C*-algebras Volume 2: Banach Algebras and Compact Operators

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

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Preface

Functional analysis plays an important role in the program of studies at the :-~wiss Federal Institute of Technology. At present, courses entitled Functional Analysis I and II are taken during the fifth and sixth semester, respectively. I ilave taught these courses several times and after a while typewritten lecture lotes resulted that were distributed to the students. During the academic year 1987/88, I was fortunate enough to have an eager enthusiastic group of students ~hat I had already encountered previously in other lecture courses. These stu~_ents wanted to learn more in the area and asked me to design a continuation _~f the courses. Accordlingly, I proceeded during the academic year, following, with a series of special lectures, Functional Analysis III and IV. for which I again distributed typewritten lecture notes. At the end I found that there had accumulated a mass of textual material, and I asked myself if I should not publish it in the form of a book. Unfortunately, I realized that the two special ecture series (they had been given only once) had been badly organized and ~ontained material that should have been included in the first two portions. And so I came to the conclusion that i should write everything anew - and if at all - then preferably in English. Little did I realize what I was letting myself in for! The number of pages grew almost impercepetibly and at the end it had more than doubled. Aslo, the English language turned out to be a stumbling block for me; I would like to take this opportunity to thank Prof. Imre Bokor and Prof. Edgar Reich for their help in this regard. Above all I must thank Mrs. Barbara Aquilino, whGwrote, first a WordMARC TM, and then a LATEXTM version with great competence, angelic patience, and utter devotion, in spite of illness. My thanks also go to the Swiss Federal Institute of Technology that generously provided the infrastructure for this extensive enterprise and to my colleagues who showed their understanding for it.

Corneliu Constantinescu

vii

Table of Contents of Volume 2

Introduction 2

xix

Ba nach Algebras 2.1 Algebras . . 2.1.1

General Results

2.1.2

Invertible Elements The Spectrum . . .

2.1.3 2. 1.4 2.1.5 2.2

Standard Examples . Complexificati on of Algebras . Exercises .. . . Normed Algrbras . . . . . . . .

2.2.1 2.2.2 2.2.3 2.2.4 2.2.5 2.2.6

Ge nera l Results . . . . . T he Stilndard Exilmples

17

32 51

65

69

69 82 125

T he Th<.'orems of Ricsz and Gelfand . Polrs of Resolvf'nts

161

Modulrs . . . . . . . Exercises . . . . . . . l nvolutiw Banach Algebras

114 153 174 197 201

2.3. 1

TnvolutiveAlg cbras.

201

2.:1.2

lnvolutivc Bannch Algebras Sesquilinear Forms . .

24 1 275

Positive Li.J1ear Forms The State Space ..

305

l nvolut ive Morlnles Exerrisrs . . . . . .

328

2.3.3 2.3.4 2.3.5 2.3.6 2.4

3

13

T he Exponential Function and tiH• eumann Ser ies I nvertible Elements of Unital Banach Algebras.

2.2.7 2.3

3 3

Grlfnnd Algebras . . . . . The Gelfa nd Transform . 2.4.2 I nvolutivc GPifand Algebras 2.4.1

287

322 331 33 1 343

Table of Contents

viii

3

.... .. .... . .. .

Eu m~~

2.<1.4

2.4.5

Locally Compact Additive Groups. Exam pl~ . . . . . . .

378

2.4 .6

T hr Fourier Transform

390

ExercisPs .

396

Compact O pc mtors 3.1 The General Theory 3.1 .1 GenN a! Results 3.1.2 Examples . . . 3.1.3 3 1.4 3.1 .5

Fredholm Operators Point S pectrum . . . Spectrum of a Compact Opera tor

Integral Op<'rators . . Exercises . . . . . . . . . . . . . . Linea.r Differential Equations. . . . . . . 3.2.1 Boundary Value P roblems for Differential Equatio ns. 3.2.2 S upple mPntary Results . . . . . . . . 3.2.3 Linear Partia.l Differential Eq uations

3.1.6 3.2

358 365

2.4.3

Exercises .

399 399 399 419 437 468 477

489 517 518 518 530 549 56.1

;\ame Index

565

Subject lndrx

568

Symhol l ndrx

588

xix

Introduction

This book has evolved from the lecture course on Functional Analysis I had given several times at the ETH. The text has a strict logical order, in the style of "Definiton - Theorem - P r o o f - Example - Exercises". The proofs are rather thorough and there are many examples. The first part of the book (the first three chapters, resp. the first two volumes) is devoted to the theory of Banach spaces in the most general sense of the term. The purpose of the first chapter (resp. first volume) is to introduce those results on Banach spaces which are used later or which are closely connected with the book. It therefore only contains a small part of the theory, and several results are stated (and proved) in a diluted form. The second chapter (which together with Chapter 3 makes the second volume) deals with Banach algebras (and involutive Banach algebras), which constitute the main topic of the first part of the book. The third chapter deals with compact operators on Banach spaces and linear (ordinary and partial) differential equations - applications of the theory of Banach algebras. The second part of the book (the last four chapters, resp. the last three volumes) is devoted to the theory of Hilbert spaces, once again in the general sense of the term. It begins with a chapter (Chapter 4, resp. Volume 3) on the theory of C*-algebras and W*-algebras which are essentially the focus of the book. Chapter 5 (resp. Volume 4) treats Hilbert spaces for which we had no need earlier. It contains the representation theorems, i.e. the theorems on isometries between abstract C*-algebras and the concrete C*-algebras of operators on Hilbert spaces. Chapter 6 (which together with Chapter 7 makes Volume 5) presents the theory of EP-spaces of operators, its application to the self-adjoint linear (ordinary and partial) differential equations, and the yon Neumann algebras. Finally, Chapter 7 presents examples of C*-algebras defined with the aid of groups, in particular the Clifford algebras. Many important domains of C*-algebras are ignored in the present book. It should be emphasized that the whole theory is constructed in parallel for the real and for the complex numbers, i.e. the C*-algebras are real or complex. In addition to the above (vertical) structure of the book, there is also a second (horizontal) division. It consists of a main strand, eight branches, and additional material. The results belonging to the main strand are marked with (0). Logically speaking, a reader could restrict himself/herself to these and ignore the rest. Results on the eight subsidiary branches are marked with (1), (2), (3), (4), (5), (6), (7), and (8). The key is

xx

Is~troductiol~

1. 2. 3. 4. 5. 6. 7. 8.

Infinite Matrices Banach Categories Nuclear Maps Locally Compact Groups Differential Equations Laurent Series Clifford Algebras Hilbert C*-Modules

These are (logically) independent of each other, but all depend on the main strand. Finally, the results which belong to the additional material have no marking and - from a logical perspective - may be ignored. So the reader can shorten for himself/herself this very long book using the above marks. Also, since the proofs are given with almost all references, it is possible to get into the book at any level and not to read it linearly. We assume that the reader is familiar with classical analysis and has rudimentary knowledge of set theory, linear algebra, point-set topology, and integration theory. The book addresses itself mainly to mathematicians, or to physicists interested in C*-algebras. I would like to apologize for any omissions in citations occasioned by the fact that my acquaintance with the history of functional analysis is, unfortunately, very restricted. For this history we recommand the following texts. 1. BIRKHOFF, G. and KREYSZIG, E., The Establishment of Functional Analysis, Historia Mathematica 11 (1984), 258-321. "t

2. BOURBAKI, N., Elements of the History of Mathematics, (21. Topological Vector Spaces), Springer-Verlag (1994). 3. DIEUDONNI~, J., History of Functional Analysis, North-Holland (1981). 4. DIEUDONN]~, J., A Panorama of Pure Mathematics (Chapter C III: Spectral Theory of Operators), Academic Press (1982). 5. HEUSER, H., Funktionalanalysis, 2. Auflage (Kapitel XIX: Ein Blick auf die werdende Functionalanalysis), Teubner (1986), 3. Auflage (1992). 6. KADISON, R.V., Operator Algebras, the First Forty Years, in: Proceedings of Symposia in Pure Mathematics 38 1 (1982), 1-18. 7. MONNA, A.F., Functional Analysis in Historical Perspective, John Whiley& Sons (1973).

Ii~troduction

xxi

8. STEEN, L.A., Highlights in the History of Spectral Theory, Amer. Math. Monthly 80 (1973), 359-382. There is no shortage of excellent books on C*-algebras. Nevertheless, we hope that this book will be also of some utility to the mathematics commutity.

2. Banach Algebras

Banach algebras are the cornerstone of this book. In fact it would be fair to say that the first chapter merely presented the pre-requisites and that the book begins properly with this chapter.

2.1 A l g e b r a s This section collects definitions and facts from algebra in preparation for the rest of the chapter. The concept of a spectrum is introduced. It plays a central rSle in this book, particularly in two special cases. The first concerns compact operators, where the spectrum is simply the set of eigenvalues of the operator, (with the possible exception of 0). The second case concerns the normal (selfadjoint) elements of a complex (real) C*-algebra, where the spectrum is the image set of an associated scalar valued function on a locally compact space. 2.1.1 G e n e r a l R e s u l t s

Definition 2.1.1.1 map

( 0 ) A multip,icatio, on a vector space E is a bilinear

E•

(x,y):

"xy

which is associative, i.e. (xy)z = x(yz) .for all x, y, z E E . xy is called the product o1 x and y . An algebra is a vector space endowed with a multiplication. A n algebra is called real (complez) if the underlying vector space is real (complex). An algebra E is called degenerate if E = {0}. Let E be an algebra. A subalgebra (a left ideal, a right ideal) of E is a vector subspace F of E such that

4

~,. Banach Algebras

xyEF for every x, y E F (.?or every (x, y) E E • F , for every (x, y) E F x E ). An ideal o] E is a subset of E which is at the same tsme a left and a right tdeal. A proper (left, right) ideal o] E is a (left, right) ideal o? E different from E . A m a z i m a l proper (left, right) ideal of E is a proper (left, right) ideal F of E , such that F is the only proper (left, right) ideal of E containing F . We define x" inductively for x E E and n E l~l by X 1 :'-

X,

X n+l

:--

X X n "-- X n X .

An element x of E is called nilpotent if x n = 0 for some n E l~l (B. Pierce, 1870). An element x of E is catted a ~e,.o-di~i,o,. i/ there is a y E E\{0} such that xy=O

or y x = O .

A subset of E is called commutative sf xy = yx for every x, y E A . A unit ol E is an element e of E such that ex

--

xe

--

x

for every x E E . Every complex algebra becomes a real algebra if we restrict the scalar multiplication to ~ . Every left (right) ideal of an algebra is a subalgebra. There is at most one unit in an algebra for if e and e' are units, then e

----- e e I ~

e I .

Because of the associativity, in expressions like x(yz) and (xy)z we usually omit the parantheses. If follows from the bilinearity of the multiplication that x0 = 0x = 0 for every element x of an algebra. Every subalgebra of an algebra is an algebra with respect to the induced structure. In commutative algebras the notions ideal, left ideal, and right ideal coincide.

2.1 Algebras

5

2.1.1.2 ( 0 ) Let E be an algebra. The intersection of a nonempty family of (left, right) ideals of E is a (left, right) ideal of E . In partscular, every A c E is contained in a smallest (left, mght) ideal of E , called the (left, right) ideal of E generated by A . II

Proposition

i

( 0 ) A unital algebra is an algebra endowed with a unit which is different from O. The unit is generally denoted by 1. Define

Definition

2.1.1.3

]K1 := {al

e lg}

and x0 : - - 1

for every element x of aumtal algebra. A unital subalgebra of a unital algebra is a subalgebra containing the unit of the algebra. We emphasize that a unital algebra is not merely an algebra which has a neutral element for its multiplication. To be unital, such a neutral element must be distinguished in the sense that the neutral element in question is an essential component of the algebra structure. We are obliged to make this distinction since in practice we may be confronted with a subalgebra F of a unital algebra E , which has a unit different from the unit of E . This unit of F is somehow accidental and has not the meaning of a true unit for the problems formulated on E . So in this case F has to be treated only as an algebra and not as a unital algebra. In particular we make the convention that F endowed with the induced structure is unital iff E is unital and F is a unital subalgebra of E . An example to clarify this is provided by the algebra ]K2,2 of 2 • 2-matrices ,RI

im

with entries from IK which is clearlv a unital algebra over IK with [ 1 0 / '

~

L

0

1

J

as unit (Proposition 2.1.4.24 b),c)). IK can be identified with E:=

{[o~ 0

0

aEIK

}

and is clearly also a unital algebra, with [ 1 0 ] as unit. Since the unit of E L0 0 ] and that of IK2,~ do not conincide, we cannot consider E to be a subalgebra of IK2,2 "in the sense of unital algebras", but only in the sense of algebras in general, i.e. algebras not required to admit a unit.

6

~. Banach Algebras

]K will be considered as a unital algebra with respect to the usual multiplication. 0 is not excluded in the above definition of x ~ , i.e. we define 0~ :-- 1. P r o p o s i t i o n 2.1.1.4 ( 0 ) Let E be a (unital) algebra. The intersection of a nonempty family of (unital) subalgebras of E is a (unital) subalgebra of E . In particular, every A C E , is contained in a smallest (unital) subalgebra of E called the (unital) subalgebra of E generated by A . Given x, y E E , the (unital) subalgebra of E generated by {x} (by {x, y}) ) is called the (unital) subalgebra o.f E generated by x (by x and y ). II P r o p o s i t i o n 2.1.1.5

( 0 ) If E is a unital algebra then

(-1)x=x(-1)=-x for every x ~. E . We have that x + ( - 1 ) x = lx + ( - 1 ) x - (1 - 1)x -- 0x -- 0, x + x(-1)

= 11 + x ( - 1 )

= x ( 1 - 1) = x o = o .

m

D e f i n i t i o n 2.1.1.6 ( 0 ) Let u" E --+ F be a linear map between the (unital} algebras, E and F . Then u is called (unital) algebra homomorphism if given x, y E E , ~(~y) = (~)(~y)

(and ul = 1 ). A bijective (unital) algebra homomorphism is called a (unital) algebra isomorphism. Two (unital) algebras E, F are called isomorphic if there is a (unitaO algebra isomorphism E ~ F . Let E, F be algebras and u 9 E ---r F be a surjective algebra homomorphism. If E has a unit 1, then ul is a unit of F . In particular, if E and F are unital algebras, u is also unital. P r o p o s i t i o n 2.1.1.7 ( 0 ) Let E be an algebra and F be the vector space IK • E endowed with the bilinear map F x F

; F,

((a,x), (f~,y)) :

; (a,B, ay +/3x + xy).

~.1 Algebras

7

Then F is a unital algebra with (1,0) as unit. {0} • E is an zdeal of F of codimension one. F is commutative iff E is commutative. If G is a unital algebra and u : E --~ G is an algebra homomorphism, then

F--+G,

(a,x)'

"al+ux

is the umque unital algebra homomorphism extending u (here E is identified with {0} • E).

The proof is a simple verification.

II

Definition 2.1.1.8 ( 0 ) Let E be an algebra. If E is not unital then, up to an isomorphism of unital algebras keeping the points of E fixed, there is a unique umtal algebra for which E is an ideal of codimension one (Proposition 2.1.1.7). This unital algebra is called the unital algebra associated to E . If E is unital then the unital algebra associated to E will be E itself.

Convention

We treat every algebra as imbedded in its associated unital algebra. Many results formulated for unital algebras then extend automatically to all algebras. In particular, the symbol 1 will be used for any algebra and is to be understood as the unit o] the associated unital algebra.

Proposition 2.1.1.9

( 0 ) Let E, F be (unital) algebras and u" E -+ F a (unital) algebra homomorphism. Let G be a (unital) subalgebra of E and A a commutative set of E . Then u(G) is a (unital) subalgebra of F and u(A) is a commutative set of F . II

Proposition 2.1.1.10

( 0 ) Let E, F be algebras, u" E --+ F an algebra

homomorphism, and G a subalgebra (an ideal) o.f F . Then u I(G) is a subalgebra (an ideal) o.f E . Hence if {0} is the only proper ideal of E , then either u = 0 or u is injective, m

Proposition 2.1.1.11

( 0 )

Let E , F

be unital algebras, u " E ~ F a

unital algebra homomorphism, and G a proper ideal of F . Then -ul(G) is a proper ideal of E .

ul(G) is an ideal of E such that 1 ~ u I(G)

II

Proposition 2.1.1.12 ( 0 ) Let E be a unital algebra and x' " E --~ ]K an algebra homomorphism. If x' 2s not identically zero, then it i e a unital algebra homomorphism and Ker x' is a maximal proper ideal of E .

g. Banach Algebras

Now x ' ( 1 ) = x ' ( 1 2 ) = x ' ( 1 ) 2.

Since x'(1) :~ 0 it follows that x'(1) = 1. Hence x' is a unital algebra homomorphism. By Proposition 2.1.1.11, Ker x' is a proper ideal of E . Since it has codimension 1, it must be maximal. I P r o p o s i t i o n 2.1.1.13 q" E --~ E / F

( 0 ) Let E be an algebra, F an ideal of E , and

the quotient map.

a) Gwen X , Y 9 E / F

and xl,x2 9 X , Yl,Y2 9 Y ,

q(xlyl) = q(x2y2). Define X Y := q(xlYl).

b) The map (X,Y):

E/F• ss a multiplication. Equip E / F

~XY

wsth this multiplication.

c) q is an algebra homomorphzsm. d) If E is a unital algebra and F is proper, then E / F

eqmpped wzth ql

zs a unital algebra and q ~s a unital algebra homomorphism. e) If E is commutative, then E / F

is commutative.

a) We have x~y~ - x2y2 = xlyt - xly2 + x~y~ - x~y2 =

= xl(yt - Y2) + (xl - x2)y2 9 F + F C F .

b) Take X , Y, Z E E / F and c~,/3 9 IK. Take further x 9 X , y 9 Y, and z 9 Z. Then

~,.I Algebras

ax + By e a X + flY, (e~x + #y)z ~ (e~X + # Y ) Z , xz E X Z ,

yz E Y Z ,

(ax + ~y)z = a x z +/3yz e a X Z + f l Y Z . Thus ( a X + i~Y)Z = a X Z + B Y Z . Similarly, Z ( a X + flY) = a Z X + ~ Z Y . From x(yz) E X ( Y Z ) ,

(xy)z E ( X Y ) Z

~(u~) = (~)~ it follows that x(Yz)

= (xY)z.

c) is trivial. d) Take X E E / F and x E X . Then Xq(1) = q(xl) - q(x) - X , q(1)X -- q(lx) - q(x) - X . q(1) is therefore a unit of E / F and q is a unital algebra homomorphism. e) follows from c) and Proposition 2.1.1.9.

Proposition 2.1.1.14

( 0 )

Let E be

a

1

unital algebra.

a) A (left, right) ideal F of E is proper sff 1 ~ F . b) The set of proper (left, right) ideals of E is inductively ordered with respect to inclusion. c) Every proper (left, right) zdeal of E is contained in a maximal proper (left, right) ideal of E .

10

~. Banach Algebras

a) If 1 E F , then x=xl

EF

(x=lxEF)

for every x E E , and so F is not proper. The converse is trivial. b) follows from a). c) follows from a),b), and Zorn's Lemma. P r o p o s i t i o n 2.1.1.15

( 0 )

1

Let E be a unital algebra, F a maximal

proper left ideal of E and take x E E \ F .

Then there is a pasr (y, z) E E x F

such that yx+z=

1.

Put G : = {~x + z l(u. z) e E • F } .

G is a left ideal of E strictly containing F . Hence G = E , proving the assertion. 1 D e f i n i t i o n 2.1.1.16

( 0 ) Let E be an algebra. Given A C E , define A c := {x E E l y ~ A ==~ xy = yx} , Ac~:=(A') c,

A~c:=(A~)'.

A c is called the c o m m u t a n t ol A and Acc the bicornmutant o l A .

For every x, y E E , 9 ~ = yx . = .

Proposition 2.1.1.1r

( 0 )

x e {~}c . = .

~ e {x} ~

Let E be a (unital)algebra and take A a

subset of E .

a) b)

A c ss a (unital) subalgebra of E . B C A ~ Ac C B c .

c) d)

A C A ~c 9 Ac = A~ .

e)

A commutatwe ~ A C A ~ r

f)

A commutative =~ the (unital) subalgebra of E generated by A is

A ~c commutative.

commutatwe.

g)

If F is an ideal of E and 1F is a unit of F then 1F E E ~.

~. 1 Algebras

II

a) Take x , y ~. A e and (~,/~ E ~ . Then (aX + ~ y ) z = a x z + Byz = a z x + B z y = z(a.r + f~y),

xyz = xzy = zxy

for every z E A. Thus a x + BY, x y ~. A e so that A c is a subalgebra of E . If E is a unital algebra, then 1 E A c , so that A c is a unital subalgebra of E. b) and c) are trivial. d) By c), A c A ~:

so by b) and c), A ~c C A e C A eee , i.e.

Ac ~

A ecr "

e) The first equivalence is obvious. Assume A c A c . By b) and c), Aee C A e C A cce .

Hence A ce is commutative. By c) Aee commutative implies A commutative. f) follows from a), c), and e). g) Let x E E . Then XlF, 1FX E F , so XlF = 1FXlF "- 1FX. Hence 1F E E e 9 E x a m p l e 2.1.1.18

I ( 0 ) Let lK[t] (resp. l K [ s , t ] ) d e n o t e the commutative

umtal algebra of polynomials in t (resp. in s and t ) with coe~cients in IK (the algebra operations are the usual ones for polynomials). Given n E IN, the set of polynomials with the property that the coeIlicients of the terms of degree strictly smaller then n vanish, is a proper ideal,

i

If2

~. Banach Algebras

Proposition 2.1.1.19 ( 0 ) wzth xy

=

Let E

be a unital algebra and take x, y 6 E

y x . The maps

~[t] ~[s, t]

E,

P:

}E ,

Q:

>

~ P(x),

t Q(x, y)

are unital algebra homomorphssm and {P(x) I P e

({Q(x,y) I Q e

lK[t]}

lK[s,t]})

is the unital subalgebra of E generated by x (by x and y ). It is commutative. II

Corollary 2.1.1.20

Let E be an algebra and take x, y E E with xy = y x .

Then

(~ + u) ~ = ~

~ u ~-~,

k=O

n-I

~. _ y. = ( ~ - ~) ~ ~ " - ~ - ' k=O

for every n E IN .

II

~.1 Algebras

13

2.1.2 I n v e r t i b l e E l e m e n t s Definition 2.1.2.1 invertible in E

I

[

0

) Let E be a unital algebra. T h e n x E E is called

if there is a y E E

such that

x y = y x -" 1. E

is called d i v i s i o n a l g e b r a if e v e r y e l e m e n t o f E\{O} is invertible in E .

Let E be a unital algebra, F a unital subalgebra of E , and x E F . If x is invertible in F then it is obviously invertible in E . But x can be invertible in E without being invertible in F . P r o p o s i t i o n 2.1.2.2

( 0 )

T h e z e r o - d i v i s o r s are not invertible.

Let E be a unital algebra and x an invertible element of E . Take y, z E E with x y = y x = 1,

xz = O

(or z x = O).

Then z = l z = y x z = yO = 0

(or z = z l = z x y = 0 y = 0 ) . Hence x is not a zero-divisor. P r o p o s i t i o n 2.1.2.3

0)

I Let E

be a unital algebra a n d take x , y , z E E

with x y = z x = 1. T h e n y = z and x is revertible.

We have z = zl = zxy = ly = y.

I

14

~. Banach Algebras

C o r o l l a r y 2.1.2.4

( 0 )

Let E be a unital algebra and x an invertzble

element of E . Then there is a unique element x -l E E with x-l:r, = x x -1 = 1. x -l is called the inverse o] x in E . Moreover

(x-')-' = x . Convention

m

If we write x - l .for an element x of a unital algebra, we impli-

citly assert that x is invertbile. Remark.

The set of invertible elements of a unital algebra is a group with

respect to the induced multiplication with 1 as neutral element. Definition 2.1.2.5

( 0 ) Let x be an invertible element of a umtal algebra.

Given n E IN, define

x-n := (x-1)n. C o r o l l a r y 2.1.2.6 ( 0 ) Let E be a unital algebra and take x, y E E . x and y are invertible sff x y and yx are mvertzble and in thss case (zy) -l = y - l x - l .

Suppose that x and y are invertible. Then (y-lx-l)(xy) = y-lx-lxy

= y-ly = 1

( x y ) ( y - l x - l ) = x y y - l x -1 = XX -l = 1,

SO x y is invertible and (zy)-' = y-Ix-'.

By symmetry, yx is also invertible. For the converse, assume that xy and yx are invertible. Set u :-- Xy)

V :-- yX.

Then x ( y u - ' ) = ( x y ) u - ' = 1, ( v - ' y ) x = v - ' ( y x ) = 1.

By Proposition 2.1.2.3, x is invertible. By symmetry, y is also invertible.

II

~.I Algebras

Corollary 2.1.2.7

15

( 0 ) Let E be a unital algebra and (x,),et a commu-

tative finite family in E . The product n x~ of this family is invertible iff each tel

x,

(t E I) is invertible.

The assertion follows from Corollary 2.1.2.6, by complete induction,

m

I

Proposition 2 . 1 . 2 . 8 t 0 ) Let E be a unital algebra. If x is an invertible element of E , then

Given y E {x} c , x - l y = x - i y x x -1 = x - l x y x -1 = yx -1 ,

Hence

P r o p o s i t i o n 2.1.2.9

( 0 ) If F is a proper left (right) ideal of a unital algebra, then the elements of F are not invertible.

Take x E F . If x were invertible, then l = x-lx E F

( l = x x -1 q. F ) ,

so E = F , contradicting the hypothesis.

m

Corollary 2.1.2.10

Let E , F be unital algebras and u 9 E --} F a unital algebra homomorphism. If E is a division algebra and F ~ {0}, then {0} is the only proper ideal o.f E , and u is injective.

By Proposition 2.1.2.9, {0} is the only proper ideal of E and by Proposition 2.1.1.10, u is injective, m Corollary 2.1.2.11

( 0 ) Let E be a commutative unital algebra. An ele-

ment of E is not invertible iff it belongs to a maximal proper ideal of E .

Let x be a non-invertible element of E . Then {yx I Y E E} is a proper ideal of E (Proposition 2.1.1.14 a)), so it is contained in a maximal proper left (right) ideal of E (Proposition 2.1.1.14 c)), which obviously contains x. The converse follows from Proposition 2.1.2.9. m

16

f2. Banach Algebras

P r o p o s i t i o n 2.1.2.12

( 0 )

If E is a unital algebra and F is a proper

ideal of E , then:

a) If E / F

is a division algebra then F is maxzmal.

b) I f E is commutative and F is maximal then E / F Let q" E ~ E / F

is a division algebra.

be the quotient map.

a) Let G be an ideal of E strictly containing F and take x E G \ F . Then qx is invertible, and so there is a Y E E / F

with

(qx)Y = ql.

Take y E Y. Then xy E G and q(1 - xy) = ql - (qx)(qy) = ql - ql = O, 1- xy E F C G ,

l = (1-

xy) + xy E G ,

G = E .

Hence F is maximal. b) Take X E ( E / F ) \ ( O } .

Take x E X . Then x does not belong to F .

By Proposition 2.1.1.15, there is a (y, z) E E x F with xy+z=l.

Thus X (qy) = q(xy + z) = ql .

Hence X is invertible and so E / F P r o p o s i t i o n 2.1.2.13

is a division algebra.

I

Let E be a unital, finite-dimensional algebra and x E

E . I f there is an y E E such that xy = 1 ( resp. yx = 1), then x is invertible.

Put ~'E

,~E,

~:

~z~

for every z E E . ~ is linear and Z 1 0 Z 2 "-- Z l Z 2

for all zl, z2 E E . We have ~oy(~) = ~(y~) = xu~ =

for every ~ E E , so that ~o~" is the identitiy map of E . Since E is finitedimensional, ~'o~ is also the identity map of E . It follows yx --- y x l = ~"xl = ~o~(1) = 1.

Hence x is invertible.

I

12.1 Algebras

17

2.1.3 T h e S p e c t r u m Definition 2.1.3.1 ( 0 ") Let E be an algebra, E its associated unital algebra, and take x E E . Put a ( x ) -=

as(x)"= {~ e ]K l al

- x ss not invertible in E } ,

r(x) := r E ( x ) : = inf{r E IR+ I aE(x) C U~(0)}.

(Hilbert, 1906) and rE(x) the

We call a s ( x ) the spectrum of x (in E ) spectral radius of x (in E ). The map ]K\a(x) , >E ,

(~1

oL ~

-

x) -1

is called the resoinent of x (in E ).

x is invertible in E if 0 ~ a ( x ) . If E has no unit, then 0 e a(x) (Pro)osition 2.1.2.9). The spectrum of an element of E remains unchanged if we reverse the multiplication. P r o p o s i t i o n 2.1.3.2

( 0 ) If E is a unital algebra, then a((~l) = {(~}

for every a E ]K.

Take f~ E IK. It follows from /~1 -

~1

- - (/Y -

~)1

that f~l - c ~ l is invertible iff f~ ~- c~. P r o p o s i t i o n 2.1.3.3

I

If E is a division algebra, then o(x) =0

for every x E E\IK1.

Since for every (~ E IK, (~1 - x :/: 0, a l - x is invertible,

i

18

2. Banach Algebras

T h e o r e m 2.1.3.4

(0)

Let E be a unital algebra. Take x E E and P E

lKit]. ~) P(~(~)) c o(P(~)). b) If P is constant and a(x) ~ 0 or if P is not constant and IK = ~ , then

P(~(,)) = ~(P(.)). a) Take a E a(x) and put P(t) := ~ a~t ~ . k=O

Then P ( a ) - P(t) = ~ a~a ~ a t t ~ -otk(otk - t k) k=O k=O k=O

= (~ - t) Z " ' ~ k=O

~t"-J-'

.

\j=O

Hence P(oO1-P(x)-(o~l-x)~ak(~o~xk-3-') k=O

.

\3=0

Since al - x is not invertible, P(a)l - P(x) is not invertibleeither (Corollary 2.1.2.6). Hence

P(a) E a(P(z))

and

P(a(z)) C a(P(x)).

b) Assume that P is constant, i.e. P = a with a E K . Then

v(~) =~I and a ( P ( x ) ) = a ( a l ) = {c~} = P ( a ( x ) )

(Proposition 2.1.3.2). If P is not constant and IK --q~, then take f~ E a(P(x)). There is an a E{C\{0} and a finitefamily (c~),et in q~, such that

~.I Algebras

~-

19

P(t) = (~H(a~ - t ) .

Then ~I - P(x) = a H(atl

- x).

LEI

Since fll - P(x) is not invertible, there is an t E I such that a,1 - x is not invertible (Corollary 2.1.2.7). Thus c~, E a(x) and f ~ - P(aL) = 0 i.e.

= P((~,) E P(a(x)), a(P(x)) C P(a(x)).

I

Remark. a) In the notation P(a(x)), we identified P with the map ]K

~]K,

a~P(a).

b) does not hold if the supplementary conditions are not fulfilled. Assume P is constant and a(x) = 0. Let E be the algebra of rational functions in the variable t. Then

o(t) =o (Proposition 2.1.3.3) and

P(a(t)) = 0 # a(P(t)) (Proposition 2.1.3.2). Assume now that IK = ]R. Let

P(t) "= t 2 E lR[t]. q~ is a real algebra,

o(0 =0 (Proposition 2.1.3.3), and

a(P(i)) = a ( - 1 ) = { - 1 } ~ 0 = P(a(i)) (Proposition 2.1.3.2).

~0

~. Banach Algebras

C o r o l l a r y 2.1.3.5 ( 0 ) Let E be an algebra. Take x E E and P E lK[t]\{0}. We assume that P ( x ) = 0 and Q(x) # 0 for every Q e IK[t] dividing P with

degree Q < degree P . Then a(x) is the set of the zeros of P . In particular, if E is unital and finite&mensional, then

Card a(x) < Dim E for every x ~_ E .

We have that P ( a ( x ) ) C a ( P ( x ) ) = a ( 0 ) = {0}

(Theorem 2.1.3.4 a), Proposition 2.1.3.2), i.e. a(x) is contained in the set of zeros of P . Now let a be a zero of P . Then there is a Q E lK[t] with P(t) = ( ~ - t)Q(t).

Since Q(x) ~ 0 and 0 = P ( x ) = (al - x ) Q ( x ) ,

a l - z is a zero-divisor and hence not invertible (Proposition 2.1.2.2). Thus a ~. a(x) and a(x) is the set of zeros of P . Now assume that E is unital and finite-dimensional. Take x E E and let n := Dim E . Then (x k I k e lNnU{O}} is linearly dependent. Hence, there is a P E lKIt]\{O} with P(x) = O. By the above, Card a(x) < n . Definition 2.1.3.6

( 0 ) (B. Pierce, 1870)

I The element x of an algebra

zs called idempotent sf x 2 = x .

C o r o l l a r y 2.1.3.7'

( 0 ) Let x be an idempotent element of the algebra E .

If x zs &fferent from 0 and 1, then

o(~) = {o, 1}.

~.1 Algebras

21

Define

P(t) := t ~ - t e ~ [ t ] . Then P(x)=O,

x#O,

x-l#O

and the assertion follows from Corollary 2.1.3.5.

I

C o r o l l a r y 2.1.3.8 Let E be a complex unital algebra. Take x E E and P, Q E r such that Q(x) is invertible, and R : a(x)

" lK,

a :

P((~)

"

Q(~)

"

If a(x) ~ 0, then a ( P ( x ) Q ( x ) - ' ) = R(a(x)). We first remark that or

o(QCx)) v Q(o(x))

(Theorem 2.1.3.4 a)), so that R is well defined on a(x). Take a E q~ and let

S := P - t r Q . The following are equivalent:

(~ q_ o ( P ( x ) Q ( x ) - ' ) , P(x)Q(x) -1 - a l is not invertible, S(x) is not invertible 0 e a ( S ( x ) ) = S(a(x))

(Corollary 2.1.2.6),

(Theorem 2.1.3.4 b)),

S(fl) = O, for some R(~/) = a ,

for some

t3 E a(x) ,

e o(x), i

~. Banach Algebras

Proposition

2.1.3.9

( 6 )

Let E be a unital algebra and take x E E . For

each a e l K \ a ( x ) , put xo : = ( a l - x) -1 . Then xa - xa = ( ~ -

a)zaxa

(the R e s o l v e n t E q u a t i o n } ,

{~o}~= {~}~ f o r every a, 13 e ~ \ a ( x )

.

Now ( a l - x)(x,~ - x a ) ( B l - x) = (/~1 - x) - ( a l - x) = (/3 - a ) l a n d so xo - xl, = (Z - a ) x o x ~ .

By P r o p o s i t i o n 2.1.2.8

{~}~= { ~ - ~}~= {~o}~ Proposition

2.1.3.10

( 0 ) ( J a c o b s o n ) L e t E a u m t a l algebra and x, y E E .

a)

a e I < \ { O } , a r a ( x y ) , z "= ( a l - x y ) - ' =} ( a l - y x ) - ' = : ( 1 + '

b)

a(xy)\{O} = a(yx)\{O}.

c)

~(~v) = ~(v~).

a) We have 1 = z(al - xy) - az - zxy,

1 = (al - xy)z = az - xyz, so t h a t z x y = az - 1, Hence

m

x y z = a z - 1.

yzx) .

I?,.1 Algebras

(1 + y z x ) ( a l

- yx) = al - yx + ayzx

= al - yx + ayzx

- yzxyx

~3

=

- y ( a z - 1)x = a l ,

(~1 - yx)(1 + y z x ) = a l - y x + a y z x - y x y z x = = al - yx + ayzx

- y ( a z - 1)x = a l ,

( a l - y x ) - l - 1(1 + y z x ) . b) follows from a). c) follows from b). Remark.

