there is an no E IN with ano # 0 (Theorem 2.2.5.4, Proposition 2.1.3.2). Given m E IN, put m
Pm := ~
ant n.
n--0
Then in the notation of Corollary 2.4.1.8, d(Pm(a(x)),a(f(x))) = d(a(Pm(x)),a(f(x)))
no (Corollary 2.1.3.4 b)).. Hence a(f(x)) = f(a(x)) (Corollary 2.2.4.5).
i
C o r o l l a r y 2.4.1.10
Take a, a, b E IR with a > 0, [a, b] C ] - 1, 1[. Let E
be a unital complex Banach algebra and take x E E such that
a(x) C [a, b]. Then
a([l 4- x]~) C [(1 + a)a, (I -+-b)~] C IR+. Let
;.@(0)
}t
~llzll
for every z E F and
aF(z) = aE(z) c la+ (Corollary 2.2.4.27). By Proposition 2.4.1.15 d),e), there is a unique y E F such that aE(y) C JR+ and y" = x. Moreover, y belongs to the closed real algebra of E generated by x and
By Corollary 2.2.4.27,
a~(~) = or(y) c rt+. By Corollary 2.4.2.5 b =~ a, y E Re F C Re E . Now take z E N o E such that a(z) C: lR+ and z" = x. Then E ( z ) is a closed involutive commutative subalgebra of E (Corollary 2.3.2.16) containing x. By the above, z = y.
~.4 Oel.fand Algebras
349
b) We put F := E(x). By Corollary 2.3.2.16, E(x) is commutative. Then
~F(u) = ~ ( y )
>_,~llyll
for every y E F and
(Corollary 2.2.4.27, Proposition 2.3.1.29). The assertion now follows from Proposition 2.4.1.15 f). cl =~ c~ follows from a). c2 =~ c3 is trivial. c3 =~ cl. We endow E with the order relation defined in Proposition 2.3.2.34 b). Step 1
x < 0 =~ x - 0.
By Proposition 2.1.3.10 b),
o(-W)\{0} = o(-y'y)\{0} c o(-x) c ~+, such that
-yy* >__O. By Proposition 2.3.2.34 c),
y*y + yy* >_0, so that
x--y*y~-yy*
~0,
x--O.
Step 2
x :> 0
By b), there are x +, x- E E+ with X -- X + -
x-
,
X+ X -
~
X-X
+ ~
0.
We get
(u~-)*(yx-)
= x-y*ux-
= ~-x~-
= ~-(x
§ -
~-)x-
=
_(~-)3 < 0
350
2. Banach Algebras
by Proposition 2.3.2.34 c). By Step 1, ( x - ) 3 = O,
such that o(~-)~ = o((~-)~) = {0)
(Theorem 2.1.3.4 b)),
IIz-II = r ( ~ - ) = o , x-=O,
x = x+ E E+.
ca ~ c4 is trivial. c4 ~ ct follows from Proposition 2.3.5.15 a ~ c.
I
P r o p o s i t i o n 2.4.2.10
( 0 ) Let E be a symmetric involuttve unital aelfand algebra and let x E E such that E = E(x, 1). Then the map a(E)
~ a(x),
x' :
: z'(x)
is a homeomorphism. If E has no unit then the above assertion still holds if E = E ( x ) and if we replace a ( E ) by a o ( E ) .
Define ~p:a(E) ~ a ( x ) ,
x' :
; x'(x).
~p is surjective (Corollary 2.4.1.7 a)) and continuous. Take x',y' E a(E) such that
x'(x) = r Then ~'(~') = ~,(~) = ~'(~) = r
by Proposition 2.4.2.3 a ::~ d and Proposition 2.4.2.2 b). Let F be the involutive unital subalgebra of E generated by x and take y E F . Then there is a P E IK[s, t] such that
•.4 Gelfand Algebras
351
y = P(x,x')
(Proposition 2.3.1.20). Thus
x'(y) = P(x'(x), x'(x*)) = P(y'(x), y'(x')) = y'(y). Hence x' and y' coincide on F . Since x',y' are continuous (Proposition 2.2.4.19) and F is dense in E , x' = y'. Hence ~o is injective and therefore a homeomorphism (Theorem 2.4.1.3 c)). The final assertion has the same proof. II P r o p o s i t i o n 2.4.2.11 Let E be a symmetric involutive quasiunital Gelfand algebra and M the set of poszttve Radon measures p on a(E) satisfying /,(cr(E)) = 1.
Given p E M , define T, E
>f
--,
Then the Gelfand transform of E is involutive, a(E) C to(E) and ~ E r(E) for every t~ e ~/1. The map .hi
", r (E ) ,
I, ~-+ T,
is mjective and {x l x E E} is a dense involutive subalgebra of r Moreover, the following are equivalent: a) to(E) C a ( E ) .
a(E).
b)
to(E) =
c)
I1~11 = I1~11 .fo," e,,~,"y x e E.
d)
Themap
is bijective.
e) to(E) c (~1~ c . ~ ) .
352
~. Banach Algebras
// R e E ' = E~ - E'+, then the above conditions are equivalent to each of the following ones:
f)
The Gelfand transform of E is an isomorphism of involuttve Banach algebras.
g)
There ts an a > O, such that r ( x ) >_ ~lixll
for all x E E .
h)
The Gel/and transform of E is bsjective.
By Proposition 2.4.2.3 a ~ c & g, and Proposition 2.4.2.2 b),d) the Gel/and transform of E is involutive and a(E) C r0(E). By Corollary 2.4.2.6, { ~ l x E E} is a dense involutive subalgebra of Co(a(E)). Take /~ E rid. Then ~(x*x) = fx~'~xd~u = f[~[2d/J >_ O,
I~(x)l---I f ~d~l_< f I~ld~ _< I1~11_< Ilxll for every x E E. ~ is thus positive and
II~ll _< 1. Let K be a compact set of a(E) and take e > 0. Since {~ I x E E} is a dense set of C0(a(x)), there is an x E E with I~1 > 1 on K. Let ~ be an approximate unit of E . We set f :E
>E ,
y:
>y'y,
~ :=/(;~). Then, by Proposition 2.3.4.9, ~ is an approximate unit of E . Hence, there is a y E E # such that
IIz*z -
z'zy*yll
< ~.
~,.4 Gel/and Algebras
353
Then
1 - y*'y _< ~ ( 1
_< II~-=~x - x-";;~ll
- y~)
_< IIx*x - ~*xy*yll < 6
on K and so
_>
>_
f
( 1 - e)d# = ( 1 - e)#(K).
K
Since c and K are arbitrary,
IlPll > #(a(E))= 1, I1~,11 = 1,
~ e r(E).
Since { ~ l z E E} is dense in Co(a(E)), the m a p
A4 --~ r(E),
#.~ .~
is injective. a =~ b follows from the inclusion a(E) C r 0 ( E ) , which was established above. b =~ c. By Proposition 2.3.5.13 a =~ c,
I!~11 >
~up ~ ' ( ~ ) =
='Ca(E)
I1~11 >_ ~up
=' eo(~:)
~'(-~)
=
~up ~'(~),
=' 6.T( E)
~up x'(-~),
=' e T(E)
such t h a t
I1~11 = I1~11 for every x E Re E . Now take x E E and ~ > 0. There is an x' E r ( E ) and a 0 E lR such t h a t
I~'(~)1 > I1~11- ~,
x'(~) = I~'(~)1~ '~.
Put y :--
Then
e-SOX.
354
$. Banach Algebras
9 '(v) = ~'(*-%) = *-'~
=
I~'(~)1
>
I1~11- *,
v
I1~11- Ilffll ~> ilre~l = IIr~ll = IIr~ll-
Ilre ~11 ~>
> reY{x') = rext(y) > II~II- e (Corollary 2.3.1.23 a =~ d). Thus
II~II > II~II since e is arbitrary. The reverse inequality follows from the fact that a(E) C
to(E). c =~ d. Take x' E r ( E ) . By d), the map {s
~.x'(x)
~:
is well-defined, linear, involutive, and continuous. Since {s x E E} is a dense involutive set of Co(a(E)), there is a bounded real Radon measure ~u on a(E) such that
du
x'(x) = f~ for every x E E (Example 2.3.4.18). By c),
If~du[ = Ix'(x)l for every x E E . Since { s
_< II~ll = I1~11
E E} is dense in C(a(E)),
llull < 1. Let ~ be an approximate unit of E and let ~u+ be the positive part of #. Then
1 - I I x ' l l - l i m x ' ( x ' x ) = lim f~'~xdl~ < lim f ~ a ~ +< Ilu+lJ < Ilull z,~
(Proposition
z,~
J
-
=,~
J
-
2.3.4.10 a)), such that
Ilu+ll- Ilull- 1,
U E M,
~, = x',
and the map
A4 --+ r ( E ) ,
I~ : ~ ~
-
~,.4 Geifand Algebras
355
is bijective. d =~ e is trivial. e =~ a. Take x' E r0(E). By f), there is a /~ e r such that ~ = x'. Assume the support of/z contains two distinct points. Then there are /~',/J" E r
and a ' , a " e R + \ { 0 } with
a' + a" = l ,
a'pt + a"p" = lJ,
I~' :/= P".
Thus
x'= a'~' + a"~",
~' # ~",
which contradicts the fact that x' is an extreme point of T(E). Hence the support of # contains exactly one point which must be x'. Hence x' G a(E) and
vo(E) c. a ( E ) . d ~ f. Let
u: E
>Co(a(E))
be the Gelfand transform of E . Since Im u is dense in C0(a(E)), u' is injective (Corollary 1.3.5.9), and since u is involutive, u' is also involutive (Proposition 2.3.2.22 d)). By d),
u'(Co(a(E))+) = E+ so that
u'(ReCo(a(E))') = E+ - E+ = Re E ' . Thus u' is surjective (Proposition 2.3.1.22, Corollary 2.3.1.23). By the Principle of Inverse Operators, u' is an isomorphism. Hence u is an isomorphism (Corollary 1.4.2.5). f =~ e & g & h is trivial. g =~ f follows from Corollary 2.4.2.7 a =~ b. h =~ f follows from the Principle of Inverse Operators.
Proposition 2.4.2.12 s~mple Gelfand algebra.
Let E be a symmetric involutive quasiunital semi-
~. Banach Algebras
356
a) There zs a umque order relatzon on E with respect to which E is an ordered Banach space and
E+ = {x e E lx' e E'+ ~ b)
E+ C R e E .
c) a(x) C R . d)
x'(x) E I~.+}.
:for every x E E+.
The following are equivalent: dl)
The cone of E' generated by a(E) is dense in (E'+)E.
d~) E . = {x E E la(x) C a&b.
Take x E E
R+}.
with
z'(x) = o for every x' E E~. Since a(E) c E'+ (Proposition 2.4.2.2 c), Proposition 2.4.2.3 a =~ e), s = 0. By Corollary 2.4.1.14, x = 0 and the assertion follows from Proposition 2.3.4.17 b) (and Proposition 1.3.5.4). c) By Corollary 2.4.1.7 a)
a(x)
c
~(a(E)) u {0}
for every x E E , and the assertion now follows from a(E) C E~ (Proposition 2.4.2.2 c), Proposition 2.4.2.3 a =~ e). d l =~ d2. Let x E E such that
a(x) r e,.+. Then x'(x) E IR+ whenever x' E a ( E ) . Hence, by dl), x'(x) E IR+ for every x' E E~. The reverse inclusion follows from c). d~ =~ dl. Let C' be the closed convex cone of E~ generated by a(E). Assume there is an x' E E'+\C'. By Proposition 2.3.2.25, there is an x E Re E with sup y'(x) = 0 < x'(x).
y'6C'
In particular,
~'(-x) > o
~.4 Gelfand Algebras
357
for every y' E a(E), i.e.
a(-x)
c ~+
(Corollary 2.4.1.7 a)). By d~), - x E E+ and we obtain the contradiction
x'(-x) < 0 < ~ ' ( - ~ ) . Hence C' = E~.