I

If we denote by u , and u t the right and the left shift of 22 , respec-

;ively (Example 1.2.2.9), then

o r o(~tu,),

o e o(u, ut).

This example shows the necessity to exclude 0 from the formula in b). P r o p o s i t i o n 2.1.3.11

( 0

)

If E

is an algebra, x E E ,

a n d ~ E IK\{0},

t h e n the f o l l o w i n g are e q u i v a l e n t :

~) ~ r o(~). b) f o r s o m e y ~_ E oly

-

1 -x

-

xy

=

O ,

xy

--

Ol

yx.

Let E be the unital algebra associated to E . Then, given y E E , ( a l - x)(11 + y) -- 1 + ( a y -

-1x -

xy)

in E . a ~ b. Put 1

y := ( a l - x) -1 - - 1 . Eg

By the above relation, 1

ay - -x

- x y - O,

xy - yx.

Ot

Hence Y=-a

b =v a is obvious.

1(1 x a

+xy)

EE.

I

2. Banach Algebras

e4

Corollary 2.1.3.12 ( 0 ) Let E, F be algebras and u" E ~ F an algebra homomorphism. Then

OF(UX) c ~ E ( ~ ) u { 0 } ,

~F(UX) < ~E(~)

for every x E E . If E is not umtal or if E, F , and u are unital , then

Take a E l g \ ( a E ( x ) U {0}). By Proposition 2.1.3.11 a =} b, there is a y E E with oty -

1 -x

ot

-

xy

=

O ,

xy = y x .

Then ~uu-

1

- = x - (=~)(uu) = 0 ,

(=~)(uu) = ( = y ) ( = ~ ) .

Hence, by Proposition 2.1.3.11 b =} a, a ~ a f ( u x ) . The final assertion follows from the fact that under the supplementary hypotheses 0 r af(UX)\aE(X). I Corollary 2.1.3.13

( 0 ) Let E be an algebra and u" E ---} ~ homomorphism. If u is not identically zero, then

an algebra

ux e o(~) for every x E E .

If E is unital, then u is also unital (Proposition 2.1.1.12) and the assertion follows from Corollary 2.1.3.12 (and Proposition 2.1.3.2). 1 Corollary 2.1.3.14 a) F o r x E F ,

( 0 ) Let F be a subalgebra of an algebra E .

aE(X) C aF(X),

rE(X) ~_ rF(X).

b) Assume E ts unttal, 1 E F , and there ts a vector subspace G of E such that the map FxG-----+E,

(x,y):

~x+y

is bijectwe and such that xy, yx E G for every (x, y) E F • G . Then

for every x E F .

12.1 Algebras

f25

a) follows from Corollary 2.1.3.12. b) First assume x invertible in E . Let (y, z) E F • G such that y-t-

Z --

X -1

.

Then xz = 1 - xy E FAG,

zx = 1-

yx E FNG,

so that XZ

--

ZX

--'- O .

It follows x y - y x --- 1

and x is invertible in F . By a) and the above considerations,

o~(~) =o~(~). Corollary

2.1.3.15

( 0 )

such that for (x,y) E E x F,

m

Let E be a n algebra a n d F a subalgebra o f E , xy E F

whenever

xy - yx.

Then

oE(~) = OF(~) for every x E F.

Take a E lK\{0}. By Proposition 2.1.3.11, 1

a ~ aE(x) r

3 y E E , ay - -x

a q~ a y ( x )

3y e F,

r

ay-

1

-x-

- xy = O , xy = xy ,

x y = O, x y = x y ,

ce

On the other hand, xy = yx, c~y-

for every y G E . Thus

I

-x

I

- x y = O ==~ y = - ( - x

I

+ xy) E F

fg. Banach Algebras

26

r OE(~) *=* ~ r o~(~). Assume that 0 ~ aE(X). Then x x -1 = x - I x -

1.

Hence 1 E F and x -l = lx -l E F . Thus 0 ~ aF(X). The converse is trivial.

ll

P r o p o s i t i o n 2.1.3.16

( 6 ) Let E be a unital algebra. Take x , y E E with

y invertible, and n E ~

such that xy = yx ,

Xn = O.

a) x ss a zero-divisor. n--I

b) ( y - x) -l = ~ x k y , k-l . k=O

c) o ( ~ ) = {o}. a) Let m be the smallest natural number with x m = O. The assertion follows from xm-lx

-- 0.

b) Use n-1

(y- z)~

n-1

xky-k-l = ~

k=O

k=O

n-I

xky -k - ~_, Zk+ly-k-I = 1 -

xny -n

"-

1.

k=O

c) Since, by a) and Proposition 2.1.2.2, x is not invertible, 0 belongs to a(x). By b), a l - x is invertible for every a E IK\{0}. Thus

o(~) = {o}. Definition 2.1.3.17

( 0 )

mm

Let E be an algebra and let E be the unital

algebra associated to E . A maximal proper (left, mght) ideal F of E zs called regular if there is a maximal proper (left, right) ideal G of E containing F and different from E .

If E is unital then every maximal proper (left, right) ideal of E is regular.

~.1 Algebras

~7

D e f i n i t i o n 2.1.3.18 ( 0 ) Let E bean algebra. The intersection of all regular maximal proper left ideals of E is called the radical of E (the intersection

is defined to be E if there are no regular maximal proper left ideals). E Is said to be semi-simple if its radical is {0}. If E is unital then, by Proposition 2.1.1.14 c), there are maximal proper left ideals of E , hence the radical of E is different from E . Proposition 2.1.3.19

( 4 )

Let E bean algebra, E the unital algebra

associated to E and R (resp. R) be the radical of E (of E ). Then R=EDR. We may assume E is not unital. Let F be a regular maximal proper left ideal of E . There is a maximal proper left ideal F of E containing F and different from E . Then E n F is a proper left ideal of E containing F and so

F=ENFDENR. Hence

RDEAR. Let F be a maximal proper left ideal of E . Define F := E A F . Then F is a left ideal of E . Assume that F is a proper left ideal of E . Let G be a left ideal of E containing F and different from F . Then F + G is a left ideal of E containing F and different from F . Since F is maximal,

E=F+G. In particular, there is a (y, z) E F x G such that

l=y+z. Then

x = x ( y + z ) = x y + x z ~. E A F + G

c G

for every x E E . Hence E = G and F is a maximal proper left ideal of E . Since it is obviously regular,

RcF=ENF,

RcEAR.

I

28

2. Banach Algebras

P r o p o s i t i o n 2.1.3.20

Let E be a commutative unital algebra and F the set

of nilpotent elements of E . Then F ts a proper ideal of E contained m the radical of E .

Take x, y E F , a, fl E IK, and z E E . There are m, n E ~1 such that x m .__ yn = 0 .

Then

--(

( a x + ~y)"+" = ~

)

m + n a~ ff,,+,,_~,xky,,,+,,_ k = 0

k=0

k

(XZ) m -- XmZm -- O,

and hence a x + ~ y , x z E F (Corollary 2.1.1.20). Since 1 r F , F is a proper ideal of E . Take x E F and let G be a maximal proper ideal of E . If x ~ G, then there is a (y,z) E E • G with xy+z--1

(Proposition 2.1.1.15). Since sty is nilpotent, z is invertible (Proposition 2.1.3.18 b)) and this is a contradiction (Proposition 2.1.2.9). Hence x E G. It follows that F is contained in the radical of E . i P r o p o s i t i o n 2.1.3.21

( 4 ) Let R be the radical of the umtal algebra E .

Gwen x E E , the following are equivalent:

a) z ~ R . b) For each y E E , there is a z E E with z(1 - yx) = 1. a =~ b. Assume that for every z E E ,

z(1 - ~x) # 1. Then, by Proposition 2.1.1.14 c), there is a maximal proper left ideal F of E containing 1 - y x . Since R C F and yx E R , we obtain the contradiction 1 =

(1 -

yx) + yx e F .

b ~ a. Assume that x r R. Then there is a maximal proper left ideal F of E with x r F . Since {yx + a I Y E E, a E F} is a left ideal of E containing F W {x}, it contains 1. Hence, there are y E E and a E F with

~,.1 Algebras

~9

a= 1-yx.

By b), there is a z E E with 1 -" z ( 1

-

yx).

But then l=zaEF,

I

which is a contradiction. C o r o l l a r y 2.1.3.22

The radical of a unital algebra is an ideal.

Let E be a unital algebra. R its radical, and take (x, y) E E x R . Assume y x ~ R . Then there is a maximal proper left ideal F of E such that y x ~ F .

Then { z y x + a [ ( z , a ) E E x F } i s a l e f t ideal of E containing F O { y x } . Hence there is a (z, a) E E x F with z y x + a = 1.

Thus x z y x -~- x a - x

(1 - x z y ) x = x - x z y x = x a ~_ F . By Proposition 2.1.3.21 a =~ b, there is a b E E with b(1 - x z y ) -- 1. Hence y x = y l x = yb(1 - x z y ) x = y b x a ~. F

which is a contradiction. Hence y x E R which proves that R is an ideal of E . m

P r o p o s i t i o n 2.1.3.23 left ideals F

( 4 ) Let E be a unital algebra and ~ be the set of

of E f o r whzch 1 - x ts invertible w h e n e v e r x E F .

radical of E is the greatest element of ~ .

T h e n the

30

~. Banach Algebras

Let R be the radical of E and take x E R . By Proposition 2.1.3.21 a =~ b, there is a y E E with y(1

-

x) = 1.

Thus 1 - y = -yx E R.

By Proposition 2.1.3.21 a =~ b, again, there is a z E E with 1 = z ( 1 - (1 - y ) ) = z y .

By Proposition 2.1.2.3, y is invertible and y-l=l-x.

Hence 1 - x is invertible and R E ~. Take F E ~ and x E F . Take y E E . Then y x E F and so 1 - y x

is

invertible. By Proposition 2.1.3.21 b =~ a, x E R . Hence F C R and R is the greatest element of ~. Corollary

2.1.3.24

I ( 4 )

Let R be the radical of the algebra E and take

x E R . Then

o(~) = { 0 } . By Proposition 2.1.3.19, we may assume E to be unital. By Proposition 2.1.2.9, x is not invertible. Thus 0 E a ( x ) . Take a E IK\{0}. Then !x~ E R , and by Proposition 2.1.3.23, 1 - ~x 1 is invertible. Hence a l - x is invertible, a r a ( x ) , and

o(~) = Proposition 2.1.3.25

( 7 ) I/E

(01.

u

is a un, tal algebra and x is an idempotent

element of E then

ix} c = x E x + (1 - x)E(1 - x) = {y E E}

3z,,z~ e

Let y, zl, z2 E E such that Xy -" ZlX,

Then

yX = XZ2 .

E, xy = z]x, yx = xz2}.

$.1 Algebras

81

y -- x y x + x y ( 1 - x) -4- (1 - x ) y x + (1 - x ) y ( 1 - x) =

--- x y x + z l x ( 1 - x) -t- (1 - x ) x z 2 + (1 - x ) y ( 1 - x ) =

= x y x -4- (1 - x ) y ( 1 - x ) e x E x -4- (1 - x ) E ( 1

- x).

Hence

{y E. E [ 3 z l , z 2 ~. E , x y = ZlX, y x = xz2} C

C x E x + (1 - x ) E ( 1

- x).

The inclusions

x E x + (1 - x ) E ( 1

- x) C { x } c c

C {y E E I ::lzl, z2 ~. E , x y = z l x , y x = xz2} are trivial.

I

32

2. Banach Algebras

2.1.4 Standard Examples

Example 2.1.4.1 ( 0 )

Let (Et)t~T be a family of algebras. The vector space I-I Et is an algebra with respect to pointwsse multiplicatwn. It ts commutative iff

tET

each factor Et (t E T) is commutative. It has a unit iff each factor Et (t E T) has a unit; then the element z E I-I Et is invertible iff each xt (t E T ) i s tET

invertible and in this case ( ~ - , ) , = (z,) -~

.for every t E T .

I

Example 2.1.4.2 ( 0 )

Let T be a set and E an algebra. The vector space E T is an algebra with respect to pointwise multiplication. E r is commutative whenever E is commutative. If E is unital, then E T has a canonical unital algebra structure, with T

;E,

t:

~1

as unit. The element x E E r is inverttble sff x(t) ts mverttble for each t E T , tn which case

x-'(t)

= ,~(t)-'

for every t E T .

Example 2.1.4.3

I ( 0 ) Let T be a set. The vector space ]I4,r ts a commu-

tative umtal algebra with respect to the pointwise multiplication, the unit being the constant Junction 1, i.e. e r . An element x of IKr ts invertible if it does not vanish anywhere, in which case 1

z-'(t) = z(t-'~ for every t 6 T .

tZC(T) and c(T) are unttal subalgebras of

x of t ~ ( T ) (of c ( T ) ) is invertible tn t~176 (in c ( T ) ) zff inf Ix(t)l > 0.

tET

eP(T) and co(T) are ideals of t~176 (p e If T zs infimte, then: e ~ ( T ) is not an zdeal o / I K r .

a)

[1, cr ).

IK T .

An element

9.1 Algebras

b) c) d) g

33

c(T) ,s not an ideal of s176176

~'(T) and co(T) are proper ideals of ~ ( T ) . co(T) is a maximal proper ideal of c ( T ) . (E := e~176 then

E := IKr

aE(X) = x ( T )

(aE(x) = x(T))

for every x 6 E .

E x a m p l e 2.1.4.4

I

( 0 ) Let T be a topological space. Then C(T) is a unital

subalgebra of e~176 The elements of C(T) are invertible in C(T) iff they are invertible in g ~ ( T ) . If T is compact, then the elements of C(T) are invertible in C(T) iff they are invertible in IKr . In particular,

a(~) = ~(T) for every x E C(T). If T is locally compact, then Co(T) is an ideal of C(T).

I P r o p o s i t i o n 2.1.4.5

Let T be a topological space and ~d the set of maximal proper ideals of C(T). Define ~'t := {x E

C(T) Ix(t) = 0}

for every t e T . Then ~t e 7d for every t E T and if T is compact, then the map T

;~,

t:

".T't

zs bijectwe.

Take t 6 T and set x':C(T)

; lK,

x ,

~. x(t) .

Then x' is a homomorphism of unital algebras and ~'t = Ker x'. By Proposition 2.1.1.12 b), ~'t 6 ~. Now assume that T is compact. By Urysohn's Lemma, the above map is injective. Assume the map is not surjective. Take :r e ld\ { Yt l t e T }.

3,l

2. Banach Algebras

For each t E T , there is an xt E ~'\~'t 9 Then Ixt] ~ = 5txt E 9r for every t E T . Put

u, := {Ix, I > o} for t E T . Each Ut is an open set containing t and (UI)tE T is an open covering of T . Since T is compact, there is a finite subset A of T such t h a t

T=Uut. tEA

Put 9= ~

I~,1 ~ e 3=,

tEA

and take s E T . There is a t E A such t h a t s E Ut. Then

x{8) ~ Ix,{8)l 2 > o, x does not vanish on T , and so it is invertible (Examl..le 2.1.4.4). This is a contradiction (Proposition 2.1.2.9). Ex~mpl~

2.~.4.6

l

( 0 ) L , E be ~ v~to~ ,p~e ~,d ~ the wao~ ,p~e o/

linear maps E -+ E . ~ is a unital algebra with respect to the multzphcation

J: • y

~ :r,

(u,v)"

~vou

An element of .T" is invertible tff it is bijective. If E is a normed space, then s is a unital subalgebra of .7r. If E is complete, then an element of s

zs invertible m s

iff it is bijective, i.e.

iff it ~s invertible in ~ . The last assertion follows from the Principle of Inverse Operators.

R e m a r k . . 7 r and s

I

are commutative iff the dimension of E is at most 1.

Since ~" is a unital algebra, we often write vu instead of v o u (u, v E .T'). Example

2.1.4.7

Take n E IN. s

n) ts a unttal algebra whose only proper

ideal is {0}. If E is a unital algebra, then every unital algebra homomorphism u: s

n) ~ E is mjective. An element of s (IK") is a zero-divisor iff st is

not invertible.

~.I Algebras

35

Let ~" ~ {0} be an ideal of s Take u e Jr\{0} and x E lKn\Keru. Let (es)ze~ be the standard algebraic basis of lKn . Take j E INn, and v E s ~) such that ~JUX --" e j .

Define n

w" IKn

} lKn

Z i=1

Uj : - - t / ~ W .

Then uj E ~" and

Hence 1=

u3E~ j=l

and ~" = s

Hence {0} is the only proper ideal of s

By Proposition

2.1.1.10, every unital algebra homomorphism u" E(IK ") --+ E is injective. The final assertion follows from the fact that an element of s n) is invertible iff the corresponding determinant does not vanish. / P r o p o s i t i o n 2.1.4.8

Let E be a normed space and take u, v E s U ' O "-- V U

with

,

P E ]K[tl, and Q e lK[s, tl. Then P(u)' = P(u') , An element of s

Q(u, v)' = Q(u', v') .

is idempotent iff it is a projection in E .

The assertion follows immediately from the Corollaries 1.3.4.3, 1.3.4.4, and 1.3.4.5. 1 P r o p o s i t i o n 2.1.4.9

(Phillips) Let E be a Banach space. Then

o(~,) = a(~,') for every u e s

2. Banach Algebras

36

Take a E IK. Then ( a l - u)' = a l - u' (Corollary 1.3.4.3, Corollary 1.3.4.4). By Corollary 1.4.2.5, c ~ l - u is invertible iff ( a l - u)' is invertible. Hence c~ E a(u) iff a E a ( u ' ) . I

Example 2.1.4.10 ( 0 ) For every x E ~ r ,

Let T be a finite set and take k ' T •

define

-kx " T ---, ~ ,

s:

~ E k(s, t)x(t) tET

and U" ]K T ~

~T

1

~1:

~" k~l 9

Let P(r) denote the characteristic polynomial of u (with the vartable r ) i.e. the determinant of the matrix

[rJ(s, t) - k(s, t)]o,t~T . a)

a E a(u) sff P ( o ) = 0 for every a E IK.

b)

Carda(u) < CardT.

e)

If n is odd or if ~ = C , then a(u) ~ {0}.

a) a l - u

is not invertible iff it is not injective, i.e. iff there is an z E ]KT\{0}

with

The assertion now follows. b) and c) follow from a).

I

L e m m a 2.1.4.11 ( 0 ) Let E be a vector space and (u,,),.,e~2 a family of linear maps E --+ E such that UsjUkl -~- ~jkUd

for every i, j, k, l E IN2. Then the vector spaces Im ull and Im uz2 are isomorphic.

~.1 Algebras

37

Now UllUI2

----- U12 ~

U22U21 : - U21

Im U12 C I m u l l ,

I m U21 C I m U22.

3efine v 9 Im u]l

} Im

w 9 I m U22

) Im u~,

U22,

X~

U21X,

y .'. ~ ul2Y,

Then WVX

- - U I 2 U 2 1 X - - U l l X = X~

v w y = u21u~2y = u22Y = y , for every x E I m u l t , y E I m

U22.

Hence v, w are i s o m o r p h i s m s of vector spaces. m

Proposition 2.1.4.12

( 0 ) Let E be a umtal algebra, (x,3)(,,3)el•

family in E such that X , . I X k t - " (~3kXal

for i, 3, k, ~ E I and

~ '~xu

= 1.

zEl

Let n be a natural number and u" E

~, s

n)

a unital algebra h o m o m o m o r p h i s m . Then n

Card I

EIN.

Define uzj = uxz~, for i , j E I . T h e n

q, = D i m I m u,,

a finite

Ig. Banaeh Algebras

38

tli3Ukl "- {~3kU d

for i, j, k, l E I and E

u,, = 1.

sEl

By Lemma 2.1.4.11, q, =qs for every i , j E I . Hence

n Card I = ql E IN. Corollary

2.1.4.13

( 0 )

I

Let n , p E IN and let (ei),els. be the standard

algebraic basis of IK n . Given i, j E INn, let x, s denote the operator on ]K n with XIjek ~

for every k E

~jkes

INn.

a) i, j, k, t e IN, =~ x,jxkt = 6~kx,t. b)

~']~x,, = 1. s--I

c)

There is a unital algebra h o m o m o r p h i s m

z:(~:")

-, s

a) For m E INn, x,jxktem

= Xljr

=

6tm6~ke, = 6jkx,tem

so that X I j X k t -~ r

b) For j E I N . , x,,

ei =

x,,e s =

z=l

I=1

so that n

Z s'-']

XH -- 1.

6,ie, = e~ i=1

~.1 Algebras

39

c) By a),b), and Proposition 2.1.4.12, if there is a unital algebra homomorphism

s

~ s

then n dvides p. The converse is easy to see.

I

C o r o l l a r y 2.1.4.14 Let T be an infinite set. Take p E [1, cx)] and n E IN. Then there is no unital algebra homomomorphism

, s

s

We may assume there are sets I, S with T=IxS,

Card/=n+l.

For x e eP(T) and i, j E I , define Xa$ " T

~]K

(k s ) ~ ~ '

x(j,s)

'

[

0

if k = i if k ~ i

and u,~:if(T)

~ffCT),

y,

)y,,.

Then Uzj Ctkt ---- ~jk ~lzt

for every i, j, k, t E I and U. = Z iEl

1.

By Proposition 2.1.4.12, there is no unital algebra homomorphism s

~s

I

T h e o r e m 2.1.4.15 Let E be a finite-dimensional complex unital algebra. If E is a division algebra, then E =(131.

40

12. Banach Algebras

Take x E E \ { 0 } . Since E is finite--dimensional, (x")n~t~ is linearly dependent. Hence there is a P E r

with

P(~) = o . There is a finite family (a,)tct in 9 and an a E r

with

,~,).

r'(t) = a 1-I(t tel

Then

H(x - a,1) = P ( x )

= O.

tel

Hence there is a t E I such that x - a~l is not invertible (Corollary 2.1.2.7). Since E is a division algebra, x - c~,l = 0 ,

x = a,1 E r Therefore E = r

II

Proposition 2.1.4.16 ( 0 ) Put [i O ] [ O 1] I := J := 0

-i

'

-1

0

,

and let E denote the real vector subspace of s162

K :=

[0 z ] i

0

generated by { 1, I, J, K } .

a) i s=J2=K s=_l IJ = -JI b)

= K ,

JK = -KJ

= I ,

KI = -IK

= J.

1, I, J, K are hnearlzt independent.

c) Ilal + ~I + 7 J + ~KII ~ and the determinant of c~l + 13J + 7 J + 5 K are equal to a s + ~3~ + 7 ~ + 6 ~ and ( a l +/31 + 7 J + 5K)(c~l - / ~ I - 7 J - 6K) = (a ~ + / ~ + 7 ~ + 62) 1

for every a, B, 7, 6 E I t .

~.I Algebras

d) E is a real unital subalgebra of the real unital algebra s

~1

It is a

division algebra unth respect to this structure.

a) is a simple calculation.

b) Take a , ~ , 7 , 6 e r

with al + ftI+TJ+

5K = 0.

Then a

+

Bi

7

-7

+

6i

a-

+

6i]

#i

=0.

Hence

a=Z=7=~=o. c) is a simple calculation. (Use Example 1.2.2.8.) d) By a), E is a real unital subalgebra of the real unital algebra s By b) and c) it is a division algebra. Corollary 2.1.4.17

I

( 0 ) (Hamilton, 1843) Let 1, i , j , k be the u m t vectors

of IR 4 . There is a umque multiplication on the vector space ]R 4 such that 1 is a unit and i2 = 32 = k 2 = - 1

ij = - j i = k ,

jk = -kj

= ~,

kz = - i k = j .

Then

tl~il~ = II~li~ll~ll~ for every x, y E IR 4 . IR 4 is a division algebra with respect to this multiplication. It is usually denoted by IH and its elements are called q u a t e r n i o n s .

lH is isomorphic to E in Proposition 2.1.4.16.

I

The real unital algebra structure of ]H cannot be extended to a complex unital algebra structure, as shown by Theorem 2.1.4.15. But Remark.

{,~ + fli l ~, f l e IR} is a unital subalgebra of lH, which is isomorphic to the real algebra q~.

4~

~. Banach Algebras

Proposition

2.1.4.18

Let E be a finite-dimensional real unital algebra and

take x E E \ I R 1 . If E is a division algebra then there are a, ~ E IR with z 2 + ax + fll =

O,

a 2 < 41~.

Put 1 y := V/4f~ _ a 2 ( a l + 2x).

Then y2 = - 1 .

Since E is finite-dimensional, (xn)ne~ is linearly dependent. Hence, there is a P E IR[t]\{0} with P(x) =0.

There are two finite families (c~,),et , (/~),et in IR and an a E IRk{0 } such that P ( t ) = ~ 1-[(t ~ + . , t +

~,).

tEI

Then 17I(~ ~ + , , ~ + ~,1) = P ( ~ ) = 0. tel

Hence, there is a t s I such t h a t x ~ + ~,x + ~,1 is not invertible (Corollary 2.1.2.7). Since E is a division algebra x 2 + a,x +/~L 1 = O. If a, _> 4/~ then there are 7, ~ E IR with (x + 71)(x + 51) = x 2 + a , x + ~, = O. Since E is a division algebra, this implies that x E IR1 (Corollary 2.1.2.6), which contradicts the hypothesis. Hence 2 4fl~, > c~,.

Then y2 =

1 (a 21 + 4ax + 4x 2) = 1 (a 21 + 4ax - 4 a x - 4/~1) = - 1 I 4f~- a 2 4f~ - ~2 "

~,.1 Algebras

~roposition 2.1.4.19

43

Let E be a f i n i t e - d i m e n s i o n a l real unital algebra. Take

x, y E E linearly i n d e p e n d e n t with X 2 ___ y 2 = _

1.

I f E is a division algebra, then 1, x, y are linearly i n d e p e n d e n t and there is an

El-

2,2[

such that xy+yx=al.

By hypothesis, x y ~ IR1. So, by Proposition 2.1.4.18, there are a,/3 E IR with (xy) 2 - a ( x y ) + t31 = O,

a 2 < 4t3.

Then 0 -- x ( x y x y 0 - y(-yxy 0 = (-yxy 0 -

~ x y +/31) = - y x y

+ ~ y + 13x,

+ a y + 13x) = x y - a l + 1 3 y x , + a y + 13x)y = y x - c~l + ~ x y ,

( x y - a l + t3yx) - 13(yx - ~1 + 13xy) -

-- x y - a l + 13yx - f l y x -t- a ~ l - fl2xy --

-- a(/3 - 1)1 - (/32 - 1 ) x y - ( t 3 - 1 ) ( a l - (/3 + 1)xy) : 0 = (/3 - 1)(cd - (t3 + 1 ) x y ) y = (i3 - 1 ) ( a y + (/3 + 1 ) x ) . Since fl+l>l and {x, y} is linearly independent,

~y + (~ + 1)x # 0. Thus /3--1,

O~2

<4

xy + yx = al .

Take a, b, c E lR with

~. Banach Algebras

44

a l + bx + cy = O.

Then a x - bl + c x y = 0

and a x - bl + c y x = O.

Then 2ax-2bl+cc~l=O,

-4a 2=(2b-ca)

2ax=(2b-ca)l,

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, s=l

I(a,j, a'.,,)l

__ ~

3=1

/=1

m~ /=1

j=l

__

3=1

m

R

/=1

3=1

for every a E Era,., so a' E (Era,.)' and t71

i=1

j=l

Let (p, q) E IN,, x IN,,, such that

II.~II = ,EINm +up II.;,II. sEINn

Further let 0 E ]0,1[ and x E E # such that l

We set a "=

(+,p+qjx)r

;,<m,, 9

Then a E Em,,~,

il~ll = ii~ll _< ~, and 171

I1~'11>-I~'(~)1 =

Itl

~5:< aij, a#i'> s=l

= I<+,~

3=1

>- o,o;,,,.

Since 0 is arbitrary we get

I1~'11 _~ lla~,ll-

~up lla~,ll.

w,iE~,m j E Isln

By Proposition 1.2.1.19 b) (and Remark of Definition 1.1.5.2), the map

E',~

.~(E~,.)',

is an isomorphism of vector spaces,

~':

;~'

i

50

~. Banach Algebras

Proposition

( 0 ) Let n E IN. If q~ (resp. ~ ) is isomorphic to a

2.1.4.26

unital subalgebra of P~n,n (resp. Cn,n, resp. l ~ , n ) (Propositson P,.1.4.~ b),c)) then ~ E IN (resp. ~n E ]IN, resp. ~n E IN ). We identify C (resp. ~ ) with the corresponding unital subalgebra. Then ]Rn and ~n endowed with the maps

are vector space over C or over Ill. If A is an algebraic basis of this vector space then

A U (iA)

(resp. A U (jA),

resp. A U (iA) U (jA) tJ (kA))

is an algebraic basis of the vector space l~ n or 4]n . Hence n = 2Card A

(resp. n - 2Card A,

resp. n - 4Card A)

so that n__CardAElN 2

n

(resp. ~ - - C a r d A E I N

n

resp. -: -- C a r d A E IN). I 4

~.1 Algebras

51

2.1.5 Complexiflcation of Algebras Complex algebras are easier to handle than real ones. In order to establish some properties of real algebras it is sometimes useful to associate to a real algebra E a canonical complex algebra /~ defined on the set E x E (called the complexification of E ) and to deduce the required properties for E using the corresponding properties of ~:. L e m m a 2.1.5.1

"( 0 )

Let E, F, G, H, I, J be real vector spaces and let

E • F --+ H , F x G --~ I , H • G -+ J , E x I -~ J be bilinear maps written in the product f o r m

(a, b): -~- ab. We define the map E 2 x F 2 -+ H 2 , F 2 x G 2 -+ 12 , H 2 x G 2 -+ j 2 , and E 2 x I s --~ J~ by

((a, b), (c, d)): a)

; (a, b)(c, d) :- (ac - bd, ad + bc).

The above defined maps are bilinear.

b) /f (ab)c = a(bc) for all (a, b, c) E E x F • G , then ((a,b)(c,d))(e, f ) = (a,b)((c,d)(e, f ) ) .for all ((a, b), (c, d), (e, f ) ) e E 2 • F 2 x G 2 The same relation holds if we define the map F 2 x G 2 --+ 12 , H 2 x G z --~ j2 by

((a, b), (c, d)):

; (ac + bd, bc - ad).

c) E 2 endowed with the map q3xE z~E

2,

((a, /~), (a, b)) : ; ( a a - f ~ b , a b + f ~ b ) ,

where ~ was identified w~th ]R 2 , is a complex vector space, called the o

cornplexifieation o l E and denoted by E .

d)

Themap

k•

((a, b), (c, d)):

~ (a, b)(c, d)

is C-bilinear. e)

If (z,),~t is an algebraic basis of E then ((z,, O)),~t is an algebraic basis o

oYE.

5~

~. Banach Algebras

a) is easy to see. b) We have ((a,b)(c,d)(e, f ) = (ac - bd, ad + bc)(e, f ) =

= ( ( a c - bd)e - (ad + bc)f, (ac - b d ) f + (ad + bc)e) =

= (a(ce - dr) - b(de + c f ) , a ( c f + de) + b(ce - dr)) =

= (a,b)(ce - df, c f + de) = ( a , b ) ( ( c , d ) ( e , f ) ) .

The second assertion follows from ((a, b)(c, d))(e, f) = (ac - bd, ad + bc)(e, f) =

= ( ( a c - bd)e + (ad + bc)f, (ad + bc)e - ( a c - bd)f) --

-- ( a ( c e + d r ) - b ( d e - c f ) , a ( d e - c f ) + b(ce + d r ) ) --

-- (a, b)(ce + dr, de - c f ) - (a, b)((c, d)(e, f ) ) . c) follows from a ) a n d b ) b y replacing E and G with lR and F , H , I , J with E . d) follows from a),b), and c). e) Take h E C (0 with

~--~h(~)(x,, 0)=0. tEI

Then (~ tEI

re h(e)x, ,

~

im h(i,)x,) = O,

~EI

so that re h(t)x, = ~ tEI

im h(i,)x, = O,

tel

and thus h(~) = 0

~.1 Algebras

53

for every t E I . Hence ((x,, O)),r is linearly independent. Take (x, y) E E 2 . There are f, g E ~(x) such that x --" ~

f(t)x=,

y -'- ~

tEI

g(t)x=.

tel

Define h "I

;qJ,

t:

~ f (t) + ig(~).

Then

tel

tel

tel

Hence ((xt,0))tel is an algebraic basis for E 2 .

I

Remark. a) We sometimes identify E with the real vector subspace E x {0} o f / ~ . By this identification we may write

(~, v)

= (~, o) + i(v, o) = 9 + iv

for every (x, y) E E . b) r is the complexification of IR. o

E x a m p l e 2.1.5.2

( 0 ) Let T be a set and W

the complexification of the

o

vector space ]R T 9 Given (x, y) E ~2~T , define (x,y) " T

- ;C,

t ~-+ x(t) + iy(t).

Then the map o

W

•T ,

(X, y) :

~, (X, y)

is linear and bijective (we may therefore identify the complexification of IRv with cT ).

It is obvious that the above map is lR-linear and bijective. Take (x, y) E o

W.

Then i(~, v)(t) = (-u,:O(t) = - v C t ) + i~(t) =

54

~,. Banach Algebras

N

= iCe(t) + iu(t)) = i(~, ~)(t) = (i(x, u))(t) for every t E T, so i(~, ~ ) = i ( ~ ) . Hence the above map is C-linear. L e m m a 2.1.5.3

I

( 0 ) Let E , F be real vector spaces, L(E,F) the real

~ a o ~ , ~ e o/ l~,,e~ m~p, o! E ~nto F, k , "~, ~,,d L(E, F) the ~ompte~fiations of E, F , and L(E, F) , respectively, and L(E, F) the complex vector 0

0

0

space of the linear maps o.f E into F . For (u, v) E L(E, F) define

(~, v'-') . k Then (u, v~) E L(E, F)

(~, ~) : ~ ( ~ - ~ , ~y + ~ ) .

, b,

for every (u, v ) ~

LCE , F ) :

fLeE, F),

LIE, F) (u,v).

~.d ~

map

~(u,v"~

is linear and bijective. (Thus L (E, F) and L ( E , F ) may be identified.} Moreover, if G is another real vector space then (with obvious notation) (c, d) o (a, b) = "(c o a - d o b, c o b + d o a) for every (a, b) E L(E, F) and (c, d) E L(F, G).

The map L(E,F) • E

~ F,

(u,x):

~ ux

is bilinear. By Lemma 2.1.5.1 d), the map L ( E , F ) x E , :; '~', ((u,v),(x,y))"

~ (u,v~)((x,y))

o F) for every (u, v) E/~(E, F) and the map is bilinear. Hence (u, v) E L (E, o

L(E, F) ~

L

o

(E, '~')

,

(u, v)"

-

-

~

; (u, v)

is linear. It is obviously injective. For w E L(E, ~ b) , we define the maps u, v of E into F by

(~, w) := ~((~, o)) for all x E E. Then u, v E L(E, F) and

~.1 Algebras

55

N

(~,,)((~,

0)) =

( ~ , , ~ ) = ~((~, 0))

for every x E E. Hence

(~, ~) = and the m a p N

L(E, F)

(u, v) : ; (u,v)

~. L(E, F) ,

is bijective. The maps L(F,G) x F L(E, F) • L(F, G)

(v,y).~ ~, vy,

; G,

(u, v) : ; y o u ,

; L(E, G) ,

L(E,G) x E

; G,

(w,x) ~

wx

being bilinear, the final assertion follows from Lemma 2.1.5.1 b). Corollary 2.1.5.4

1

( 0 ) Let E be a real vector space, E its complexifio

cation, E' (resp. (E)') the algebraic dual of E (resp. of E ) and ~ o

the

complexification of E'. Given (x', y') E E' , define

(~',~') k - - , r

(~,~) : ,~~'(~)

- y'(u) + i(~'(u) +

~'(~)).

o o

Th~

(~', y') e (E)' /o~ ~.~r~

(~', ~') e " ~

~ . d the map

O

O

is linear and bijective. (Thus (E)' and ~

Example 2.1.5.5

may be identified.)