I
~. Banach Algebras
358
2.4.3 Examples Example 2.4.3.1
( 0 ) Let T be locally compact. For t E T , define t ' C o ( T ) --4 I~,
x"
Ix(t).
Then t E a(Co(T)) for every t E T and the map T
, a(Co(T)),
t:
~t
is a homeomorphism. The first assertion is trivial. Since the map
T ~
K,
t"
~ (x,t'}=x(t)
is continuous for every x E C0(T), the map
f'T
,a(C0(T)),
t:
;t"
is also continuous. By Urysohn's Lemma, this map is injective. Take x' E a(Co(T)). By Example 1.2.2.10, there is a bounded Radon measure p on T such that
Xt(2)
= f z d~
for every x E C0(T). Since x' is an algebraic homomorphism, it follows that the functions of C0(T) are constant on the support of p . Hence x' --- t for some t E T and thus f is surjective. If T is compact, this map is a homeomorphism. Assume that T is not compact and let ~ be the filter on T converging to the Alexandroff point of T . Then x(~) converges to 0 for every x E Co(T). Hence the filter f(~) has no points of adherence in a(Co(T)). It follows that f is a homeomorphism. E x a m p l e 2.4.3.2
I
( "[ ) Let T be a set. Given t E T , define t" co(T)
; IK,
z:
~ z(t).
Then t E a(co(T)) .for every t E T and the map T
~. a(co(T)),
t"
~. t
ts a homeomorphism, where we regard T as a discrete space.
~.4 Gelfand Algebras
359
The assertion follows from Example 2.4.3.1 by taking T with the discrete opology. I ~xample 2.4.3.3
Let ~ T be the Stone-Cech compactification of the comple-
"ely regular space T and ~x the continuous extension of x to ~ T for every x E d ( T ) . Given t E B T , define t "d ( T )
~ IK,
x"
~ ~x(t).
Then "{ is a character of d ( T ) whenever t E ~ T and the map ~T
" a(d(T)),
t~
~t
is a homeomorphism.
In fact, the map C(T)
; d(~),
x:
" f~x
's an isometry of unital Banach algebras and the assertion follows from Example 2.4.3.1. I Example 2.4.3.4
Let T be a set, 13T the Stone-C_,ech compactification of T
viewed as a discrete space. For x E ~ ( T ) , let 13x be the continuous extension of x to 13T. Given t E 13T, define t" t ~ ( T )
~ IK,
x:
" ~x(t).
Then each t is a character of ~~176 and the map /3T
~ a(C(T)) ,
t:
"
is a homeomorphtsm.
The assertion follows immediately from Example 2.4.3.3, by endowing T with the discrete topology. I Example 2.4.3.5
( 7 ) Let T be the Alexandroff compactification of the
set T viewed as a discrete space. Given x E c ( T ) , let ~ denote the continuous extension of x to T . Gwen t E T , define t" c(T)
~ IK,
x"
" 5(t).
Then each t is a character of c(T) and the map T
is a homeomorphism.
~a(c(T)),
t ; .... >t
~. Banach Algebras
360
In fact, the map c(T)
~C(T),
x.:
'~ x
is an isometry of unital Banach algebras and the assertion follows from Example 2.4.3.1.
II
Proposition 2.4.3.6 ( 0 )
Let S,T be nonempty compact spaces and u "C(S)
~C(T)
be a unital algebra homomorphism.
a)
There is a unique map ~o" T --, S such that UX -- Xo~p for every x E C ( S ) , and this map ~a is continuous.
b)
u is continuous and
Ilull
= 1.
c) ~o is injective (surjective) iff u is surjective (injective). d)
I f u is bijectwe, then ~o :s a homeomorphism.
a) Let ~ol (resp. ~o2) denote the homeomorphism s
(resp. T
~.~(c(s)),
~ a(C(T)),
~:
;'~
t:
; t')
defined in Example 2.4.3.1. Define (Proposition 2.4.1.17 a)) r
~ a(C_.(S)),
y' :
; y' o u ,
:= ~oi"l o r o ~o~. By Proposition 2.4.1.17 c), ~ is continuous. Take x E C(S). Then xor
= Xo r
o r o ~P2(t) = X o r 1 o r
= ~ o ~ i - ' ( ? o u) = ~'o u(~) = t ( ~ )
for every t E T , so
= (u~)(t)
=
~.4 Gel.land Algebras
Uo~o
361
-- U X .
Take ~o'" T ~ S such that XOCpI -- UX for every x E C(S). Assume that ~o(t) r ~'(t) for some t E T . By Urysohn's Theorem, there is an x E C(S) such that
~(v(t)) # ~(v(t')). It follows that (u~)(t)
=
~(vCt)) # ~(v'(t)) = (,~)(t),
which is a contradiction. Hence ~o = ~o', which proves the uniqueness of ~o. b) follows from a). c) If u is surjective, then r (defined in the proof of a)) is injective (Proposition 2.4.1.17 d)) and so ~ is also injective. Conversely, if ~ is injective, then by Tietze's Theorem, u is surjective. If ~o is surjective, then u is obviously injective. If ~o is not surjective, then S\~o(T) is nonempty. Since ~o(T) is compact, there is an x E C(S)\{0} vanishing on ~(T) (Vrysohn's Theorem). Hence
UX-'XOCp--01 and so u is not injective. d) follows from c).
Example 2.4.3.7
I
We assume IK = ~ . Let K be a compact nonempty set of
ff3. Put .4 .= {x e C.(K) l xlt( is analytic) and := {x E .A I :ly rational function on r
Assume that .7r is dense in .4 and define t . ,4
for t E K .
~r ,
x ~
z(t)
YIK = x } .
361~
~. Banach Algebras
a)
.4 ts a closed unital subalgebra of C ( K ) .
b) t'E a ( A ) for every t E K .
c)
The map K
~a(A),
t.~
"t
is a homeomorphism.
d) .,4 ts semi-simple. e)
Assume that "5 E K whenever a E K and put x~ K .for every x E A .
~q~,
a:
~ x(-5)
Then ,4 is an involutive commutative unital Banach
algebra with respect to the involution .,4
~ .A ,
X:
~ X* .
But ,4 is not :ymmetric.
f) If (x,)tet is a finite family in .A such that .for every a E K , there is a t E I with x,(a) # 0, then there ts a (Yt),et E .A t with
Z
xtYt
---- e K 9
tel
a) and b) are trivial. c) Define x" K---~r
a"
~a.
Then a(x) = K .
By Corollary 2.4.1.12, the map a(A)
~ K,
x' :
" x'(x)
is a homeomorphism. It is obviously the inverse of the map K
>o(A),
t: ;t.
~.4 Gel]and Algebras
363
d) Take x E .A. By c), ~(a(r
= x(K),
so that ~=0~x=0. By Corollary 2.4.1.14, ~4 is semi-simple. e) is easy to see. f) We set Jr := { ~e~1x'Y' (Y,),el e .,41} . Then ~" is an ideal of j4. If 9r # ,A then ~r is contained in a maximal proper ideal of ,4. By c) and Proposition 2.4.1.5, there is an a E K such that x(a) = 0 for every x 6 j r , and this is a contradiction. Hence there is a (Y,),et E .At with
E xLyL = eK.
i
tel
Remark. Many useful conditions on K are known which ensure that 5v is dense in .A (e.g. if q~\K has only a finite number of connected components). The Theorem of Vitushkin (1966) gives necessary and sufficient conditions for ~" to be dense in ,4. E x a m p l e 2.4.3.8
Let E be an mvolutive Banach algebra.
a) If we replace the multiplication in E by the multiplication x 9 y := --xy for all x , y E E , then we get an involutive Banach algebra, which we denote by E . b) If E zs unital, then - 1 is the unit of E . c) If x is an invertible element o.f E , then x is an invertible element of E and x -l is its inverse m E . d) ~ e E ~ ~ ( ~ ) = e) r(E) - -r(E).
-o~(~).
1~. Banach Algebras
s~4
f) If u is an algebra homomorphism defined on E , then - u
is an algebra
homomorphism defined on E .
g)
t! (x,, u,), ( ~ , u~) e k , t h e .
(~,, u,) * ( ~ , u,) = - ( x , , u,)(x,, u , ) . h) I f E is a Gelfand algebra then E is also a Gel]and algebra and o(E) = -,,(E) .
a) For x , y , z E E ,
(~ 9 ~) 9 z = - ( ~ 9 ~)~ = (~y)z = x 9 (~ 9 ~), (x 9 ~)" = ( - x u ) " = - u ' x " = u" * x ' .
b) For x E E , (-1),x--lx--~x,
x , (-1) = xl = x.
c) We have X * X - 1 ---- - - X X - 1 - - - - - - 1 1
x -1 , x = - x - I x
= -1.
d) By b) and c), for a E E , a r --aE(X) ~
r
--a ~ aE(X) r
--otl -- X is invertible in E r
a ( - 1 ) - x is invertible in E ~
e) For x E E , a n d
a r a$(x).
x ~ E E ~,
(~" 9 ~, -~') = (~'~, x'). f) Put V :-~ - - ~ .
For x, y E E ,
~(~ * u) = - ~ ( - ~ )
= ~(~u) = ( ~ ) ( ~ u )
= (~)(~u).
g) We have (x~, ~ ) 9 (x2, ~2) = (x~ 9 x2 - y~ 9 u2,x~ 9 ~2 + y~ * x2) = = - ( ~ , x 2 - ~u~, x~u~ + u ~ 2 ) = -(~,, ~,)(~, ~ ) .
h) E is obviously commutative. By d) and g), o ~ ( ( x , 0)) = - a ~ ( ( x , 0)) = - o ~ ( ~ )
The last assertion follows from f).
= o~(x).
m
~.4 Gelfand Algebras
365
2.4.4 Locally Compact Additive Groups In this subsection we assume IK = r One important application of the Gelfand transform is in the study of locally compact additive groups by means of the associated involutive Banach algebras L 1 (Proposition 2.3.2.29 a)). We do not pursue this theory in this book, as interesting as it is. For us, it serves only as an illuminating example. Definition 2.4.4.1 ( 4 ) Let T denote the subgroup {a E IEI I~ I = 1} of the multiplicative group C\{0} endowed with the induced topology. Given a locally compact additwe group T , let a(T) denote the group of continuous group homomorphisms T -+ T endowed with the topology of compact convergence (z. e. the topology of uniform convergence on the compact sets of T ). Let A be the Haar measure of T . Given p E [1, c~], define LP(T) := LP(A) and identify L'(T) with the closed involutwe subalgebra
{z.A I x e LI(T)} of JVlb(T) (Theorem 12.3.12.29 a)).
We have U
--1
m
--?I
for every u e a(T). If the topology of T is discrete then
t'(T) P r o p o s i t i o n 2.4.4.2 u E a(T), define
LX(T).
( 4 ) Let T be a locally compact additive group. Given
..
;
Then ~ E a(.helb(T)) for u E a(T) and the map a(T) lS c o n t i n u o u s .
~. a(.A4b(T)),
u:
~.
366
~. Banach Algebras
Take u E a ( T ) . Then
~ ( p , ~) = f ~d(~, , ~) =
f n(s + t)d(t, o ~)(s, t) = f n(s)n(t)d(t, O ~,)(s, t) =
=
for every /~, v E .Mb(T) (Theorem 2.2.2.10 f)), so that g E a(.Mb(T)). Now take u0 E a ( T ) . Let ~ be a finite subset of There is a compact set K of T such that
.]i,4b(T)
and take e > 0.
C
Ittl(T\g) < 5 for every ?t E iV'. Take u 6 a ( T ) with lu - uol < 3(1 + sup It'l(g)) ssE/V"
on K . Then
O, there is a O-neighbourhood U in T such that
.for every u E a(T) and s , t E T with s - t E U.
~.~ GelJand Algebras
367
By L e m m a 2.2.2.12,there is a 0-neighbourhood U in T such that
II~,- ~II,< for every t E U. Then
- ]'1~,-,- xldA -IIx,-, for every u E a(T) and s , t E T with s -
T h e o r e m 2.4.4.4
c
xlll
t E U.
m
( 4 ) ( S ) L~t T b~ ~ to~ny ~o~po~t ,dd~,,~ ~o~p
and let A be its Haar measure. Given u E a(T) , define ~" LI(T)
~(~,
x ~
/~xdA.
Then ~ E a(La(T)) for u E a(T) and the map a(T) .... ; a ( L I ( T ) ) ,
u ,=-~
is a homeomorphzsm. By Proposition 2.4.4.2, ~ E a(LI(T)) for every u E a(T) and the map
a(T)
" a(LI(T)) ,
u ~
is continuous. Given x E Ll(T) and t E T , define
xt " T
Step 1
"r
s:
" x(s - t).
The map a(T) --+ a ( L l ( T ) ) ,
u:
" ~ is bijective
Take x' E a ( L l ( T ) ) . There is an x E LI(T) with x'(x) ~ O. We may even assume that x is continuous with compact carrier, since the set of these functions is dense in L1 (A). Take y E L1 (A) and t E T . Then
(x, 9 ~,)(~) = f:~,(r)y(~
=
- ,-)da(O = f:~(,- - t)u(~ - ,-)d~,(,-) =
=
=
368
~. Banach Algebras
for every s E T (Theorem 2.2.2.10 e)), so that
xt . y = x * yt , 9 '(z,)~'(y) = ~'(~, 9 v) = ~'(~ * ~,) = ~'(x)~'(y,). In particular,
:,:'(x,)x'(x_,) = :,:'(~)~ # o,
x'(:,:,) r o. Hence, by complete induction,
X'(X,))"-t = x'(x.,)
9 '(~) for every n E 77,,. Thus f { ~ ) " ) k z'(g)
~'(~,) '~
is bounded, and
SO
go(X)
/ I1~ 77,
Define
u:T
;T,
t:
~'(~,)
~'(~)
Then u ( s + t) = ~'(~'§
x'(~)
= ~'(~') ~'(~') = u(s)~,(t) x'(~)~
for all s, t E T . Hence u is a group homomorphism. Take t E T . Then lim,__,tIIx. -
fl .
x, II = lim,_,,
- xtldA = 0.
Hence u(t) = x'(xt)
x'(~)
= lim x'(x.),, = limu(s) "-*'
~'(~)
"-"
and u is continuous at t. Hence u E a(T). Since LQ~
is isometric to the dual of LI(T), there is a z E L~
that
x'(y) = / y z d ) ~ whenever Y E Ll (A). Thus
such
,~.4 Gelfand Algebras
369
1
x'(x) / (/xt(s)z(s)dA(s)) y(t)dA(t) = 1
x'(x) / ( I x ( s - t)y(t)dA(t)) z(s)dA(s) = 1
x'(x) f (/x(t)y(s - t)dA(t)) z(s)dA(s) = = ~ 7or every
if
(~ 9 y)zd~ = ~
y e LI(T)
1
~'(~ 9 ~) = ~
1
~'(~)~'(~)
=
~'(~)
(Theorem 2.2.2.10 e)), so that ~---
Xs "
Hence the map
u: "
a(T) ---+a(L'(T)),
is surjective. It is easy to see that this map is injective, showing t h a t it is bijective. Step 2
The map
a(LI(T)) x T
~ffJ, (~, t) ~-+ u(t)
is continuous.
Take (uo, to) E a(T) x T and let ~ be a filter on a ( T ) x T such t h a t lim ~ = (u,t),~
uo,
lim t = to. (u,t),~
Let x 9 T --+ q~ be a continuous function with compact carrier such t h a t uo(x) 0. Then
=
(s + t)~(~)d~,) = ~(t) J
for every u E a ( T ) and t E T . Thus
.
.
.
.
.
.
370
I~. Banach Algebras
~(t)~(~) a(~)
luCt) - uo(to)l =
~(~,) ~o(~) ~(x) ~o(x)
u
uoCto)~o(~) ~o(~)
~(~,)- ~(~,o) ~(x)
~o(~)
for every t E T and u E a(T) with ~(x) # O. Now lim (u,O,~
~(~) ~(x)
~(~,o) ~,o(x)
=0
and
~(z,)
lim (.,t),~
- ~o(x,.o)
=0,
~(~)
by Proposition 2.4.4.3. Hence lim u(t) = Uo(to). (',,0,~ Step 3
The map G(T)
~, a(Ll(t)),
u:
; ~ is a homeomorphism
Take uo E G(T) and let ~ be an ultrafilter on G(T) such that lim ~ = ~o. u,~
Let K be a compact set of T . For u E G(T), take t . E K such that l u ( t . ) - u o ( t . ) l = sup
sEK
lu(s) - uo(s)l.
Define to = lim t . . u,~
By Step 2, lim u(t.) = uo(to) u,~
so that lim l u ( t . ) - u o ( t . ) l < 111~
< lim l u ( t . ) - u o ( t o ) l + lim luo(to) - u o ( t ~ ) l = 0 .,~ u,~
--
'
~,.4 Gelfand Algebras
371
lira sup lu(s) - u0(s)[ = 0 . u,~ s E K
Hence ~ converges to uo uniformly on K . Since K is arbitrary, ~ converges to u0. It follows that the map
a(L'(T))
~,a(T),
~:
;u
is continuous. Assume a(L'(T)) is non-compact and let @ be an ultrafilter on a(L'(T))
converging to the Alexandroff point of a ( L ' ( T ) ) . Then
liraf ~xdA = 0 ~,~ J for every x ~. L~(T) (Theorem 2.4.1.3 e)). It follows that the image of ~ with respect to the map
a(LI(T))
~ a(T),
~: ~ u
does not converge. Thus the map
a(T)
~ a(LX(T)),
u:
~ "~
is a homeomorphism.
1
C o r o l l a r y 2.4.4.5 ( 4 ) If T is a locally compact additive group, then a ( T ) is a locally compact abelian group. By Theorem 2.4.4.4 and Theorem 2.4.1.3 b), a ( T ) is locally compact. We need only to show that composition in a ( T ) is continuous. Let K be a compact set of T and take e > 0. Take u, v E a ( T ) such that g , Ill -- e T [ 0 f o r every ~
wo~~ 7 E - I - f2 ~ 2~-[ with ..... e,~ = u ( e ' a ) .
388
~,. Banach Algebras
Take n E IN. Then reu (e'~) k = reu ( e ~ ) > 0 for every k E INn and so (e.') =e.
.
Hence
u(e'~ for every p E 7/,,. By continuity,
u(e '~ = e,'ra for every /3 E IR. In particular, el-1~
-u(e'~ k
-u(e~")--u(1)-
/
1,
n'= --7 E71., Ot
and
for every /3 E 1R. Hence n---B
and the map 7z.---,a(l'),
n"
.'a
is surjective. By the above considerations and by Theorem 2.4.4.4, the maps }a(l'), }a(Ll(1')),
n.
~, n:
~n,
are bijective. They obviously are continuous. Moreover, for every sequence (kn)ne~ in ~ with lim Ikn[ = oo
n--too
•.4 Gelfand Algebras
389
the sequences (kn).er~, (k.).~r~ have no points of adherence and so the above maps are homeomorphism. Turning to the last assertion, let x 6. LI(T). Then ~(n) = "~(~t) =
for every n 6. ~ .
Example 2.4.5.11
1 Take n 6. 1N and let A be the Haar measure of T " . Define k " lF~
~T ,
(%)jerk."
~
% fi "'k$
3=1
k" L (IF") - - - - ~ , for k = ( k ~ ) j ~ . 6. ~ " .
x : =~ ;
dA
Then k 6. a ( T " ) , k 6. a(L~(IF")) Ior every k 6. 7Z"
and the maps
~" --~(T"),
~'~
), a(LXOV')),
k, ~
k:
~, k ,
~k
are homeomorphssms.
The assertions follow from Example 2.4.5.10 and Proposition 2.4.4.9.
II
390
2. Banach Algebras
2.4.6 The Fourier Transform
In this subsection n is a fixed natural number, A always denotes the Haar measure on I t n , normalized by A([0, 1]n) = (21r)-~J, and ('1") denotes the inner product n
('1")" R" • rt"
, n:t,
(,,t):
; (sit):= ~ 8,tk. k=l
Moreover, we take 1/: = C .
Proposition 2.4.8.1
(4)
Take n = 1. D e f i n e
S" it
~s" L l (1~.) for every s 6. I t .
Then ~ 6- a ( i t ) ,
) T,
t:
~ IC,
x."
~ e Ist ,
; f ~xdA
~ 6- a ( L t ( i t ) ) .for every s 6- IR and the
maps
R
~o'(l:t),
s"
I t ----, a ( L l ( i t ) ) ,
,~',
s .~ ~ s ,
are homeomorphisms.
It is obvious that ~ 6- a ( i t ) for every s 6- I t and that the map Ft
~ a(l~),
s:
; "~
is injective and continuous. Take u 6- a ( i t ) . Then there is an c > 0 such that re u([-e, e]) C I t + . Take s' 6- [ - ~F, F] ~ with
'.(E) = ~'". Define
~.~ Gelfand Algebras
8 :m_
391
1 t --8 . E
Then for p E I N and k E I N v, reu( ) k = r e U ( p ) _ > O , so that p ---
Hence ~/(Ep)
= ~(~)
-- U ( ~ ) m "-- "~ ~:)mS(p
for every (m, p) E 7Z • IN. Thus, by continuity, U'-8,
so that the map
is bijective. Let (s~)~e~ be a sequence in lit with
oo.
lim I s ~ l -
k--too
Then (sk)ke~ has no points of adherence. It follows that the map IR
~a(n~),
s=
;
is a homeomorphism. By Theorem 2.4.4.4, the map I:t ~
a ( L ~( lFt) )
,
f~ ~ 8
8:
II
is also a homeomorphism.
Theo,em 2.4.6.2 ( 4 )
G,v~,~ ~ ~ ,
~" IR n
s " L l ( l R ") Given x E L1 ()~), define
~T ,
>r ,
d~,~
t:
x:
~ e i(slt) ,
;
d~.
39~
P,. Banach Algebras
R"
s:
f
~~ 7 ( x ) = j e - ' ( ' l ~
.
Then "gE a(lR"), ~ E a(Lt(IR")) .for every s E IRn and the maps
are h o m e o m o ~ h i s m s .
We identify
a(Lt(R")) with
IR" via the last h o m e o m o r -
phism. Then,/or z E LI(IR"), ~ is the Gel/and transform of x . The integral
f e -'{~ x(t)dA(t) is called the Fonrier integral oJ x (Fourier, 1 8 ~ ) and the map
L~(Ft") .... ~ co(rt"),
x~
is called the Fourier franc/otto. It is an in#dive, involutive, continuous algebra homomorpMsm of norm 1 and its image is dense in C0(1R").
The first assertion follows from Proposition 2.4.6.1 and Proposition 2.4.4.9. By Proposition 2.4.4.8, the Fourier transform is an injective, involutive, continuous algebra homomorphism of norm at most 1 and its image is dense in Co(l~."). Now
II~ll >_ I~(0)l--If=da
= II=ll,
for every positive real function x E Lt()0, and so the norm of the Fourier transform is 1. II P r o p o s i t i o n 2.4.6.3
( 4 ) Take x E Lt(lR"), a E IR", and a > O. Define y " ]R"
~r ,
t :~ ~ z ( t + a) ,
z.rt"
-~'r
t'~ ; z ( ~ t ) .