II

Let m, n E lN. For a E IKm,n, define "d " ] K n

~ lK m,

x ~

ax ,

and identify IKm.n with the vector space L(IK n, I~ m) of hnear maps of IK" into I~ m via the map ]Km, n

~ L ( ] K n, ] K m ) ,

a :

~ "a.

If L(]l:tn, IRm) denotes the complexification of L(]R n, ][:tm) then the map o

L(IR n, ]Rm) is linear and bijective.

}

L(ffJn,ffJm),

(a, b):- ; a + ib

56

~. Banach Algebras

The assertion follows from Example 2.1.5.2 and Lemma 2.1.5.3. Proposition 2.1.5.6 o

o

m

( 0 ) Let E , F be normed real vector spaces and

o

,y,,.?,

o

E t) be the complexifications of the unE, ~', , , so F) , and s derlying real vector spaces E, F, E', F', s F) , and s E') , respectively. We assume that E and F are endowed with norms such that the canonical bijectwns o

~ExE,

~"

~FxF

are isomorphisms of real normed spaces. Given (u, v) E ~:(E, F), define

(~, ~)" ~ - - , ~ ,

(~, v)~ ; ( ~ - ~ , ~ + ~ ) .

o

For (u, v) ~. s

E') put

(,,,~) . (~)'

; (~)' , (~',r . ~ (u~' - ,,r ,,r + ,=') .

0

Then (u, v) e s

F) and (with the identification of Corollary ~.1.5.4) (u, ~,) = (u', r o

.for every (u, v) e s

and the map

s

F)

~s

(u, v) : ; (u,v)

,

is an isomorphism of complex vector spaces.

The relation (u, v) E s

0

follows easily from Lemma 2.1.5.3. Take

o

(x, y) E/~ and (x', y') E " ~ .

((~,~), (~',r

r

Then

= ((~,~),(~'~'_

(~,~'~')_ (~,r162-

=

Cy', ~'y' + r

(y, ~'y') _

=

(~,r

+i((x, u'y')+ (,, v'x')+ (y,u'x') - (y,v'y'}) = =

(~, ~') -

+i((~,

( w , y') - ( ~ , y') - (vy,

r + (~, x') + (~, ~') -

= ((,= - vy,,,y + vz),

(~', r

~')+

(~y, r

= ((,,, v)(~,

y),

=

(~', r

Hence (~', r The last assertion follows from L e m m a

= (~, ~) 2.1.5.3

B

~.1 Algebras

P r o p o s i t i o n 2.1.5.7

57

( 0 ) Let E be a real algebra and k the complexifi-

cation of the underlying vector space (Lemma 2.1.5.1). Then E endowed with the bilinear map E• k

~E,

((a,b),(c,d)):

~ ( a c - b d , ad+bc)

is a complex algebra, called the cornpleziflcation of E . If E is unital, E is unital too. I f 1 is a unit of E , then (1, O) is a unit of E . If E is commutative, then E is also commutative. If every left (right) invertible element of E is invertible, then the same holds for E . It is easy to see that the above map is bilinear. By Lemma 2.1.5.1 b), this map is associative. Assume that /~ is unital and let (a, b) be the unit in /~. Take x E E . Then

(x, O) -- (x, O)(a, b) - (xa, xb) , (x, O) = (a, b)(x, O) = (ax, bx) , so that

x=xa=ax. Hence a is a unit of E . The final assertions are trivial. C o r o l l a r y 2.1.5.8

II

( 0 ) Let E be a real vector space, L(E) the unital

real algebra of linear maps E -+ E , (Example 2.1.~.6), E and L(E) the complexification of E and L(E), respectively, and L(E) the complex unital algebra of linear maps k ~ E. Given (u, v) E L(E), put

Then the map N

L( E )

~ L( E~ ,)

(., v ) : " (., v)

(Lemma 2.1.5.3) is an isomorphism o.f unital algebras. Thus the unital algebras o L o L(E) and (E) may be identified. The assertion follows immediately from Lemma 2.1.5.3 and Proposition 2.1.5.7. II

58

~. Banach Algebras

o

Example 2.1.5.9

For n E IN, let R,,,= be the complexification of ]R.n,n.

Then the map o

IR,,,n 9 } C=,n ,

(a, b) : ~ a + ib

is an isomorphism of unital algebras.

Proposition 2.1.5.10

1

( 0 ) Let E be a real algebra and E its complexifi-

cation.

a) If F is a subalgebra of E , then

is a subalgebra of E . b) If F is a (left, right) sdeal of E , then

{ (~, y) I ~, ~ ~ F} is a (left, righO ideal of E.

c) Let F b~ a,, ide~t o! E a,,d p" E --+ E/F, q " k quotient maps. Define r" E ---+ E / F ,

(x, gt) ~

-, k/:

the ,'~,p~ai,,~

(p~',py).

o

Then there is a unique u " ~ - + isomorphssm of complex algebras.

Elk

such that q = u o r

The proof is a simple verification.

1

Proposition 2.1.5.11

( 0 ) Let E , F be real algebras, E , F xificationn, and u" E --~ F an algebra homomorphism. Then

~ ~ --, ~,

and u is an

their comple-

(~, ~)~-, (~, .y)

is an algebra homomorphzsm, surjective iff u is surjectwe. The map u ~ u is injective.

It is obvious that fi is linear. If (a, b), (c, d) E/~, then ~((a,b)(c,d)) -- u ( ( a c - bd, ad + bc)) = = ((ua)(uc) - (ub)(ud), (ua)(ud) + (ub)(uc)) = -- (ua, ub)(uc, ud) - ~((a,b))~((c,d)) , o

so u is an algebra homomorphism. The last assertions are easy to see.

1

~.I Algebras

Proposition

2.1.5.12 ( 0 )

Let E

be a unital real algebra and k

59

its

complexification. Then x E E is invertible in E iff (x, O) is invertible in E .

Assume (x,0) invertible in /~ and let

(v, z) := (x, 0) -t . Then

(1, o) = (~,,)(~, o) = ( w , z~), (1, 0) = (~, 0)(~,,) = (~v, = ) . We get xy = yx = 1,

so x is invertible in E . The converse is trivial. P r o p o s i t i o n 2.1.5.13

I

( 0 ) Let E be a real algebra and F its associated

unital algebra. The F is the unital algebra associated to E .

The map ((,,,0~), (O,v)) .

. (,~ + i~,~: + iv)

is an isomorphism of unital algebras. Corollary 2.1.5.14

I

( 0 ) If E is a real algebra and x E E , then o~(x) = la n a~((~, o)),

where E denotes the complexification of E .

By Proposition 2.1.5.13 we may assume that E is unital . Take c~ E IR. By Proposition 2.1.5.12, a l - x is not invertible in E iff a(1,0) - (x, 0) is not invertible in E so e ~(~) ~ ~ e %((~, 0)). n P r o p o s i t i o n 2.1.5.15

( 4 ) Let E be a real algebra and take x e E . Then

the following are equwalent:

a) x belongs to the radical of E .

~. Banach Algebras

60

(x, O) belongs to the radical of E .

b)

In particular, if E is seres-simple, E is also semi-simple. By Proposition 2.1.3.19 and Proposition 2.1.5.13, we may assume E unital. a =~ b. Take (a, b) E E . By Proposition 2.1.3.21 a =~ b, there are c, d E E such that

c ( 1 - ax) - l ,

d ( 1 - (a - bxcb)x) - l .

Then

(d, dbxc)((1, 0) - (a, b)(z, 0)) -

= (d, dbxc)(1 - a x , - b x ) = (d(1 - az) + dbxcbz, dbxc(1 - ax) - dbx) = = (d(1 - (a - bxcb)x),dbx - dbx) = (1, 0). By Proposition 2.1.3.21 b =~ a, (x, 0) belongs to the radical of E . b =~ a. Take y E E . By Proposition 2.1.3.21 a =~ b, there is an (a, b) E such that

(a,b)((1,0)- (y,O)(x,O))= (1,0). It follows that a(1 - yz) = 1. By Proposition 2.1.3.21 b ~ a, x belongs to the radical of E . P r o p o s i t i o n 2.1.5.16

I

Let E (resp. F ) be a real (resp. complex) algebra,

the complexification of E (Proposition ~.1.5.7), and jr (resp. ~ ) the set of real (resp. complex) algebra homomorphssms of E (of E ) into F . For u E Y , define o

u 9 E

.~F,

(x,y):

;ux+iuy.

o

Then u E ~ whenever u E jr and the map jr

;~,

u:

~uo

is bijective. If E, F , and u are unital, then ~ is also unital.

~,.1 Algebras

61

Let u E ~r. Let a, b, c, d E E and a,/~ E ]Ft. Then u((a, b) + (c, d)) = u((a + c, b + d)) = u(a + c) + iu(b + d) =

= ua + uc + iub + iud = u((a, b)) + u((c, d)), u(~ + i~)(a, b)) - ~((~a - 13b,ab + 13a)) '- u(~a - ~b) + iu(ab + 13a) = = ~

- Z~b + i ~ b + i a ~

= (~ + i a ) ( ~

+ iub) = (~ + ia)~,((~, b)),

~((a, b)(c, d)) = u ( ( a c - bd, ad + bc)) = u ( a c - bd) + iu(ad + bc) = = (~)(~)

- (~b)(~d) + i ( ~ ) ( ~ d ) + i ( ~ b ) ( ~ ) = (u~ + i u b ) ( ~ + i~d) = = ~,((a, b));,((c, d)).

Hence u E {7. Given v E ~ , define

u'E

}F,

x~v((x,

0)).

Then u E ~" and ~,((x, 0 ) ) = v((x, 0)) for every x E E . Hence ~ = v and the map Jr---rG,

u:

~ uo

is surjective. Since it is obviously injective, it is bijective. The last assertion is trivial.

II

C o r o l l a r y 2.1.5.17 Let E be a real algebra and E its complexification. Let jr and ~7 denote the sets of algebra homomorphisms of E into IR and of into ~ , respectively. Further, given u E Y:, let ~ denote the element of

defined by /~ ----~C,

(x,y):

~,ux+iuy

(Proposition 2.1.5.11) and define ~'Y:~ Then the following are equivalent:

}~,

o

U: -~,

U.

6~

~. Banach Algebras

a)

~ is bijective.

b)

~o is surjective.

c)

I f u is an algebra h o m o m o r p h i s m of E

into the real algebra C , then

~(E) c R . These conditions are fullfilled if

d)

a ~ ( ( x , 0)) C lit for every x E E .

a =~ b is trivial. b =~ c . By Proposition 2.1.5.16, the map

~k

~r

(x,v):

~uz+iuv

belongs to ~/. By b), there is a v E ~" such that vx + t v y = u x + iuy

for all x, y E E . Thus ux = v x E

for every x E E , i.e. u ( E ) C Ft. c =~ a . Take v E ~ . By Proposition 2.1.5.16, there is an algebra homomorphism w of E into the real algebra C such that v(x, y) = w x + i w y

for all x, V E E . Using c), define u'E

~ ]R,

x:

~ wx.

Then u E Jr and ~u = v. Hence ~ is surjective. By the last assertion of Proposition 2.1.5.11 ~o is bijective. d =~ c . Let u be an algebra homomorphism of E into the real algebra C,

ok

u"

~r ,

(x, y) :

~ u x + iuy ,

and take x E E . By Proposition 2.1.5.16, fi is a homomorphism of complex algebras and

~ e {~} = or

= ,,c(;,(~:, 0)) c a~(Cx, 0)) u {0} c ~t

(Proposition 2.1.3.2, Corollary 2.1.3.12), i.e. u ( E ) C I R .

I

~.I Algebras

63

P Proposition 2.1.5.18 real algebra E.

( 0 ) The following are equivalent .for every unital

a) 9 ~ E ~ o~((~, 0)) = OE(~). b) x e E ~ as(x 2) C ]R+. a ~ b. P u t P ( t ) := t ~ E IK{t]. By a) and T h e o r e m 2.1.3.4 b), aE(X ~) = a~(Cx2, 0 ) ) = a~((X, 0) 2) = P(o'~(x, 0)) = P(O'E(X)) C F(,+. b =~ a. Take c~,/~ E IR,/~ # 0. By b ) , y "= ( a l - x) 2 + f]21 is invertible. From (cxl - x, f/1)((~1 - x)y -1, _/~y-1) =

: ((~I - x)2y -I + f~2y-x, f~((~l - x)y -I - f~(~l - x)y - l ) = (I, 0),

( y - l ( a l -- X ) , - - ~ y - l ) ( C d -- X,/~1) = = (y-l(oll _ X)2 + f~2y-1, ~y-l(c~l _ X) -- ~y-l(c~l -- X)) = (1, 0) it follows t h a t (a + if~)(1, 0) - (x, 0) = ( a l - x, f~l) is invertible (Proposition 2.1.2.3). Hence

~ + ia r ,,k((x,o)),

~,~((~,o)) c ]a, o~((~, o)) = o~(~) (Corollary 2.1.5.14).

I

6~

~. Banach Algebras

Corollary 2.1.5.19

( 0 ) Let E be a unital real algebra such that

Or~((X,0)) =OrE(X) ,for every x E E and let F be an ideal of E . Then

~ . ~ ((x,0)) =aE/F(X) ElF

.for every X ~ E / F .

Take X E E / F and x E X. By Corollary 2.1.3.12 and Proposition 2.1.5.18 a=~b,

a~/F(X2) c aE(X2) c

R+.

The assertion follows from Proposition 2.1.5.18 b =:, a.

I

P,.l Algebras

65

Exercises E 2.1.1

A monoid is a set T endowed with a map

TxT

~T,

(s,t)"

;st

and an element 1 E T such that (rs)t = r(st),

It = fl = t

for every r, s, t E T . The monoid T is called commutative if

st = ts for every s, t E T . Let T be a monoid. Given x, y E ~(T), define

xy" T-= ; lK,

t:

~~

x(r)y(s)

r,sET ra--t

Show: a)

(et)teT is a basis of the vector space IK(T) (Proposition 1.1.2.6 a)).

b) The vector space ]K(T) is a unital algebra with :espect to the multiplication ]K (T) x ]U (T)

~ ]U (T) ,

( x , y) :

'; x y

c) eset = e,t for every s, t E T . d)

The unital algebra ]K(T) is commutative iff T is commutative.

e) The unital algebras lK[s] and ]K[s, t] are special cases of the above construction. E 2.1.2

Let a(lK[t]) be the set of unital algebra homomorphisms lK[t] --+ lK.

For a E ] K , define

I~: lK[tl

~ ; I(,

Show:

a) % E a(H,~[[t]) for every a E ]K.

P,= ~; P(c~)

66

~. Banach Algebras

b) The m a p ~(K[t]),

~ ~

fo

is bijective. E 2.1.3 Let 1 and j be the standard basis vectors of I~ 2 , and for every a E {-1, 0,1} let Eo be the unital algebra obtained from the vector space I< 2 by defining a multiplication such that 1 is the unit and j2 = a I . Prove the following: a) If lK = lR then: al) No two of the algebras E-l, Eo, El are isomorphic. as) Every two-dimensional real unital algebra is isomorphic to one of these three unital algebras. a3) The group of unital algebra isomorphisms Eo ~ Eo is isomorphic to the multiplicative group IR\{0} . a4) The group of unital algebra isomorphisms E_I ~ E_I (El ~ El) is isomorphic to the multiplicative group {-I, 1} .

as) E-I is isomorphic to the real unital algebra C. a~) El is isomorphic to the real unital algebra PL {~

(Example 2.1.4.3).

az) The set of unital algebra homomorphisms Ea ~ IR is empty if a = -1, has exactly one element if c~ = 0, and has exactly two elements if c~ = 1. b) If IK = C

then:

bl) The unital algebras E-l and Et are isomorphic. b~) The unital algebras E0 and El are not isomorphic. ba) Every two-dimensional complex algebra is isomorphic either to E0 or to El.

b4) The group of unital algebra isomorphisms E0 ~ Eo is isomorphic to the multiplicative group ~\{0}. b5) The group of unital algebra isomorphisms E1 --~ El is isomorphic to the multiplicative group {-1,1}. be) E] is isomorphic to the complex unital algebra ~{o,]} (Example 2.1.4.3).

$.1 Algebras

67

bT) The set of unital algebra homomorphisms Ea --+ C has exactly one element if a = 0 and exactly two elements if lal = 1. c) Find the zeros of the polynomial x 2 - 1 in Ea. d) Find the spectrum of j in E a . e) Find the group of invertible elements of E~. f) For which a E {-1, 0,1} is there a monoid T such that Ea is isomorphic to IK(r) (E 2.1.1). E 2.1.4

Let K be a subfield of lR, such that the dimension of the K-vector

space IR is infinite (e.g. K = ~ or K the set of algebraic numbers of ~ ) . Show that there are maps

~o'~x~

;IR,

~)'RxR

"~

such that lR (with the usual addition and with these maps) is a complex unital algebra with a) a,/~ E K ==~ ~o(a,/~) = r

fl) = a/~ ;

b) IR is isomorphic to C as complex unital algebra. E 2.1.5 Let E a unital algebra, F a maximal proper ideal of E , and IK • E the unital algebra defined in Proposition 2.1.1.7. Show there is a maximal proper ideal G of lK x E containing {0} x F other than {0} x E . E 2.1.6 Let E be a unital algebra and take x, y, z E E with x invertible and y + z = x. Prove that yx-lz

E 2.1.7

~ ZT,-ly .

Let E be a unital algebra, G the set of its invertible elements, and

~" the group of isomorphisms of unital algebras E ~ E . Show: a) The map

GxG renders G a group.

~G,

(x,y):

;xy

~. Banach Algebras

68

b) For every x E G, the map ~" E

~E,

y:

~x-~yx

belongs to 9c. c) The map G

~',

x:

~

is a group homomorphism. d) a(x) - a(ux) for every x E E and u E ~'. E 2.1.8 Let E be a unital algebra, (x,),~! a finite family in E , (P~),EI a family in lK[t] and let -1

A, := P,(O) for every ~ E I . Assume 1) , ~ I ~ O ~ A , ,

P,(x,)=O, x,~{allaEA,).

2) ~ , ~ S , ~ # ) ~ x , x ~ = 0 . Prove that c ~EI

~EI

E 2.1.9 Let E, F be algebras and u 9 E --+ F an algebra homomorphism, which does not vanish identically. Prove the following (generalization of Proposition 2.1.1.12)" a) If E and F are unital and 0 the only zero-divisor in F , then u is unital. b) If the only ideals of the subalgebras of F are trivial, then Keru is a maximal proper ideal of E .

~.~ Normed Algebras

2.2

Normed

69

Algebras

This section lays the foundations for the entire theory. The completeness of Banach algebras allows the construction of the inverses of many elements by means of Neumann series. This has powerful consequences for the spectrum and the resolvent, culminating in a fascinating mathematical theory with the theorems of Riesz, Gelfand, and Gelfand-Mazur as highlights. Applications are numerous including to the algebra of operators on a Banach space, to quotient spaces of such algebras (for example the Calkin algebra), to C*-algebras, and to harmonic analysis ( whose fundamentals appear in this book in summary form).

2.2.1 General Results Definition 2.2.1.1

( 0 ) (M. Nagumo, 1936)

A normed algebra is an

algebra E equipped with a norm such that

Ilxyll _< Ilxll Ilull .for every x, y E E . A normed unita! algebra is a normed algebra which is unital and for which

Illll = 1. A normed (unita 0 algebra is called real or complez when the underlying (unital ) algebra is real or complex., respectively. A Banaeh (unitaO algebra is a complete normed (unitaO algebra. Let E, F be normed (unital) algebras and u : E ~ F a (unitaO algebra isomorphism, u is called an i s o m o r p h i s m of normed (unitaO algebras if u and u -1 are continuous, i.e. if u is an isomorphism of the underlying normed spaces, u is called an i s o r n e t ~ of n o r m e d (unitaO algebras if

Iluxll = Ilxll for every x E E , i.e. if u is an isometry o.f the underlying normed spaces. E and F are called isomorphic or isometric if there is an isomorphism or an isometry E ~ F o] normed (unitaO algebras, respectively. Let E be a (unital) Banach algebra. Then E can be considered as a (unital) Banach category with a single object.

70

~. Banach Algebras

P r o p o s i t i o n 2.2.1.2

( 0 ) Let E , F be norrned unital algebras and u "

E --, F a continuous unital algebra homomorphism. Then

Ilull ~ 1. Note that

II

1 = 11111 = Ilulll _< Ilul111111 = Ilull.

P r o p o s i t i o n 2.2.1.3

( 0 ") Let E be a normed algebra and let K x E the unital algebra defined in Proposition ~.1.1.7. Then K • E ----+ R+,

(~,x)'

; lal + Ilxll

is a norm and ~, x E is a normed unital algebra with respect to this norm. It is complete iff E is complete.

By Proposition 1.1.5.1, the above map is a norm and ~ x E is complete in this norm iff E is complete. Given (a, x), (/3, y) E E • E ,

I~1 I~1 + I~l IIz~ll+ I~1 Ilxll + Ilxll IIz~ll = = (lal + Ilxll)(l~l + Ilzlll) = I1(~, x)ll I1(~, y)ll and

[[(1,0)11 = 1. Hence ~ x E is a normed unital algebra with respect to the above norm. Remark.

I

There could be other norms on • • E with similar properties.

Definition 2.2.1.4

( 0 ) Let E be a normed (Banach) algebra and F the

unital algebra associated to the underlying algebra of E .

F endowed with a

norm extending the norm of E (identified with {0} • E ) and making F to a normed (Banach) unital algebra is called a normed (Banach) unital algebra associated to E (Proposition 2.12.1.3).

~.$ Norrned Algebras

71

Proposition 2.2.1.5

Let E be a normed algebra and ]K x E the normed algebra defined in Proposition $.Y2.1.3. Define a norm on the vector space IK x E' by

IK x E'

" n',

(a',x')"

; sup{la'l, IIx'll}

(Proposition 1.1.5.1) and set

(a',x') " I K x E

~IK,

(a,x):

"aa'+(x,x')

.for (a',x') E lK x E ' . Then (~',x') e (]K x E)' .for every (a',x') e l K x E' and the map

IK•215

(a',x'):

"(a',x')

is an isometry of Banach spaces.

The assertion follows from Proposition 1.2.2.13. Proposition 2.2.1.6

I

( 0 ) If E is a normed algebra, then the map E x E

>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 <- -

lirax , ~ inf

llxU Ilull = Iluli

limz , 9 inf

i1~1', <- -

< Ilyll lira sup Iixll < Ilull since E # E ~, and so lim IIx II = 1. If E has a unit then o = lira llxl-

111 = lira llz-

1ll.

Hence lim ~ Proposition

2.2.1.17 ( 0 )

1.

I

Let E be a Banach algebra. For each x E E ,

define 5"E---> E,

y:

~xy.

76

~. Banach Algebras

Then "~ E s

and

I1~11 _< I1~11 ]or every x E E and the map

E ~

s

x~

is a continuous algebra homomorphism. I] E is quasiunital (unital), then

II~ll = II~ll, and

,,.~(~)\{o} r ,,.,:r

c ,,.~(x)

(,,.~(x)= o,:c~(~))

.for every x E E . In particular, if E is finite-dimensional then Card aE(X) <_ Dim E . Since

II~ull = II~ll < ll~ll Ilull, E f..(E) and

IJ~ll < II~ll for every x E E . Take x, y E E . Then z ~ z = x y z = 5 ( y z ) = "~(~z) = "~z

for any z E E . Thus xy = z y . Hence the map

E ---~ s

,

z:

; "~

is a continuous algebra homomorphism. Let ~ be an approximate unit of E and take x E E . Then }lxl} - lim g,;} ]}xYll = lira y,;~ lily}} -< JI~lJ since E # E ~ . Hence, by the above result,

~.~, Normed Algebras

77

ll:ll= ll~ll. By Corollary 2.1.3.12, {7s

C aE(X).

Take a E IK\ (ar.(E)(5)U {0}). Then a l - 5 is surjective. Hence there is a y E E with J

(al

-

1

~)y =

- x .

C~

It now follows successively that ay-

1

xy = -x,

y : (al - 5 ) - : 1 5

(al - ~)~ = a~-

-- 15(c~I - 5)-',

~

1_

= -x,

~(~I - 5) - - 5 .

X y -- y x ,

and xy = yx.

Hence a ~ a E ( x ) by Proposition 2.1.3.11 b =~ a. Thus

o~(~)\{o} c ~(~)(~). Assume that E is unital. Assume that 5 is invertible. Then there is a y E E with x y = 5y = l s .

Thus xy = I s

y=X

,

~ ' ~ = 1 E(E),

and SIx :

IE.

Hence x is invertible. Thus

Remark.

If Il:ll = II~ll

for every x E E and if we identify E with {5 [ x E E } , then the closed subalgebra of s generated by {1E} U E is an associated unital Banach algebra of E .

78

2. Banach Algebras

P r o p o s i t i o n 2.2.1.18

( 0 ) Let E be a real Banach space. Put

H(x, y)[[ := sup I](cos O)x + (sin O)y[[ eER

Io~ ~ , ~ 2.1.5.7).

(~,v) e ~, w h ~ ~ i~ the ~ompt~$c~tio. o! E (P~opo.itio.

a)

(x, y) E E ~ sup{llxll, IlYlI} < II(x, Y)[I < Ilx[I + IlYll-

b)

Themap

,, ~+,

(x,v)-

~ I1(~,~)11

is a complete norm.

a) is easy to see. b) Take (xl, yl), (x2, y2) E E . For 0 E • ,

IIcos e(z~ + x2) + sin 0(yl + ~2)11 < _< II(cos e)z, + (sin o)~ll + II(cos e)x2 + (sin e)~ll _< _< II(x~, u,)ll + II(x~, y~)ll 9

Hence

II(x,, u,) + (x,, ~)11 -< I1(*,, ~,)11 + II(x,, ~,~)11 9 Take ( z , y ) E ~: and re '~ E l . Then re'~(x, y) = ((r cos ~)x - (r sin ~)y,

(r sin ~)x + (r cos ~)y).

For 0 E R ,

II cos e((~ cos ~)~ - (~ sin ~o)y) + sin 0((r sin ~)x + (r cos ~)~)11 = = rJJ(cos 0 cos ~o + sin 0 sin ~o)x + (sin 0 cos ~o - cos 0 sin ~o)y[I = -- rll(cos(O- ~o))x 4- (sin(O- ~o))y[[. Thus

life's(x, U)II = rll(x, U)II. By the above, the map

is a seminorm. By a), it is a complete norm.

m

~.~ Normed Algebras

Proposition 2.2.1.19

79

( 0 ) (Goodearl, 1982) l.f E is a real Banach algebra o

then there is a norm on E making it into a complex Banach algebra and such that

I1(~, o)11 = I1~11 for every x E E . E endowed with such a norm is called a eomplezifieation oyE.

Put

I1(~, ~)11o := ~up II(r eEIR

o)~ + (sin O)yli

for every (x, y) e/~. By Proposition 2.2.1.18 b), the map

~ n~+,

(x,~)~

~ II(x,y)llo

is a complete norm. We set

II(x,y)ll

I (a,b) e /~, II(a,b)llo = 1}

"=sup{ll(a,b)(x,y)llo

for all (x, y) e~/~. Step 1

I1(',')11 is a complete norm

Take (x,, yl), (xa, y2) E E. For (a,b) E/~, II(a,b)llo = l ,

II(a, b)((x,, ~) + (~, Y~))llo <- Ii(a, b)(~,, Y,)llo + II(a, b)(~, ~)11o <_< IIC~,, y,)ll + I1(~, y~)ll, so that

IIC~,, ~,) + C~, ~)11 _< I1(~,, y~)ll + I1(~, ~)11 9

Take (x, y)E/~ and

re iw e •.

Iir

For (a, b) e/~, II(a, b)llo = 1,

b)llo = II(a, b)llo =

(Proposition 2.2.1.18 b)). It follows that

life's(x, ~)11 = sup{ll(~,

b)(re'~(x,

~)11o I (~, b) E E, IICa, b)llo = 1} =

80

~. Banach Algebras

= ,- sup{ll(e'~(a, b))(x, ~)11o I ('2, b) E/~, I1(~, b)llo = 1) = rll(~, Y)II. Moreover, by Proposition 2.2.1.18 a),

II(a, b)(x, Y)llo = II(a~

by], ay + b~)llo <_

-

_< I1~ - byll + lily + bxll _< I1~11 + I1~11 + Ilavll + Ilbxll < _< 11~1111xll + Ilbll I1~11 + I1~11 Ilull + Ilbll I1~11 = (11~11 + Ilbll)(ll~ll + Ilull) -< 411(x, ~)11o, so that

I1(~, ~)11 _< 411(~:, ~)11o. By Proposition 2.2.1.18 a), we may assume

1 It follows that

I1(~, ~)il -> I1(~, o)(~, ~)11o = 1 = II(ax, a~')llo -> sup{llaxll, Ilayll} -> ~ sup{llxll,

Ilyll}

9

By the above and by Proposition 2.2.1.18 b), the map

-~ ~+,

(x,~)-

~ II(x,~)ll

is a complete norm. Step 2

/~ is a Banach algebra

Take (x,l, Yl), (x~, y2)E E. For (a,b)E E, II(a,b)llo = 1,

11(,2, b)((xl, yl)(X2, W))IIo = I1((~, b)(xl, yl))(xa, v2)llo -< _< I1(~, b)(xn, ~)11o IICx2, ~,~)11 _< I1(~, ~)11 I1(~, y~)ll, so that

I~.~ Normed Algebras

81

II(x,, y,)(~, ~)11 <_ I1(~,~, ~,)11 II(x~, y~)ll 9 x e E => II(x, o)11 =

Step 3 Take (a, b) e / ~ ,

Ilxll

II(a, b)llo = 1. For 0 E IR,

II((cos e)a § (sin O)b)xll

_

II(cos o)a + (sin O)bll Ilxll _< II(a, b)ilollxll = Ilxll.

It follows that

II(a, b)(~, o)11o = II(a~, b~)llo = sup II(~os O)a~ + (sin O)b~ll < I1~11, efilR

so that

II(x, o)11 ~ Ilxll. In order to prove the reverse inequality, we may assume E unital (Proposition 2.2.1.3). Then

I1(1, O)llo- Illll- 1, so that

II(~, o)11 .>. I1(1, o)(~, o)llo = II(z, o)11o = I1~11 (Proposition 2.2.1.18 a)).

I

8~

~. Banach Algebras

2.2.2 The Standard Examples Example 2.2.2.1

( 0 ) Let T be a set. t ~ 1 7 6 a commutative unital Banach algebra with respect to pointwise multiplication, c(T) is a closed unital subalgebra of t~176 co(T) is a closed ideal of g~176 If T is a topological space, then C(T) is a closed umtal subalgebra of Coo(T). If T is locally compact, then Co(T) is a closed ideal of C(T) and the Banch algebra Co(T) is quasiunital, m Example 2.2.2.2

( 0 ) If (T,9t, p) is a measure space, then Loo(p) is a

commutative unital Banach algebra. For every x E Loo(p), a(x) = {a e ~ l e

> 0 ==:, ?z'(Uy(a)) is not a p-null set .}

If there is an e > 0 such that x1(U~(O)) is a p-null set, then T

~K,

t:

r

1

-

~(t)

is the inverse of x in Loo(p). If x is invertible then xt(U~(O)) is a p-null set, where 1 , :=

m

Ilx-'ll "

Example 2.2.2.3

( 0 ) Let (Et)teT be a family of normed (umtal} algebras

and

E := {x E H E t # (#lx(t)ll),er e g~(T)}. tET

Then E

is a (unital} subalgebra of the (unita O algebra I'I Et

(Example

tET

~.1.4.1), E endowed with the norm E ---+ IR+,

x ~-+ sup tET

IIx(t)[I

(Proposition 1.1.2. 7 a)) is a normed unital algebra, and

{x e E l(llx(t)ll),eT E co(T)} ss a closed sdeal of E . If all Et (t E T) are complete then E zs complete too. If T is a topological space and all Et (t E T) are equal then {x E E l x is continuous} is a (unital) closed subalgebra of E .

~.~ Normed Algebras

8:i

The last two assertions follow from Proposition 1.1.2.7 b) and Corollary 1.1.2.8.

I

Example 2.2.2.4 ( 0 )

If E is a normed (Banach) space, then s

is a

normed (Banach) semi-simple algebra, unital if E ~ {0}.

By Example 2.1.4.6, Corollary 1.2.1.5, and Corollary 1.2.1.11, s

is a

normed (Banach) algebra. We now show that s

is semi-simple. Take x e E \ { 0 } and put

l u x = 0}.

jr := {u e s 9v is a left ideal of s

Let ~7 be a left ideal of s

strictly containing ~'.

Put F := ( u x l u ~- {~}. Then F is a non-zero vector subspace of E such that u(F) C F

for every u E s

Take y ~. E \ F .

Then there is a z E F and a u E s

such that y = u z . Thus y E F , which is a contradiction. Hence F = E . Therefore there is a v E G such that VX--'-X.

If u ~. s

then (uv - u)x = ux - ux = 0,

so that uv - u E .T'.

= u v - ( u v - u) e ~ ,

= s

Hence ~" is a maximal proper left ideal. It follows immediately that {0} is the radical of s

.

I

84

~. Banach Algebras

E x a m p l e 2.2.2.5

(

1)

L a T be a set. Take p E [1, oo] U {0} and let q be

the conjugate exponent of p.

a)

The Banach space P"q(T, T) endowed with the multiplication ff'q(T,T) x f f " ( T , T )

(h,k):

~ ff'q(T,T),

r hok

(Proposition 1.2.3.5) is a Banach algebra and the map ff'q(T,T)= ~ s

k:

n

~k

(Proposition 1.2.3.4 b)) is an injecttve continuous algebra homomorphism.

b) If p = oo then the above algebra and the above map are unital. n

c) If k E (2'2(T, T) then a(k) = a ( k ) . a) and b) follow immediately from Proposition 1.2.3.5, Proposition 1.2.3.4 b), and Example 2.2.2.4. c) By a), the map n

s

u" t2'2(T,T) ~

k:

~, k

is an injective algebra homomorphism. Im u is thus a subalgebra of s so we have that n

~r(k) = ~,m.(k). By Proposition 1.2.3.4 g), (v, w) E (Im u) • s

vw = wv ==~ vw E I m u.

Hence, by Corollary 2.1.3.15, n

n

Olmu(k) -- a(k). Thus

~(k) =,,(~). E x a m p l e 2.2.2.6

a

If A is a (unital) Banach category, then A(E) is a (umtal) Banach algebra for each ob3ect E of A. I

~,.~ Normed Algebras

Theorem

2.2.2.7 ( 4 )

85

Let T be a group and let 1 be its neutral element.

Given z, y E t I (T), define

sET

(the convolution of x with y ).

a) z , y

is in t ' ( T )

whenever z , y E s

and s

equipped with the

multiplication

•'(T) x iX(T)

; t'(T),

(x,y)"

~x , y ,

is a unital Banach algebra with el as unit.

b)

The map t:

; et

is a group homomorphism of T into the multiplicative group of invertible elements of el(T).

c)

T is commutative iff el(T) is commutative.

d)

Define s ,.~ t :r

3r E T, r-lsr = t

,for all s, t E T . Then:

dl)

,,~ is an equivalence relation on T ; its equivalence classes are called

conjugacy classes of T .

d2) d3)

s,t E T =~ st ,,~ t s . An element o.f e'(T) belongs to e'(T) ~ iff st ,s constant on the conjugacy classes of T .

d4)

If T is the permutation group of a finite set then two elements of T are equivalent with respect to ,,~ iff they have the same cycle structure (i.e. the complete factorizations into disjoint cycles have the same number of n-cycles for each n ).

ds)

If Q ts a conjugacy class of T then {t-'It ~ Q} is a conjugacy class of T and so it is either equal to Q or disjoint .from Q .

YL Banach Algebras

86

e)

I f S is a subgroupf of T then et(S) may be canonically identified with a unital subalgebra of t I (T) and

ortt(5,)(X ) -~- O'tI(T)(X ) f o r every

:r,

E eI ( S ) .

a) Take x, y E e l ( T ) . Since el(T) C ~ ( T ) ,

I(x 9 ~)(01 <_ ~

Ix(ts-')l I(~(s)l -

tET aET

tET

~-

~

x 9 y is well-defined. Now

~ ly(s)l ~ Ix(ts-')l - ~ ly(s)lllxll, mET

tET

Ilxll,llyll~,

sET

and so x 9 y E e I (T) with

IIx * uli~ <-- Ilxllt Ilyll~ 9 Take x, y, z E e I (T) and a, ~ E lK. Then ( a x + ~y) 9 z = a x 9 z + ~y 9 z ,

z 9 ( a x + ~y) = a z 9 x + ~ z 9 y.

Take t E T. Since

sET

sET

qET

sET

vET

qET

= ~

sET

~(tq-')(~ 9 z)(q) = (= 9 (~ 9 z))(O,

qET

it follows that ( z 9 y) 9 z = z 9 (y 9 z ) .

Finally,

(el 9 x)(t) -- Z e1(ts-1)x(8) sET

= x(t),

~.~ Normed Algebras

87

(X * el)(t) -- ~ x ( t s - l ) e l ( 8 ) = x ( t ) , aET

for every t E T . Thus el ,x=x,el

Since

Ile~ll~

= 1,

--x.

el(T) is a unital Banach algebra with el as unit.

b) Take s, t E T . Then

(es * et)(r) -----~

e,(rq-1)et(q) = e,(rt -1) = est(r)

qET

for every r E T . Thus es * et ~ est

and the map t:

~et

is a group homomorphism of T to the multiplicative group of invertible elements of T . c) Suppose that T is commutative and take x, y E e I (T). Then

sET

sET

rET

rET

for every t E T . Thus

x,y=y.x and

el(T) If s

is commutative. is commutative, then est --

Hence

es * e t - -

et * es --

st = ts

for s, t E T and T is thus commutative. dl) and d2) are easy to see. d3) Take x E e l ( T ) . Then for s, t E T ,

ets 9

88

~. Banach Algebras

(~ 9 ~,)(~) = ~

~(~-~)~,(~)

= ~(a-'),

rET

rET

Hence if x is constant on the equivalence classes of ~, then, by d2),

x , et = et , x for every t E T . It follows that x E l l ( T ) c . Now assume x E l l ( T ) c and let s, t E T , s ,,~ t. T h e n there is an r E T such t h a t r - 1 8 r --- t .

By the above, ~(~) = ~ ( s r ~ - ' ) = (x 9 ~ , ) ( ~ )

=

= C~, * ~)(s,') = z(,--'s,') = x ( t ) . Thus x is constant on the equivalence classes of ,~. d4) If r is a transposition and s is a cycle then it .s easy to see that r - l s r is a cycle of the same length. Hence s and r - l s r have the same cycle structure for every s E T . Since every p e r m u t a t i o n is a product of transpositions, it follows that s ~ t implies that s and t have the same cycle structure. Assume now that s and t have the same cycle structure. Place the complete factorization of s over that of t, so t h a t cycles of the same lenght correspond and let r be the permutation sending the b o t t o m to the top. Then

r - l s r = t. ds) follows from

(s-Its) -l = s - i t - I s for all (s, t) E T x Q . (y E s

e) For y E s

T

;lK,

put

t'~: ~ ~ y(t) 0

(

if t E S

( if t E T \ S )

if T \ S

( if t E S )

P,.~ Normed Algebras

and identify s and y ~ ~. Then the map

eI(T\S) with

e'(S) x e'(T\S)

vector subspaces of

} e'(T),

(y,z):

el(T) using the

89

maps

" y+z

is bijective. Let y , z E el(S) and t E T \ S . Then

( v , z)(t) = ~

v(t,-')z(,)

sET

= ~ v(ts-')z(~) = o, sES

so that Y * z ~ e l(S). Hence, by the above identification, e l(S) is a unital subalgebra of t I (T). Let (y,z) E el(S) x e l ( T \ S ) and t E S. Thus

(v 9 z)(t) = ~ v(t~-')~(~) = ~ seT

v(t~-')z(~) = o,

sET\ S

(z 9 v)(t) = ~ , ( t s - ' ) u ( ~ )

= ~

sET

z(t:')y(~)

"o,

sES

so that Y * z , z 9 V q e l ( T \ S ) 9 By Corollary 2.1.3.14 b),

at,(T)(X ) E x a m p l e 2.2.2.8 rem 2.2.2. 7).

"

-

*tl(S)(X) .

The following assertions hold for every x G el(~2) (Theo-

a) x is invertible iff

x(0) ~ # z(1)'. b)

m

The spectrum o/ x is

{~(o) + ~(1)}. a) Given y E e l ( ~ ) , x 9 y = (x(O)y(O) + x(1)y(1))e0 + (x(O)y(1) + x(1)y(O))el. Hence x is invertible iff it is possible to solve the system of equations x(O)y(O) + x(1)y(1) = 1 x(1)y(O) + x(O)y(1) = 0

90

2. Banach Algebras

with y(0) and y(1) as unknown. This occurs iff z(O) 2 ~: x(1)2. b) By a), an a E IK belongs to the spectrum of x iff (x(0) - a) 2 = x(1)2 i.e. iff a = x(0) • x(1) .

1

2.2.2.9 Let ~T be the Stone-G'ech compactificatzon of the group T with respect to the discrete topology. Take so, to E /~T. Let el(T) be the unital Banach algebra defined in Theorem ~,.~.IL7. Identify s canonically with e~~ (Example 1.e.e.3 d)) and e~176 with C(I~T) - by identifying each element of e~176 with sts continuous extension to # T . FmaUy, Identify e~176 ' with the Banach space o.f Radon measures on BT (Example 1.2.e.11). l/6,o,6to denote the Dwac measures on ~T at So and to respectively, then, (using the notation of Definition 1.5.~.10}, Example

(6,o --I 6to,X') =

<6,0 I-- 6to,x') =

x'(st) ,

lim

lim

lim

lim x ' ( s t )

Tgt-~to Tgs--+so

Tg,-*so Tgt-+to

,

for every x' E t~176 Now

(6, 0 -I 5to, z') = (5,0, x'6,o) =

=

=

lim (6,0,5tx') =

Tgt--+to

lim

lim (6,, 6tz') =

lim

Tgt--*to Tgs-~,o

lim ( 6 , 6 t , z ' ) =

Tgt~to Tgs~,o

lim (~to, x'~s) =

Tgs--4 so

lim

lim

lim x'(st),

Tgt~to Tgs~so

(5,o I- 6to, x') = (6, o, 6fox') =

=

lim (6t, x'6,o) =

Tgt~to

lira (6,, 6to x') =

Tgs~ao

lira (6t, x'~,) =

Tgs--~ so T~t--~to

IL$ Normed Algebras

=

lim ((is(it, x,')=

lim

T~s-,so T~t~to

lim

lim x ' ( s t ) ,

T~s~so Tgt-~to

where (it denotes the Dirac measure on fiT at t (5 T .

Remark.

91

II

Take T := 7Z. Let ~ be a free ultrafilter on 7/, with IN (5 ~ and put So := lim n n,~

'

to := l i m ( - n ) . n,~

Then

Supp(5so -{ (ito) C IN,

Supp(~,o I- (ito) C ~TZ\IN.

Thus ~o -~

~to # ~o ~- ~to .

Theorem 2.2.2.7 admits the following generalization. Theorem 2.2.2.10

( 4 ) La T b~ a lo~~

~o~pa~t ~o~p .~d Io~ ~ve~y

t E T let 6t be the Dirac measure at t E T . a)

For every #, v E ,A/lb(Z) there is a unique element of M b ( T ) , denoted by # , u (the convolution of tz with t,, ), such that

for e v e n x E Co(T). b)

The multiplication fl/tb(T) • .h/tb(T)

~ ~lb(T),

(p, ts).~ ~ # * v

renders ~/Ib(T) a unital Banach algebra with 61 as unit. c) M n ( T ) is commutative iff T is commutative. d)

The map u " t ' ( T ) ----+.Mb(T),

x:

>E a ( t ) ~ t tET

is a unital algebra homomorphism presermng the norms (Theorem 2.2.g. 7). In particular

~. Banach Algebras

9~

6,~ = 6, 9 &

for every s, t E T . Let x E tl(T) . If # E fl/lb(T) :S atomless then (ux) 9 I~ and p 9 ux are atomless. In particular, O'tlCT)(X) -----O'.h4br

9

e) If A is a left invariant Haar measure on T , then {x.), I x e L'(u)) i, ,, closed subalgebra o.f A4b(T) and II~.Xll = Ilxll,,

(x.x) 9 (u.~) = (x 9 y).~

for every x, y E L l (~), where x 9 y" T ~

]K,

fz(s)u(s-'t)d~(s)

t ~

(the c o n v o l u t i o n o f x w i t h y ). L n(A) has a unit iff T is discrecte.

f)

The function

TxT~IK,

(s,t):

;x(st)

is in s (~ | u) and

for every t~, v E .h4b(T) and x E s

g)

The map J~b(T)

~" ~(,

is a unital algebra homomorphism.

h) p(T) C o(~) for every ~ E .A4b(T). i)

u).

f x d A E aLt(T)(X ) for e~)ery x E LI(T).

a) and b). The map Co(T) - - , ~ ,

is linear and

1~: ~ l~(T)

$.~, Normed Algebras

f~(st)d(~ | for every x E

I

<_Ilxllooll~

~,)(8.t)l

<

|

I1~11=ii~1111~'11

Co(T). Hence there is a unique element /~ 9 • of Mb(T), such

that / x d ( # 9 t,) for every

=/x(st)d(.

| v)(s, t)

x E Co(T). Moreover, tt~,. ,,tt < tl~,tt It,,tl 9

It is obvious that the map .A/Ib(T) • ~/lb(T) ~

J~b(T) ,

(#, v) ~

p 9v

is bilinear. Take ,~, p, v E J~ib(T). Then

/xd(()~ . t~) * v) =/x(st)d(()~ 9 ?t) | v)(s, t)

for every

93

x E Co(T). Thus

(,~, ~) 9 ~, = a 9 (~ 9 ~,). It is easy to see that 61 is the unit of .Adb(T). c) is trivial. d) The map u't'(T)

, ,~ A,4b(T),

x"

; Ex(t)6' tET

is obviously linear and preserves the norms. Take s, t E T . Then

f yd(~, , n,)

=

f y(,'t')d(n, | n,)(~',t') = y(~t)

for every f E Co(T), so that 6,, * 6t = 6 a .

94

~,. Banach Algebras

By linearity, ,,(~ * u) = (~,x) 9 (~,u) for every x, y E lK (T) . By continuity,

u(x 9 u) = (,=) * (~,u) for every x, y E el(T). Hence u is a unital algebra homomorphism. Now we prove the last assertion. 6t * p and /~ 9 6t are obviously atomless. Hence (ux) 9 I~ and p 9 ux are also atomless. By Corollary 2.1.3.14 b), O'/'(T)(X) = O'.Mb(T)(U~,).

e) Given f (5 C0(T),

f .fd((x.A) 9 (y.A)) =

f f(st)d((x.A)|

(u.,~))(s, t ) ) =

= f (f.f(t)Y(s-tt)dA(t)) x(s)dA(s) =

= ff(t)(/x(s)y(s-'t)da(s))da(t) = f f ( ~ , y)dA and so

(~.~) 9 (y.A) = (~ 9 ~ ) . A . If T is discrete, then LI(A) = .Mb(T), so that LI(A) has a unit. Assume that LI(A) has a unit x. Let K be a compact set of T containing 1 E T. By approximating eK with a downward directed family of real functions in C0(T), we obtain 1=

eK(1) "- (x,eK)(1) = f x(s)eK(s-~)dA(s) - f x dA, K-I

and this implies that x -----0

A-a.e.

ILe Normed Algebras

95

on T\{1 } and that

A({1}) > O. Hence {1} is an open set of T and T is discrete. f) The assertion holds if x is the characteristic function of a compact set of T, since in this case x may be approximated by a downward directed family of real functions in C0(T). The Lebesgue Convergence Theorem enables us to extend the assertion to arbitrary x. g) follows from f). h) and i) follow from g) and Corollary 2.1.3.13. L e r n m a 2.2.2.11

( 4 )

II

Let T be a locally compact group and take x E

C0(T). Given t E T , define s:

xt " T ' - ' - + I K ,

~ x(st)

(resp. x ( t s ) ) .

Then

lim I l x t - xlloo = 0. t-41

Given t E T , let st be a point of T such that IIx,- xll~ -Ix,(s,)

- x(s,)l 9

Let ~ be an ultrafilter on T converging to 1. If a := lim st E T , t,~

then lim s t t = a . t,~

Hence lira I 1 ~ , - ~1too = lira I~(~,t) - ~(~,)1 - I~(~) - ~ ( a ) l = o .

If lim st is the Alexandroff point of T , then lim stt is also the the Alexandroff t,~

t,~

point of T and so lira ]]xt- x]loo = lim Ix(sit) - x(st)] = ] 0 0 l t,~

t,~

= O.

Since ~ is arbitrary, lim I l x t -

xll~ =

0.

II

96

~. Banach Algebras

L e m m a 2.2.2.12

( 4 ) Let T be a locally compact group, A a left invariant

Haar measure on T , and take p E [1, c~[. For x E / f l ( A ) and t E T , define xt " T

~ ~~,

s ~

(resp. x(st) ).

x(ts)

Then, for every compact set yz of L p(A),

lim IIz,- xllp = o t~l uniformly in x E ~ . Take e > O. There is a finite set {~ of continuous functions with compact support on T such that

cU yE~

Let K be a compact neighbourhood of 1 in T containing the carriers of the y E ~ . By Lemma 2.2.2.11, there is a neighbourhood V of 1 in T such that s

Ily,- ~11~ < 1

1

IA(t) ,~ - 1 1 < -

+ 3A(KK)~ '

3

for every y E ~ and t E V, where ,4 is the module of G. Then

KK C t

, + 3,~(KK);

AIKKIp <-

C

3

for every yE{7 and t s Take x s .7r. There is a y E {7 with E

IIz- yllp < ~. Then IIx, - y, llp = IIx - yllp < 5'

, 1 e e (resp. I1~,- v, llp < A ( t ) ~ i l z - VlIp < (1 + ~)~ = 5 ) IIx,- xllp ~ IIx,- y, llp § Ily,- yll § Ily- xllp < ~ § ~ § ~ = for every t e V N K . Hence lim [Ix, t--,l

uniformly in x E .7r.

-

xll~ = 0 1

~.~ Normed Algebras

97

proposition 2.2.2.13 ~ " 4 ) Let T be a locally compact group, A a left i;nvariant Haar measure on T , and ffl the neighbourhood filter of 1 in T . Given V E ~3, let blv denote the set of positive real functsons x of L 1 ( A ) n L~176 such that Supp x C V,

f x(s)dA(s) = 1

(resp.

f

1).

Let ~ denote the filter on L~(~) generated by the filter base {lgv [ V ~ ?0). Then

(~esp. lim Ily * = -

lim IIx * u - ull, = o

yll~ = o )

for every y E L1 (A). In particular, L ~(A) has an approximate unit. I.f x E

LI(A) and "~ " L'(A)

- ~ Ll(A) ,

y ~

x . y,

then

II~II

=

llxll,

c

a(x)\{0}

a(~)c

a(x).

Take ~ > 0. For t E T , put yt " T

, ,~ IK,

s:

~ y(ts)

(resp. y(st)).

By Lemma 2.2.2.12, there is a symmetric neighbourhood V of 1 such that l l y ~ - yll, <

for every t E V. Then, for every x E/r

/

))~ * y - y)), =

I~ * y - y ) d ~ =

= f [fx(t)u(t-'s)d~(t)-y(s)f~(t)d~(t) dA(s) <_

~. Banach Algebras

98

= f (f,,,,-, ~=)- ,,~=),d;,I=))xl

-/

-/

-fl

I1~,-* - yll~x(t)dA(t) < e

IlY * x - VIII =

(resp.

0=

fly, x-

yJdA =

I f y(t)x(t-ls)dA(t) - y(s) dA(s) =

f y(st)x(t-l)d)~(t) -

f

y{s)x(t-l)d)~(t)J dX(s) <

< f (flu,(s)- Y(s)lx(t-l)d,~(t))dA(s)= = /(fJyt(s)-y(s)jdA(s))x(t-l)dA(t)=

= fllu,-

ylltx(t-~) dt <

~).

Hence lz,~ imJJx,y-yJJ=O

(resp. lz,$ imJJy,x-yJJ=O)

and LI()~) has an approximate unit. The last assertion now follows from Proposition 2.2.1.17. 9 Proposition 2.2.2.14 ( 4 ) Let A be a left invariant Haar measure on the locally compact group T and take p E [1, oo]. Assume that p ~ oo or T is compact and define A(/~,x) := {t E T I T

#,x

:T

~, IK,

~ IK, t~

s"

~ x(s-lt) is not p-integrable},

~ I fx(s-lt)dp(s)

t

o

i f t q~ A(p,x) z/ t e A(~, z)

for every # E .A/Ib(T) and every Borel function x : T ~ IK.

~.12 Normed Algebras

99

a) A ( U , x ) is a A-null set, # , x is a Borel function of s

I1~ * xll,, <_ Ilull Ilxll,,, and the function T x T----+ IK, is i n / . 2 ( # |

(s,t) '

~ x(s-lt)

.for every I~ E .~lb(T) and every Borel function x e s

For p E fl/lb(T), put ~ " LP(A) - ~ L~'(A) ,

x ~

l~ * x .

b) If T is compact, then I~ * x e C(T) for every p e fl,4b(T) and every x e C(T).

c) ~ e/:(LP(/~)) and I1~11 <~ Iluil for every u e J~b(T). d)

(/~,x) E fl4b(T) • L I(A) =~ (/~ * x).A =/~ * (x-A).

e)

The map Mb(T) - - ~ s

p:

,"

is an mjective umtal continuous algebra homomorphism {'Theorem 2.2.2.10 b)).

f) For every x E L I ( T ) and every Borel function y E s

for A-almost all s E T .

g) If T is compact, then {x.~ I x e L,'(~,)} and {~.a I 9 e C(T)} are ideals of M b ( T ) .

a)

Case 1

p--f- c~

For t E T, define xt " T

Then, by HSlder's Inequality,

", IK,

s:

" x(s-lt).

I00

~. Banach Algebras

/'

/'

/'

'(/

it

f for every t E T , so that

f" ( f" a~(.-',)l~l.l(.))'~(.)<_ ( Ix'lPdl~l)dA(t)

< I1,11p-

-_ I1~11, - t

--

f"

Ix(s-lt)lPdA(t))dl~l(s)

--II~llP-~llxllgll~ll

--

= II~ll~llxllg

by Tonelli's Theorem. Hence, A(/z, x) is a A-null set, /~ 9 x E s

I1~ * xll, _< I1~11 Ilxll~. and the function TxT~~, is in

s

(s,t):

rx(s-tt)

| A). Case 2

p = e~3, T compact, and x = 0

A-a.e.

By Tonelli's Theorem,

f" ( f" ~(,-~,)~(~.~(,))~(,)=f" ( f* Iz(s-lt)ldA(t))dll~l(s) = O, so that /J 9 x = 0

Case 3

A-a.e.

p = oo, and T is compact

We may assume by Case 2, that Ix[ _< [[x[Ioo. Then, obviously

I~ 9 xi _< I1~11 Ilxll~.

~.~, 3formed Algebras

I01

b) follows from Integration Theory. c) follows from a). d) Now

for every compact set K of T (by a)). e) The map is obviously linear and continuous. Take #, v 6 .A4b(T). Then, by d), for x 6 Lt(A)N/2(A), (ffr&).~

= ((,,

~ ) , x ) . ~ = ( , 9 ~) 9 (~.~) =

= ~ 9 (~ 9 (~.~)) = ~ 9 ((~ 9 x ) . ~ ) = ~ 9 ( ( ~ ) . A )

= (,,

(~x)).a = (~).a,

so that ~'~x

= ~v~.

Since LI(A) N LP(A) is dense in /2(A),

h'~=~*~. For x e LP(A) and t e T,

(al 9 ~)(t) = f~(~-~t)d,~(~) = ~(t), so that {~lX --" X

for every x 6 LP(A). Hence the map A4b(T)

~s

#:

~

=

lOg

g. Banach Algebras

is a unital homomorphism of Banach algebras. Take p E ,A4b(T) with ~ = 0 and let K be a compact set of T. Then

~(K).

o =/~ * eK-,(1)=

Since K is arbitrary, /~ = 0. Hence the map .Mb(T) ----+ s is injective. f) By a),

A(x.A,y) 9 y)(,)

ft:

~

is a A-null set and =

=

(t)x(st}dX(t) f

for every s E T\A(x.A, y). g) If T is compact, then A is bounded. Hence //'(A) C LI(A). Take x E/./'(A) (x E C(T)) and /z E .Mb(T). Then 9 (~.~) = (, 9 ~).~

b y d ) and p , x E //'(A) by a) ( p , z E C(T) by b)). Since A is also right invariant, {x.A ] x E//'(A)} and { x . A I x E C(T)} are ideals of .A4b(T). I P r o p o s i t i o n 2.2.2.15 ( 4 ) Let T be a locally compact group, A a left invariant Haar measure on T , and ~ the neighbourhood filter of 1 in T . Given V E ~ , let llv denote the set of real positive functions x of s Iq L~176 such that SuppxCV,

fxdA=l

and let Jr be the filter on LI(A) generated by the filter base {Llv I V E ~7}. Then

lim(y, (p 9 x).A -/~) = 0

lim 11/~* xllt = I1~,11

.for every ~ E A4b(T) and y E Co(T). Take e > 0. For t E T, put

yt'T

;IK,

s:

~y(st).

By L e m m a 2.2.2.11,there is a V E ~I, such that

~,.~,NormedAlgebras Ilu,-

103

ulloo <

for every t E V. Then

.
=

=

f (fx(t)u(st)dA(t))dt~(s)-f (fu(s)x(t)da(t))a~(s)[ =

f <_f ( f [yt(s)-y(s)[x(t)dA(t)) dll~l(s)<_ _< f

(/]]Yt-y]loox(t)dA(t))dl#](s)

< e][#[[

for every x E Uv. Hence lira(y, (# 9 x).A - #) = O. Thus [(Y,/~)1 = lira 1(~/, (/~ * x)'A)] < lira inf

II(t, 9 x).AII

= lira inf

I1~ * xll~

for every y E Co(T) # (Proposition 2.2.2.14 d) ) and

I1~11 - sup { I(y, ~)11 V e Co(T) ~ } < lim,~nf I1~ * xll, _< < lim sup [[,u * x[[1 _< lim sup II~ll Ilxll~ - I1~11, lim I1~ * xll~

-

I1~11 9

I1

104

12. Banach Algebras

( 4 ) Let )~ be a left invariant measure on the compact group T. Using the notation from Proposition ~.~.IL14 a)b), the map

Corollary 2.2.2.16

.A/Ib(T )

~ '~ s

U:

~

is injectwe. Take p E fl'lb(T) with ~ = 0. Let ~ be the filter introduced in Proposition 2.2.2.15 and ~ its trace on C(T). Then, by Proposition 2.2.2.15, f y d~ = lz,C) i m ( y ,(U , x).A) = 0 for every y E C(T). Thus p = 0. Proposition 2.2.2.17

I

Let T be a locally compact group. Take t~ E .A/Ib(T)

and t E T. Then lu * 6,I(A) = lul(At-')

.for every Borel set A of T. Let B be the set of bounded Borel functions on T with compact support. Then

f xd(u , 6t) = f x(rs)d(u | 6t)(r,s) = = f (fz(rs)d6,(s))dp(r)= fx(rt)du(r) for every x E C0(T) so that, by Lebesgue's Convergence Theorem,

f zd(u , ,,) = f x(rt)du(r) for every x E/3. Hence

lu,,~,l(a)=sup{fzd(t,,,,5,, I xE t3, Ixl_<ea} = =sup{Ifx(rt)du(r)llzeY,

Ixl < c a / =

= s u p { / y a t , l l y E B, lYl< eAt-l } = lPl(At-l) .

I

~.~ Normed Algebras

P r o p o s i t i o n 2.2.2.18

105

Let ,k be a left invariant Haar measure on the locally

compact group T , and (I,tt)t~t a finite family in .Mb(T)\{O}. Then there is a compact set K of T such that

lit,,- ~,. * ,~,11 > inf I1#,11 tel for every t, tr

I, t E T\K.

Put a "= inf I1~,11 > o. tel

There is a compact set L of T with 1 E L and sup I#~I(T\L) < t~l

-

4

"

Now define K := L - ' L "= { s - ' t i s ,

t E L}.

Then

I#,I(L) = I # , I ( T ) - I#,I(T\L) > a -

a 4

3~ 4

for every t E I and LN(Lt-')

for every t E T \ K .

--0

By Proposition 2.2.2.17,

I#~ * 5 t i ( L ) = I#,~I(Lt -1) _< I#,~I(T\L) < ~ , I#~ * JtI(T\L) = I#~ * 6tl(T) - I#~ * (ftl(L) > a

a

>" " l ~-l ( ~ ' t - ' ) - ~ > ~ - ~ =

3~

T'

so that

I1#~ - #,~ *$tll = ]~ - #,~ * $tl(L) + I#L - #,~ * $tl(T\ L) >_ [#~[(L) - 1#~ 9 (ft](L) + I#~ * 6tI(T\L) - I#,](T\L) > 3a 4

~ 3~ + 4 4

4

-~.

I

106

~. Banach Algebras

C o r o l l a r y 2.2.2.19

Let T be a locally compact group and take

p E Mb(T)\{0} 9 l,f T is not compact, then there is a sequence (tn)nEt~ in T such that

I1~ * 6,. - ~ 9 a,. II > I1~,11 .for distinct m, n E IN.

We construct the sequence inductively, starting with tl = 1. Take n E IN\{1} and assume that the sequence has already been constructed for all m E IN,-I 9 Then

I1~ * 6,11 = I1~11 for every t E T (Proposition 2.2.2.17). By Proposition 2.2.2.18, there is a tn E T such that

I1~ * 6,. - ~ 9 6~. II > I1~11 for every m E ~ n - l . L e m m a 2.2.2.20

I Let A be a right invariant measure on the locally compact

group T . Take p E [1, oo[ and let (Xt)tEt be a finite family in Lv(A)\{0}. For x E 12'(A) and t E T , define xt : T

~ IK,

s:

r x(st).

Then there ts a compact set K of T such that

IIx, - x'~ll~ > inf IIx, llp tel

for every t, tr

I and t E T \ K .

Define a := inf IixtIl~ > 0. tel

Take /~ E ]1, c~[ such that

There is a compact set L of T such that

107

~.~ Normed Algebras

f

Ot

Ix, l~'dA <

T\L for every t E I . Define K := LL -1 "= {st -l I s, t E L}. Then

IL

Im,lPdA-

f

T

/ Ix, lPdA- T \ L

Ix, lPdA > 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) <

Lt

f

IxtlPdA-

T\L

f

Izt IPdA-

T

I x , - z~t lPdA

f

Ix~IPdA> ~ - -~ 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~)~ <-Ilxll~llYllp for all y E LP(#) if p # oo. This inequality also holds for p = oo. Hence E s and

110

f2. Banach Algebras I1~11_< II~lloo.

In order to prove the reverse inequality, we may assume that

Ilzll ~

0. Take

a E [0, JJxJ[oo[. There is a p-integrable set A with

f~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'
2) W ' < R . 3)

The sum of any countable family of car&nal numbers strictly smaller than W' is strictly smaller than W'.

Denote by yc the set of u E s .for whicL the topological cardinality of Im u is at most equal to R' (is strictly smaller than R" ). Then .T is a closed proper zdeal o.f s (Example e.2.~.4). Take u, v E ~" and a,/~ E IK. Then Im (au + f~v) C Im u + I m v so that c~u +/~v E ~'. Hence ~" is a vector subspace of s Take u q. s

and v E ~'. Then Im (uv) -- u(Im v),

Im (vu) C i m v ,

so that uv, vu ~..T'. Hence ~" is an ideal of s

and it is obviously proper.

Let u E ~" and let (u,)ne~ be a sequence in 5r converging to u. Then Im u C U Im Un nEFN

so that u E ~" and 5r is closed.

I

114

~. Banach Algebras

2.2.3 T h e E x p o n e n t i a l F u n c t i o n and t h e N e u m a n n Series

P r o p o s i t i o n 2.2.3.1

Let E be an algebra and f a positive real

( 0 )

function on E such that

/(~u) _< f(x)l(u) for x, y E E . Then, given x E E , f ( x n) < f(x) n

for every n E IN and

f(xn)~

lim n-~oo

f(xn)~ < f(x).

= inf nE~

--

The first inequality follows by complete induction. Take k E IN. For every n E IN, there are uniquely determined

p(n), q(n) E IN U {0} such that

n = kp(n) + q(n) ,

q(n) < k.

Clearly, lim q(n) n~oo

O,

lim pCn)

n

n ~

n

1 k

Then

f (x n) = f(xkp(n)+q(n)) _< f (xk)n(n) f (x)q(') , f(xn)~ < f ( x k ) ~ f ( x ) ~ for every n E IN and therefore n

1

lim sup f ( x )'~ < f(x~') ]; n--}oo

Since k is arbitrary, we see that lim sup f(xn)88 < inf f(xn)~ < lim inf f(xn)~. Hence lim f(xn)~ = inf f(xn) ! < f ( x ) . n~oo

nEIN

--

il

~,.~ lVormed Algebras

Proposition 2.2.3.2

115

OO

( 0 ) Let E be a unital Banach algebra and E ~ tn n--O

a power series in IN. whose radius of convergence is r . Gwen x (5 E with

lim llx"ll ~ < r,

ll--~OO

the family (a,x"),e~u{0} is absolutely summable and the map oo

UrE(O) ~

E,

x~

Z ~ n--O

is continuous.

Now lim sup ll---tOO

Ila.x"ll~

sup

-lim

lt--I, oo

I~.l~llx"ll ~ =

= \(lim,~oosuplan]~) n~limI]xnl]-~ = -rl n--,OOlimIlx"]l~ < 1. By the Cauchy Root Test, the family (anXn)ner~u{o} is absolutely sumlnable. Take x (5 U~(0). By Proposition 2.2.3.1,

lim IIx"ll k _ Ilxll < r.

It--bOO

Take ~ > 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 <

and so

Ilxll

r + Ii~ll + flail =

2. Banach Algebras

116

"

(j)

+2 ~

"
I'~,,I r.+lxll

n=p+l

for every y ~. U~(x). Hence the map co

uE(o)---~ E ,

x:

~E

ol.xn

n'-O

is continuous.

II

Proposition 2.2.3.3

( 0 ) Let E be a unital Banach algebra, oo

co

n=O

n----O

power series in IK with radii of convergence r and r', respectively, and oo

~

7.t" the product of the two power series. Then

n-'O

Olnxn

~nxn

n=6

= E

n=O

"Tnxn

n--O

for every x q. E , so that lim IIx"ll ~ < inf(r,r'}.

ll---t oo

By Proposition 2.2.3.2, the above series are well defined. By Proposition 1.2.9.6 (and Proposition 2.2.1.6),

a,,x"

#,x"

n=O

Corollary 2.2.3.4 oo

=

n=O

oo

ak#.-k n=O

k=O

z" = y ~ 7,,x".

II

n=O

Let E be a unital Banach algebra. Take x E E . Let

a . t " , ~ /~,,t" be power series m IK with radii of convergence r and r', n--O

n=O oo

respectively, and ~

7n tn the power sertes in 0 corresponding to the analytic

n--O

functzon t~

~ n=O

~,,

c~t ~ k=O

.

~.~, Normed Algebras

If

Ilxll

117

< r and

then ~_, ~.

akx ~

n=O

7,x ~ .

=

k=O

n=O

By Proposition 2.2.3.2, the above series, are well defined. By Proposition 2.2.3.3 and Proposition 1.1.6.11, the above relation holds whenever {n 6 IN ] /~n # 0} is finite. The general result now follows by passing to the imit. II Definition

2.2.3.5 ( 0 )

Let E be a unital Banach algebra and take x s E .

Define (Proposition ~.~.3.~) oo 1 xn

eZ := E

(E.N. Laguerre, 1867).

~..

n=O

For every ~ 6 IK with

lim IIx"ll~ < a

Tt---kOO

define (Proposition ~2.2.3.2) 1 O~ - - X

(C. Neumann, 1870).

O~n t l

The map E

-~E,

x:

;e z

is called the ezponential f u n c t i o n (on E ) . Remark. Let F be the closed unital subalgebra of E generated by x. Then F is commutative (Corollary 2.2.1.10) and

ex ,

1

6 F.

2. Banach Algebras

118

Proposition 2.2.3.6

Let E be a unital Banach algebra and take x, y E E .

I/ ye ax = eaZ y for every a E ~ ,

then xy - y x .

Given a E K , ----X n

Y

n! n--O

)

_~

--X

n

n!

Y'

n--O

so that y X ---- y X

I

by Corollary 2.2.1.12.

Proposition 2.2.3.7

( 0

)

Let E

be a unital Banach algebra and take

x,y E E ~th xy = y x .

a)

e x+y--exe y.

b)

( e : ) - ' = e-=.

c) : # 0 . a) Given n ~ IN tJ {0}, define xky n-k "

z. .=

k ! ( n - k)! k=O

Then (z + y)n =

n,

xky n-k =

k=0

k=0 k ! ( n - k)!

1

z. = ~.,(~ + u)" for every n E IN U {0}. It follows that

x~y n-k = n!zn '

~.~ Normed Algebras

oo 1 oo (~ lxn~ e ~+~ = ~ ~ ( x + y)" = ~ z,, = n=O n=O n=O n! /

119

(~.~ 1 n ) ~y

= e~ey

n=O

(Proposition 1.2.9.6 and Proposition 2 . 2 . 1 . 6 ) . b) By a), eZe -z = e z-~ = 1 and so eX) 1 --

e -x

"

c) follows from b). Remark.

II

e ~+~ is not always eXev . For example, let E := s

X :---

[Ol] 0 0

,

y :-----

[oo] 1 0

2) and take

.

Then x 2 = y2 _. 0,

xy --"

[lo] 0 0

[Ooo]1

'

and so (x + y)2n = 1,

(x + y)~,-1 = x + y

for every n E IN. Hence eX=l+x,

eY=l+y

e ~e~ = 1 + x + y + x y ,

e Z + ' = ~l n = o~ . 1 + . ~ o =

C o r o l l a r y 2.2.3.8

(2nl+l)!(x+Y)=(chl)l+(shl)(x+Y)5~eZeY

(Le Page, 1967) Let E be a unital complex Banach alge-

bra. If there is an a E IR+ such that

Ilxyll _< ~llyxll for every x, y E E , then E is commutative.