Then ~'(s) =
for every s E ]Rn.
e'*(s),
~.4 Gel~and Algebras
393
By Theorem 2.4.6.2,
~(s)
+ a)dX(t) =
fe
'dA(t> =/e-'x(t)d~(t)) y(s)d)~(s) =
fx aa,
= f (fe-iq~,
t
:
;
e-89
then x E S(IR n) and ~ = x. It is clear that x E S(IR"). In order to prove the other assertion we may assume that n = 1. Then
dx ds (S) = -sx(s) , O0
O0
d'~
d
1
e - ' t e - - f dt =
d~ (~) = d~ v ~
9
2~
e - ' t t e - T dt = ~00
--00
= i2s fr e t~ i e_,t dt = -s~(s) v / ~ J-oo
)
for every s E ~ , by partial integration. Hence x and ~ are proportional. Since oo
~ ( 0 ) = v~~
J e--~dt = 1 = x(O), --00
it follows that
~--x.
m
394
2. Banach Algebras
Proposition 2.4.6.'/
(4)
Take x E LI(A) which is continuously differentiable in the first variable such that the partial derivative giTt~ belongs to L 1(A). Then Ox
for every s E I t " .
By the method of partial integration and Fubini's Theorem,
9
II
Theorem 2.4.6.8 ( 4 ) (L. Schwartz) a) ~ E 8(R") for every x E S ( R " ) . b) T h e m a p 8(rt")
>8(rt"),
x.
>
is linear and bijectwe and A
x(s) = ~ ( - s )
.for every x E ,.q(Ft") and s E IR".
c) If P is a polynomial in n variables with coe.Oicients in K and D is the associated partial differential operator, then A
Dx(s) = P(is)9(s) for every x E S ( R " ) and s E R " .
a) and c) follow from Proposition 2.4.6.7 using complete induction. b) Take a > 0 and x, y E S(]R"). Define
Then
xo.~ ~
)r
t: ;x(~t),
~o.R"
)c,
t: ~u(~t).
1~.4 Gel]and Algebras
/
395
~yd~ = f xyd)~
(Proposition 2.4.6.4), so that
= / ~ ( s ) y ( l s ) d A ( s ) = /~y~dA by Proposition 2.4.6.3. By Lebesgue's dominated Convergence Theorem we see that for c~ ~ oo,
x(o) f ~dA = u(o) f ~dA . If we replace y by the function introduced in Proposition 2.4.6.6, then we obtain
x(o) = f ~d~. Define
z.lR"
~q~,
t:
~x(s+t).
By Proposition 2.4.6.3,
~(-~)
X(8) for every s E IRn . It follows immediately that the map 8 ( ~ ~) - - ,
8(~"),
x .~ ~
is bijective.
I
C o r o l l a r y 2.4.6.9 ( 4 ) Let .~ denote the involutive subalgebra 8(]R '~) of C0(IR~) uath the induced structure. 8(]R '~) is an involutive subalgebra of
L~(IR") and the map 8(IR n) -~ .~ defined by the Fourier transform is an isomorphism of involutive algebras. Take x, y E 3(IR"). Then z := z 9 y = zy ~ 8(IR") (Proposition 2.4.1.2, Theorem 2.4.6.2, Theorem 2.4.6.8 a)) and (z 9 ~)(~) = z ,~"~(-~) =
~(-~)
for every s E IRn (Theorem 2.4.6.8 b)). Thus x , y E S(lR") by Theorem 2.4.6.8 a). Hence ~q(]Rn) is a subalgebra of Ll(lR ") and it is obvious that it is an involutive subalgebra of LI(IR"). By Theorem 2.4.6.8 a) and Corollary 2.4.4.8, the map S(IR n) -+ ~- defined by the Fourier transform is an isomorphism of involutive algebras. II
396
~. Banach Algebras
Exercises
E 2.4.1 Let E be the unital Gelfand algebra of E 2.2.6. Given t E [a, b], define "~" E - - - - , f , ,
x:
~ x(t) .
Prove: a) t'E a(E) for every t E [a, b] and the map
f " [a, b] --4 a ( E ) ,
t:
~t
is a homeomorphism. b) ~ o f = x
for every x E E .
c) The Gelfand transform is injective but not surjective. d)
{ ~ l x E E} is dense in C(a(E)).
E 2.4.2 Let T be a compact space, S a closed set of T , S ~ T, S ~ 0, and put
.F" := {x E
C(T) I =
= o on 5'}
Show: a) ~" is an involutive closed ideal of C(T). b) The factorization of the map C(T)
; C(S),
=,
;=IS
through C(T)/.7c is an isomorphism of involutive unital Banach algebras. c) S is homeomorphic to a(C(T)/.T).
E 2.4.3 Let E, F be unital Gelfand algebras, such that the Gelfand transform of E is injective but its image is not closed and the Gelfand transform of F is injective but its image is not dense in C(a(F)) (Example 2.4.3.7 for K = ]K# ). Show that the Gelfand transform of E x F is injective and its image is neither closed nor dense.
~.4 Gel]and Algebras
397
E 2.4.4 Let E be a unital Gelfand algebra. If E contains a finite subset A such that E is the closed subalgebra of E generated by A, then a ( E ) homeomorphic to a subset of C n where n = Card A
is
E 2.4.5 Assume ]K - q ~ . Let T be a locally compact additive group and )~ its Haar measure. Take p e [1, cx)[, x E I2()~), and define xt " T --'~ q~ ,
s:
" x(s-
t)
for t E T . Show that the map T
~. LP()~),
t:
~ Xt
is continuous.
E 2.4.6 Assume ]K = r
Take ( x , ) n e z E s y.77.
"C ,
n:
and define > nx.
.
Show that if y E t I ( ~ ) , then ~ is continuously differentiable and its derivative is
-i~'.
E 2.4.7 Assume IK = q~. Let x be a continuously differentiable function on T . Show that ~(n)-
-n~(n-
1)
for every n E ~ .
E 2.4.8 Let E be the underlying real algebra of r x ~'E:
"E,
x:
>x,
y"E
>E,
x:
~
Show that
are algebra homomorphisms such that Ker x' - Ker y' -- {0},
x' ~- y' .
398
~. Banach Algebras
E 2.4.9 Let E be the involutwe, commutative, complex, unital Banach algebra obtained by endowing the complex Banach algebra tl(7"Z,) (Theorem 2.2.2.7 a),c)) with the involution
Take n E 1~I tO {0} and put
F := {x E E l m E ~ , m < n ==} x,~ = O} and 9F
~lI3,
x:
~ Ex'na'n rtl ~ ' n
,for every a E C # 9 Show:
a) F is an involutive Banach subalgebra o,t E . It is a unital subalgebra iff n--O. b)
I] n = 0 (it n # O) then a E a(F) for every a E r ,~ e r and th~ map
r
(r
, a(F),
a:
(for every
~ "~
is a homomorphism.
c) F is semz-simple but it is not symmetric. d)
If we restrict the scalars of F to R then Re F is a strongly symmetric involutive unital Banach subalgebra of F .
e) If n = 0 and if we take K = r
in Example 2.4.3. 7, then ~ E A for
every x E F and the map F
;.4,
x:
t.~
is an injective continuous involutive algebra homomorphism.
3. Compact Operators
There are two main reasons for developing a mathematical theory of compact .~perators: they have strong properties which lead to numerous theorems and _.heyhave proven to be useful in many branches of mathematics. The first section _~fthis chapter presents the theory and the second provides an example of its Jsefulness in the study of linear differential equations.
3.1 T h e G e n e r a l
Theory
3.1.1 General Results Definition 3.1.1.1 ( 0 ) (F. Riesz, 1918)Let E , F be normed spaces. A linear map u : E --~ F is called compact if u(A) is relatively compact whenever A is bounded (or, equivalently, if u(E #) is relatwely compact}. We write /C(E, F) for the set of compact linear maps of E into F and define ~C(E) := ~C(E, E) .
Proposition 3.1.1.2 ( 0 ) Let A , B be relatively compact sets of the normed space E . Take ~, [3 E IK. Then a A + ~B is relatively compact. Let (zn)ne~ be a sequence in aA +/~B. Given n E IN, there are x, E A, y, E B such that z, = a x , + ~y~.
By Lemma 1.1.2.11 b =~ c, there is a strictly increasing sequence (kn)ne~ in IN such that (xk,)ne~ and (Y~,),e~ converge in E. It follows that (Zk,)~Er~ converges. By Lemma 1.1.2.11 c =~ b, aA +/3B is relatively compact. 1 Theorem 3.1.1.3 ( 0 ) Let E , F be normed spaces. Then IC(E,F) is a vector subspace of s F). If F is complete, then IC(E,F) is closed in s
~00
3. Compact Operators
Take u E K:(E, F ) . Then u(E #) is relatively compact and therefore bounded. Hence u is continuous and K:(E, F) C s
f).
Take u, v E K:(E, F) and a, ~ E ~(. Let A be a bounded set of E . Then u(A), v(A) are relatively compact sets of F . By Proposition 3.1.1.2,
au(A) +/3v(A) is a relatively compact set of F . Since
(an + ~v)(A) C an(A) + ~v(A) , (an + ~v)(A) is relatively compact. Hence au +/~v E K:(E, F) and K:(E, F) is a vector subspace of s F). Suppose F is complete and u E K.(E,F). Let ~ > 0. There is a v E K:(E, F) such that I1~- ~11 < ~. Since v(E #) is relatively compact, it is precompact. Therefore there is a finite subset B of F with
~(E#) c U u , ( ~ ) . u
Take x E E # . There is a y E B with vx E U~(y). Hence
Ilux- yll -< Ilux - vxll +
Ilvx- yll
-
< IJ~ - ,11 II~ll + ~ < ~ + ~ = ~, i.e. ux E Ue(y). Hence
u(E #) C U Ue(y). yEB
Thus u(E #) is precompact. By Lemma 1.1.2.12, u(E #) is relatively compact. Thus u E K:(E, F) and so ]C(E, F) is a closed set of s F). I
( 0 ) If E, f are normed spaces, then s vector subspace of IC(E, F). If F is complete then
P r o p o s i t i o n 3.1.1.4
s
F) C K:(E, F ) .
F) ts a
3.1 The General Theory
4o~
Take u E s and let A be a bounded set of E . Then u(A) is a bounded set of the finite-dimensional space Im u and therefore relatively compact (Minkowski's Theorem). Hence u is compact and
s
F) C IC(E, F) .
It is easy to see that s F) is a vector subspace of K:(E, F ) . The last assertion follows from the fact that K:(E, F) is closed (Theorem 3.1.1.3). II /
C o r o l l a r y 3.1.1.5
~, 3 ) If E is a normed space and F is a Banach space,
then s
C IC(E,F).
By Theorem 1.6.1.3 a),
s (E, F) C s
F)
and by Proposition 3.1.1.4
s P r o p o s i t i o n 3.1.1.6
F) C IC(E, F) .
II
If E is a normed space and T is a compact space, then s
= IC(E,C(T)).
Take u e IC(E,C(T)) and ~ > 0. By Ascoli's Theorem (Theorem 1.1.2.16), u(E #) is equicontinuous. For t E T , let Ut be an open neighbourhood of t such that
for y e u(E #) and s e Ut. finite subset of T such that
(Ut)teT is an open covering of T . Let A b e a
T=UUt. tfi A
Let (Yt)tEA be a partition of the unit on T subordinate to the covering (Ut)tEA, that is, (Yt)tEA is a family in C(T) with
~-~yt= 1 tEA
~02
3. Compact Operators
such that each yt (t E A) is positive and vanishes on T \ U t . Define
v'E
~C(T),
x:
~E(ux)(t)y,. tEA
Then v E s
and
,,ux,,s, _ ,vx,,s,, - ,lz,ux,,,,y,,s,
Z,ux,,t,y,,s,]
< ~ I(,,x)(~) -(ux)(OI ~,(s) < c tEA
for x E E # and s E T . Hence
for x E E # and
Ilu - vii _< ~. Thus
IC(E,C(T)) c s
.
The reverse inclusion follows from Proposition 3.1.1.4.