1~0

E Banach Algebras

Take x, y E E and consider f :q~ - ~ E ,

~:

~ e~Zye - a z .

f is analytic (Propositions 1.3.10.3 and 1.3.10.10) and

II/(~)ll-

Ilea~ye-a=ll <- allYe-a=ea=ll -

~IlYll

for every/~ E r (Proposition 2.2.3.7 b) ). By Liouville's Theorem, f is constant. Hence y = f(O) = f ( ~ ) = ea'ye -a~ ,

ye a~ = eaXy

for every ~ E r (Proposition 2.2.3.7 b) ), and so yx = xy

(Proposition 2.2.3.6).

1

Remark. l-I (Corollary 2.1.4.17) is a counterexample to the above Corollary for the case ~ = IR.

P r o p o s i t i o n 2.2.3.9

Let E be a unital Banach algebra. Given x ~. UE(O),

put log(1 -

-

! n

n--i

(Propositwn 2. I~.3.1~). Then

= 1- x

e I~

for every x E uE(0) and

log e z := log(1 - (1 - e x)) = x for every x ~ E , so that e x ~ UE(1).

Now Z

_

n=0 ~.l

~k

= e l~176

---

1 -cr

k=l

for every a ~ Ul~ (0), and ~

1

=

n

1-

~a

=log(1

(1-ea))=loge

c'=~

k=l

for every a E lK. Thus Jl - e~ < 1. The assertions now follow from Corollary 2.2.3.4. ii

~.~ Normed Algebras

D e f i n i t i o n 2.2.3.10

( 0 )

I~I

Given oh E C and n E IN, define

(o), (:)-(.)_0

~(c~ - 1 ) . . . (c~ - n + 1) n!

We have

for every n, p E IN, with n > p.

Lemma 2.2.3,11

( 0 ) (Raabe'sRatioTest)A series ~ c~. in IK is n--O

absolutely summable if lim inf n n--*oo

(~. 1) 1 (1o. OZn+ 1

:=oo

if a,+a=O).

OZn+l

Choose

-1)[.

1

c~ E 1,1ira inf n Since

lim (1+/3) a 0--*0

1

f~

0>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 <

(:)

for every n E IN and x E E # , it follows from Lemma 2.2.2.12 that the series

~

x~

n----0

converges absolutey and uniformly on E # . Moreover, the radius of convergence of the power series

~

t~

n'-0

in IK is at least 1. By Proposition 2.2.3.2, the family ( ( ~ ) x n ) , e ~ is absolutely summable whenever

lim IIx"ll~ < 1. c) By Proposition 2.2.3.3 and Lemma 2.2.3.12

for all x E E with lim llx'~[l-~ < 1.

n-~oo

In particular, the above relation holds for every z E E with [[x[I < 1 (Proposition 2.2.3.1). By continuity, the relation holds for every x E E # .

1

~.~ Normed Algebras

1~5

2.2.4 I n v e r t i b l e E l e m e n t s of U n i t a l B a n a c h A l g e b r a s ( 0 ) Let E be a unital Banach algebra. Take x E E

P r o p o s i t i o n 2.2.4.1 and a ~. ]K such that

lim IIx"ll A < I~1.

n---}oo

Then 1

(al - x) -1 =

r(z) < n--boo lim I1~-II~ _< I1~11. Now (~1

-

x)

oo __l-l--xn a-x~-=l = (al - x) Z o~n+l

--" Og

n=O

:Z

oo

~lxn-Z

n----O

oo

1 ~x

oo 1 x. oo __1_l_xn = ~ ~ -- X ~ o~n+1 n=O

n+l

oo

=1+

n--O

Z

n=O

L x n

~n

n=l

oo

Z

kX n

an

=1

n=O

(Corollary 2.2.1.11), so that (c~l -

X) -1

=

1 O~--X

It follows that

r(x) < ll--~ lim IIx"ll~. OO The inequality lim I1~-II~ < Iixll

n--too

was proved in Proposition 2.2.3.1. C o r o l l a r y 2.2.4.2

( 0 ) Let E be a Banach algebra and take x e E. If

~(~) = I1~11 the,, r(x) n = r(x n) = [Ixnll = IIx[In for every n E IN.

II

I~6

I~. Banach Algebras

We may assume that E is unital (Proposition 2.2.1.3). By Theorem 2.1.3.4

a), ~(x)" < r ( : ) and by Proposition 2.2.4.1,

r(:)

_< I1:11.

Thus r(x) = Ilxll implies that Ilxll" = r(~)" <_ ~(x")<_ I1:11 <-Ilxll"

(Proposition 2.2.3.1). Hence II

r(x)" = ,'(x") = I1:11 = I1~11".

Corollary 2.2.4.3

(Le Page, 1967) Let E be a complex unital Banach alge-

bra. If there is an a E R+ such that

I1~11 _< ~r(~) for all x E E , then E is commutative.

Given x,y E E,

IIx~ll _< ar(xu) = ar(ux) _< alluxll (Proposition 2.1.3.10 c), Proposition 2.2.4.1). By Corollary 2.2.3.8, E is commutative. II Corollary 2.2.4.4

( 0 ) Let E be a u m t a l Banach algebra and take x, y E

E . If x is invertible and

IIx - yll <

1

IIx-'ll '

then y is also inverttble,

II1- x-'ull

1,

<

and oo

-l

Y

1

= l-(l-x-ly)X

-l

- ~ n=O

(1 - x - l y ) n x

-l .

1127

12.12Normed Algebras

Now

II1- ~-'yll = I1~-~(~-

y)ll _< I1~-'111ix- yll <

1.

Since

x - l y = 1 - (1 - x - l y ) , we deduce that

(x-ly) -' = 1 - (1 - x - l y ) (Proposition 2.2.4.1). From

it follows further that oo

y-l = (x-lu)-lx-1

=

1

1-(1-

--1

= Z

x - ' y ) "z

(1 - x - i y ) " x -l

rt--O

It

(Corollary 2.1.2.6, Corollary 2.2.1.11). C o r o l l a r y 2.2.4.5

( 0 ) Let E be a Banach algebra. Take x E E . Then

a(x) is compact. We may assume that E is unital (Proposition 2.2.1.3). Choose a E IK\a(x) and put 1 "=

11(o~1

-

x)-'ll

"

Take 7 E U ~ ( ~ ) . Since a l - x is invertible and

11(71-

x)

-

(al

x)ll = I~ - al <

-

a,

it follows that 71 - x is also invertible (Corollary 2.2.4.4). Hence

@(~)

c

~\o(~),

]K\a(x) is open and a(x) is closed. Being bounded (Proposition 2.2.4.1), a(x) is compact. C o r o l l a r y 2.2.4.6

II

Let E be a unital Banach algebra and x an element of

E . Let f " IK\o'(x)

>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 ~
By Proposition 2.2.4.1,

r(:) <

llxll

= 1,

so that

a(=) c (~ e H
(X).

n'--:'~U

By Corollary 2.2.4.4, x - a l is invertible, i.e. a r a(x). Hence

a(x) c {a e IKI I,~1 = 1}. C o r o l l a r y 2.2.4.8

( 4 )

II

Let E be a Banach space and u " E ~ E an

isometry. Then

a(,,) c {~ E IKI Ir = 1}. Now I1~11 = I1~-'11 = 1

and the assertion follows from Corollary 2.2.4.7 (and Example 2.2.2.4).

II

~.I~ Normed Algebras

Corollary

2.2.4.9

1~9

Let E be a Banach space. Take u E E(E) and 6 > 0 ,

such that

for every x E E . Then either

@ ( o ) c o(u) or

Us~(0) n a ( u ) = q t . Assume there is an

z @(o)\o(~). Then

II~x - uxll ~> Iluxll- II~xll ~> ~llxll- lal Ilxll- (,~ -Ic~l)llxll for every x E E. By Proposition 1.2.1.18 b), I1(~1 - u ) - ' l l

1

_<

Take /3 e U~_tol(a ) . Then 1 11(/~1 - u ) - - ( a l

- u)l I = [/3 - a I < 6 - ]a I < ii(a 1 _ u ) - ' H

"

By Corollary 2.2.4.4, /31- u is invertible, i.e. /3 ~ a(u). Hence

u ~ , ~ ( , ) n o(=) = ~. It follows that

@(o) no(~)=~, Corollary 2.2.4.10

tt

( 0 ) Let E be a unital Banach algebra. Take x ~ E

and let (Xn)nelN be a sequence of invertible elements of E , converging to x . If (x~l),,e~ is bounded, then x is invertible.

Put

:= sup I1~' II. nq~

There is an n E IN with 1

I1~. - xll < -C~. By Corollary 2.2.4.4, x is invertible.

1

130

2. Banach Algebras

E x a m p l e 2.2.4.11

( 4 ) Let T be a group. Take s E T

and P E t l , c ~ ] .

Let E be one of the following Banach spaces: @(T), co(T), or c ( T ) . Given x E E , set "~'T

} •,

t"

Ix(st)

and define u'E

a) /f s n # 1 for every n

~E,

y:

~.

e IN, then

a(u) = {a e :':: I lal = 1}. b) If n is the least natural number such that s n = 1, then

a(u) = { a e K l a

n = 1}.

u is an isometry, so that

aC~,) c {,~ E IKI I,~1 = 1} (Corollary 2.2.4.8). Take a E I/: with ]a I = 1, and x, y E E such that otx

-- ux

--

y.

Then

for every k E ~ . Hence, by complete induction, k-I

x(s*) = a~x(1) - ~

aiy(s ~ - l - ' )

j=O

for every k E IN.

a)

Case 1

E = g~176

Define y " T -----* IK ,

Then

t:

; ~ ak t 0

if t - - s k fora k E I N otherwise

~.~ Normed Algebras

131

k-I

x(s ~) =

~x(1) - ~

(:rTc~~-l-a = ~ x ( 1 ) - k~ k-t

$=0

"or every k E IN which is impossible since x is bounded. Hence a l - u is not surjective and so ~ E

Case2

a(u).

E=c(T)

and a = l

P u t y "= e r . T h e n

x(s k)

=x(1)-k

7or every k E lN, which is impossible since x is bounded. Hence a l - u is not

a(u).

surjective and therefore (~ E

Case 3

T h e other cases

P u t y := e l . T h e n

x(s -'c)

= a-~x(1)

for every k E IN by complete induction, which is impossible since x E E . Hence o l - u is not surjective and so a E

a(u).

b) P u t y := el and assume t h a t c~n = 1. T h e n n-1

x(1) = x(s")= (~"x(1)-

aZe,(s"-'-')= x(1) -

~

an-',

3--0

which is a contradiction. Hence ( ~ 1 - u is not surjective and therefore ~ E a ( u ) . Thus {c~ e l K i a "

a(u).

= 1} C

Now let a e ] K with (~" # 1. Given y E E , define n--1

x'T-

;IK,

t'-;~

1 -

:

- -

My(s"-l-~t) .

3=0

It is easy to see t h a t x E E . T h e n ((,,:

- ~)x)(t)

= :~(t)

- ~(~t)

=

P,. Banach Algebras

n-I

an

1

1Z ~

-

j=O

3=0 1

,~--

n-1

a n - 1 y~ ~

1

N

a"-

1 ~.1=1 ~u('"-Jt)

n-I

1 y~ ~

1

1

,e-

, ~ - - 1 ~(~"t)=

1 ,"u(t)

=

,~=0

u(t)

for every t E T , so that ( a l - u)x = y. Hence a l - u is surjective. Take z E E with (al -u)z

=0.

For t E T , z(st) = a z ( t ) , so that

by complete induction. Hence

~(t) = o . z=O,

and a l -

u is injective. By the Principle of Inverse Operators, a l -

u is

invertible. Thus a ~ a(u) and a(u) = {a E

C o r o l l a r y 2.2.4.12

(4)

~[a"

= 1}.

I

Let T be a group. Take t E T and let el(T) be

the Banach algebra defined in Theorem 2.2.2. 7 a). If t n ~ 1 for every n E IN, then

,,(e,) = {,~ e ~:1 I,~1 = 1}. If n is the least natural number such that t n = 1, then a(et) = {or E ] K l a n

= 1}.

~,.~ l~ormed Algebras

Given x E

s

133

define ~" e'(T) ---+ t ' ( T ) ,

y:

"x,y.

By Proposition 2.2.1.17, aCe,) = ~(~,).

For x E/~l(T) and s e T,

(~,~)(~) = (~,, ~)(~) = ~ ~,(~-~)~(~) = ~(t-'~). rET

The assertions now follow from Example 2.2.4.11. Corollary 2.2.4.13

m

( 4 ) Let T be a locally compact group, Mb(T) the

unital Banach algebra defined in Theorem 2.2.2.10 b), and 6r the Dirac measure at the point t E T . If t n ~ 1 .for every n E IN, then

o(a,) = {~ e ~ 1

lal = 1}.

If n is the least natural number such that t '~ = 1, then a(6,) - {~ E IK I ~ n -- I } .

The assertion follows immediately from Theorem 2.2.2.10 d) and Corollary 2.2.4.12. m T h e o r e m 2.2.4.14

( 0 ) Let E be a unital Banach algebra. Then the set

U of invertible elements of E is open and the map U----~U,

x:

~, x -1

is a homeomorphism. In particular, U is a topological group with respect to the multiplicatzon.

Take x E U. By Corollary 2.2.4.4

U~, (x)

c

U.

Hence U is open. Since l i m ( 1 - x - l y ) = 1 - 1 = O,

y--+x

it follows that

e. Banach Algebras

134

1

lim

y--,x 1 -

(1 -

x-ly)

=1

(Proposition 2.2.3.2) and so 1

lim r-,z

y-~

=

lim

y--,z 1 - (1 - x - l y )

x -l

= x -1

(Corollary 2.2.4.4). Remark.

Ill

a) Let

T:-{mer

I~1-< 1}.

It is easy to see that T

)C,

a:

;a

is an interior point of the set of non-invertible elements of C(T). Hence the set of invertible elements of a unital Banach algebra is not necessarily dense. b) I f x E U then lim "~"

IlY-I - x - l - x - l ( Y - x)x-lll = O. Ily - xll

Indeed, let x, y E U such that

I1~ - yll <

1 I1~-'11 "

Then

1 - x-'y = x-'(x - y),

II1 - x-'Yll _< IIx-'ll IIx - Yll

so, by Corollary 2.2.4.4,

n----2

oo

<_ ~

IIx-'ll~+'llx _ yll - =

n----2

IIx-'llSllx- yll ~ 1 -IIx-'ll IIx - yll

and the assertion follows. C o r o l l a r y 2.2.4.15 ( 0 ) Let E, F be Banach spaces. The set LI of isomorphisms E -4 F is an open set of s F).

I~.~ Normed Algebras

135

Take v E / / . The map w's

~s

u:

"v-lou

is linear and continuous. If l) is the set of invertible elements of s

then

-1

u = w(v).

Since l) is open (Theorem 2.2.4.14),// also is open.

I

C o r o l l a r y 2.2.4.16 ( 0 ) Let E be a unital Banach algebra. The closure of a proper (left, mght) ideal of E is a proper (left, right) ideal of E . The maximal proper (left, right) ideals of E are closed. m

Let F be a (left, right) proper ideal of E . By Corollary 2.2.1.8, F is a (left, right) ideal of E . Let U be the set of invertible elements of E . Then

FnU=O (Proposition 2.1.2.9). Since U is open (Theorem 2.2.4.14),

-fnu=O, and hence F is a proper (left, right) ideal of E . Let G be a maximal proper (left, right) ideal of E . By the above considerations, G is a proper (left, right) ideal of E . Hence

G=G, i.e., G is closed.

I

C o r o l l a r y 2.2.4.17 The regular maximal proper (left, mght) ideals and the radical of a Banach algebra are closed. Let E be a Banach algebra and F a regular maximal proper (left, right) ideal. Let E be a unital Banach algebra associated to E (Definition 2.2.1.4) and let G ~ E be a maximal proper (left, right) ideal of E containing F . Since F is maximal,

F=ENG. By Corollary 2.2.4.16, G is closed, so that F is also closed. It follows that the radical of E is closed.

I

136

2. Banach Algebras

Let E be a unital Banach algebra, U the topological group of invertible elements of E (Theorem ~,.2,.4.14), and V the connected component of U containing 1. Then V is a clopen normal subgroup o.f U and

P r o p o s i t i o n 2.2.4.18

V = {,el~te~:' (x,),e, f i n i t e f a m i l y i n E } .

If E is commutative, then

V={e~IxEE} and no element of (U/V)\{1} has fimte order if ~ =C,. Take x E V and y E U. Then x-iV, y-tVy are connected sets of U (Proposition 2.2.1.6), containing 1. Hence

x-IV C V,

y-IVy C V.

Thus V is a normal subgroup of U. Since U is obviously locally connected, V is open. Being a subgroup of U, it is a closed set of U. By Proposition 2.2.3.7 b),

{e ~ I x e E} c U. Since E is connected and the map E

DE,

x:

,~ex

is continuous (Proposition 2.2.3.2), {ex Ix E E} is connected. Hence

{e~lx e E} c v . Thus

W : = { H, e Z, ' l~( x ,' ) ~

finitefamilyin E } c V .

By Proposition 2.2.3.9,

u~(1) c {e~lx e E} c w . Since W is obviously a subgroup of V, it is a clopen set of V. Hence

W=V.

~.~ Normed Algebras

137

Assume that E is commutative. By Proposition 2.2.3.7 a), W = {eZ[x E E}.

~et lK = C and take x E U with x n E V for some n E IN. By the above 9esult, there is a y E E such that X n --. eY.

3iven c~ E it, put Z "'--- Xe - ~ y

za := a z + (1 - a ) l .

Then Z n = x n e -It = 1

so that a(z) is finite (Theorem 2.1.3.4 a)) and {a e r ] z~ e U} is connected. Since ; E,

a:

.~za

is continuous and zo=l,

zl = z ,

it follows that z E V and 1

x---- z e ; Y E V ~

which proves the last assertion.

I

P r o p o s i t i o n 2.2.4.19 ( 0 ) Let E be a Banach algebra and x " E --~ lK an algebra homomorphism. Then x' is continuous with norm at most 1. I f E ts quasiunital, then x ' = 0 or IIx']l = 1.

First assume that E is a unital Banach algebra. Given x E E ,

9 '(~) e {x'Cx)} = G~C~'(x)) c a~(~) (Proposition 2.1.3.2, Corollary 2.1.3.13), so that ]xt(X)I _< r E ( X ) <_ tix]l

(Proposition 2.2.4.1). Hence x' is continuous and IIx'll < 1.

138

~. Banach Algebras

Now suppose that E is not unital. Let I( x E be a unital Banach algebra associated to E (Definition 2.2.1.4). The map y " IK • E

~ lK,

(a,x) ~

a + x'(x)

is a unital algebra homomorphism (Proposition 2.1.1.7). By the above proof, y' is continuous with norm at most 1. Hence z' is continuous with norm at most one. Now suppose that E is quasiunital and x' ~ 0. Take x E E with

z'(z) # o and let ~ be an approximate unit of E . Since x' is continuous, x'(x) = limz'(xy) = lim x'(x)x'(y) = x ' ( x ) 1 ~ x'(y) lJ,~ II,~ , '

lim x'(y) = 1.

IIx'll > limlx'(y)l--- 1 --

Definition 2.2.4.20

it,~f

IIx'll- i .

m

( 0 ) The element x of the normed algebra E is called

quasinilpotent if

lim IIz~ll~ - o. Every nilpotent element is quasinilpotent. Proposition

2.2.4.21

( 0 ) Let E be a Banach algebra, x a quasinilpotent

element of E and take y E {x}e.

a) xy is quasin@otent. b) If E is unttal and y mverttble then x + 1/ is invertible.

c) o(~ + ~) = o(u). d) o(~)= {o}. a) For n 6 1 ~ , ii(zy)-ii ~ -iiz-y=ll~ < IIx=ll~lly"llk _< IIx"ll~llyll (Proposition 2.2.3.1), so that

~.~ Normed Algebras

139

lim II(xv)"ll ~ - 0.

n.-.4.oo

b) Since x ~. { y } C , x ~_ {y-1}c (Proposition 2.1.2.8) and y-l E {x} c. By a), x y -1 is quasinilpotent, so that 1 + xy -1 is invertible (Proposition 2.2.4.1). Since x + y = (1 + xy-1)V, x + y is invertible (Corollary 2.1.2.6). c) We may assume E to be unital. Take a E E:. By b), a l - y iff ( a l - y) - x is invertible. Thus

is invertible

a(~ + v) = ,,(v).

d) By c), ~(~) = o ( x + 0) = o(0) = {0}

(Proposition 2.1.3.2). E x a m p l e 2.2.4.22

I (Volterra, 1896) Let T : - {(s,t) e [0,1121t <__s}

and take k E C(T) . Given x E C([0,1]), set

kx" [0,1]

.. ~ IK,

s'

~

I"

k(s,t)x(t)dt.

Then

k'C([0,1l) ~

C([0,1]),

z:

" kx

is a quasinilpotent element of s and given ce E IK\{0} and y E 17([0,1]), there is exactly one x E C([0,1]), such that ax-

kx = y.

This equahon is called the Volterra Integral Equation.

It is obvious that kx is continuous for every x E C([0,1]) and that k is linear. Set := Ilklloo.

140

e. Banach Algebras

Then

Ik(s, t)l Ix(t)ldt

Ikx(s)l <

< allxlloo

for every x E C([0,1]). Therefore, k is continuous. We define the sequence (kn),,r in C(T) inductively by putting k~ := k and k.+t " T

} IK ,

(s, t) ~

i'

k(s, r)kn(r, t)dr

for every n E IN. By complete induction, knx(s)

-

=

kn(s,t)x(t)dt

f"

for every n E I N , x E C([0,1]), and s E [0, 1]. Furthermore,

Ik.(s, t)l

otn 8 n - I

<

( n - 1)!

for every n E IN and (s, t) E T. It follows that 0 n

IIk~ll < ( n - 1)! for every n E IN and that lim I1~-11~ = 0.

I1.--~oo

Hence k is quasinilpotent and a(k) = {0} (Proposition 2.2.4.21 d)). The final assertion now follows. 1 E x a m p l e 2.2.4.23

Given x , y E C([O, 1]), set

x , y .[o, 11

, ~,

s ~

I'

x(s -

t)~(t)dt.

The Banach space C([0,1]) endowed with the bilinear map

C([O, 11) x C([O,I])

"C([O,I]), (x, y).~ ; x 9 y

is a commutative Banach algebra all of whose elements are quasmilpotent.

Take x, y, z E C([0,1]). Then (x 9 (y 9 z))(s) =

x(s - t)(y 9 z)(t)dt =

~.~ Normed Algebras

141

= fo" x ( s - t) ( fo' y ( t - r)z(r)dr) dt =

=

/o'(f"

)

x(s - t ) y ( t - r)dt z(r)dr =

= fos ( foS-r x ( s - r - u)y(u)du) z(r)dr = $

=

fO

(x * y)(s - r)z(r)dr = ((x 9 y) 9 z)(s)

for every s E [0, 1], so x 9 (~ 9 z) = (~ 9 y) 9 ~.

Hence {7([0,1]) is an algebra with respect to the above bilinear map. It is easy to see that it is a commutative Banach algebra. Now, by complete induction, I~(~)1

<

8n-1

hxd"Cn'-' 1)!

for every s E [0, 1] and n E IN, so that lim Ilxnll -~ < aim -.-,oo

.-,~

Iixli = 0

(~!)~-

"

Hence x is quasinilpotent.

I

( 0 ) Let E be a normed algebra. An element x of E is called a topological zero-divisor if there is a sequence (xn)ne~ in E such that

D e f i n i t i o n 2.2.4.24

lim sup IIxnll > 0 71---I'00

and either (xxn).~r~ or ( x n x ) . ~

converges to O.

Every zero-divisor of 0 is a topological zero-divisor. I

"t

( 0 ) Let x be a topological zero-divisor m the norreed unital algebra E. Then x is not invertible.

P r o p o s i t i o n 2.2.4.25

14~

~. Banach Algebras

There is a sequence (xn).et~ in E , with lim sup n--~oo

I1=.11 > 0

and either (xXn)ne~ or (x,x)ne~ converges to O. Assume x to be invertible. Since =. = (=.=)=-~ = =-'(==.)

for every n E IN, ( x . ) . e ~ must converge to 0 (Proposition 2.2.1.6), which is a contradiction. P r o p o s i t i o n 2.2.4.26

II ( 0 ) Let E be a unital Banach algebra and (Xn)ne~

a convergent sequence of invertible elements of E . If x :=

lira xa is not n-too

invertible, then there is a sequence (Yn),,e~ in E such that lim Xyn = lim ynX

n--too

n--too

= 0

and IIv.II = z

for every n E IN. In particular, x it is a topological zero-dwiaor. Now

"-",=.t, -

sup

oo

nEIN

(Corollary 2.2.4.10) and we may assume that lira [[x~'{I = oo.

n---~oo

Given n s IN, set ~ X1 - I

Then lly.Ii = z

for every n E IN, lim y . x . = O,

n.--4,oo

1~.~ Normed Algebras

~43

and lim

sup

n-too

Ily.(x

~.)11 <

-

lim

1"1-..I,oo

I1.: - .:.11

= 0.

Thus lim yn(x - xn) = O,

n--~oo

lim ynx = lim yn(x - :r,n) + lim y , x , = 0. The relation lim x y . = 0 is provided similarly. Hence, x is a topological zero-divisor.

II

C o r o l l a r y 2.2.4.27 ( 0 ) (~ilov) Let F be a closed subalgebra of the Banach algebra E . Take x E F . Then the boundary of aF(X) is contained in the

boundary of aE (x). In particular, rE(Z) = ~F(X),

oE(x) c IR+ = ~ a~(x) c lit+. If a~(x) has not interior points or if IK =q~ and (rE(x) C IR, then o.(x)

= OF(*).

We may assume that E is unital (Definition 2.2.1.4) and that F is a unital subalgebra of E . Let a be a boundary point of O'F(X). Let (an)ne~ be a sequence in

~\aF(x)

converging to a . Then (anl - x)ne~ is a sequence of invertible

elements of F converging to a l - x. Since a l - x is not invertible in F (Corollary 2.2.4.5), it is a topological zero-divisor in F (Proposition 2.2.4.26). A fortiori, it is a topological zero-divisor in E . Hence a l - x is not invertible in E (Proposition 2.2.4.25). Thus a belongs to aE(X). Since O'E(X ) C O'F(X)

(Corollary 2.1.3.14 a)), a is a boundary point of aE(x). Assume that aF(X) has no interior points. Then

~,. Banach Algebras

144

ok(x) C aF(X) = boundary OfaF(X) C boundary ofa~(x) C as(x) and hence

OE(~) = ~ F ( : ) . Now suppose that lq = C and aE(X) C JR.. Since the boundary of CrF(X) is contained in the boundary of aE(X), aF(X) C JR. By the above result,

aE(X) =

aF(X),

m

Proposition 2.2.4.28

( 2 ) Let (Y2,A) be a unital Banach category. Take E, F E ~2, and x, y E A(E, F). If x is invertible and

II~- ~ll <

1

II~-'ll'

then y is invertible, II1E- x-ly[I

1,

<

and oo

I

Y-' -- IE - - ( I z -- x--;y) x

-i

-- ~

(IE --

x-:

y)"X-t.

n=O

Now IIIE - x-~ll

= IIx-'(x - ~)11 ~< IIx-Sll IIx - yll -< 1.

Since

x-ly = 1E -- (1E -- x - l y ) , it follows that

(x-ly) -1 =

1E--(1E--x-ly) (Proposition 2.2.4.1). From

y=x(x-ly) we deduce further that

~-~ =

oo

(x-ly)-I=-~

=

1E -- (1E -- x-ly) (Corollary 1.5.1.7, Proposition 1.5.1.3 b)).

X

--I

- ~

(IE - ~-~,)"~-~

n=O

i

~.~ Normed Algebras

145

T h e o r e m 2.2.4.29 ( 2 ) Let (12, A) be a unital Banach category. Take E, F E I2. The set of invertible (left invertible, right invertible) morphisms of A(E, F) is open. If U denotes the set of invertible elements of A(E, F) then

U---~ A ( F , E ) ,

x"

~ x -1

is continuous. Let x0 be a left invertible morphism of A(E, F). Take y ~_ A(F, E) with

yXo : 1E . Take x ~. A ( E , F ) with

llx- xoll<

1

llyll"

Then ilYx --

1Ell = ilYx -- yXoII = flY(x -- Xo)il ~ IIY[IIIx -- Xoll < 1.

By Proposition 2.2.4.1, yx is an invertible element of A(E). Thus

(yx)-lyx = 1E, i.e. x is left invertible. Hence the set of left invertible morphisms of A(E, F) is open. Similarly, the set of right invertible morphisms of A(E, F) is also open. Hence the set of invertible morphisms of A(E, F) is open. Now take x E U. From

"" "'a~m.lt- x - l y ) ' = l - l = 0

y-..~z

(Proposition 1.5.1.3 a)), we deduce t h a t 1 lim - 1 ~-~x 1 - (1 - x-ly) (Proposition 2.2.3.2). Thus lim y - l =

lim

1

x-l=x-t

(Proposition 1.5.1.3 a), Proposition 2.2.4.28). Hence the map

U is continuous,

~A(F,E),

x~x

-1 m

146

~. Banach Algebras

P r o p o s i t i o n 2.2.4.30

( 0 ) Let E be a unital Banach algebra. Take x E E ,

and a E a ( x ) . Then there is an x' E E' such that

IIx'll = x ' ( 1 ) -- 1 ,

x'(x) = a .

Let F be the vector subspace of E generated by {x, 1} and consider y':F~K,

3x+71:

,~a3+7.

y' is well-defined even if F is one-dimensional. Indeed, in this case, there is a 6 E IK such that x = 61. Since a E a(x), 6 = a . Given 13,7 E IK, 3x + 71 = (a3 + 7)1, so that the expression a 3 + 7 is uniqely determined by 3 x + "71. Take ~, 7 E l ( . Then a 3 + 7 e 3 a ( x ) + 7 = a ( 3 z + 71),

[y'(3x + 71)l = [a/3 + 71 < r(3x + 71) < 113x + 71[[ (Proposition 2.2.4.1). Hence

i1r ~< x. By the Hahn-Banach Theorem, there is a continuous linear extension x' of y' to E with II~'ll = I1r

We have 9 '(~) = r

=

and [[x'[I ~ 1 - y'(1) - x'(1) ~ IIx'll Ilxll = IIx'll, i.e.

Iix'[I = x'(1) - 1.

I

~.~ Normed Algebras

Corollary 2.2.4.31 and put

147

( 0 ) Let E be a unital Banach algebra. Take x E E ,

K := {x'(x) lx' E E', IIx'll = x'(1) = 1}.

t/ IIx § ~111 -- r ( x § a l )

for every a E IK, then K is the convex hull of a(x). By Proposition 2.2.4.30, a(x) c_ K . Put A := { (a, r) E IK x ]0, oo[ I a(x) C U~ (a) }. Take (a, r) E A and x' E E' such that

Ilx'll

= x'(1) = 1.

Then ~(~ - ~1) = o(~) - ~ c uy(0),

so that IIx - ~111 = r ( x - c~l)

< r.

Hence Ix'(z) - ~1 = i x ' ( z - a l ) l

< IIz'll I1~ - ~111 < r ,

9 '(~) e u y (~). Since x' and (c~,r) are arbitrary, it follows that g C

N

U~(a).

(a,r)EA

But

N

U~(~) is the convex hull of a(x). Since K is convex it must, in

(,~,r)EA

fact, be the convex hull of a(x).

I

2.2.4.32 Take p E {0} U[1, ~ ] and let ur, ut be the right and the left shift operators of if, respectively. Then

Example

o ( u , ) = ,,(,,t) = ur

This relatwn still holds for c in place of ~ .

~4S

~. Banaeh Algebras

Take cr E IK, with [cr[ < 1 and put := (a"),~

e ~'.

Then (al - ut)y = ay-

u t y = 0,

so that a l - ut

is not injective. Suppose t h a t y E I m ( a l - ur). Then there is an x E ff such that (al

-

u,)x = y.

By Example 1.2.2.9 d), 1 n Xn ~- OCn+1 Z ~ 2 k k=l

,.

for every n E IN. Hence

lim Ix,,I = o o ,

n---}oo

which is a contradiction. Thus r I m ( a l - u,.) and a l - ur is not surjectiv. It follows t h a t

{,~ e X
it follows, by Proposition 2.2.4.1 and Corollary 2.2.4.5, that a ( U r ) = a ( U t ) = IK ~ 9

II

$.lg Normed Algebras

149

Definition 2.2.4.33 Let E be a unital Banach algebra and let (xn).e~ be a sequence in E . We put

IhE0- [ x n

:=

1.

/f A E ~s(IN) then

H

Xn

nEA

is defined inductively using the order mduced on A by the natural order of IN. If the limit lim H x.

A,~

nEA

exists, then the sequence (xn),er~ is called multipliable and the above limit is called the product ol the sequence (xn),et~ and it is denoted by H x,,

(in E ) .

nEIN

Take x E E . The relation

z= 1-Ix,, heir,

is equivalent to the following: For every e > 0 there is a finite subset A of lN such that for every finite subset B of IN containing A,

xEB

Let (x,),er~ be a sequence in the umtal Banach algebra E . Let (C) denote the following statement: For every e > O, there is a

P r o p o s i t i o n 2.2.4.34

finite subset A o.f IN such that

II H

x.

- 111 <

nEB

whenever B ss a finite subset of IN\A. a)

If (x,,),,er~ is multzpliable and I-I x,, is invertsble, then (C) is fulfilled. nEIN

b)

If (C) is fulfilled, then (xn),,e~ is multipliable.

150

~. Banach Algebras

c) If (C) is fulfilled and each x. (n e IN) is inuertible, then

1-I x. is nEIN

invertible. Given A e gll(lN), put T"T

ZA :'- J.J. Z n . nEA

a) Put 27::

~ - [ :;l:n 9 n61N

Take e > 0. By the definition of the product and by Theorem 2.2.4.14, there is a p E iN such that for every A E ~/(lN) with INp C A, XA is invertible and

IlxX'I1~ - ~11 <

x-ill < 1,

2(1 + IIx-'ll)"

Thus

I1~11 _< IIx~'

-

~-tll

+

IIx-~ll

<

1 +

IIxA - x~,ll _< IIxA - xll + IIx - x~,ll <

IIx-'ll,

1+

llx-~ll

for every A E ~I(IN) with lNp C A. Take B E ~I(IN\INp). Put A := lNp O B. Then

<_ IIx~lll IIXA- ZUnlI < (1 + [ix-Ill) 1 + IIx-'ll

=

r

Hence (C) is fulfilled. b) By (C), there is a p E IN such that

I[XA- 1[[ <

1

for every A E q3/(iN\lNp). Hence, if A E ~I(IN) and INp C A, then

~.~ Normed Algebras

151

I[XA - x~,[ I = IIx~,(XA\~,, - 1)l I < Ilx~,ll IlXA\~, - Ill <~ [Ix~,,ll ,

IIXAII --< IIxA - ~,11

+

I1~,,11 _< 211x~,ll 9

Take e > 0. By (C), there is a q E IN such that

IIXA

-

111 <

2(1 +

I1~,,11)

br every A E ~ ! (IN\INq). We may assume that q _> p. Then IIxA

-- XBII

<_ IIXA --

111 + II1 - ~,,11

<

1 + I1~,11

_'or all A, B E ~I(IN\INq) and

for all A, B E ~f(lN) with lNq C A 0 B. Hence the image of ~ map ~[II(IN )

~ E,

A;

under the

" XA

is a Cauchy filter. Thus the family (xn),e~ is multipliable. c) By (C), there is a p E IN such that

II~A

-

111 <

1

for every A E ~I(IN\INp)" By b), the sequence (xn+n)~e~ is multipliable and so, by the above inequality, 1

II I-[ Xp+,,- 111 < ~ < 1. nEll

By Corollary 2.2.4.4, I-I xp+,~ is invertible. From nE~

nell

we deduce that H xn is also invertible.

nE~i'

Ill

nEl~l

Corollary 2.2.4.35

Let (x,,),,e~ be a multiphable sequence in a unital Banach algebra the product of which is invertible.