I
Let E, F be normed spaces, 15 an upward directed set of fimte-dimensional vector subspaces of F , and for each G E t5 let PG be a projectwn of F onto G such that P r o p o s i t i o n 3.1.1.7
inf Ily -
9E~
PGYll = 0
for every y E F and Ily - p-yll
0. Put K := u(E#) and re: K
~ R,
y-
"~lly-payll
for G E ~ . By hypothesis, (re)Gee is a downward directed family of continuous functions with infimum O. Since K is compact, there is a G E ~ such that [lY - PaYII < whenever y E K (Theorem of Dini). Hence Ilu - pG o ull -- s u p Ilux - pGuxll -" sup Ilu - pGyII < e. yEK
zEE#
Since PG o u E s
F) and e is arbitrary,
uEs
F),
/C(E, F) C s
F).
By Proposition 3.1.1.4, IC(E, F) = s
F)
whenever F is complete.
I
Remark. It is not always true that IC(E, F) = s
F)
(see Per Enflo, A counterexample to the approximation problem in Banach spaces, Acta Math. 130 (1973) 309-317). Proposition 3.1.1.8 ( 3 ) Let E, F be normed spaces. Take ~ E IR+. Let be an upward directed set of finite-dimensional vector subspaces of F such that (J G is dense in F , and for each G E ~ let PG be a projection of F GEq)
onto G with
I1~ - pGxll
0. Put
K = u(E#). Then (dG)oeO is a downward directed family of continuous real functions on F (Proposition 1.1.4.5) with infimum 0. Since K is compact, there is a G E such that 6
dG(U)
0. There is a finite subset C of B such that
u(E#) C U U~(y) yEC
(Lemma 1.1.2.10 a :=~ c). Take m E ~I with t
I~'(~) - ~.(~)1 < whenever y E C and n E ~ , n There is a y E C such that
r
_> m. Take x E E # and n E ] N , n
_> m.
3(~ + 1)" Hence
I(*, C z " -
r
= ](., r
~'))1 = I(~*, ~ " -
~')l
a - e , kl the existence of j,~ is clear. Take n E IN, n > 1, and assume the sequence was constructed up to n - 1. Then
kn-I
km=l
_ - _ _ 1 Card (A N l ~ k . _ ~ ) kn-I
k-~_lCard(bl(AN]km_lkm_l+jm])) 0 with U3r(t) C T . Take further x e ~'# and s E U~r(t). By Gauss's Mean Value Theorem and H51der's inequality,
3. Compact Operators
4ca
I*(~)1
--
t xdA --- a(u,(o))
f
~(u,(o))
U,(s)
1
f IxldA< U,(s)
~(u,(s)),*
} E/C(/n'(T)). Define
r e)
} K:(ff(T)),
x:
~ 5.
r ,s linear and the map
c0(T)
;Imr
x~
;r
is an isometry.
f)
The map K:(ff(T))
~K:(ff(T)),
is a projection of K:(ff(T)) onto Im r
u~
u
of norm 1.
g) Im r is a complemented subspace o.f IC(~'(T)). a) is easy to see. b) Take 6 > 0. Since u(P'(T) #) is relatively compact, there is a finite subset S of T such that IleT\SltXllp < e
4~8
3. Compact Operators
whenever x E if(T)# (Proposition 1.1.2.13 a =~ b). Thus
lu(t)l = I(ue.)(t)l 0. There is a finite subset S of T such that Izl < 6 on T \ S . Thus
IleT\S~UlI, = Iler\szull, < ellull, < e for all y E if(T) # . By Proposition 1.1.2.13 b =~ a, ~(ff(T)) # is relatively compact, so E is compact. If E is compact, then, by c) and b),
x = x E cO(T). e) follows from d) and a).
f) By a), b), and d), the m a p is a well-defined operator of norm at most I. By c), it is a projection of K~(ff(T)) onto Im ~ and so it has norm 1. g) follows from f) and Murray's Theorem.
Corollary 3.1.2.12 then
IC(e(S),if(T))
II
( 0 ) If S, T are mfimte sets and/f p E [1, oo[ U{O}, is not a complemented subspace of s e(T)) .
First consider S = T . By Proposition 1.2.5.16 b) and Proposition 3.1.2.11 d), f), there is an operator u : fr~
,~ s
such that the m a p
F~
~Imu,
x:
~ ux
is an isometry,
u(co(T)) C ~(ff(T)), and there is a projection of/C(lP(T)) on u(co(T)). Since c0(T) is not a complemented subspace of g~176 (Corollary 1.2.5.14), it follows from Corollary 1.2.5.10 that /C(ff(T)) is not a complemented subspace of f..(~'(T)). Now let S and T be arbitrary. We may assume that
4e9
3.1 The General Theory
R:=SnT
is infinite. Given x E if(R), define
_
x'T
~ IR
Given u E s
t:
'
"
f x(t) ~
0
if if
tER tET\R.
define ~ if(T),
~'ff(S)
x:
~ u(xlR).
Given v e K:(ff(S), if(T)), define ~o" s
;/:(if(S), if(T)),
v" f f ( S )
;if(T),
x,
u~
~,
; envCenx )
and r
K:(ff(S), if(T))
.~ K:(ff(S), if(T)),
v:
" v.
Then ~(K:Cff(R))) c IC(F(S), i f ( T ) ) , the map s
~
Im~o,
u':
;~ou
is an isometry, and r is a projection of K:(eP(S), ~(T)) onto ~(/C(~(R))). By the above considerations and by Corollary 1.2.5.10, K:(gP(S),/~(T)) is not a complemented subspace of s ~'(T)). II E x a m p l e 3.1.2.13 and define
Let (aL),et be a summable family in the Banach space E ,
u'~(I)
L ~ E,
x:
" ExCe)aL tel
(Corollary 1.1.6.8). Then u is a compact operator.
Take e > 0. There is a J E ~S(I) such that
I! II
~30
3. Compact Operators
for every K E ~ l ( I \ J ) (Proposition 1.1.6.6). Thus
x(0a,
< T
whenever K E ~ l ( l \ J ) and x E ~ ( I ) # (Proposition 1.1.1.5). Hence
tEl\J
for every x E s176176 If we now put v't~176
;E,
x:
;~x(Oa,, ~EJ
then v E f../(e~176 E) and, by the above considerations, I1~' - "11 < e.
Hence
uEs
F) C IC(E,F) I
(Proposition 3.1.1.4).
Take u E s
Proposition 3.1.2.14
goo) and let ~ be a free ultrafilter on
IN. Then Card{n E IN I I lim(uen)k] > 1} - O, there is an A E ~ and an no E ]N such that
no (n ~_ IN).
Since u(c#o) is relatively compact, there is a finite subset ~" of s
such
that
-(4) xUE ~ Define f " IN
; IK '
n:
; lim(uen)~ , k,~
{7 := {x E Y ll lim~,~x kl < _~_}.2~ Then there is an A E ~ such that 2e
< T for every x E {7 and k E A. By Proposition 3.1.2.14, there is an no such that
If(n)l
C
no (n E ]No). Take n E IN, n >_ no. There is an x E ~" such that
4se
3. Compact Operators
ue. E
U~|
Thus E
Ixkl < Ixk - (~-)~1 + I(ue.)kl < ~ + I(ue~)kl for every k E IN and so
I~,~ I _< g6 + f ( n ) < ~6 + ~6 = --~-, l-imzk 26 xE~7. It follows that r
2r
for every k E A.
E x a m p l e 3.1.2.16
i
Take p E {0} U [1, oo]. Let q be the quotient map
s
,
s
and let u., ut be the rzght and left shift operators of i f , respect:vely. Then Ilqu, II = Ilqudl = 1,
( q u , ) -~ = q,,t
and this stall holds when @ is replaced by c. That (qUr) -l = qut, follows immediately from
utur-" 1,
Case1
UrUt- 1 E IC(ff).
pE{0}U[1,~[
Take v E ~(ff) and r > 0. Since v((ff) #) is relatively compact, there is a finite subset ~" of ff such that
zE.~"
There is an no E IN such that
3.1 The General Theory
Ix.I
- I1(~, - ,)~.11 = I1~.+~ - w . I I _>
>
I1 -
( ' ~ . ) . + , 1 _> 1 - I ( w . ) . + , l
> 1 - 6,
Ilut- vii _> II(u~ - v)e~ll _> IieT,-~ - venll >_ > I1 - ( v e . ) . - l l > 1 - I C w . ) . - , I
> 1 - 6.
Since v is arbitrary, it follows t h a t
Ilqu, ll _> 1 - 6,
Ilqu, II >_ 1 - c .
Since 6 is also arbitrary, we deduce t h a t
Ilqu~ll- Ilqutll- I. Case 2
/~oo and c
Take v E K:(s ~176(resp. v E K:(c) ). Let ~ be a free ultrafilter on IN and take 6 > 0. By Proposition 3.1.2.15, there is an A E ~ and an no E IN such that I(ve,)kl < e whenever k E A and n E IN, n ___ no. Take k E A with k > no. T h e n II~, - ~II >_ II(~ - ~)~-~II = II~ - v~k-,ll >_
>_ 11 -
('~-~)~I > 1 - I ( ' ~ - , ) ~ I
> 1 - E,
434
3. Compact Operators
Ilue - vii _> II(ue - v)ek+lll -- lie,, - vek+tll _>
>
I1 -
(vel,+,)kl
_> 1 - ] ( v e , + l ) k ] > 1 -- e.
Since v is arbitrary, it follows that
Ilqu, II ~
1
-
E,
Ilquell >_ 1 - E.
Since e is also arbitrary, we deduce that
Ilqu,-II- Ilquell- 1.
I
Let p e {0}uIa,oo]. Let (Et)tET, (Ft)tE T be]amilies of Banach spaces, and E (resp. F ) the Banach space defined in Proposition 1.1.2.7 b) for p and (Et)teT (resp. (Ft)teT). I]
Proposition 3.1.2.17
(~,),~T e I I It(E,, F,) tET
such that
(Ilu, II),~T e co(T), then the map u'E
~F,
x~(utxt)t~v
is a compact operator. It is easy to see that u is an operator. Let (x(n))ne~ be a sequence in E # . Since {t E T I ut # 0} is countable, the diagonal procedure provides a subsequence (x(t:*))ncr~ of (x(n))ne~ such that (utx~))ne~ converges whenever t E T (Proposition 3.1.1.19 a =~ b). Given t E T , put Yt
:= lim n-coo
utxl k')
Then y := (Yt)teT e F and (llY, ll),eT ~ co(T). Take E > 0 and p E [1, oo[. There is a finite subset K of T such that sup Ilu~ll < ~, tEF\K
~
Ily~ll" < ~.
tET\K
Furthermore, there is an no E IN such that for every n E IN with n > no,
3. Compact Operators
436
a) Take y E e l ( T ) . Then
I(x 9 u)(t)l < Ilxll=llyll, for every t E T . Hence x 9 y E l~162 and the map is continuous. Take t E T . Then
(x 9 et)(s) = ~ f(sr-', r)x(sr-')et(r) = f ( s t - ' , t ) x ( s t - ' ) rET
for every s E T . Take to E T such that
Ix(to)l = I1=11=. We construct a sequence (t,)ner~ in T inductively such that 1
I1= * ~,., - x 9 ~,. II >_ ~ Ilxllo,, for all m, n E IN, rn < n . Take n E IN and assume the sequence to be constructed up to n. Then for every t E T and m E INn
IIx *
e,,~ -
x 9
e, ll= >_ I ( =
*
e..)(totm)
- (x , et)(tot.,)l
>
> If(to, tm)x(to) - ]'(tot,.t -l, t)x(totmt-1)l >
_> II=lloo
-Ix(tot.,t-~)l
9
Since there is an S E. ~Js(T) such that 1
I~(t0tmt-')l _< ~ Ilxll~ for every t E T \ S and rn E INn we can take t,+l arbitrarily in T \ S . By Proposition 3.1.1.19 a =~ b, the map is not compact. b) Assume the map is compact. By Proposition 3.1.1.11 (and Proposition 1.1.2.6 b),e)), the map e'(T) ---~ e~176 is compact which contradicts a).
y:
~x,y 1
3.1 The General Theory
437
3.1.3 F r e d h o l m O p e r a t o r s Definition 3.1.3.1 ( 0 ) Let E, F be Banach spaces. The operator u" E --+ F is called a $~edholm operator if Ker u and Coker u are finite-dimensional. The number Ind u := Dim Ker u - Dim Coker u is called the indez ol u . We write by .T(E, F) .for the set of Fredholm operators E --r F and define .~(E) := ~ ( E , E ) .
The right and left shift operators on @ (p E [1, c~] U {0} ) are examples of Fredholm operators (Example 1.2.2.9). The index was introduced by F. Noether (1921) in the case of singular integral equations. For selfadjoint Fredholm operators on a Hilbert space it was defined by Carleman (1923) under some special suppositions and by J. yon Neumann (1929) in the general case. P r o p o s i t i o n 3.1.3.2 ( 0 ) Let E, F be Banach spaces and u E Y:(E, F) . Then Im u is a complemented subspace (and therefore a closed subspace) of F and there is a projection of E onto Ker u and a projection of F onto Im u. The existence of a projection of E onto Ker u follows from Corollary 1.3.3.2 b). By Corollary 1.4.2.7, Im u is a closed subspace and a complemented subspace of F . By Murray's Theorem (Theorem 1.2.5.8 a =~ b), there is a projection of F onto I m u . I
Proposition 3.1.3.3 ( 0 )
Let E, F, G, H be Banach spaces and take
u e s F), v E s G), w e s H ) . If ger (w o v) and G / I m (v o u) are finite-dimensional then v E ~(F, G). The assertion follows from Ker v C Ker(w o v), P r o p o s i t i o n 3.1.3.4 ~r(E, F) . Then
( 0 )
Imv :3 Im(v o u).
Let E , F
I
be Banach spaces and take u E
Dim Ker u' = Dim Coker u,
438
3. Compact Operators
Dim Coker u' = Dim Ker u, u' E ~'(F', E ' ) ,
Ind u' = - I n d u ,
Im u = ~
u').
If Ind u = O, then
Dim Ker u = Dim Ker u' = Dim Coker u = Dim Coker u' < co. Now Imu = ~ follows from Proposition 1.3.5.8 and Proposition 3.1.3.2. We have Dim Coker u = Dim(Coker u)', Dim Ker u = Dim(Ker u)' (Corollary 1.2.4.10), Dim(Coker u)' = Dim Ker u' (Corollary 1.3.5.10), and Dim Coker u' = Dim(Ker u)' (Proposition 1.4.2.9 c)) which proves the assertion. C o r o l l a r y 3.1.3.5
II
( 0 ) Let E, F be Banach spaces and take u E s
such that Im u i8 closed or E is reflexive. Then u is Fredholm iff u' i8 Fredholm.
Assume that u' is Fredholm. By Proposition 3.1.3.4, u" E .F(E", F"). Now jF O U = U" ojE
(Proposition 1.3.6.16), so that -l
Ker u C jE(Ker u") and Ker u is finite--dimensional. Moreover, if E is reflexive then -1
Im u = j f ( I m u") so that Im u is closed. By Corollary 1.3.5.10, ( F / I m u)' is isomorphic to Ker u', which is finite-dimensional. Hence F / I m u is finite-dimensional (Corollary 1.3.6.4) and u is thus Fredholm. The converse was proved in Proposition 3.1.3.4. m
3.1 The General 'l heory
I
Ssa
%
L e m m a 3.1.3.6 ~, 0 ) Let E , F , G be vector spaces. Take u" E--+ F linear and in3ectwe and v " F --~ G linear and surjective such that I m u = Kerr.
Then F is finite-dimensional iff E and G are finite-dimensional and in this case
Dim F = Dim E + Dim G. Let (x,)Let be an algebraic basis for E . There is a family (Y~),eJ in F such that (Y,),sJ together with (ux,)~et forms an algebraic basis for F . We show that (vy,),ej is an algebraic basis for G. Let (a,),ej be a family in IK such that {,eJl~,~0} is finite and ~
oL~vy~ = O.
~EJ
Then
so that ~'-~ c~y~ E Kerv = Im u. eEJ
Hence aL = 0 for all ~ E J , i.e. (vy,)~ej is linearly independent. Take z E G. There is a y E F such that vy = z . There are families (flL),et, (%)~eJ in IK such that {ee I1~,~0},
{Le J l % r
are finite and
~EI
LEJ
Thus z -- vy -- 2..., %vy'"
4,~0
3. Compact Operators
Hence (vy~)Lej is an algebraic basis for G. Thus Dim E = Card I ,
Dim G = Card J ,
D i m F = Card I + Card J,
proving the assertion.
1
P r o p o s i t i o n 3.1.3.7 ( 0 ) (Atkinson, 1951) Let E , F , G be Banach spaces and take u E Jr(E, F ) , v E Jr(F, G). Then v o u E Jr(E, G) and Ind(v o u) = Ind u + Ind v. First consider the maps Keru ~
(Imu) n (Kerv)
Ker(v o u)---~(Im u) O (Kerv),
~ Kerv
~(Kerv)/((Imu)n(Kerv)),
where ~ is the map defined by u and the others are inclusion maps or quotient maps. By Lemma 3.1.3.6, all vector spaces appearing above are finitedimenisonal and Dim Ker(v
o
u) = Dim Ker u + Dim((Im u) N (Kerv)),
Dim Ker v = Dim((Im u) n (Ker v)) + Dim(Ker v)/((Im u) O (Ker v)). Now consider the maps Kerv
Im v
; F V~Imv,
~G
~ G / I m v,
where ~ is the map defined by v and the others are inclusion maps or quotient maps. By factorization, we get the maps (Ker v)/((Im u) n (Ker v))
Im v / I m (v o u) ~
~ F/Im u
G / I m (v o u)
~ Im v / I m ( v o u),
~ G/Im v .
3.1 The General Theory
~(41
By L e m m a 3.1.3.6, all vector spaces appearing above are finite-dimensional and Dim F / I m u = D i m ( K e r v ) / ( ( I m u) N (Kerv)) + D i m I m v / I m ( v o u ) ,
Dim G / I m ( v o u) = Dim Im v / I m ( v o u) + Dim G / I m v . Hence v o u is Fredholm and Ind(v o u) = Dim Ker(v o u) - Dim G / I m ( v o u) =
= D i m K e r u + Dim((Im u) N (Ker v)) - D i m l m v / I m (v o u) - D i m G / I m v
=
= Dim Ker u + Dim K e r v - Dim F / I m u - Dim G / I m v = = Ind u + Ind v. Theorem
3.1.3.8 ( 0 ) (F. Riesz, 1918) If
I E is a B a n a c h space, then l - u
is a Fredholm operator w h e n e v e r u E I C ( E ) .
Define v:=l-u,
F:=Kerv,
G:=Imv.
Then x -- ~X --'--VX -- O,
x = u x e u ( F #) C u ( F # )
for every x E F # , such t h a t F # C u(F#).
Hence F # is compact and F is finite-dimensional (Theorem 1.1.4.4). By Corollary 1.3.3.2 b), there is a projection w of E onto F . Define H := K e r w ,
vo'H
~G,
x:
;vx.
Then H is a Banach space and vo is linear and continuous.
3. Compact Operators
Step 1
Vo is injective
Take x E H with v o x - O.
Then vx---O,
xEF,
x-wx-O.
Hence vo is injective. Step 2
vo is surjective
Take x E G . There is a y E E with x -- v y .
Put z :=y-wy.
Then w z = wU - w ( w u ) = O,
zEH.
Since w y E F , VoZ = v z = v y -
vwy = vy = x.
Hence vo is surjective. Step 3
vo 1 is continuous
Let (Yn)ne~ be a sequence in G converging to 0. For n E lN, define x n := V g l y , , .
3.1 The General Theory
$4s
Assume that (Xn)ne~q does not converge to 0. Replacing (Yn)ne~q with a subsequence if necessary, we may assume that there is an e > 0 such that
for every n E IN. Since
~x,~
is a bounded sequence, we may assume nE~
f
i
X
\
(by replacing (Yn)n~. withasubsequenceifnecessary), that xx//(u(~'~xn))n~_~ converges (Proposition 3.1.1.19 a =~ b). Put x
:= lira u
xn
9
Then
IIxn
jl Irv(xn)II 1
1
=
I1~.!1
IIv0(x~)ll = ~ 1 1 ~ . 1 1 ][Xn[[
"
1
whenever m, n 6_ IN are distinct. For each n 6_ IN, choose x, 6_ X, such that
II~.ll _ l and P
E ( t _ a)_n( u _ a l ) . _ l v + E ( t _ a) n ( - 1 ) " wn+, ~n+l
n=l
n=O
is the Laurent series of the resolvent of u at a . It converges in tee
o < lt-~l
< ~
o}
9
Thus the resolvent of u has a pole of order p at a and v is its residue.
Since Uo is not invertible, p # 0 by the Principle of Inverse Operators. By the last part of Theorem 3.1.3.17, p
{x}
B--I
n'O
E t-nuon-lv-Etnwn+l is the Laurent series of the resolvent of uo in 0 and it converges in o~< Itl < I - ~
"
Now (31 - u) -1 = ((/3 - a ) l + ( a l
_ -
-
1 --~
a-fl n=l
- a
)
-~
n'-I
,//,))-1
__
1 It~-lv
p
= ~(~
-
"l- ~
(~
1
~-,6 n=O
- {~---
oo
- ~)--(_l).-,~.-,~-,v
-31_uo
+ ~(z n---O
-
)"
)-1
wn+l_ --
~)- (-1)~ ~+, _ an+i
3. Compact Operators
458
p
oo
= E ( t 3 - a)-n(u - a l ) n - ' v + E ( t ~ - a) n ( - 1 ) n wn+, an+l n+l
n=O
for every # E ~r, so that
0 < Ia - a a
I.
1 _ no. It now follows from (*) and Lemma 3.1.3.26, that
Ix.I ___Iflrl
~ - ~ [flql qfiR
k=l
, 1(
_ > ] f i r [ ~ ~-
~
~9',~(q)
1-
>
k=l
)
>
k=l
n
1
1
]fir]
---I~1Y:~ ~; (1- ~)_> 2(1- ~)((n +
1) 1-r
-
1)
k=l
for every n E IN, n > no. Thus (x,),et~ r t~ and this is the contradiction sought. Hence flqyq f~ I m ( ~ l
-
ur) .
q~Q
Now suppose that c ~ 1 - Ur E ~ ' ( ~ ) . Then there is a finite-dimensional vector subspace F of ~ such t h a t s = F ~ Im(c~l - u). Let v : ~ --+ ~/Im(exl - u) be the quotient map and d the dimension of F . Let Q be a finite subset of A with Card Q > d. Then ('tJyq)qEQ is linearly dependent. Hence there is
a
(flq)qEQ E ]KQ\{0) such
that
y~ &vya= 0 . qEQ It follows that ~qyq G...I m ( a l - - u ) qEQ
which contradicts the above result. Hence a l - ur ~ Y ( f f ) . From "~1 - Ur = ('~ue - 1)Ur = -~(c~1 - u t ) u ,
~66
3. Compact Operators
(Example 1.2.2.9 e)), it follows (Step 1 and Proposition 3.1.3.7) a l - u t By Proposition 3.1.3.25 a), {a e K I I,~1 = I} C oe(ur)n,:(ut).
Step 3
ae(ut) =
: ( ~ , , ) = {,~ e I< I I~1 = 1}
Let q be the quotient map
~(e) ~
s
Since utur = l ,
urut - l E K~(f f ) ,
we see that (qu~)(qu,) = q(,,:,,) = I,
(qur)(qut) -- q(urut) - 1.
By Example 3.1.2.16,
II(qu,)-~ll = Ilqutll
=
x,
II(qut)-~ll Ilqu, II =
=
x.