15~

a)

2. Banach Algebras

lim z . = l .

n--~oo

b) Every subsequence of (x.).z~ is multiphable. a) follows from Proposition 2.2.4.34 a). b) follows from Proposition 2.2.4.34 a), b).

II

Corollary 2.2.4.36

Let E be a unttal Banach algebra and (x,,),,zt~ an absolutely summable sequence in E . Then (1 + x,,),,~r~ is a multipliable sequence in E . If in additwn 1 + z,, is invertible for every n E IN, then I-I (1 + z,,) nElg

is also invertible.

Given A E ~I(IN), put XA : - - H X n . nEA

Take c > 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_<

nEB

CCB nEC

the.

CCB Cr162

nEB

and the claim now follows from Proposition 2.2.4.34 b), c).

l

~.~ Normed Algebras

153

2.2.5 T h e T h e o r e m s of Riesz a n d G e l f a n d T h e o r e m 2.2.5.1 ( 0 ) (F. Riesz, 1913) Let E be a unital Banach algebra and take x E E . Then the resolvent of x

lK\a(x)

>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 <

1 - X-1Y

= X-'(X

IIx-'ll'

- Y) = (ao -o0X

-1 ,

111- X-~YII < 1.

By Corollary 2.2.4.4, oo

(al - x)-' = Y-' = Z

oo

(I - X - ' Y ) n X - ' = Z

n=O

n=O

where the radius of convergence of the power series oo

Z

tnX-n-I

n'-O

is 1

lim sup

IIX-"-'ll~

71-'I'00

(Proposition 2.2.4.1). Hence the map

> -

1

-r

(a - ao)"X -n-' ,

154

~. Banach Algebras

IK\a(x)

;E,

a:

; ( c ~ l - x) -I

is analytic. Given ~ E K \ a ( x ) , c~ ~ O, (c~l-x) -1-

1

1-

1 )-1 x

Since lim [ [~l x [ I - a--,~ lim "IIxll a-.oo ~ - = O, we see that lim ( 1 -

a~oo

1 ) -l = 1

~x

(Theorem 2.2.4.14) and so lim (~1 - x) -t = 0.

m

Q--~OO

Coronary 2.2.5.2 ( 0 )

Let t

be ~ ~,,t~t B ~ n ~ h ~tg~b~. Take ~ ~

and let U be an open set of ]K containing a ( x ) . Then there is an 6 > 0 such that

o(~) c u for every y e U[ (x). Since

lim (al - x) -l = 0 (Theorem 2.2.5.1), /~ := sup I I ( a l - x ) - ~ l l < o o . aE~\U

Set 1 m

.

Take V E U~(x) and a E lK\U. Then 1

1

[[(al - x) - (al - y)[[ = [Ix - y[[ < E = ~ _< [[(c~1 - z)-i[[ "

By Corollary 2.2.4.4, c~l - y is invertible, i.e. a ~ g(y). Hence

o(~/) c u.

II

I~.~ Normed Algebras

155

2.2.5.3 ( 0 ) Let E be a unital Banach algebra and ( x n ) n ~ a sequence m E converging to x E E . Then

Corollary

is compact.

Put

Let tl be an open covering of K . Then there is a finite subset ~ of tl with a(x) c U v. VE~

(Corollary 2.2.4.5). By Corollary 2.2.5.2, there is an m E IN, such that

U oct.) U v. n=m

VE~

m

Since U a(xn) is compact (Corollary 2.2.4.5), there is a finite subset ~J of II m=l

with 1TI

U w. n=l

WEeVIl

Thus KC

U

v,

VE~II

and hence K is compact. T h e o r e m 2.2.5.4

I

( 0 ) (Gelfand, 1941) Let E be a Banach algebra and

x E E . For ]K = ]Ft, we assume in addition a~((x, 0)) C IR. Then

rCx) = lira IIx"ll ~ _< Ii,~11, and

o(x) # 0.

156

2. Banach Algebras

We may assume that E is unital (Proposition 2.2.1.3) and 1K = q~ (Proposition 2.2.1.19, Corollary 2.1.5.14). Put U := ~/ a Eq~]la] > lim ] ] x " ] ] ~) , n-.-))oO

v : - {~ E C l lal > r ( x ) } ,

n-'O

g'V----cE,

a:

;(al-x)

-l

(Proposition 2.2.3.1, Theorem 1.1.6.23). By Proposition 2.2.4.1, U C V and ] = glU, and by Theorem 2.2.5.1, g is analytic. Take x' E E ' . Then x' o g is analytic (Proposition 1.3.10.4) and oo

~'(~")

~'o g(,,) = ~' o 1(,~) = ~ n----O

for every a E U (Proposition 1.2.1.16). By the classical theory of analytic functions, it follows that k/'x-2-~ a - * . / n e ~ is summable for every a E V. Take a E V. Then Xe(X n )

lim

n-~oo O~n + l

-- 0

so that sup

z'( 1

xn ]


nEi~l

Since a is arbitrary, 13 "= sup

nEIN

)J=x-ll ~n+l

(Corollary 1.4.1.4). Given n E IN,

so that lim IIxnil 88< c~.

nEIN

Since c~ is arbitrary, we deduce that

< co

~.~ Normed Algebras

157

lim IIx"ll~ <__~(~)

n--~oo

and

r(x)-

lim IIx"llk < Ilxll

n--t, o o

(Proposition 2.2.4.1). Assume that a(x) is empty. Set f'r

-> 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~ <_ ~11~11 for all x E E , then E is commutative. Take x E E . Then, by complete induction

II~ll ~ _< ~ ' - ' l l ~ ~" II, II~ll-< v ~ l l ~ ' l l ~-" for every n E IN, so that

Ilxll _< nlim v/-dllx2"ll 2-~ - v ~ r ( x ) -too (Theorem 2.2.5.4). By Corollary 2.2.4.3, E is commutative.

I.

C o r o l l a r y 2.2.5.7 Let E be a finite-dimensional complex Banach algebra. If no element of E\{O} is nilpotent, then E is commutative. Let {r x E be a unital Banach algebra associated to E . Take (a, x) E C x E . Assume that (a, x) is nilpotent. Then there is an n E IN such that (~, x)" = O. Then

kx

)on = ,o x,n = 0

~.~ Normed Algebras

159

so that, successively, a n=O,

a=0,

xn=O,

x=0,

and

(a,x)=0.

Put

/~ := inf { IIx~ll I x e r x E , Ilxll = 1 } . Since q~ x E is finite-dimensional, {x E q~ x E [ /~ > 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<1~-,~1<

~

,

and y is its residue.

We first assume that a is a pole of order p of the resolvent of x. Let y be the corresponding residue. By Proposition 2.2.6.1 a), b), c), y is idempotent, x y = y x , and p is the smallest number p E l~l such that ((x - al)y)p

= 0.

In particular, ( x - a l ) y is nilpotent. Let z be the residue of r

" E,

/3:

~ - ( / 3 - a ) - l ( / ~ l - x ) -1

at a . Then yz ---- zy = O,

~,.~ Normed Algebras

167

z ( x - a l ) = ( x - c~l)z - 1 - y (Proposition 2.2.6.1 a), b)). Now a s s u m e t h a t the above conditions 1), 2), 3) are fulfilled. Let p be the smallest n a t u r a l n u m b e r with

((x

-

al)~)~ = o.

Then p

( ( f ~ - a ) l - (x - a l ) y ) -1 = ~ - ' ~ ( f / - a ) - n ( ( x - a l ) y ) "-~ n--1

for every /J E C\{c~} (Proposition 2.1.3.16 b)). Define f "~\{c~) ---+,

g'Ur

~ ~-+ ((/~ - e~)l - (x - a l ) y ) - l y

(a) - - ~ E ,

f~:

~ - z ( 1 - (/J - c~)z) -1

(Proposition 2.2.4.1). T h e n (1 - y ) f ( ~ ) - - 0 ,

((~

for every f~ E r

-

(~)1

-

(x

-

~l)y)f(f~)

=

y

and

ug(~) = 0,

g(/J)((/~

= -z(1

- (~

-

og)z)-l(,~

-

y)

-

(x

-

o~)(1

-

o~1)(1

-

a ) ( 1 - y) + z(1 - ( ~

-

-

y)) =

o~)z)-l(x

-

al)(1

- y) =

-- (1 - y)(1 - (f~ - a ) z ) - l ( 1 - (/~ - c~)z) ------1 - y for every

/J E Ur (a) (Proposition 2.1.2.8).

It follows t h a t

(f(/~) + g(/~))(/~l - x) -- (f(f~) + g ( / ~ ) ) ( ( f / - c~)l - (x - c~l)) --

= (f(/~) + g ( / ~ ) ) ( / ~ - c~)y - (x - a l ) y + ( / ~ - ~)(1 - y) - (x - ~1)(1 - y)) =

168

~. Banach Algebras

-- f(/3)((/~ - c~)y - (x - al)y)

-4- g(/~)((/3 - a)(l - y) - (x - al)(l

- y)) -

=y+l-y=l for every ~ 9 Ur ~ (a)\{a}. Therefore -p

"~

y ~ ( t - ol)n((x --

clzl)~) - n - I

-- ~

n=-I

(t -- ot)nz "+1

n=0

is the Laurent series of the resolvent of x at a. All the other assertions now follow. II Theorem 2.2.6.6 ( 6 ) Let E be a B a n a c h space and take u E s G w e n n 9 IN u {0}, put Fn := Ker u n,

Gn := Im u n .

a) F._, n 9 IN ==~

F.,

u(Fn)c F._t,

G,, C G , , _ t ,

u(G,,_t) c G,,.

c

b) I f F t, = Fp+l f o r a p 9 IN, then Fp = F,, .for every n 9 IN, n > p , and the map

Gp

} Gp,

x~--~uz

ts :njective.

c) I f Gp = Gp+l f o r a p E IN, then Gp = Gn f o r every n 9 IN, n > p , and the map Gp

",. Gp ,

z:

~ ux

is surjective.

d) Given p E IN, the following are equwalent: dl) Gp is closed and Fp = Fp+~, Gp = Gp+~. d2) There are v, w E s

such that

v2 = v ,

uv=vu,

vw = wv = O ,

(uv)P=O,

uw = wu = l - v.

~,.~, Normed Algebras

169

If these conditions are fulfilled, then E = F,,~Gp, v is the projection of E onto Fp with G n = Ker v , the map Gp ~

Gp,

x:

; ux

~ ep~

x:

~ ~JX

is an isomorphism, and

Gp is its inverse. Defining i'Gp

~E,

x:

~ x,

j " Gp

~ Gp,

x:

; ux,

~ Gp,

x:

~ (1-v)x,

and

(l-v)0"

E

it follows that w = i o j-1

o (1 - V ) o .

In particular, v and w in 2) are unique. e)

If IK = r

and 0 E a ( u ) , then 0 is a pole of the resolvent of u iff there

is a p E IN for which the above equivalent conditions d l) and d~) of d) are fulfilled. In thzs case, the order of the pole is the smallest p E IN for which these assertions hold and

p oo Z t-nun-lveZ tnWn-I-I n=l

n=0

is the Laurent series of the resolvent of u in O. It converges in

{ I and v is its residue.

1}

170

~. Banach Algebras

a) is obvious. b) We first show, by complete induction, that Fp = Fn for every n E IN with n > p. Assume that Fn-l = Fp. Then, for x E Fn, u n - l u x -" l~nx "- O,

ux e F~_l = Fp,

~ ~ F,+, = F , ,

i.e. Fn = Fp. Now take x E G~ with u x = 0. There is a y E E such that uPy = x. Thus UP+Iy ----UX ~ O,

x -- uPy - - O .

Hence the map Gp

~ Gp,

x :

~ UX

is injective. c) Take x s Gp. There is a y E E such that un+ly = x. Then uPy E Gp

and = ~"y ~ ~(G,). Hence the map Gp

~. Gp,

u:

,~ux

is surjective. It follows that G . = u"-P(Gp) = Gp for every n E IN with n > p. d) First suppose that dl) is fulfilled. By b) and c), the map Gp ~ ~ Gp,

27:

; ux

is bijective and therefore an isomorphism (Corollary 1.4.2.4). Take z E Fp N Gp. Then uPX ----O.

~.~, Normed Algebras

171

Thus x -- 0. Hence

Fp n Gp = {0}. Now take x s E . There is a y E Gp with

Since UP(X

-- y)

-- UPX -- UPy

-- O,

x - y s Fp, so that the map

FpxGp ; E ,

(x,y)'

;x+y

is surjective. By Corollary 1.4.2.6, E = Fpr

Let v be the projection of E onto Fp with K e r r

= Gp (Theorem

1.2.5.8 a =~ b). Take x E E . There is a (y, z) ~. Fp x Gp such that x=y+z.

Then My E Fp, u z ~_ Gp by a), and so u v y --- u y -- v u y ,

Z t V Z - - 0 "-- V Z t Z ~

uvx = uvy + uvz = vuy + vuz = vux ,

i.e. U V --" V ~ .

Thus ( u v ) p = v P u p = O.

Define w := i o j - t o (1 - v)0.

172

2. B a n a c h A l g e b r a s

The map Gp

,

Gp,

X:

~ WX

~, G p ,

x:

~ ux ,

~

is obviously the inverse of the map Gp

and

= wv = O,

vw

uw=uozoj

wu=ioj

-l

-l

o ( 1 - v)o = l - - v ,

o(1- v)0ou = 1-v.

Now suppose that d2) is fulfilled. Set F:=Imv,

G:=Kerv.

Then

E = F ~ G

(Theorem 1.2.5.8 b =~ a) and vuP = uPv = uPvP = ( u v F = O.

Thus F C Fp, Gp C G. Now w u p+~ = w u u p --

(1 -

v)u p = u p-

vu p = u p .

We deduce that UPX --.--wuP+Ix ----O, so that

x ~ Fp

whenever x ~ Fp+l. Hence Fp = Fp+~.

From vw=O, and

~.~ Normed Algebras

uw=

173

l-v,

it follows that Imw C G C u ( G ) ,

G C u~+I(G) C Gp+l C Gp C G . Hence G

=

Gp

=

Gp+l.

Thus, dl) is proved. From the first part of the proof, we see that E - Fp (~ Gp ,

so that

F=F~. From

uw

=

wu

=

1

- v,

it follows that the map Gp

~> 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 <_ Ilali Ila'll,

Ila%ll-< lla'll llall

and for (z, a, a') 6 E x F x F' (xa)a' = ~(aa'),

(,,'a)~ = a'(ax ),

(aa')x = a(a'x,),

( ~ ' ) ~ = ~(a'a),

(a'z)a = a'(za),

(a~)a' = a ( = r

The first assertion is easy to verify. Take y 6 E . Then (u, (~.)a') = (u(~.),.') = ((~).,.') = (~, a-') = (u, ~(..')), (u, (a'a)x) = (~y,,,'a) = (a(xu), a') = ((a~)u, a') = (u, a'(az)), (u, (a.')~) = (~u,..') = ((~u).,.') = (~(u.), a') = (y.,.'~) = (u, ~(.'~)), (u, (~.').) = (au, ~.') = ((.u)~,.') = (.(u~), a') = (~,.'.) = (~, ~(a'.)),
(u, (ax)a') = (u(a~), a') = ((u")~, ~') = (ua, ~a') = (u, ~(~')) (Proposition 2.2.7.2) and the last assertion now follows,

P r o p o s i t i o n 2.2.'/'.10 then

( 0 ) l/E

m

is a Banach algebra and F an E-module

OFa)a' = aa',

a'OFa) = a'a

/or an~t (a, a') e F • F'. Take z 6 E . Then

(x, (jFa)a') = OFa, a'x) = (a, a'x) = (xa, a') = (x, aa'),

Hence OFa)a' = aa'

a'OFa) = a'a.

m

~,.~ Normed Algebras

Definition

2.2.7.11 ( 0 )

179

Let E be a Banach algebra and F an E-module.

Given (x", a') E E" x F ' , define a'x" : F

~, ]K,

a:

; ( x" ,a a ') ,

x"a" F

~ lK,

a'

~ (x",a'a)

(Proposition 2.2. 7.9).

It is easy to see that if E = F then the above definition coincides with Definition 2.2.7.8. P r o p o s i t i o n 2.2.7.12

( 0 )

Let E be a Banach algebra and F an E -

module. Then, given x" E E " , a E F , and a' E F' :

~) ~'~",~"~' e r ' ,

II~'~"ll < II~'ll I1~"11, II~"a'li < II~"ll II~'ll.

b) a ( a ' x " ) = (aa')x",

x"(a'a) = ( x ' a ' ) a .

c) If u is defined by u :F

~ E' ,

a:

; aa'

( resp. a' a) ,

then

u' z" = a' x"

( resp. x" a')

for every x" E E".

a) is easy to see. b) Given x E E ,

(~, ~(a'~"))= (~a, a'~")= (~", (~a)a')= (~", ~(aa'))= (~, (a~')~"), (~, ~"(a'a))= (~", (a'a)~)= (~",~'(a~))= (~, ~"a')= (~, (~"~')~) (Proposition 2.2.7.9). Thus

a(~'~") = (~')~",

~"(~'~) = (~"~')~.

c) Given a E F ,

(a, ~'~") = ( ~ , ~ " ) = (aa',~")= (a,a'~") (resp. (a, u'x") -- (ua, x") -- (a'a, x") -- (a, x" a') ). Thus u'x" - a'x"

(resp. x"a').

m

2. Banach Algebras

180

Definition define

2.2.7.13

( 0 ) Let E be a Banach algebra. For x',y" E E",

x" -~ y" : E' ~

IK,

x" ~- y" : E'

~K,

x"

~, (y",x'x"),

x' ~

(x", y"x')

(Proposition 2.2.7.12). q and ~- are called the right and the left Arens multiplication, respectwely (Arens, 1951). Remark. -~ and F may be different (see Remark to Example 1.5.2.15). It is easy to see that x" -~ y", x" ~- y" E E" and

IIx" ~ y"li _- IIx"ll Ily"ll,

IIx" e- y"ll < IIx"ll Ily"ll

for every x', y" E E" and that the map

E" x E"

> 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 _<

_< I1~11I1~. + xzll = Ilall II(a, ::,:)(o, z)ll < Ilall I1(~, ~)11 I1(o, ~)11 _< I1::,:11(~,~)11, II~((a, z)a)ll = IIz(,~a + ~a)ll = Ila~a + ~xall = I1(~r + ~)all -<

_< IIo~ + ~11 Ilall = I1(O, ~)(~, x)ll Ilall _< I1(o, ~.)11I1(,:,, x)ll I1~11_< I1(~, x)ll Ilall for every z E E # . Hence

I1~(~, x)ll

= lim

Ila(a, x)z)ll < Ilall II(a, x)ll

II(,~,z)all

= l Zt i m~

I1~((~ ' ~)a)ll -<- I1(-,,:,:)11 Ilall

and the norm axioms are also fulfilled. Proposition

2.2.7.18

I

Let E be a Banach algebra, E an associated unital

Banach algebra, ~ an approximate unit of E , and F an E-module such that for every a E F , lim xa = lim ax = a. x,i~

x,~

a)

lim IIxa'll = lim Ila'xil = Ila'l[ for every a' E F ' .

b)

F' admits a unique E-module structure extending its canonical E-module structure (Proposztion Z.2. 7.2). a) Take a E F # . Then, given x E E ,

I(a, a')l _< I(a - ax, a')! + i(a, xa')l _< Ila- axll Ila'll + Ilxa'll,

~,.~ Normed Algebras

189

I(a, ~')1 <- I(a- ~, ~')1 + I(~, ~'x)i <_ Ila- xall Ila'll + Ila'xll, so that

[(a, a')l < lim inf I1~'11 < lim sup IIx~'ll _< I1~'11, I(a, a')l _< limx , ~inflla'xll < limx , ~sup Ila'xll _ Ila'll. Since a is arbitrary, it follows that lim

Ilxa'll

= lim Ildxl[ = [[a'[I.

b) follows from a) and Proposition 2.2.7.17.

Remark.

II

Assertion a) cannot be strengthened to lim xa' = lim a'x = a' for

every a' E F ' as shown by the counterexample E := c0,

F := ~1.

Let E be a quasiunital Banach algebra and E an associated unital Banach algebra. Then E' admits a unique unital E-module structure extending ~ts canomcal E-module structure. I

C o r o l l a r y 2.2.7.19

P r o p o s i t i o n 2.2.7.20

( 0 ) If E is a quasiunztal Banach algebra, then

Ix'(x)l ___ Ilxx'll,

and

Ix'(x)l <_ IIx'~ll

for every (x, x') e E x E ' . Let ~ be an approximate unit of E . Then

I~'(u~)i = I(~,~')1 <_ li~'ll, I~'(~)1 = I(~,~'~)l <_ I1~'~11, for every y E E # . Thus Iz'(z)l = limlx'(yx)l < Ilxz']l y,~ -

i~'(~)l = Jim i~'(~u)l < II~'~ll-

,

II

lgO

~. Banach Algebras

2.2.7.21 ( 0 ) Let E be a (unital) Banach algebra, of E-modules, p E [I,c~] O 10},

Example family

F : = { ~EHEttET

(Et)tE T a

(,lti(t)ll),~rE "(T)},

and

~ II(ll~(t)ll)t~rllp.

l~.+, ~:

q: F ~

a) F is a vector subspace o / I ' I Et and q is a complete norm. tET

b) F endowed with the maps E x F - - + F,

(x,~):

; (x~(t))tET,

F xE ~

(~,x):

~ (~(t)x),er

F,

is a (unitaO E-module.

a) was proved in Proposition 1.1.2.7. b) is easy to verify.

I

E x a m p l e 2.2.7.22 ( 0 ) Let E be a (unital) Banach algebra, F a Banach space, and G a (unital) E - module. For all x E E and u E s G), put

Then xu, ux E s

xu : F ---+ G,

y ~

ux : F ~

y:

G,

x(uy),

~ (uy)x.

G) for all x E E and u E s

G) and

s

G) endowed

with the maps E • s s

G) • E

~s

G),

~ Z:(F,G),

(x, u) ~

xu,

(u,z):

; ux

is a (unital) E-module.

I

Definition 2.2.7.23

( 0 ) If E ,s a Banach algebra, F an E-module, and m, n E lN, then we denote for every x E E and a E F,n,n by xa and ax the elements of Fm,n defined by

(~),~ := ~ , , , for all (i, 3) E lNm • INn, respectively.

( ~ ) , , := a,,~

$.~ Normed Algebras

191

?roposition 2.2.7.24 Let E be a quasiumtal Banach algebra. For every n E IN, we denote by E n the corresponding E-module endowed wsth the euchdean norm (Example IL2. 7.21). Furthermore, we denote for every m, n E IN by ~ ( E n, E m) the set of u e s n, E m) such that , , ( ~ ) = (~,~)~ for all (~, x) E E" • E and set

3"-1

IE~m

for every a E Em,n. For all m, n, p E IN, we have:

a)

F(En, E m) is a closed vector subspace of s

x e E , u ~ ~:(E ~, E m) ~

b)

E m) such that

zu e ~=(E", Era).

For all a E Era,n, the map "5 belongs to .T(E n, E m) and m

m

11

sup Z Ila'+li~< Ilallm< Z E Ilat+ll2"

j e a n I=1

c)

z=l 3=1

Themap Er,,,.

~ . r ( E" , E " ) ,

a :

; "6

is linear and m3ective and its smage is a closed vector subspace of ~r(En, Era). If E is unital the this map is bijective. We endow Em,n wzth the norm induced by E(E n, Era).

d)

x E E , a e Em,,~ ::v ~'6 = x'5.

e) Em,n endowed with the maps ExEm,,~

~ Em,.,

(x,a):

:xa,

E,,,,nxE

~ Era,n,

(a,x):

~ax

is an E-module. It is unital if E is unital.

f)

a E Era,n, b E E,,,p ~ ab = "5b.

19~

g)

~. Banach Algebras

We set ,~,

171

a' " Em,n ~ ~ If,,

a:

11

~.

%,(%) s = l 1=1

/or all a' ~ E',m. Then a' E

(Era,n)'

and

s = l 1=1

for every a' E E ' , .

=

1=1

a':

~a'

and the map

E;,m

>(E.,n)',

is linear and bsjective.

h) En,n is an E-algebra and for all (a, a') e E.,, x E ' . (aa'), 1 =

a,~a~,

(a'a), 1 =

,

k-I

and i , j e INn

'

atkakl

9

k--1

If E zs unital then the matrix [6,~l],aE~. ss the unit of E . , . .

a) is obvious. b) ~ is obviously linear. For all ~ E E" ,

s=l

<-

E s=l

1=1

It~,,Ittl~,it 1--1

so ~ is continuous and m

n

S iJo,, S=I 1=1

Let j E INn. For every x E E , denote by ~alx the vector

$.~ Normed Algebras

193

(0,...,0,x,0,...,0) of E " , where x is on the j t h _ place. T h e n for every x E E # and j E ]Nm, m

I1~11~ >_ I1~(,,,:,.,~)11~= I1(~,.,~),~.,,11~ = ~

I1~,.,~11~.

1"-'1

Since E is quasiunital, we get m

I1~11~ >_ ~

Ila,jll ~ ,

I--'1

SO m

sup ~

Ilall ~ >_

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',,<~ _<E E:,.;,It

t=l

j=l

i=1

<_ ~: s--1

it.;,, ~

tt.,,II'

_<

3=1

3--1

_< licit ~

It-',,It'

s-----I

so a' E

, . , . , _<

3=1

3"-1

(Era)' and

iI.~ii<__~

II.;,II ~ 3--1

s=l

Let 8 E ]0,1[. For every (z, j) E IN~ x ~I., there is a b,j E E # such that I

0

aj,(b,3) _> ell%ll. We set

.,~ = ll.;,llb. for all (i, j) 6 ~ , . x ]hl.. By b), rn

I--1

We get

3"-1

195

~.~, Normed Algebras

lla~ll

~ t=l

=

ila',,ll'

>_ 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-ll<~ on K. Define

f:]K

~]K,

(~:

;S

/

1

if Ic~l>l

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 <_ x _< 1}. The considerations above show that ~'(K, e) is not empty. Let ~ denote the filter on .~" generated by the filter base {~'(K,e) I K compact set of T \ S , ~ > 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 <- ~(11~11 + IIx'll) = I1~11.

m

Proposition 2.3.2.8

( 0 ) Let E be an mvolutive normed (Banach) algebra. Then the normed (Banach) unital algebra IK x E defined in Proposition 2.2.1.3 is an mvolutive normed (Banach) unital algebra with respect to the involution IKxE

;~:xE,

(a,x):

;(~,x').

The assertion follows from Proposition 2.2.1.3 and Corollary 2.3.1.9. Definition 2.3.2.9

1

( 0 ) Let E be an mvolutwe normed (Banach) algebra.

The associated involutwe unital algebra F of E (Proposition 2.3.1.9) endowed wi:h an extension of the norm of E whwh renders F an mvolutive normed (Banach) umtal algebra is called an invotutive normed (Banach) unitat algebra associated to F (Propositzon 2.3.2.8).

Proposition 2.3.2.10

( 0 ) Let E be an mvolutwe normed space (normed

algebra, normed unital algebra) and F a closed mvolutive subspace (ideal, proper ideal} of E . Then E / F ss an involutive normed space (normed algebra, normed unital algebra) (Proposstzon 2.2.1.14, Proposition 2.3.1.21).

Let X E E / F and x E X. Then x* E X*, so IIx'll _< I1~'11 = II~ll.

We get IIX'll < IlXll, IlXll = IIX"ll < IIX'll,

IlXll- IIX'll 9

l

~.3 lnvolutive Banach Algebras

e45

Proposition 2.3.2.11

( 0 ) If E is an involutive normed space, then its involution is an isometry o/real normed spaces and Re E is a closed real vector

subspace of E .

Corollary 2.3.2.12

II

( 0 ) If E is an involutive normed space, then A" = (A)*

for every A C E . Hence, if A is an involutive set of E , then A is also an involutive set of E .

II

Corollary 2.3.2.13 ( 0 ) Let E be an involutwe normed algebra and F an involutive subalgebra (ideal) of E . Then F is an involutive subalgebra (ideal) o/E.

The assertion follows immediately from Corollary 2.2.1.8 and Corollary 2.3.2.12. II Corollary 2.3.2.14 ( 0 ) Let E be an mvolutive normed (umtal) algebra, A a subset of E and F the involutwe (unital} subalgebra of E generated by A.

a) -F zs the smallest closed involutive (unital) subalgebra of E containing A . It is called the closed involutive (unital) subalgebra of E generated by A.

b) If A t.J A* is commutative, then F is also commutatwe. c) If A is countable, then F is separable. a) follows from Corollary 2.3.2.13. b) follows from Proposition 2.3.1.19 and Corollary 2.2.1.10. c) follows from Proposition 2.3.1.19 and Corollary 2.2.1.9.

II

Definition 2.3.2.15 ( 0 ) Let E be an mvolutwe normed (unital) algebra and x an element of E . The closed involutive (unital) suba!gebra of E generated by {x} is called the closed involutive (unital) subalgebra of E generated b~ x . It is denoted by E(x) (by E(x, 1)).

Corollary 2.3.2.16 ( 0 ) If E is an involutwe normed (unital) algebra and x a normal element of E , then E(x) (E(x, 1)) is commutative and separable. ii

~46

~. Banach Algebras

Corollary 2.3.2.17' Let E be the completion of the involutive (unital) normed algebra E (Corollary 2.2.1.13). I.f we endow E wsth the continuous extention of the involution of E , then E becomes an involutive (unital) Banach algebra. If u is a continuous involutive (unital) algebra homomorphism of E into an involutwe(unital) Banach algebra, then the continuous extention of u to E is an involutive (unitaO algebra homomorphism. II Corollary 2.3.2.18 ( 0 ) If E is an involutive normed (unital) algebra, then No E (Un E) Is a closed set of E . The assertion follows from Proposition 2.3.2.11 and Proposition 2.2.1.6. 1 Corollary 2 . 3 . 2 . 1 9 x E Re E , then:

( 0 ) If E is an involutive unital Banach algebra and

a) eZ6. R e E . b)

lim [Ix"[[-x < 1 and a e ~ = ~ [ l + x ]

~

n--~oo

The assertions follow immediately from Proposition 2.3.2.11. Corollary 2.3.2.20 Let E be a complex involutme algebra. If the interior of No E /s nonempty, then E is commutative. Let x be an interior point of No E and take y E E . Then x + a y is normal for all sufficient small a E Ft so that o = (~ + ~ u ) ' ( ~ + , ~ ) -

= (x" + ~ ' ) ( ~

(~ + au)(~ + ~ ) " =

+ c~u) - (~ + ,~u)(~" + ~u') =

= x*x + ax*y + ay*x + a2y*y - xx* - axy* - ayx* - a2yy * =

= ~ ( ( ~ ' u + u'~ _ ~u. _ u~') + ~ ( y ' u - uu')).

It follows that y*y - yy* = 0,

i.e. that y is normal. By Corollary 2.3.1.25, E is commutative.

1

~.3 lnvolutive Banach Algebras

e47

P r o p o s i t i o n 2.3.2.21 ( 0 ) Let E be an involutive Banach algebra. Then the following are equivalent: a) E is symmetric. b) Every closed involutive subalgebra of E is symmetric. c) Every closed involutive commutative subalgebra of E is symmetric. a =,, b . Let F be a

closed involutive subalgebra of E and take x E No F .

Then

a~(x*x) c la+. By Corollary 2.2.4.27,

aF(X'X) c la+. Thus F is symmetric. b =,, c is trivial. c =~ a. Take x e No E . By Corollary 2.3.2.16, F := E(x) is commutative. By c), it is therefore symmetric. Then

II

(Corollary 2.1.3.14 a)). Hence E is symmetric.

e opo ition 2.3.2.22 ( 0 ) a) s

F

i,,ol t ve no,

,d

F) is an involutive normed space (Example 2.3.1.6).

b) E' is an involutive Banach space. c) Re s

F) is closed with respect to the topology of pointwise convergence.

d) u * ' = u'* for every u E s e)

F).

The evaluation jE : E --+ E" is involutive.

f) If G (resp. H ) is an involutive vector subspace of E (resp. of E'), then G ~ (resp. ~ is an involutive set of E' (resp. o.f E ) , the map EH

~ EH ,

x:

'ix*

is continuous, and Re E is a closed set of EH.

~8

~. Banach Algebras

g) Let G be a closed involutive subspace of E , q: E ~ E / G the quotient map, and i : G ~ E the inclusion map. Then the map

~G ~

(E/G)'

x' :

~ q'x'

and the factorization E'/G* ~ G' of i' through E'/G* are involutive isometries (b), f ) and Proposition 2.3.2.10).

h)

If G is a closed involutive subspace of E~ and q : E --+ E/*G is the quotient map, then the map (E/*G)' ~

G,

x' :

~ q'x'

ts an involutive tsometry (b), f), and Proposttion 2.3.2.10).

i) If G is an mvolutwe normed space, then D 0 U)* ~ D* 0 U*

for every u E s

j)

F) and v E s

G).

For every x' E Re E ' ,

IIx'll = sup{Ix'(x)l l x e Re E ~ } . k) Re E '# is a compact set of E ~ . 1) For every x E Re E\{0} there is an x' E Re E' such that

IIx'll-

1,

x'(x)---Ilxll.

In partwular,

Ilxll = sup{Ix'(x)l m)

I x' e

Re E'#}.

Take IK =q3 and x E E such that x'(x) E IR for every x' E Re E ' . Then x is selfadjoint.

n)

If u is a projection in E and Im u is an involutive set of E , then u* and re u are projections of E onto Im u. If m addition Ker u ts also an mvolutive set of E , then u is selfadjoint.

o)

/stu E Re s

F ) , then the algebraic isomorphism ~ : E / K e r u -+ Im u associated to u is involutive.