By Proposition 2.2.4.7, ae(ur)
=
a ( q u , ) C {or e ~ I I,:,1 = 1},
, : ( , e ) = o(qu~) c {~ e n~ I I,~1 = 1}. By Step 2,
,,'(~,~) = ,,'(,t) = {~ e K I I,~1 = x}. Step4
aelK,
Jal 5' and set
1
~n "= i i ~ n x n l l ~ . X . . Then 0 = lim
1 ....
(cd - u)unx,., = lim (c~l - u ) y , ,
contradicting the above inequality. Hence a is an eigenvalue of u . Example 3.1.4.8
Let E
I
be a n o r m e d space and u a projection in E .
~) o,,(u) = o(,,)c {o, I}. b)
Im u zs the eigenspace of u corresponding to 1.
c)
Ker u is the eigenspace of u corresponding to O. Now
o(,,) c {o,1} follows from Corollary 2.1.3.7. Furthermore, ux = x r
x E Imu,
ux = 0 ~
x E Keru
for each x E E and all the assertions now follow.
I
472
3. Compact Operators
Example 3.1.4.9
If p e {0} U [1, cx:)] and if ur is the right shift operator of
i f , then
o~(u) =0. The same assertion holds when P' :s replaced by c.
By Example 1.2.2.9 e), a l - ur is injective for any a E IK.
i
E x a m p l e 3.1.4.10 Take E := c or E := ff with p E [1,c~] LJ {0} and let ut be the left shift of E .
{a e ~I lal< 1} a) ap(ut) =
if E
~' unth p E [1, cr
i . f E = s ~176
~ 0 We have ue A ~
so that c~ is an eigenvalue of u.
xe A ~
~eA~
1
3.1 The General Theory
477
3.1.5 Spectrum of a Compact Operator Theorem 3.1.5.1
( 0 ) (F. Riesz, 1918) Let E be a Banach space. Take
u E IC(E) and (~ E IK\{0}.
a)
Dim Ker (~1 - u) = Dim Ker ( a l - u') = Dim Coker ( a l - u) = = Dim Coker ( a l - u') < cx:), I m ( a l - u) = ~
b)
( a l - u')).
The following are equivalent: bl)
a E o'(u),
b2) a l - u is not injective (s.e. ~ E ap(u) ), b3) a l -
u is not surjective,
b4) a is an isolated point of a ( u ) . c) Either a(u) is finite or there is a null sequence (an)ne~ in IK such that
o-(,,) = {o,.In ~ Z,T} u {o}. d)
If E is infinite-dimensional, then 0 belongs to a ( u ) .
e) o,.(~)'\{o} = ~ ( = ' ) \ { o }
and th, ,,,=~tiVli~iti,~ of any ,~ ~ x 0. There is a neighbourhood U of s, such that
Ilk(s, . ) - k(g, ")lla
_ t.
~$,l
3. Compact Operators
Take s E [a, b]. Then
(gx)(s) =
g(s, t)x(t)dt =
g(s, t)x(t)dt +
= ~ (f,"bi(t)x(t)dt+
g(s, t)x(t)dt =
fsba~(t)x(t)dt)x~(s).
l--I
Using complete induction on j , we prove that
(gx)tJ)(s) =
~(f'
b,(t)x(t)dt +
,=t
=
f,
/~
a,(t)x(t)dt
~
)
x~')(s)- ,5~, x(s) = ' p.(s)
~(~)
g~)(s, t)x(t)dt - 6n,j pn(s)
holds for every j E INn U {0}. Take j E INn and assume the above equalities hold for j - 1. Then I1
(g~)C,(~) =
~,(b,(~)~(~)
-
~,(~)~(~))~'-~(s)+
i---1
+
~(/'
b,(t)x(t)dt +
/'
a,(t)x(t)dt
)
x~J}(s)=
b,(t)x(t)dt +
/.
a,(t)x(t)dt
)
x~')(s).
t=l
n(f.
+~
s-'l
Then, by Theorem 3.2.1.3,
(gx)(1)(s) = -e,,,pn(s i +
b,(t)x(t)dt + l-'l
a,(t)x(t)dt
x~')(s) =
3.~ Linear Differential Equations
=
gO)(s, t)x(t)dt +
=
5l~5
g(~) ( S, t)x( t )dt - ~,,~;(~~) =
f bg(J)(s, t)z(t)dt
- ~,,j
X(S)
p.(s) "
This completes the induction. It follows from the above result that gx is n-times continuously differentiable and that
f,(gx) --'- ~ ot,,(gx)O-l)(a) -tt~,j(gx,)(J-l)(b) -" j=l j=l
J=l
j=l
b
=
f~Cg(., t))x(t)dt = 0
for every i E INn. Hence gx E D . Moreover n
(ugx)(s) = ~ p i ( s ) ( g x ) ( i ) ( s ) = I-'-0
=
p,(s)g(')(s, t)x(t)dt - x(s) \ t=O
= -x(s) I
for every s E [a, b]. C o r o l l a r y 3.2.1.5
)
( 5 ) u is bijectwe, ~ I - l x ---~ - - g X
for z E C and
for z E D , where g is the Green function.
5~6
3. Compact Operators
The assertions follow immediately from Proposition 3.2.1.4.
Coronary 3.2.~.6
( 5 ) L~t g b~ th~ a,~,, I,,,a~o,. T~k~ a ~ ~ ~nd
put . r := { x ~ Z~ l u x = a x } ,
g := { a z - u z l x a)
e z~}.
~" = {x ~ C I z + a a z = o } .
b) (7 = { x + a g x l x E C } . c) Dim.T = DimC/g < n. d)
1
Wehave
Dim .T" = Dim {z E C I z + a g ' z = 0} and the following are equivalent/or every x E C : dl )
x = t r y - uy f o r some y E lg.
d2)
I f z E C such that z + a g ' z = O, then , b x ( t ) z ( t ) d t = O.
a) Take z E ~'. By Proposition 3.2.1.4, x + a g x E :D,
u ( x + a g x ) = u x - a x = 0,
so that x + a g x = O.
Hence ~'c
{xECIx+agx=O}.
On the other hand (Proposition 3.2.1.4) x = - a g x E :D,
5~7
8.I~ L i n e a r D i f f e r e n t i a l E q u a t i o n s
ux -
--~ugx
- - OtT,
"or every x E C for which x + a g x = 0 and this proves the reverse inclusion. b) Take x E 29. By Corollary 3.2.1.5, there is a y E C such that gy=x.
~ence ax - ux = agy - ugy = y + agy
(Proposition 3.2.1.4) which establishes the inclusion
G c (x + a g x l x e C}. Now take x E
C.
By Corollary 3.2.1.5, uy
=
--X
for some y E 29. Hence x + agx
= -uy
- aguy
= ay - uy
(Corollary 3.2.1.5), which proves the reverse inclusion. c) We may assume that a r 0, since, by b), the assertion is trivial for a = 0. The operator
C
~,C,
x:
; gx
is compact (Corollary 3.1.6.22) and so Dim{x E C Ix +
agx
= 0} = D i m C / { x +
agx
I x E C} < co
(Theorem 3.1.5.1 a)). By a) and b), it follows that Dim .T = Dim C/G.
Dim.T < n is a consequence of the general theory of differential equations. d) For a = 0 the assertion is trivial (Proposition 3.2.1.4), so we may assume a ~ 0. By a) and Corollary 3.1.6.7, Dim~" = Dim Ker(1 + a g ) = DimKer (1 + a g ' ) = Dim{z E
C lz +ag'z
= 0}.
5~,8
3. Compact Operators
dl =~ d2. By b), there is a t/E C such that
z = y + agy = - a
--~y-
gy
.
Since 1
glZ - - - - - - Z ,
f x(t)z(t)dt = 0 by Corollary 3.1.6.7 a
~
b.
d2 =~ d l . Now
f x(t)z(t)dt = 0 for every z E C with 1
g'(z) =
--~.
Hence by Corollary 3.1.6.7 b =~ a, there is a go E C with 1 --x
1 = - - Y o - gYo,
x =yo +agyo. By b), there is a y E / ~ with
x = ayC o r o l l a r y 3.2.1.7
( 5 ) v'•
Take a E ~ ;C,
x:
uy.
II
and ~ax-uz.
The following are equivalent: a)
v isinjectwe.
b)
v is sur3ective.
c)
For each y E C. and each famdy (a,)~et~, in K , there is a unique n times continuously differentiable function z on [a, b] such that n
-~p,x (i) - a z = y I--0
and
y,(~) = a , for every i E IN,,.
3.~ Linear Differential Equations
5~9
a r b follows from Corollary 3.2.1.6 c). a & b =~ c. The uniqueness follows from a). By Lemma 3.2.1.1, there is a family ( / ~ j ) j ~ . in K such that
for all i E ]N,. By b), there is a z E Z) such that 11t
.z - az = u + a ~
a~x~.
j=l The function n
3=1
has the requested properties. c =~ a is trivial.
Proposition 3.2.1.8
m
( 5 ) G~v~. ~, ~ ~ C, d~.~ (xly) :=
z(t)y(t)dt.
Then the .following are equivalent: a)
The Green function is selfadjoint.
9b) x, y E ~D =~ c)
(~,xly)
- (xluy).
The sesquilinear form
ts Hermitian. a =~ b. By Corollary 3.2.1.5 and Proposition 3.1.6.18 a),
(~ly)--(uxlguy)--(guxluy)-
(=luy) 9
b =~ c. Let x, y E C. By Corollary 3.2.1.4,
(gxlv) =-(gzlugz/) =-(ugzlgy) = (zlg~)= (gYl~)c =~ a. Let A be Lebesgue measure on [a,hi, Since C is dense in L2(A), it follows that the sesquilinear form
L2(A) x L2(A)
>~,
(=, y) ~
f(gz)~dA
is Hermitian. By Proposition 3.1.6.18 d), g is selfadjoint,
m
530
3. Compact Operators
3.2.2 Supplementary Results
In this subsection we use the notation from Subsection 3.2.1, but do not assume u injective. Given x, y E C, define (~1~) =
Proposition 3.2.2.1
~(t)y(t)dt.
( 5 ) Given a,,6 E 14,, define ua : :D
>C ,
x ,----+ ux + a x ,
.r',,,, := {z ~ v I , , . x = a ~ ) ,
Ao := {~ z x< 1 7o,~ # { o } } . Take a , ~ E ~ .
a)
.~',.,a = ~'o,~-o 9
b) A,~ = a + Ao. c) Either Ao = ~
or Ao has no pomts of adherence (i.e. Ao is finite or it
consists of a sequence which converges to 0r
d) If lK = r and (,,xl~) = (~1"::) .for every x E Z}, then Ao is contained in Ft and has no points of adheFence.
a) Given x E/:),
r
(uz + ,~x = f~z) r
b) By a), if ~ E ~ , then
(uz = ( Z - a ) z ) r
(x E ~'o,,-o) 9
3.~ Linear Differential Equations
(/3 E Ao) ~
(Jra, a # {0}) ~
(Jro,a_a :~ {0}) r
531
(f~- a e Ao).
c) Assume that Ao # IK. Take a E IK such that - a r Ao. By b), 0 ~ Ao. Hence ua is injective. Let g be the Green function associated to us (Theorem 3.2.1.3) and y the compact operator
C---+C, x:
~gx
(Corollary 3.1.6.22). We have 1
As = {/3 e I K [ - ~ e a(g-)} (Corollary 3.2.1.6 a) and Theorem 3.1.5.1 bl r b2). By Theorem 3.1.5.1 c), A,, has no points of adherence. By b), Ao has no points of adherence. d) Take /3 E Ao. Then there is an x fi D\{0} such that UX "- ~X.
Then
Since b
Ix(t)12dt# o,
(xlx) =
it follows that
~=~. Hence /3 E IR and Ao CIR. By c), A0 has no points of adherence, Proposition 3.2.2.2
m
( 5 ) Let p be a real continuous function on [a, b]
and define v " l) ---~ g ,
x:
~ ux + px.
Then the following are equivalent:
b) x , y e I) =~ (vxly) = ( x l v y ) . Hence, if u and v are injective and g, h are the Green functions associated to u and v , respectively, then g is selfadjoint iff h is selfadjoint.