12.3lnvolutive Banach Algebras

a) Take u E

s

F). Then

Ilu*xll- II(ux*)*ll = Ilux*ll __< Ilull IIx*ll = I1~11I1~11 for every x E E and so

II~'II< II~II. It follows that

Ilu'll = Ilull. b) follows from a). c) follows from Re

s

F) = A {u E s

F) I ux* = (ux)*}.

xEE

d) Given (x, y') E E x F ' ,

(u'~, r

((~')', r

(=~', u'*)= (~*, u'u'*)= (z., (u,*u').)= (~, u'*r

and so U *f

_~_ U t *

by Theorem 1.3.4.2 a). e) Given

(x*,x') E E x E',

((jEx)',x') = (jEx, x'*)= (x,x~,*) = (x*,x')= (jEX*.X'), so t h a t

(j~x)* = ~sz* for every x E E . f) Take x' E G ~ . Then

(x, x'*) = (z*, x') = o for every x E G . Thus x'* E G ~ and G ~ is an involutive set of E ' . Take x E ~

Then

(x*, x') = (x, x " ) = 0 for every x' E H . Hence x* E ~

and ~

is an involutive set of E .

e49

~. Banach Algebras

~50

We have (~', ~')= (~, ~,.)

for all x E E and x' E H . Hence the map Ell

~ EH ,

X:

~ X*

is continuous and so Re E is a closed set of EH. g) The first assertion follows from d) and Proposition 1.3.5.2. The second assertion follows from the first one, d), and Theorem 1.3.5.12 c). h) follows from d) and Corollary 1.3.5.5. i) Since (vou)'~

= (vux')" = ~'(ux')" = v'~,'~

for every x E E , U 0 U)* :

U*

0

U* 9

j) First take I~ = lq.. Given x E E # ,

x'(x) = re(x'(x)) -- x'(rex) (Proposition 2.3.1.7). Since

Ilrexll ~ Ilxll _< 1 (Proposition 2.3.2.7),

Ix'(x)l _< sup{Ix'(y)l I y e

IIx'll ~

Re

E#},

sup{Ix'(x)l I x 9 R e E #} < s u p { l x ' ( x ) l l x e E #} =

IIx'll = sup{W(~)l I x e Re E # } N o w consider IK = ~ . Since re(x'(x)) = x'(rex) for every x E E (Proposition 2.3.1.7),

I1~'11 - life o ~'11 - sup{lx'(rex)l I x e E #} _<

IIx'll,

~.3 lnvolutive Banach Algebras

<_ s u p { I x ' ( x ) l l x e Re E #} _< sup(Ix'(x)l

I x e E #} = Iix'll

(Corollary 1.2.1.6). k) follows from c) and the Alaoglu-Bourbaki Theorem. l) The image of x with respect to the evaluation E --.+ E" belongs to Re E" (e) and Proposition 2.3.1.7). By j),

Ilxll = s u p { I x ' ( x ) l l x ' e Re E ' # } . The additional assertion follows from k). m) Since

(j~x, x') = (x, x') e la for every x' E Re E ' , jEX is involutive by Corollary 2.3.1.23 b =~ a. By e), jE is involutive and so jE:(imx) = im j E x = 0 (Corollary 2.3.1.23 a =~ c, Corollary 2.3.1.24 a)), and hence imx=0,

xE ReE (Corollary 2.3.1.24 a)). n) By i), u* is a projection. We have

u*x = (uz*)" e (Im u)* = Im u for every x E E

so I m u * c l m u .

IfxEImu

then

~.~= (~.). =~.. =~, so u* is a projection of E on Im u. It follows that re u is also a projection of E on I m u . Assume now in addition that Ker u is also an involutive set of E . Then for every x E E u*x = (ux*)* =0 iffxEKeru,so

K e r u = K e r u * and u = u * .

252

2. Banach Algebras

o) Let q: E --~ E / K e r u be the quotient map. By Propositions 2.3.1.21 b) and 2.3.2.10, E / K e r u is an involutive normed space and q is involutive. We get for every x E E ~(qx)* -- ~qx* -" ux" = (ux)* = (~(qx) ) ' .

I

Since q is surjective, ~ is involutive. Remark. algebra.

s

is a unital Banach algebra, but by g), it is not an involutive

C o r o l l a r y 2.3.2.23 ( 0 ) Let E be an involutive normed space and F an involutive subspace of E . Take x' E Re F' (Proposition ~.3.2.1~ b)). Then there/s a y' E Re E' with y ' [ F - x' and

IIv'll--IWII.

By the Hahn-Banach Theorem, there is a z' E E ' , such that

9 'IF = ~',

IIz'll = IWII.

Put y' := re z' E Re E ' . Then y'IF = x' and

Ilx'll _< IIv'll _< IIz'll- IIx'll (Proposition 2.3.2.7), so that

I1~'11 = Ilu'll.

I

P r o p o s i t i o n 2.3.2.24 ( 0 ) Let E be an involutive normed space, A a nonempty closed convex set of Re E and take x E (Re E ) \ A . Then there is an x' E Re E' (Proposition 2.3.2.22 b)), such that

~up ~'(v) < ~'(~). yEA

~,.3 lnvolutive Banach Algebras

~53

By Corollary 1.3.1.7, there is an y' E E' such that sup re (y'(y)) < re(y'(x)). yEA

Put x' := re y' E Re E ' . Then 1

9 '(~) = ~(y'(y) + ~"(~)) =

(y'(~) + y'(y)) = re~'(~)

for every y E Re E , so that sup x'(y) = sup re (y'(y)) < re(y'(x)) = x ' ( x ) . yEA

I

yEA

( 0 ) Let E be an involutive normed space, A' a nonempty convex set (cone) of Re E' (Proposition 2.3.2.22 b)) which is closed in E'~. Take x' E (Re E ' ) \ A ' . Then there zs an x E Re E such that

P r o p o s i t i o n 2.3.2.25

~.p ~'(~1 < ~'(~) yOE A~

( ~up ~'(~)= 0 < yt E A'

~'(~)).

By Proposition 1.3.1.8, there is a y E E such that 9sup re(y'(y)) < re(x'(y)). y' EA'

Put x := r e y E R e E . Then sup y'(x) = sup re (y'(y)) < re(x'(y)) = x'(x) y~ E A ~

y~ E A r

(Proposition 2.3.1.7). If A' is a cone, then sup y'(x) = 0 < x'(x).

I

yt E A p

P r o p o s i t i o n 2.3.2.26

(

4 )

Let E, F be involutive Banach algebras such

that IIx'xll = Ilxll ~ for every x E E and u 9 F ~ E an injective involutive algebra homomorphism. For IK = IR, we assume in addition that

:for every x E E . Then F is semi-simple.

2. Banach Algebras

e54

Take x E E . By induction, I1(~'~)~'11 = I1~'~11 ~"

for every n E IN. Thus

rE(X'X) = nlim II(x*x)"ll:-= --tOO

lim II(~*x)~'ll~-" = IIx*xll

111-"11'00

by Theorem 2.2.5.4. Let y be an element of the radical of F . Then y*y also belongs to the radical of F . By Corollary 2.1.3.12 and Corollary 2.1.3.24, ~E((~)'(,,y))

= ~E(~(U'~)) < ~F(U'U) = o.

By the above, I1,,~,11' = II(u~,)'(,,u)ll = ,'E((UU)'(uU)) = o,

uy=O,

y=0.

Hence the radical of F is equal to { 0 ) a n d F is semi-simple.

II

Remark. An involutive complex Banach algebra such that Ilz*xll = IJxl? for all its elements x is called a complex C*-algebra. They will be studied in Chapter 4 (also in the real case). The above map u is continuous (even if it is not injective), but the full story will be told in the above mentioned Chapter 4. We anticipated it a bit here in order to be able to prove that some important involutive Banach algebras are semi-simple. Let (T,'I',#) be a measure space. We identify L~(#) ' canonically with L~(#) and define E x a m p l e 2.3.2.27

( 0 )

for x, y E L~(#) and u" " L2(~)

~ L2(~),

z:

~ u'-~

for u E s (This definition of u* differs from the defimtion of u* obtained by combining Definition 2. 3.1.1 with Example 2. 3. 2. 3). a) ~" e s

~nd I1~'11 = I1~11 .to," e~'e~'~, ~ e s

~55

~.3 lnvolutive Banaeh Algebras

b) <~ly> = <~l~'y>/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 ~ ) =

<~'~l~> _< }lu'~ll II~ll _< II~'~tl licit ~

for every x E L2(p). Hence

II~l}~ _< II~'~ll. The reverse inequality follows from a) and b). d) By Example 2.1.5.8, the above map is an isomorphism of unitai algebras. o

Take (u,v)E ~(L~(p, IR)) and (x,y) E ' ~ ( / ~ , ] R ) . By Proposition 2.1.5.6,

()"

(~, ~) (~, ~) =

()'

(~, ~) (~, ~) = (~,, v')(~, -y) =

= (u'x + v ' y , - u ' y + v'x) = (u'x + v'y, u'y - v'x) =

= (~'~ + ,'~, u'~ - .'~) = ~u', -.'~(~, ~) = (~, ,).(~, ~ ) Hence

= (~, v). and the above map is involutive.

1

~.3 lnvolutive Banach Algebras

~57

Proposition 2.3.2.28

( 4 ) Let T be a locally compact group and Mb(T) the unital Banach algebra defined in Theorem ~2.P,.2.10 b). For # E fl4b(T) let p* denote the element of .Mb(T) with

f xa~,*=fx(t-')~(t) for every x E Co(T).

a) .Ma(T) is an involutive unital Banach algebra with respect to the involution .,~la(T)

~ Jk4a(T),

t~:

~ p*.

b) e'(T), canonically identified with a subset of .A~b(T) (Theorem e.~2.YL10 d)), is a closed involutive subalgebra of .Ma(T) and et is unitary.for every tET. c) If A is a left invamant Haar measure on T , then the in3ective unital continuous algebra homomorphism

.Mb(T) ~ ~ s

p:

;~

defined in Proposition 2.2.2.1~ e) is mvolutive (Example 2.3.~,.e7 c)).

d) Let .Mb(T, IK) be the involutive unital algebra.Mb(T) with respect to IK and ~Ab(T, IR) the complexification of .Mb(T, IR). Then the map o

.A,4b(T, JR)

~ .A/Ib(T,r

(~, v) ~ / ~

~- it/

zs an isomorphism of involutive unital complex algebras. e) J~b(T) is semi-simple.

f) The map .Mb(T) .... .~ IK,

/~:

;~u(T)

is an involutive unital algebra homomorphism.

g) Every algebra homomorphism el(T) ~ IK is mvolutive.

a) Take p,v E Adb(T). Then

(~,~,..~,')--f ~(st)d(~"|

~*)(s,t)--f(fx(st)d~,'(s))d~,'(t)--

t~ N,"

&l

"I::

II ,.....,.,

v

II

~,,

~ ~-

~

v

~

v

II

t,,t

--

A

0

"

--

~.

~..

v

t,,,l

~

II

r~

t,.t

I

I

II 9

.

II

~

N

~,

"

0

0 t.-I

v

t'.,l

II

II

"

**

"

~

II

I

v

II

II

I

I

I

II

t.,t

II

II

t.,t

II

o

t~

~.3 lnvolutive Banach Algebras

~59

f~(t)(~*,u)(t)d~(t)=

=

= (~1~" * u)= (~l~'y) 9 Hence ~*=~* and the map is involutive. d) This requires a long verification. e) By d) and Proposition 2.1.5.15, we may assume ]K = q~. In this case, however, the assertion follows from c), Example 2.3.2.27 c), and Proposition 2.3.2.26. f) follows from Theorem 2.2.2.10 g). g) Let x t : el(T) -+ IK be an algebra homomorphism. By Proposition 2.2.4.19, x t is continuous with IIx'll = 0 or IIx~ll = 1. We may assume llx~ll = 1. Take t E T . By b), 1 = x'(1) --- x'(e;et) = x'(e;)x'(et). Since

Ix'(e,)l <__ IIx'll Ile, II =

1,

Ix'(e;)l ~ IIx'll ile;ll--IIx'll Iletil---

1

it follows that

x'(e~) = ~'(~,). Hence x ~ is involutive.

II

P r o p o s i t i o n 2.3.2.29 ( 4 ) We use the notation of Proposition e.3.yLes. In addition, we denote by A a left and right invariant Haar measure on T . a)

LI(A), canonically identified with a subset of ,h/lb(Z ) (Theorem $.$.e.lO e)), is a closed mvolutive subalgebra of .,~4a(T).

b) x E LI(A), t ~. T =~ x'(t) = x ( t - l ) .

~,60

2. Banach Algebras

c) /.f we identify L~176 with L~(~) ' canonically, then

x'(x* * x) = 9

//

x'(s-'t)x(s)x(t)dA(s)dA(t)

for every x E L l (T) and x' e L ~ ( T ) . Let Lt (T, IK) be the involutive algebra Lt (T) with respect to K and

d)

o

~L~'(T, Ft) the complexification of L t ( T , R ) . Then the map o

" ~ ( T , IL)

~ Lt(T,r

( x , y ) ~ "~ x + zy

is an isomorphism of involutive complex algebras. e) LI(A) is semi-simple. a & b. For y E C0(T),

f yd(z.,~)" =

fu(t-')dx. ,~(t) = fy(t-')Z(t)da(t)= = f~(t)x(t-,)d~(t).

Hence

x'(t) = x(t-') for every t E T. Thus x* E L1(T) and the assertion follows from Theorem 2.2.2.10 e). c) By b) and Theorem 2.2.2.10 e),

(x" 9 x)(t)

=/:~'(~)~,(~-'t)d~(~)

so that

z'(x* , z) = f z'(t)(z" , z)(t)d~(t) = f x'(t)( f x(s)x(st)dA(s))dA(t) =

=

JJ

x txst

t x

s =

JJ

x's-ltxt

t))x--~dA()s =

~.3 lnvolutive Banach Algebras

~61

= ffx'(s-lt)xis)x(t)d)~(s)d)~(t). d) follows from a) and Proposition 2.3.2.28 d). e) By d) and Proposition 2.1.5.15, we may assume that IK = 113. By Proposition 2.3.2.28 c), there is an injective involutive algebra homomorphism of Lt(A) in s By Example 2.3.2.27 c) and Proposition 2.3.2.26, LI(T) is semi- simple. II E x a m p l e 2.3.2.30 ( 4 ) Let T be a finite group, eI(T) the involutive unital Banach algebra defined in Proposition ~.3.2.~8 b), and E the involutive unital algebra of T • T-matrices (Example 2.3.1.32). For x e e~(T), let ux denote the matrix [x(st-1)]s,teT. a) u: el(T) --+ E is an injective involutive umtal algebra homomorphism. b) a~(uz) = at~(T)(X) for every x E el(T). c) el(T) is strongly symmetric. a) u is obviously linear and injective. Given x , y E el(T) and s,t E T (x 9 y)(st -1) = E

x(st-lr-1)Y(r) = E

rET

x(sr-t)y(rt-1) '

rET

~'(~t-')=~(ts-'), 51(st -1) = ~ , t .

Hence u is an injective involutive unital algebra homomorphism. b) If x is invertible in el(T), then, by a), ux is invertible in E . Assume that ux is invertible in E. Then the determinant of ux does not vanish. Thus there is a unique y E el(T) such that

tET

for every s E T. Hence x*y--5l. (ux)(uy) = u(x 9 ~) = u(~,) = 1,

26~

~. Banach Algebras

y,x=61 (since u is injective), i.e. x is invertible. Hence O'E(UX ) "-- O'tl(T)(X) .

c) follows from a), b), and Example 2.3.1.32.

m

Remark. If T is a compact group, then the involutive Banach algebra LI(T) is symmetric (Theorem 3.1.6.11). This is an generalization of c) above. E x a m p l e 2.3.2.31

The map s

~s

k;

f'l

~k

is an injective involutwe contmuous algebra homomorphism (Example ~.3.2.5, Example e.3.e.e7 c)). By Example 2.2.2.5 a), the map 12'2(T,T) - - , s

t3

k:

~k

is an injective continuous algebra homomorphism. Let k E ~'2(T, T). Then N*

13

(k ~)(~)= (k'~)(~)= ~ k'(~,O~(t) = ~ k'(~,Ox(t)= tET

tET

N

= ~

k" (s, t)~(t) = (k'~)(~)

tET

for every x E ~(T) and s E T, so A*

A

kx=k*x for every x E s

and A*

N

k =k*. Hence the above map is involutive.

1

'2.3 lnvolutive Banach Algebras

Example

2.3.2.32

( 0 )

Let ( T , T , p )

~63

be a measure space. Given x e

L~176 define "~" L2(#) .....~ L~(#),

y , ......~. x y .

Then the map LCr

~

s

,

x ;: ~ ~

(Example ~.~.2.22 a)) is a continuous homomorphism of involutive unital algebras (Example 2.3.2.3, Example 2.3.2.27 c)) and has the property that ~ is normal.for every x ~ L ~ ( # ) . I f x E L ~ ( p ) , then ~ is selfadjoint (unitary) iff

x(t) e ~

(I.(t)l = 1)

for p-almost all t E T .

By Example 2.2.2.22 b), the map Lr

.... ,~ s

x ~

is a continuous unital algebra homomorphism. Take x E L~176 Then

for all y, z e L2(#) (Example 2.3.2.27 b)), so ~*

x*

and the above map is involutive. The last assertion follows immediately from the first one.

I

Proposition 2.3.2.33 ( 0 ) Let E be a unital algebra. Take x E E with a(x)CFt. Then the following are equivalent:

a) (r(x) c ~.+. b) r(c~l- x) < a for every a E F t wsth r(x) < a . c) r(cd - x) < a .for some a E Ft.

~. Banach Algebras

e64

a =, b. We have that

o(~)

Io,~],

c

a ( a l - x) = o t - a(x) C [0, a],

r(al-

x) < a .

e is trivial. c =~ a. We have that

b ~

a - a ( x ) = a ( a l - x) C I - a , a],

so that

,,(x) P r o p o s i t i o n 2.3.2.34

c ,~-

[-,~,,~l = [o, z,~]

c

rt+.

m

( 0 ) Let E be a symmetric involutive unital Banach

algebra such that

Ilxll =

r(x)

for every x E Re E .

a)

{x E Re EJa(x) C Ft+} ts a closed sharp convex cone of E .

b) E admits a unique order relatton rendermg it an ordered Banach space such that

E+ = {z E Re E I a ( x ) c I t + } . c) If ~ =ffJ and n E IN, then x e E+ ==~ x" e E § x E Re E ==, x zn E E + , x E E ==v. x*x + xx* E E + .

a) Put

C := {x E Re Ela(x) c n+}. Take x,y E C. Define

:= Ilzll,

~:= II~,ll.

~.3 lnvolutive Banach Algebras

~65

By Proposition 2.3.2.33 a =v b,

II(c~ + fl)l

-

(x + y)ll = II(al - x) + (~1

-

y)ll <

<_ I1~1 - zll. + 11,Ol - ~11 __ ~ + ~, so that

a(x + y) c la+ by Proposition 2.3.2.33 c =~ a (and Proposition 2.3.1.29). Hence x + y E C . Since obviously

a(~x) c u~+ for every 7 E IR+, C is a convex cone of E . Take x e C n ( - C ) . Then - x E C , so that

-oCz) c o ( - x ) c

n~+.

Since

a(x) c ~ + , as well, we see that

o(x) c {0},

~(x) = 0 , X--'0.

Hence C is sharp. By Proposition 2.3.2.33 c =~ a =~ b,

C = { x e Re EIII I1-111 - -11 < Ilxll}. Hence C is closed. b) follows from a) and Lemma 1.7.1.3. c) If x E E + , then, by Theorem 2.1.3.4 b)

o(x~)-

o(,)" c n~+,

~. Banach Algebras

~66

so x" e E+.

If x E Re E, then a(x 2") = a(x) 2" c rt+,

by Theorem 2.1.3.4 b). Thus x 2" E E+. Take x s E. Define a:--rex,

b:=imx.

Then

x*x + xx* = (a - ib)(a + ib) + (a + ib)(a - ib) = = a s + l a b - tba + b~ + a s - iab+iba + b2 = 2a 2 + 2b2 s E+

by the above considerations and b). Example 2.3.2.35

II

( 0 ) Let E be an involutive complex Banach (unital}

algebra, E the complexification of the underlying involutive real algebra of E o

(Proposition 2.3.1.40) and E' the eomplexificatwn of the underlying involutive real vector space of E' (Proposition 2.3.1.38). For all x, y E E , define xSy:=yx t4

and let E denote the involutwe (unital) Banach algebra obtained from E by replacing the multsphcation with

(x,y):

E•

~xSy.

a) E endowed with the norm

; n+,

(x,~).

; sup(llx + wll, IIx- wll}

is an mvolutwe complex Banach (unitaO algebra.

b) If we endow E • E with pomtwise multiphcation, pointwzse involution, and supremum norm (Example 2.8.2.4), then the map

/~~ExE,

(x,y):

;((x+iy),(x-iy)*)

is an isometry of involutive Banach (unita 0 algebras.

~.3 lnvolutive Banach Algebras

~67

o

c) ' ~

endowed with the norm o

"~

, IR,+,

(x', y')'

;llx' + iu'll + IIx'-iy'II

and involutive Banach space. o

d) For each (x', y') E E' , define

(~', y'). Then

k

~r

(~, y).

, (~ + i~,, ~' + i~,') + (:~ - iu, ~ ' - i~').

(x', y') ~. (E)' and the map o

E'

~ (/~)',

(x', y')::: ; (x', y')

is an isometry of involutive Banach spaces. e)

The map E-==~E,

"

x:

;

X*

is an isometry of mvolutwe real (unital) Banach algebras which is conjugate linear.

f) If u is a conlugate revolution of E preserving the norms, then 4-4

E

~E,

x;

;(ux)*

and E ----+ E x E ,

(x,y):

~ (x + zy, ux + iuy)

are isometrtes of involutive complex (unita 0 Banach algebras, where E • E is endowed with pointwise multiplication and involution and with supremum norm.

a) Take (x, y) E/~ and a, ,6 E IR. Then (,~ + / Z ) ( z , y) = ,~(z, y) + iZ(~:, u) = (az, m,) + (-Z~,, Zx) = (c~x - Z~, Z~: + ~y), i1(,~ + @ ( x , t~)ll = sup{ll(,~ - ~u) + i(~x + ~ , ) l l , II(,~x - ~u) - i(~x +,:,u)[I} =

2. Banach Algebras

268

= sup{ll(,~ + i a ) ( x + ~u)ll, II(c~ - i#)(x - iu)ll} =

=

s u p { l a + it/I I1~ +

iyll, I~

- i#l I1~ - i u l l } = I~ + ,Zi I1(~, :u)ll,

I1(~, y)'ll = I1(~:',-u')ll = ~up{ll*" - ,u'll, I1~:" + / u ' l l } = = sup{ll(x + iy)*ll, I1(* - ,y)*ll} = ~up{ll~ + ,ull, IIx - iyll} = I1(~, y)ll.

Now take (xt, Yl), (x2, Y2) E E . Then I1(*~ + ~,) ~: i(u, + u~)ll <- Ilxl + i~,ll + I1~, ~- iy211 <

<_ II(x,., y,)ll + I1~, y,)ll,

_< IIx, 9 iu~l111~2 9 iy:,ll _< I1(:,:,, y,)ll I1(~, u2)ll 9 Hence

II(x,, ~',)+ (*~,~)11 <_ I1(~,, u,)ll + II(z2, u~)ll, II(x,, u,)(*2, u2)ll < II(z,, y,)ll II(z~, :~2)11 9 b) Denote the map by u and take (x, y) E/~. Then u ( i ( , , y)) = , , ( ( - y , x)) = ( ( - u + i~), ( - u - i x ) ' ) = = i((~ + iu). ( . - iu)') = . . , ( ( . . u ) ) .

,,((x, y)') = u ( ( x ' , - u ' ) )

= ((x + i u ) ' , ( * -

= ((x" - iu'), (~" + , u ' ) ' ) =

,u)) = u((~:, u ) ) ' .

Hence u is linear and involutive. Take now (xl,yl), (x2,y2) q- E,. Then

~,.3 lnvolutive Banach Algebras

((,~,~ - y~)

+ i(,,~

+ y~x2) , ( ( x ~ x , - ~ y , )

- i ( ~ , ~ , + y~x=))') =

= ((.~ + iy~)(x= + i u , ) , ( ( ~ , - i u , ) ( . ,

- i~,2))') =

= ( ( ~ + iy~)(~= + i y , ) , (~2 - iy=)'(x~ - i ~ , ) ' ) =

((x~ + i~)(x~ + i ~ )

( ~ - i~,)* $ ( ~ - y~) )

= u ( ( ~ , ~))~((~=, v,)).

Hence u is an isometry of involutive Banach (unital) algebras. o

c) Take (x',y')E E' and a, f l e I R . Then It(~ + i,)(~', y')ll = I1(~,' - a r

a~' + ~y')ll =

= I1(~:' - Zy') + i(Z~' + ~y')ll + I1(,~*' - ~,') - i(Ox' + m,')ll =

= I1(~ + iO)(~' + iy')ll + i1(~ - , ~ ) ( . ' - iy')ll = : I~ + ~/~l IIx' + w'll + I,~ - i/~l Illlx' - i y ' l l -

I~ + ~/~1 II(-~', y')ll,

and II(x', y')*ll = II(x'*,-y'*)ll--IIx'*

- ,y'*ll + II~'" + iy'*ll =

= [l(x' + iy')*ll + I[(x' - iy')*ll = IIx' + iy'll + Ilx'-'y'l[

= ll(x', y')]l 9

d) (x"~y') is obviously ]R-linear. Take (x, y ) E / ~ . Then (,(~, ~1, (~,-~7r = ((_~, ~), (~,, r

=

= { - y + i~, ~' + i y ' ) + ( - y - i~, ~ , - iy') =

= i(~ + iy,.' + iy') + i(~ - iy,.,

= i((~, ~), (.,, r

- ir o

Hence (x', y') is C-linear and thus belongs to (E)'.

1269

'270

`2. Banach Algebras

We now prove the final assertion and denote for t h a t the m a p by u. It is o

easy to see t h a t u is l~-linear. Take (x', y') E E '

and (x, y) ~_ E . Then

((.. y)...(i(.', u'))) = ((x. u). ~.((-u'. x'))) = = (x + i y , - y ' + ix') + (x - i y , - y ' -

ix') =

= i(x + iu, x' + iv') + i ( . - iy,.' - iu') = ((., ~,), i,,((:e, u'))). Hence u is G-linear. We have

((z. u). u'((x', u'))) = ((.. u). ,,((x'. u')')') = ((~, u), ~ ( ( x " , - r = {(:,,, z,)', u ( ( . , ' , - r

= ((x.,-~,.),,,((x,',-r

=

= (x* - iy', x'* - iY") + (x" + ,y', x'* + iy") = = ((x + iy)*, (x' + iy')*) + ((x - iy)', ( x ' - iy')*) =

= (x + iu,~' +iu') + ( ~ - , u , ~ ' - i y ' )

= ((x, u), u((~', r

Hence u * = u and u is involutive. Moreover,

I((*. u)..,((.'. r

= I(. + iy, x' + iy') + (x - , y , x , -

_< I(z + iy, z' + iu')l + I(* - i ~ , . ' -

_< I1- + iyll I1-' + ir

+ I1- - iyll I1-'- ir

iu')l <

_< II(z, v)ll(ll*' + iu'll + IIz'- ir

= I1(~, ~,)11 I1(~', r Hence

Ilu((~', r

u,')l <

~ II(x', r

Take e > O. There are Xo, yo E E # such t h a t

=

~.3 lnvolutive Banach Algebras

(xo, x' + zy') > Ila:' + iy'll- 6,

(yo, a:'- iy') > IIx'- iu'll- e.

Put 1

1

"= ~(~o + ~o),

y := ~(~o - uo).

Then

x + i y = Xo,

x-

iy = Yo,

so that

II(x, u)ll_ 1 and

IIx' + iy'll- 6 + IIx'- iy'll- e < (xo, x' + w') + (yo, x'- iu') = 1~ + i~..' + iy')+ (~- i~. ~,- dy,) = ((.. ~)..,((~'. y'))) <_ Ilu((*'. ~,'))11 9 Since e is arbitrary,

I1(~', y')ll = I1~' + ,~'11 + I1~'- i~'il < II~((~'y'))ll, Ilu((x', ~'))ll = II(x', ~')ll 9 Hence u' preserves the norms. o

Finally, we show t h a t u is surjective. Take z' E (E)'. Define

{

~"E

"r

~-~

b" V

;~,

x : ..... ~. 89

89 zx)).

a' and b' are obviously lR-linear. For any x E E we have

{ ~'(,~)= 89

89

~z'((~,-i~))= ia'(~),

1 It b'(ix) = ~z'((ix,--"x)) = ~I i( - - i - ~ , ' ~ x ) ) = ,(~Pi"~,~x)j) = ib'(x).

Hence a' and b' are qNlinear and therefore belong to E ' . P u t XI

1 = 5(a' + b'),

y'= •2s Then

b')

".

272

12. Banach Algebras

x' + iy'

a'

gCt _

b'.

in t =

Thus for any (x, y) E E ,

((.. ~). (~'. u')) = (.

+ iy,.'

+ w') + (x - iu,.'

- iu') =

= a'(x + iy) + l/(x - iy) = ~z'((x + i y , - i x + y)) + ~z'((x - , y , ix + y)) =

=

2~'((2~,2~))= z'((~,~)).

Hence z' = (z', ~') =

=((.', ~'))

and u is surjective. e) is easy to see. f) follows from e) and b).

I

Proposition 2.3.2.36

Let E be a unital Banach algebra, U (resp. ll ) the group of invertzble elements of E (resp. s ) and ~ be the set of elements of bl which are umtal algebra homomorphzsms. Gwen a E U, put "5" E ~

E,

x:

; axa -1 .

Then .7r zs a subgroup of bl closed m the topology of pointwise convergence. For each a E U, "5 E .7r, the map U

~.~,

a:

~'5

is a group homomorphzsm and {~la E U} is a normal subgroup of Y . If E zs an involutwe unital Banach algebra, then "5 is involutive for every unitary aEE.

It is obvious that Jr is a subgroup o f / 4 . It follows from "%':= ( N

{u E Lllu(xy)= ( u x ) ( u y ) } ) N { u E/4,ul = 1}

Kz,yEE

that Y is a closed set of/4 in the topology of pointwise convergence. Take a E U. Then ~ (5 s and

2.3 lnvolutive Banach Algebras

~73

'5(xy) = a x y a -1 = a x a - l a y a -1 = ('~x)('sy),

"51 = a l a -1 = 1

for every x, y E E . Hence '5 is a unital algebra homomorphism, i.e. '5 E ~'. Take a, b E U. Then abx = (ab)x(ab) -1 = abxb-~a -1 = ~'bx

l x = lx1-1 = X for every x E E , so that ab='5b,

l=lu.

U

a:

Hence the map ~,.T,

''5

is a group homomorphism. Take a E U and u E ~'. Then ( u - l ' s u ) x = u - l ( a ( u x ) a - l ) = ( u - l a ) x ( u - l a -1) = ( u - l a ) x ( u - l a ) -1 = u-l~ax

for every x E E , so that

u-l-au = u-l~a and {'51a e U} is a normal subgroup of ~'. Now suppose that E is involutive and let a be a unitary element of E . Then ~* 9 = ( ~ * ) *

= (.~*.*)'

= a~a* = ~

for every x E E . Thus '5* - ' 5 and so '5 is involutive'

II

P r o p o s i t i o n 2.3.2.37

( 6 ) Let E be a complex involutive unital Banach

algebra and x a normal (selfadjomt) element of E . Take t~o E C ((~o E IR), rl, r2 E lit+ such that

~. Banach Algebras

e74

u := {~ e e l f , < la - aol < ~2} c r and let O0

Ct

-

,~o)"~.

n--'--O0

be the Laurent series of the resolvent of x in U (Theorem ~.f,.5.1 and 1.3.10.8). Then x . is normal (selfadjoint) .for every n E 2g and

x.,~;, = x:,x,. .for m, n E ~ . Now

v := { . e e l s , < I ~ - nol < ~ } c r and OO

(t - ~ o ) " ~ : Yt= --00

in the Laurent series of the resolvent of x* in V (Proposition 2.3.1.14, Proposition 2.3.2.11). If x is selfadjoint and c~0 E IR, then x , is selfadjoint for every n E (Theorem 1.3.10.8). Now suppose that x is normal. Then X m X n --. X ~ X m

for every m, n E 7Z, by Corollary 2.2.6.3. In particular, x, is normal for every nETZ. I P r o p o s i t i o n 2.3.2.38

If E zs an involutive normed space, then Re E ''# zs

the closure of jE(Re E #) in E [ , . By Proposition 2.3.2.22 e),k), the closure of jE (Re E #) in E~, is contained in Re E "# . Let x" E Re E "# . By Corollary 1.3.6.8, there is a filter ~ on E # such that jE(~) converges to x" in E~,. By Proposition 2.3.2.7, re(~) is a filter on E # such that x" = r e x " = lira re (jE(~)) = lim jE(re (~)) (Proposition 2.3.2.22 e)).

I

~.3 lnvolutive Banach Algebras

~75

2.3.3 Sesquilinear Forms

Definition 2.3.3.1

( 0 ) Let E, F, G be vector spaces. A map f "ExF

~G

is called sesquilinear if for every (x, y) E E x F the map f (., y) is linear and the map f ( x , .) ss conjugate linear. If E = F , then we define f " E --+ G,

x : -~ f ( x , x )

,..,,

and call f the associated quadratic map of f . If, in addition, G is an involutwe vector space, then we define f*'ExE

~G,

(x,y):

~ f(y,x)*

and call f* the adjoint of f . If G = IK then f is called sesquilinear f o r m on E • F . A sesquilinear f o r m on E zs a sesquilinear.form on E • E .

Every sesquilinear map is bilinear over ]R. Proposition 2.3.3.2

( 0 ) Let E , F be vector spaces, f " E • E --4 F a sesquilinear map, x, y E E , avld a, ~ G IK. a) f-(.~ + ~y) = I.l~f-(~) + ~ f ( ~ . ~) + ~M(y. ~) + I~l~f(y) b) f(x + u) + f ( x - y) = 2(/(~) + I(y))

(P~.~U~zog.~mL~w)

c) f ( x + y) - f ( x - y) -- 2 ( f ( x , y ) + f ( y , x ) ) . d) I f l K = r t4

t4

= 1~ 4

,k f ( x + i~y ) (Polarization Identity)

k=l

a) f (ax + 13y) = f (ax + ~y, a x + fly) = a f (x, a x --I-13y) + ~Of (y, a x --t-~y) -- a(~f(x, x) + ~f(x, y)) + f~(~f(y, x) + -~f(y,y)) =

~76

2. Banach Algebras

=

1~127(x)+ a - ~ f ( x , y )

+ "~/~f(y,x) + 1/312f(y).

b) By a), I ( x + v) + l ( x - v) =

-- f ( x ) + f ( x , y ) + f ( y , x ) + f ( y ) + f ( x ) -- f ( x , y ) - f ( y , x ) + f ( y ) --

= 2(f(x) + f(y)). c) By a), f ( ~ + v) - f ( x - v) =

-- f ( x ) + f ( x , V ) + f ( y , x ) + f ( y ) - f ( x ) + f ( x , y ) + f(y, z) - f(y) --

= 2(f(x,y)+ f(y,x)).

d) By c), f ( x + iy) - f ( x -

iy) = 2 ( f ( x , ,y) + f ( i y , x)) -- 2(-if(x, y) + i f ( y , x ) ) ,

SO

-~(f(x + y) - fCx - y)) + ~ ( f ( x + iV) - f ( x - iy)) = 1

,

-- ~ ( f ( x , y ) + f ( y , x ) ) + ~ ( - i f ( x , y ) Remark.

+ if(y,x)) = f(x,y).

II

If f is sesquilinear and IK = ~, then, by the polarization identity, f'-Oc==~ f--O.

Proposition 2.3.3.3

I

( 0 ) Let E be a vector space, F an involutive vector space, and S the set of sesquihnear maps E • E --+ F . a) $ is a vector subspace of FE•

b) f* E S for every f E ,5.

~,.3 lnvolutive Banach Algebras

~77

c) S is an involutive vector space with respect to then involution 8

'~8,

f'

~, f * .

The selfad3omt elements of $ are called H e r m i t i a n .

d)

If F = IK, then the set

{f e R e S I f ( E ) C R+} is a sharp convex cone of S . e)

If F = IK, then there is a unique order on S , with respect to which S is an ordered vector space such that S+ = { f ~_ R e , S I f ( E ) c R + } .

a), b), and c) are easy to verify. d) P u t K: := (f e R e S I f ( E ) c IR+}. It is easy to see that K: is a convex cone. Let

f e K~N (-K:). Then f(~) = o

for every x E E . If IK = IR (resp. lK =(13 ), then

I(z, y)=0 for every x, y E E , by Proposition 2.3.3.2 c) (resp. d)). Hence f = 0 and K: is sharp. e) follows from d) and Proposition 1.7.1.3. Co~.e!'*.r:, :? 3.3.4

( O )

I

Let E be a vector space, F an involutive vector

space, ,.. i f . E )< E -, ~ a Hermitian sesquilinear map. Then

f~(otx -+-t~Y) --1~I2Y(~) for all x, y E E and ~ , ~ E IK.