3. Compact Operators
53#
a =~ b. We have
(v=ly) = (,,= + v=ly) = (u=l~,) + O~=ly) = = (=luy) + ( = 1 ~ ) = (zluy + ml) = (xlvy).
b =~ a follows from a =~ b. T h e last assertion follows from a r D e f i n i t i o n 3.2.2.3
( 5 )
b and Proposition 3.2.1.8 a r
b.
II
If p, is i-times continuously dtfferentiable for
each i E I N . , then tl
;C,
u'' V
x~
Z(-1)'(~,x)(') t=O
is called the adjoint differential operator o f u. u is called selfadjoint if U-- U*.
If x and y are Lebesgue integrable Junctions on [a, b] then we say that ux = y i n t h e s e n s e o ] L2-distributions if
(ylr for all C~176
= (zl,,'r
r on ]a, b[ with compact carrier.
If ~ = lR then ux = y in the sense of L~-distributions conincides with ux = y in the classical sense of distribution theory. Take
v " :D
>C ,
x ~
px ("),
where p is n - t i m e s continuous differentiable. Then for x E ~),
I----O
(p"-')x) (') --
(v*)*x = ( - 1 ) " ~ ( - 1 ) ' t----O
p("-J)x O) :
-- ( - 1 ) = ~ ( - 1 ) ' t=O
.7=0
533
3.~ Linear Differential Equations
" n! i! P(n-a)x(i) Y ~ ( - 1 ) ' i! (n - i)'------~,j ! ( i - j)---'---~.
= (-1)" j=O
:=j
p(n_j)x(j) ~ ( _ l ) j + s :
= (-1)" j=o
-- ( - 1 ) " Z ( - 1 ) '
n!
(n - j - k)!j!k!
k=o
P("-J)xO) Z ( - 1 ) k k ! ( n
$=0
(o3,
- j - k)!
k=O
= px (") + ( - 1 ) " Z ( - 1 ) ' $=0
(:)
. p(n-J)x(i) Z ( - 1 ) k=O
( )
~ n - j
= px(. )
k
"
Hence (it*)* -- 'it.
( 5 ) Take n = 2. If p, is real and i-times continuously differentzable for each i E IN2 U {0}, then u is selfadjoint iff
E x a m p l e 3.2.2.4
Pl "--P2.
Take x E D . Then ,,*~ = (po~) - (r,,~)' + ( p ~ ) " = po~ - p i ~ - p , x ' + pg~ + 2 p i ~ ' + p ~ "
=
= (po - pl + p~)~ + (2pi - p , ) ~ ' + p ~ " ,
~*~ - ~
= (pg - p l ) ~ + 2 ( p l - p ~ ) ~ ' ,
~ ( = ' ~ - =~) = (pg - p l ) ~
+ 2(pl - p,)~'
= ((pl - p , ) ~ ) ' .
Since x is arbitrary, it follows that u = u* iff pl = p~.
m
( 5 ) If p, is i-times continuously differentiable for each i E INn, then given any n-times continuously differentiable functions x, y on [a, b] we have
P r o p o s i t i o n 3.2.2.5
~-x
- 1)'(p,~) (')
- -
5~
3. Compact Operators
dt ~=t \~=o
,=1
3=0
(t'-'
_
~(_
t=l
)
l)J0~,~)(,)~,-,-,)(.).
j=O
W e have, for each i 6 INn,
d (~ (_1).#(p,~)(,)x(,_j_,))= dt \~=o
t--I
t--I
= E(-1)#(p.~)O+l)x('-'-:)+ E(-1)a(p.~)O)z('-')= .7=0
J=O
t
I--I
= - ~(-1)J(p,~)(,)~('-~) + ~(-1)J(p,~)(,~('-,~ = J=l
3=0
= - ( - l)'(p,~)O)x + p,~x (') .
It follows that p,x(')
~ = pox~ +
s--O
= vo~+ ~
(p,~)z(') = "=
(-1)'(p,~)(% + ~
(-1)J(p,~)(,):,:('-J-')
z----1
p , x (') s--O
d dt
~-
x
~ ( - 1 ) ' (pi~) (')) = t=O
,
3.~ Linear Differential i~,'quations
535
Thus
-
(-1)J (pty)(i)x(i-J-t)))(b)
-
-
(-1)J (pi~)(J)x ('-'-l)
(a).
II
i=l
Corollary 3.2.2.6
( 5 ) If p, is i-times continuously differentiable for each i E INn, then the following are equivalent for all x, y E I) :
a) Z(-1)J(p,~)CJ)x 0-j-l) t=l
(a)=
j=O
Z(-1)J(p,~)O)x ('-'-1) t=l
(b).
3=0
b) (uxly) = (xlu*v). If in addition u is injective then the following are equivalent: c) u is sel/adjoint.
d) The Green function for the given boundary problem is selfadjoint. a r b follows from Proposition 3.2.2.5. c ca d. Take Coo-functions ~o,r on ]a, b[ with compact carrier. By a =r
b~
(uv I r = (v1r162 so that the assertion follows from Proposition 3.2.1.8 a r b. Corollary 3.2.2.7 C ~176
II
( 5 ) Suppose that u is injective and its coeJ~cients are
If x is a Lebesgue integrable function on [a, b], then gx E C
and
~gX -- --X
in the sense of L2-distributions.
536
3. Compact Operators
By Corollary 3.1.6.4, gx E C. Let (xn)nCm be a sequence in C such that [xn(t) - x(t)[dt = O.
lira
n.,,4,~
Let ~o be a C~176
on ]a, b[ with compact carrier and u* the adjoint of
u. By Corollary 3.2.2.6 a =~ b and Proposition 3.2.1.4, (gz.)(u*v)dt =
(ugz.)~dt = -
xn~dt
for every n E lq. It follows that
(gxlu*~o)=
(gx)(u*~p)dt= lim
(gxn)(u*~o)dt=
}l--too
=
-
lim n--~oo
f.'
x.~dt
=
f.'
-
~at
= (-~1~,).
Hence
UgX -- --X II
in the sense of L2-distributions. Corollary
3.2.2.8
( 5 )
Take n = 2, ]K = R , and let u be selfadjoint.
Put
Let x, y be twice continuously differentiable functions on [a, b] and take D := xy'
a) (=xly)
-
(~l~y) = p2(a)D(a)
b)
x , y e 1) ==~
c)
If x, y E1) and
-
-
x'y.
p2(b)D(b).
a((uxly) -
(zluy))=-(ap2(b)
Z((~Iu)-
(x[uy))=(ap2(b) + t~p2(a))D(a).
+/~p2(a))D(b),
a=~=O
and if the boundary conditions f, = 0 then
( i E INn) are linearly independent,
D(a) = D(b) = O.
537
3.1~ Linear Differential Equations
a) By Example 3.2.2.4, 2
s--1
E
Z
t=l
3=0
(-1)(')(p'y)(')x('-3-') -- (p]y)x + ( P 2 y ) x ' - (p2y)'x --
= plyx + p2yx' - p~yx - p2y'x = - p 2 D .
We deduce from Proposition 3.2.2.5 (uxlY) - (xluy) = 1~(a)D(a) - p2(b)D(b) .
b) We have
[oll ol2][x,~
~12] I x ( b ) y ( b ) ] #22 x'(b) y'(b)
0.
It follows from this that o~D(a) + ,6D(b) = O.
From a) we deduce that ot( (uxly) - (xluy) ) = o~p2(a)D(a) - c~p2(b)D(b) = = -~1~(a)D(b) - o~p2(b)D(b)
-(otp2(b) + ~p2(a))D(b),
~((uxly ) - ( x l u y ) ) = ~ p 2 ( a ) D ( a ) - ~p2(b)D(b) = = ~p~(a)D(a)+ ap2(b)D(a) = (ap2(b) + ~p2(a))D(a).
c) Since the boundary conditions are linearly independent, it follows that the rank of the matrix ~21 ~22 &l
&2
is 2. Hence there are i , j E {1, 2} such that D e t [ (~1/~2, /~" 0 . 1 ]7 2 :/: j From
3. Compact Operators
538
[ol ]
u'(a) + ~
/31:2] Ix qo(tl)x(tl) , t~z(t2) = ux(t~) = q~(t~)z"(tg) + qo(t~)x(t~) < qo(t~)x(t~) .
If
z(t~) > o, then qo(tl) E IR and a > qo(tl) > infreqo(t). - tqr
x(t~) inf re qo(t). --
te'r
Hence {a e R I~'o :~ {0}} c [infreqo(t)
oo[.
t~l'
II
E x a m p l e 3.2.2.12
( 5 ) Assume IK = C . Let (cck)k~.u{0} be a family in C with an ~ O. Given k E IN U {0}, let C,(~) denote the set of k-times continuously differentiable functions on T . Define I1
u " C (")
~ C (~
x '
~~
a~x (k) ,
k=0
and let A be the set of those c~ E C for which there is an x E C(n)\{0} such that UX
---- O l X .
If A is infinite, then it has no points of adherence and
lira a = in c~n ~
for n even, and •
are the only possible points of adherence of the filter
(~/o~a,lol-oo for n odd. By Proposition 3.2.2.1 c) (and Proposition 3.2.2.10 e),f)), A has no points of adherence. Given a e r define
Po(t) := ~ akt~ -
~
e
r
k=0
Bo .= {# e r I P~,(#')= o},
C:= {,8 e~r
n
} ka~/~ k-~ = 0
.
k--I
C is finite. Take a E A \ C . Then Ba has precisely n elements and (ea')~EB~ is a fundamental system of solutions of the differential equation
3.~ s
1)ilterential l~quations
54s
u x - a x = O.
Hence, if x E C(n)\{0} satisfies
then there is a family (Ta)aeBo in r such that
B6Ba
Thus, if k E INn, then
BEBo
BEBh
so that (1 - e a ) # ~ - ~ , , / a = o . ,06Bo
Since not every 7~ (/3 E B~) vanishes, the determinant of the above system of linear equations must vanish. Hence there is a /3a E B~ with eao = 1. There is therefore a jR s 2g, so that 13o, = 27rij,~.
It follows that n
o = P~(/~) = ~
~k(2~iy~) k -
~.
k=O
Hence lim IJ,,I = c~,
Iol-.-*co aEA
and therefore limoo a
lim ( ~ ) n
if n is even. If n is odd, then :t:i" ~ of the filter
(~l-a-i aEA,lal--,oo"
an .n
in an
are the only possible points of adherence
54~
3. Compact Operators
Proposition 3.2.2.13
( 5 ) Take n E IN, define
A:={jE~,U{O}Ij
is even},
B := {j E IN, I j
is odd},
and denote .for every j E IN. U {0} by CO) the set of j-times continuous differentiable functions on T . Let (Pj)jet~.u(o} be a .family in C(~ N lFtx such that P3 E CO) for every j E A and put
u ' C (")
)C (~
z:: ; ~ p j x
0).
3=0
Then for each k E A there are families (qj)jzA and (rr that qj E C(~) for every j E A and
Z
sEA
./EB
~EA
.~>n-k
in C(~ flIRT such
j o,
55~
3. Compact Operators
/~r -- Xo "- - - ~ u p + u ( x ' p ) -- u ( ( x -- ~ e T ) ' p ) >_ 0
and a r < xo < # r
on U. Since U, a,/~ are arbitrary, x0 = x r . Thus u(x'p) = -rx
and so x E ft. Step 2
If (xn)nr
is a monotone sequence in ~" converging
pointwise to some x E 9r, then x E (xn'p)nei~ is a monotone sequence converging pointwise to x . p . Thus u ( x . v ) = lim u ( x , . p ) = n -4 00
lim ( - r x , ) = - r x . 111"400
Hence x E ~. Step 3
Every Borel function in ~r belongs to
This follows immediately from the first two steps. Step 4
Every bounded x E ~', belongs to
There are bounded Borel functions y, z on T such that y<x