+ 2re(c~f(x, y))-4-I/~12f(y)

lg78

~. Banach Algebras

The assertion follows from Proposition 2.3.3.2 a). Example

2.3.3.5 ( 0 )

I

If T is a set, then the function

e~(T) • e~(T)

~~,

~~(t)yit)

(~,y):

tET

is a positive sesquilinear form. If (T, ~E,p) is a measure space, then the map

1

is a positive sesquilinear form.

Example 2.3.3.6 ( 0 )

Let n E IN, 8 the involutive vector space of sesquilinear forms on I(n (Propositwn ~.3.3.3 c)), and IKn,n the involutive unital algebra of n x n-matrices (Example e.3.1.3~). Gsven a := [%1 ~ IKn,n, put

a.~"•

,~:,

~, ~ a , , x , ~ , .

(x,~):

,d=l

Then "~ E $ for every a E ~n,n and the map E

--}$,

a:

;~

is an isomorphism of mvolutive vector spaces. If a ss a selfadjoint element of lKn,n, then "6 ss positwe iff a(a) C IR+. It is easy to see that ~ E $ for every a E Kn,n and that the map

is linear and bijective. Take a E IK~,~. Then

r ( x , ~ ) - ~(~,~) = ~ a,, y,~, = ~(a'),,~,~, = a'(~,~) :d=l

I,j=l

for every (x, y) E K " • IKn . Hence

and the above map is involutive. Now take a E IKn,n such that ~ is positive and a E a ( a ) . Then there is an z E lKn\{0} with n

~ j-_!

atjxj

--- o x l

~.3 lnvohtive Banach Algebras

1279

?or every i E INn. It follows n

II

I1

~ I~,1~ =

~(~,)~,

1=1

t=l

=

~ ~,,~,~,~ R + 1,7=1

and so a(a) C fit+. Let now a be a selfadjoint element of IKn,n with a(a) c fit+ and I:IK n

~IK,

x:

;~(x,x).

We have n

s(~) = ~(~, ~)

n

~

= a,j~,~, = ~

t,l=l

aj,~,~, =

s,$=l

i,3=1

for every x E IK", so f is real. Let x be a point of (llxll - 1 } , where f takes its infimum. By the Lagrange Theorem, there is an a , so t h a t

~

_ ailxl ---- otxi

$=1 for every i E IN,,. We get a E a ( a ) , so a E IR+, and

I ( x ) = Z..., a"x'x3 = cr t,$--I

x,'~, = or. t----I

Hence

~(y, y) >__Ilyl12~(~, x) = allyll ~ > o I

for every y E lKn and ~ is positive. P r o p o s i t i o n 2.3.3.7

( 0 )

Let E be a real vector space, F an involutive

real vector space, and f'E•

~F

a sesquilinear map. Then the follounng are equivalent: a)

f is Hermitian.

280

b)

2. Banach Algebras

Gwen x , y E E , 1 f ( x , V) = -~(f(x + y) - f ( x -- y)). a =~ b. By Proposition 2.3.3.2 c), f ( z + v) - f ( x - v) = 2 ( f ( x , v ) + f ( v , x ) )

=

4f(x, v).

b =~ a. Given x,y E E ,

1

1

f ( x , y ) = -~(f(x + y ) - f ( x - y)) = -~(f(y + x) - f ( y -

x)) = f ( y , x )

(Proposition 2.3.3.2 a)), so that

f*=f.

l

P r o p o . i t i o . 2.3.3.8 ( 0 ) Le, E be. give complex vector space, and f :ExE

F on ,nvol -

~F

a sesqutlmear map. Then the following are equivalent: a)

f is Hermittan.

b)

f(E) C ReF.

c)

re f ( x , y) - 88

d) imf(x,y)

=

+ y ) - f ( x - y)) for every x, y e S .

1 ( f ( x + ,y)

i

-

f ( x - i y ) ) /or every x, y E S

In particular, if F =q3, f ,s positive (Propos,tion 2.3.3.3 e)) iff I ( E ) C IR+. a ~ b. Since

f(x) = f(x,x)= f(x,x)* =/(x)*, ,.,.,

f(x) C Re F, for every x E E. b ::> c & d follows from the polarization identity (Proposition 2.3.3.2 d)). c :=> d. Given x,y E E,

im f ( x , y) = re ( - z f ( x , y)) = re f ( x , iy) - l ( f ( x + iy) - f ( x -

zy))

~.3 lnvolutive Banach Algebras

~81

(Proposition 2.3.1.22). d =~ c. Given x, y (5 E , re f(x, y ) = im (if(x, y)) = im f ( x , - i y ) = ~ ( f ( x + y ) - f " ( x - y)) (Proposition 2.3.1.22). c & d =~ a. Given x,y E E , 1 1 re f ( x , y ) = -~(f(x + y ) - f ( x - y)) -- -~(f(y + x) - f ( y - x)) -- re f ( y , x ) 1

im

= 1 ( f ( y _ ix) -- f-(y + ix)) = - i m f ( y , x ) ,

f(x,y)-

f ( y , x ) * = f*(x,y)

(Proposition 2.3.3.2 a)). Thus

S=f*. The last assertion follows from a r

b (and Proposition 2.3.3.3 e)).

I

P r o p o s i t i o n 2.3.3.9 ( 0 ) Let f be a posztive sesqmhnear form on the vector space E (Proposition 2.3.3.3 e)). Then <_

,

]or every x, y E E . Take a E lR. Then

0 <_ f"(ax + f ( x , y ) y ) = a 2 f ( x ) + 2alf(x,y)l 2 + If(x,y)12f(y) (Corollary 2.3.3.4). Thus If(x, y)]4 ~ If(x, y)]2f(x)f(y),

~. Banach Algebras

~,82

,f(,,

<_

.

Hence, by Corollary 2.3.3.4, f ( x + y) = f(x) + 2re f ( x , y) + f ( y ) < f ( x ) + 2If(x,

<

i(v)=

v)l +

+

f(y) _<

,

Let T be a locally compact group and .Adb(T) the involutive unital Banach algebra defined in Proposition 12.3.2.28 a). Let A be a left invarmnt Haar measure on T and S the mvoluttve vector space of sesquihnear forms on L2(A) (Proposition 12.3.3.3 c)). Define E x a m p l e 2.3.3.10

for x, y E L2(A) and fu : L2(A) x L2(A)

' Y<,

(~,v) ,

~
for p E Mb(T) (Proposition 12.*.*.14 a)). Then fu E ,S whenever 1~ E Mb(T) and the map U : Mb(T)

~S,

U"

~I.

is an mjective involutive linear map such that fu ts Hermitian iff /~ ts selfadjoint and fu..j, ts posztwe for every i, E .Adb(Y) (Proposstion 12.3.3.$ e)). It is easy to see that fv E S whenever ~ E J~fb and that u is linear. By Proposition 2.3.2.28 c) and Example 2.3.2.27 b),

f ; ( x , y ) = fu(y,x)---

(u* vl~)=
for every x, y E L2(/J) and V E .herb(T). Hence f ; -- fu" for every ju E .A4b(T). Thus u is involutive. Take p E .hdb(T) with f~ = 0. Then (~ 9 ~lv) = L ( ~ , v) = o

fu'(x,Y)

~.3 lnvolutive Banach Algebras

~83

for every x, y E L2(/J). Hence p.x=0 for every x E L2(/~). By Proposition 2.2.2.14 e), /J = 0 and u is injective. Let /z E A4b(T). If /~ is selfadjoint, then f~ is Hermitian, since u is involutive. If f~ is Hermitian, then

f._..= f . - f.. = y . - f ; = o , p-~*=O since u is injective and /z is selfadjoint. By Propositions 2.2.2.14 e), 2.3.2.28 c), and Example 2.3.2.27 b), ((#* 9 #)~1~) = (#" * (~, * ~)1~) = (~, * ~1~, * ~) e rt+

II

for every x E L2(A). Hence f~,.,~, is positive.

Propositiou 2.3.3.11

( 1 ) Let T be a set, t2'2(T,T) the involutive Banach algebra defined in Example 2.3.2.5, and S the involutive vector space of sesquilinear forms on t2(T) (Proposztzon e.3.3.3 c)). For x,y E t2(T) and k E e2'2(T, T), define t6T

and f~" t2(T) x t2(T) f'l

a)

; IK,

(x,y):

N

~,
1"3

(kxly) = (xlk*y) for every x, y ~. e2(T) and k e g2'2(T,T).

b) A E ,5 for every k E e2'2(T, T) and the map ~2'2(T,T)

~S,

k:

~A

is mjective, linear, and involutive. c) For every k E t2'2(T, T) the sesquilinear form f~ is Hermitian sff k is selfadjoint. d)

The sesquilinear form f~.o~, is positive whenever k E ~2'2(T, T ) .

~. Banach Algebras

es4

e)

The map t2'2(T, T) x e2'2(T, T)

~ ]K ,

(h, k) :

E(ho k')(t,t) tET

is a positive sesquilinear form with the property that E ( k o k')(t, t) = Ilkll2 tET

for every k E 12'2(T, T). a) Now

aET

tET

tET

sET

tET

sET

tET

A

$ET

tET

A

sET

b) It is obvious that fk E S whenever k E g2,2(T, T) and that the map e~'~(T,T)

~8,

k~A

is linear. It is injective since if k E ~2'2(T, T) and fk = 0, then n

n

0 = h(~,, ~,) = = (k~,)(~) = ~

k(~, ~)~,(~) = k(~, t)

rET

for every s, t E T. Take k E g2'2(T, T). Then, by a), A

A

n

.f;(z,y) = A ( y , z ) = (kylz)= (ylk'z)= (k'xly) = A.(z, y)

for every x, y E ~2(T), so that ;=A..

Hence the above map is involutive. c) follows from b) and Proposition 2.3.1.7. d) For x E f2(T),

~.3 lnvolutive Banach Algebras

~85

N

~'~ nn n n .h,.o~.(z,x) = (k* o kxl:r.) = ( k * k x l x ) = ( k z l k x ) e IR+

~y a) and Example 2.2.2.5 a). Hence, by c), fk.ok is positive. e) Given h, k e g~,~(T, T),

~ ( h o k')(t, 0 = ~ ~ h(t, ,)k* (,, 0 = ~ h(t,,)k(t,,) tET

tET sET

s,tET

I

~nd the assertions now follow. Remark.

This proposition will be generalized in Proposition 3.1.6.18

P r o p o s i t i o n 2.3.3.12

(M.H. Stone, 1932) Let f, g be sesqmhnear forms on

a vector space E such that g is positive. Then the set N

{f(x)lx

e E,

~ ( x ) = 1}

is convex.

Take x, y E E with "~(x) = "~(y) = 1.

f (x) ~ f (y) .

Define

.f"]KxiK---+~,

(a,/~),

>f(ax+Zv),

g'-]Kx]K-----~iK,

(~,~):

,~(~x+~v),

f ' - y(v)g' f (x) -- f (y) Take a, b E IK such that h(c~, f/) = Ic~l2 + ac~fl + b~f] for all a,/3 E IK (Proposition 2.3.3.2 a)). Define

7"----{ b-a1 ib--al

if

a -- b,

if

a ~ b,

2. Banach Algebras

286

6:=a~+bTe

R,

e := r e ~ g ( x , y ) 9 [-1, 1]

(Proposition 2.3.3.9) and Da : - - e a + (1 - (1 - e2)a2)89 9 R for every a 9 [0, 1]. Then g'(a, ~aT) = "~(ax + ~oTY) = a 2 + 2ea~a + ~a2 = 1,

h ( a , ~ o 7 ) = a ~ + aaf~o~ + ~ f ~ o 7 = a 2 + 6aZo e

for every a 9 [0, 1] (Corollary 2.3.3.4). We get h(O,&7)=O,

h(1,/~17)=1.

Take A 9 [0, 1]. Then there is an a 9 [0, 1] with h(a, ZoO) = A. It follows that

f(c~x + ~a~/Y) -- f ' ( a , ~ o ' / ) = ( f ( x ) - f ( y ) ) h ( a , ~aT) + f ( g ) g ' ( a , f l a T ) -

=Af(x)+(1-A)f(y).

Hence

{f(x) l x ~ E , ~(x) = 1} is convex.

II

12.3 lnvolutive Banach Algebras

1287

2.3.4 Positive Linear Forms D e f i n i t i o n 2.3.4.1

( 0 )

Let E

be an involutive algebra. The involutive

linear form x' on E is called positive if

x'(x*x) e la+ for every x E E . In this case, we set

N~, := {x e Elx'(x*x)= 0}. Care should be taken with the word "positive" above, since, in general, F , the vector space of linear forms on E , does not admit an order relation rending it an ordered vector space for which F+ is the set of positive linear forms for example take the case when x*x = 0 for every x E E . If, however, E is an involutive quasiunital Banach algebra, then E' does admit such an order (Proposition 2.3.4.13 b)) and this is the only case which we pursue in this book. We find it convenient to avail oursleves of the expression "positive" in the more general setting defined above, rather than using more elliptic expressions. The expression "positive linerar form" was already introduced in Proposition 1.7.1.8 in another context. This will produce no confusion in this book, since in the case in which both expressions are defined they will coincide.

Example 2.3.4.2 Let T be a unimodular locally compact group and A the Haar measure on T . Identify canonwally L ~176 with L1 ()~), and take x' E Re L1 (A),. Then the following are equivalent: a) x' is positive (Proposition 2.3.~.~9 a)). b) x ~ LI(~) =~ f f x ' ( s - l t ) x ( s ) x ( t ) d ~ ( s ) d ) ~ ( t ) E IR+. The assertion follows immediately from Proposition 2.3.2.29 c).

Example 2.3.4.3 equwalent: a)

I

Take n E IN and [a,j] E IKn,,. Then the following are

The sesquilinear form n IK n •

IK n

~ ~ } IK ,

i,3=1

(Example 2.3.3.6) is positive.

288

b)

I~. Banach Algebras

The hnear form n t,3=l

is positive (Example 2. 3.1.2~). If these conditions are fulfilled, then a([a,j]) C R + . a=~b. a,'~c% s,j=l

%,=

k=l

~,,a,j , =

li

11

k=l

1,3=1

%,=

k=l

b =~ a. Take z E IK" and put Ott3 : - - ~ 3 ~ t l

for i, j E IN,,. Then .

.

OttkOtk3 = "

=g k=l

"~'~tOtkl =

E k=l

"X,f~klX3f~kl = " ~ , X l , k'-I

so that

~-~_ %z,z-S = ~ t,3=l

a,,(a'a)~, ~ la+.

t,3=l

The last assertion follows from Example 2.3.3.6. E x a m p l e 2.3.4.4

Given a b]E~2,~ ' c d

the following are equivalent: a)

The sesquilinear form IK 2 x IK 2

is positwe.

~ IK,

((x, y), (X, Y)) ~

axX + bxY + cyX + dyY

II

~.3 Involutive Banach Algebras

b)

P,89

The linear form ]K2,2

" aa + 7 b + ~ c + ~d

) ]K,

7 is posztive (Example c)

a, d ~ n % ,

2.3.1.32).

b - ~,

Ibl2 <

ad.

a r b follows from Example 2.3.4.3. a r c is easy to see.

I

Proposition 2.3.4.5 ( 0 ) If E is an involutive algebra, then every involutive algebra homomorphism of E in IK zs positive.

Let x" E

~ IK

be an involutive algebra homomorphism. Then x'(x*x) = x'(x*)x'(x) -- x'(x~x'(x) - I x ' ( x ) l 2 >_ o

for every x E E and x' is therefore positive. P r o p o s i t i o n 2.3.4.6

I

( 0 ) Let E be an mvolutive algebra, x' a linear form

on E , and f " E • E : ; ]K,

(x,y) ~--~ x'(y*x).

a) f is sesqmhnear. b) f ss nermitian (positive) if x' is involutive (posihve); the converse implication also holds if E is unital.

c) If x' is positive, then Ix'(x*u)l ~ < z'(x*x)x'(u*u)

for every x, y E E and Nx, is a left ideal of E .

d) If x' is mvolutive, then rex'(x) = x ' ( ~1( x + x*)) for every x E E .

290

2. Banach Algebras

e) If IK =q3 and E is umtal, then x' is positive iff x'(x*x) ~. IR+ for every xEE.

a) is trivial. b) For x , y e E , f ( x , y ) = x'(y*x),

f ( y , x ) = x'(x*y) = x'*(y*x),

f ( x ) = x'(x*x)

and the assertions now follow. c) The inequality and the fact that Nx, is a vector subspace of E follow from b) and Proposition 2.3.3.9. The inequality implies that Nz, is a left ideal of E . d) The complex case follows from Proposition 2.3.1.22 and Corollary 2.3.1.23 a =~ d. A~ume that lK = 1~. Then

x'(~') = ~'(~), so that

, 1 1 (~,(~) + ~,(~.)) = ( ~ ( ~ + ~')) =

~,(~)

e) follows from b) and the last assertion of Proposition 2.3.3.8.

II

Corollary 2.3.4.7 ( 0 ) Let E be an involutive unttal Banach algebra and x' a positive hnear form on E . Then x' ts continuous and

I[x'll

= x'(1).

Now

x'(1)=x'(l*l)>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 <__ x'(1). Since x'(1) < I1='1111111 = I1='11,

I1~'11-- x'(1),

mm

We cannot replace "Banach" by "normed" in the above corollary, as the following example shows. Let E be the involutive unital Banach algebra consisting of the unital Banach algebra gl(~) (Example 2.2.2.7) together with the involution Remark.

E--+

E,

x:

~.~

and set x " l K (z)

~ IK,

x:

; ~2"x(n). nEZ

]K(~) is an involutive unital subalgebra of E and x' is an involutive linear form on it, but it is not continuous. For x 6 ]K(z) ,

nEZ

nETZ

pq2Z

2 pEZ

nE~

pEgZ

nE~

nETZ

and so x' is positive. C o r o l l a r y 2.3.4.8

( 0 ) Let x' be a positive linear form on the involutive

unital Banach algebra E . Take x, y E E . Then

Ix'(y*xy)l _< Ilxll~'(y*y) Let F be a unital involutive Banach algebra associated to E and take y'

9

F

> ]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 <_ I1~-

~11 I1~'11 + I1~'~-

ull,

Ilu~'~ - ~11 _< Ilu~'~ - ~11 + Ily~ - ~11 _< I1~" - ~11 $$z$1 + Ily~ - ~$1

~.3 lnvolutive Banach Algebras

~,93

for every x E E . Thus lim 9 ,9(~)

Ilxu -

lira Ilyx - Yll = 0

ull = o '

lim Ilxu - yll - o

z,9(~)

'

lira Ilyx - ull -- o

by the above considerations. Since y is arbitrary, g(~) and h(~) are approximate units of E (Proposition 2.3.2.7). Moreover

IIx'yx - yll ~ IIx'yx- yxll + Ilyx- yll ~ IIx'y- yll Ilxll + Ily~ - yll for every x E E , so that lira "llz*yx - yll = O.

l

x,$

P r o p o s i t i o n 2.3.4.10 ( 0 ) Let E be an involutive Banach algebra, ~ an approximate unit of E , and x' a continuous posstive linear form on E . a) l~,~ i m x ' ( x * x ) - II~'ll. b)

Ix'(x)l ~ _< IIx'llx'(x*x) f o r e~ery x ~ E .

c) Nx, + N~r, c K e r x ' . d)

x'(x*x).

I1~'11-- sup xEE#

a) Take x, y E E # . Since

I~'(~*y)l ~ <_ ~'(~*~)~'(y*y) <_ ~'(~*~)11~'11 (Proposition 2.3.4.6 c)), Ix'(y)! ~ = lim Ix'(x*y)l 2 < II~'ll lim inf x ' ( x ' x ) for every y E E # (Proposition 2.3.4.9). Hence IIx'll 2 ~ IIx'll lim inf x ' ( x ' x )

IIx'll < U m inf x'(x*x) <_ lim sup x'(x*x) <_ IIx'll (since E # E ~), and

e94

g. Banach Algebras

IIx'll = limx'(x*x). b) By Proposition 2.3.4.6 c),

Ix'(y*x)l 2 <_ x'(x*x)x'(y*y) for every y s E . Hence

I~'(~)1 ~ = lim I~'(u'~)l ~ < ~(~'x)limx'(y*y) = il~'ll~'(x*~) by a) and Proposition 2.3.4.9. c) follows from b). d) By b), I1~'11~ = sup Ix'(~)l 2 < IIx'll sup ~'(x*x) < I1~'11~, xEE#

zEE#

sup x ' ( x * x ) = IIx'll.

m

zEE#

P r o p o s i t i o n 2.3.4.11 ( 0 ) Let E be an ,nvoluttve quasiunttal non-unital normed algebra and F an associated involutive unital Banach algebra. Take x' E E', a E IK, and let y' be the linear extension of x' to F defined by

y'(1) =a. Then the followmg are equivalent:

a) x' is positive, a E IR+, a >_ [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 _

<_ IIz'- z'll + Ilu'- z'll = IIz'll- IIz'll + I1~'il- IIz'll = = IIx'll § Ily'll- 211z'll, IIz'll = 0,

z'=O.

m

Proposition 2.3.4.16

( 0 ) Let A' be an upward directed set of E+ for the involutive quasiunital Banach algebra E . Then the following are equivalent: a) A' is upper bounded. b) A' is bounded (in norm). c) The upper sectzon filter of A' converges. d) The upper sectwn filter of A' has a po,nt of adherence. e) A' has a supremum. If these conditions are fulfilled, then the upper section filter of A' converges to the supremum of A'. In particular, E' is order complete and every operator on E is order continuous.

By Corollary 2.3.4.14, IIx' § y ' l l - IIx'll § Ily'll

for every x' y' 6 E'+ , and the assertions follow from Proposition 1.7.2.5. II

Proposition

( 0 )

2.3.4.17

Let E be an involutive quasiunital Banaeh

algebra and put

C - = {x e Re E lx' e E'+ ~

x'(x) e ~+}.

a) C is a weakly closed convex cone of E and El

~

m?

12.3 lnvolutive Banach Algebras

b)

~99

The following are equivalent: bz)

E~+ - E~+ is dense in Re E' with respect to the topology of pointwise convergence on Re E .

b2)

C issharp.

ba)

There is an order relation on E with respect to which E is an ordered Banach space with E+ = C .

If these conditions are fulfilled, then the order relation of b3) is unique. If IK = ~ , then C:={xEE

I x' E E ' + ~

x'(x) e

n+}

a) Define C' := {x' e Re E'I x e C = ~ x'(x) e IR+}.

It is obvious that C is a weakly closed convex cone of E , that C' is a closed convex cone of E~ and that

E'+ C C' 9 Take x' ~_ C'\E'+. By Proposition 2.3.2.25, there is an x E Re E such that sup v'(x) = o < ~'(x), y~E E~.

i.e. - x E C . This implies the contradiction

o < ~ ' ( - x ) = - ~ ' ( x ) < o. Thus E~ = C ' . bl =~ b2. Take x E C N ( - C ) . Then

z'(x) = 0 for every x' E E'+ - By bl),

x'(x) = 0

300

~. Banach Algebras

for every z' E Re E' and, by Proposition 2.3.2.22 1), this implies that x=0. Hence C is sharp. b2 =~ bl. Take x E Re E such that

x'(x) = 0 for every x' E E~ - E~. Then x E C N ( - C ) , so that, by b2), x=0. Hence E~ - E~ is dense in Re E' in the toplogy of pointwise convergence on Re E. b2 r b3 and the uniqueness of the order relation on E follow from Propositions 1.7.1.2 and 1.7.1.3. Now suppose that ~ = C and take x E E such that

x'(x) E R+ for every z' E E~. By bl) and Proposition 2.3.1.22, E~ - E~ is dense in Re E' in the topology of pointwise convergence on E. Then

x'(x) ~ R for every x' E Re E' and so x E Re E by Proposition 2.3.2.22 m). Hence xEC. I

E x a m p l e 2.3.4.18 ( 0 ) If T ,s a locally compact space, then Co(T) ,s an involutive commutative quasiunital Banach algebra, which possesses a unit iff T is compact (Example ~.~,.~.1, Example ~.3.1.4). If we zdentify the Banach space A4b(T) of bounded Radon measures on T canonically with Co(T)', then #.---6 for every I~ E .A4b(T). In particular, # is selfadjoint iff it is real as a measure. The element # E .A4b(T) is positive in the sense of Defimtion 2.8.4.1 iff it is positive as a measure. For every # E Re.Mb(T), there exist uniquely determined

#1, #2 E .MIb(T)+ such that - m - m,

I1~11- IImll + 11~,211.

In particular, any linear form x' on A4b(T) such that x'(A4b(T)+) C IR+ is continuous. The order relatzon on Co(T) introduced in Proposition ~.3.4.17 b) coincides with the natural order relation on Co(T).

Y2.3 Involutive Banach Algebras

301

Since

I1~11= Ilzll for every x E Co(T), Co(T) is an involutive Banach algebra. Take # E .A,4b(T). Since

=f x

=f

d, = f x'd, = <x', ,) = (x, ,')

for every x ~. Co(T), we deduce that #* = ~ .

Moeover,

(x*x,~)= f z*xd~= f lxl2d~ for every x E C0(T). Hence # is positive in the sense of Definition 2.3.4.1 iff it is positive as a measure. The last assertions follow (Corollary 1.7.1.8). II

( 4 ) Let T be a group and e'(T) the involutwe Banach algebra defined in Proposihon ~.3.2.28 b}.

E x a m p l e 2.3.4.19

a)

Given x' e t ' ( T ) such that 9 '(t) = ~ ' ( t - ' )

for every t E T and

x'(1) ~ ~

Ix'(t)l,

tET\{1}

the linear form

x"~" g'(T)

) IK,

x:

,~ E x ( t ) x ' ( t ) tET

zs positwe. b)

There is a umque order relation on s (T) which renders t' (T) an ordered Banach space w~th

It(T)+ - {x E Rel'(T) [ x' E t'(T)~ ~=~ x'(x) E IR+}.

IL Banach Algebras

30~

a) By Proposition 2.3.2.29 b), ~' is selfadjoint.

Take x E el(T). Then

(~" 9 ~)(t) = ~ ~'(t~-')~(~) = ~ ~(~t-,)~(~), sET

sET

x"(x* * x) = ~(x"

*

x)(t)x'(t) =

tET

= x ' ( 1 ) ~ I~(~)1~ + ~ seT

>1 -2

sET

Ix'(t)l ~ Ix(st-')l Ix(s)l

tET\ { 1}

~

seT

tET\{I}

- ~ 1

~ ' ( t ) ~ ~(a-')~(~) >

t e T \ {l }

=

xET

lx'(t)l ~ (Ix(s)l 2 + Ix(st-')l 2 21x(st-')l I~(s)l) -

,eT~|l}

=

seT'

1

~

Ix'(t)l ~ (Iz(s)l- Ix(st-')l) 2 _>o.

teT\{l}

seT

Thus the linear form ~'

e'(T)

, ~ ~,

x ~

~

x(t)x'(t)

tET

is positive. b) Identify ~I(T)' with s176 c), s

canonically. By a) and Proposition 2.3.2.28

N Re s176176C F~

- fr~

Since el(T)f3 Ree~176 is dense in Ret~176 in the topology of pointwise convergence, b) follows from Proposition 2.3.4.17 b). Ii Proposition 2.3.4.20

( ~ ) Let E , F be involutwe quasiunital Banach

algebras, u" E --+ F a continuous involutive algebra homomorphism, and y' E

F;.

~.3 lnvolutive Uanach Algebras

3'03

a) u'u' e E'+. b) If E, F , and u are unital, then

Ilu'y'll- Ilu'll 9 a) Given x E E ,

Since u'y' is involutive, it is positive. b) By a) and Corollary 2.3.4.7,

Ilu'y'll- (1, u'y')

=

(ul,

y') =

(1, y')

= [[Y'II

m

P r o p o s i t i o n 2.3.4.21 ( 7' ) Let E, F be involutive unital Banach algebras such that E is a dense involutive unital subalgebra of F and such that the inclusion map E ~ F is continuous. If the set r of finite-dimensional mvolutive unital subalgebras of E is upward directed and if its union is dense in E , then u'(F~.) = E'+. By Proposition 2.3.4.20,

~' e F~ ~

~'y' e E~, II~'Y'll = IlY'II.

Put H:=UG. GE~

For every involutive unital subalgebra K of E, denote by Ko the corresponding normed involutive unital subalgebra of F . Let x' 6 E~.. For G 6 ~ , put F

xG " G o ~

IK,

yv

",-x'(y).

Then x'G E G~+ and, by the above,

I1~11 = il~'lVll = II~'ll. Hence the map

Ho---~lg,

y:

"x'(y)

is continuous. Since H0 is dense in F , this map is the restriction to Ho of an element y ' E F'. Since H0 is dense in F , y ' E F~.. u'y' and x' are equal, since they coincide on the dense set H . Thus x' E u'(F~) and E~ = u'(F~). II

304

~. Banach Algebras

E x a m p l e 2.3.4.22

(

0

) Wzth the notation o.f Example 2.3.2.35 el), o

I

(x', y') E (E)+ .'r

Xl

x' + iy' ,

I

-

iy' e E+

o

]or every (x', y') ~ E' .

Fo~ (.,~)s ~,

((~, ~)'(~, ~ ) , ~ )

= ((~',-y')(.. y). .(~~ )

= ((x'x + y'y,x'~ - y'x),(~,

=

)) =

= ( ( x ' x + y ' y ) + i ( x ' y - y*x),x' + iy') + ((x*x + y ' y ) - i(x*y - y * x ) , x ' -

= ((x + iu)'(~ + iu),x' + iu') + ((x - , u ) ' ( x - i u ) , . ' -

Hence, if x' + iy', x ' - iy' E E'+, then

iy') =

ir

E (E)~. Assume that

E

o

(E){~ and z e E. If we choose 1

z.=~z,

1

y.=~z,

then x + iy = z ,

x - iy = O

and therefore

(z'z,x' + iu') = ((~, u)'(x, y), ~ )

e R+.

I we choose 1

x:=~z,

i

y:=~z,

then x+iy=O,

x-~y=

z

and therefore

(z'z..'- iv')= ((*. ~)'(~. ~ ) . ~ ) Hence x' + zy', x' - iy' E Et+.

e ~+ I

~.3 Involutive Banach Algebras

305

2.3.5 The State Space Definition 2.3.5.1

( 0 )

Let E be a involutive quasiunital Banach alge-

bra. A state oI E ss a continuous positive linear form x t on E (Proposition e.3.4.13 b)) with I1"11 = 1.

The set oJ states of E endowed with the topology of pointwise convergence (i.e. with the topology induced by E~ ) is called the state space of E and is denoted by "r(E). By Corollary e.3.4.14, it is a convex set of E'. Gwen x E E , define Y

x.r(E)

~, IK,

z' :

" x'(x).

The extreme points of -r(E) are called pure states oJ E . We use to(E) to denote the subspace of r(E) consisting of the pure states and call it the pure state space of E .

Example 2.3.5.2

Let T be a locally compact space and identify .A/[b(T) canonically with Co(T) ~. Then r(E) = {p E .k4b(T) l l~ positive, #(T) = 1},

to(E) = {6t l t 6 T } . It follows from Example 2.3.4.18 that

Ig(T)= 1}.

~'(E) = {l~ e r

Hence, by Example 1.2.7.14, To(E) = { 6 t i t 6 T } .

1

Example 2.3.5.3

Let s 2) be the involutive unital Banach algebra defined in Example ~,.3.2.6. Take a 2 x 2-matrix

The following are equivalent:

a) The linear form s

2)

=7IK,

~

7

is in ~ ( s

(~n ~o(s

aa + 7b + ~c + 6d

306

I~. Banach Algebras

b)

a, d E l R + ,

a+d=l,

b=~,

( a , d e ]R+, By Example 2.3.4.4 b r

a,d

[bl2 < _ a d

a + d - 1,

b - "~,

Ib[2 -- a d ) .

c, the above linear form is positive iff E

•+,

b = e,

Ibl2 < ad.

Hence the assertion for r follows from Corrolary 2.3.4.7. We now consider the assertion for "to(E). The implication a =~ b is easy to see, so we prove the implication b =~ a. Take t E ]0,1[ and

,

E

I~2,2

such that a t , d l , a 2 , d 2 E R:~+,

al §

-" a2 § d2 --- 1,

Ibm[2 S a t d l ,

[a, b,] [o2b2] [a b] +(I-O

g,

d,

=

g

d2

Ib2[2 ~_ a2d2,

.

~

d

Then a = tat + (1 - t)a2,

b -- tbt + (1 - t)b~,

d = tdt + (1 - t)d2, so that 0 = ad-

Ibl 2 = t2aldl + (1 - t)2a2d2 + t(1 - t)(ald2 + a 2 d l ) -

-t2lb;[2 - (I - t)21b2l2 - t(1 - t)(btb2 +

= t2(aldl - lbll~) +

(I -

t)2(a2d2 - ]b~l~) +

t(l -

bib2) =

t)(ald2 + a2dl - bl~ - blb~).

We have that

a~d~ -Ib~l 2 ~ O,

a2d2- Ib212~ 0

~.3 lnvolutive Banach Algebras

307

a~d2 + a2d~ - b,b~ - b, b2 >_ a~(1 - a2) + a2(1 - a , ) - Ib~l ~ - Ib212 >

> a, + a2 - 2 a l a 2 - a , ( 1 - a l ) - a2(1 - a2) -- a~ 4- a~ - 2 a l a 2 =. (a, - a2) 2 > O.

Hence Ib~l 2 - -

aid1,

Ib212 = a2b2,

a~ - a2

and this implies a~ = a2 = a ,

d~ = d2 = d ,

Ib~l = ]b21 = Ibl.

Let 01,0~ E ]R such that

bl---Ible ''~ ,

b2 - I b l e 'a~ .

Then 0 -- aid2 + a2dl - b l ~ - bib2 - 2 a d - 21bl 2 cos(8~ - 0~),

cos(01 - 0~) = 1,

bl = b2 = b. Therefore the linear form of a) belongs to Vo(s

ll

If we identify s 2) with its dual via the map defined in Example 2.3.1.33, then, by Example 2.3.1.36, every element of To(s is idempotent and selfadjoint. This result will be generalized in Example 4.2.3.4 b). Remark.

Proposition 2.3.5.4

(

0

)

Let

E

be an involutive quaszunital Banach

algebra and take x' E v ( E ) . Then the following are equivalent:

~) ~'e ~o(E). b) y ' E E ' § , y ' < _ x ' ~ y' - I 1 r c) y ' E E ' + , y ' < x '

x' 9

=~ 3 a E ]K , y ' = ~x' .

~. Banach Algebras

308

a =~ b. We may assume that y' # x' and y' # O. Set ZI : = Xt _ ye. Then

Ily'll + I1~'11 = IIx'll = 1 (Corollary 2.3.4.14),

1 yl 1...l_ ze

I1r

'llz'll

er(E),

,,,',,

:

-

,

so that ~ y1 l

=

I1r

V=

XI ,

Ilu'llx'

bya). b ~ a. Take a , ~ E ]0, 1[ and y',z' ~ r(E) with a+/~=l,

ay' + Bz' = x'.

Then c~y' E E~. and ay' < x'. Hence, by b),

. ~ ' = I1~r

= .~',

ye = Xl and thus x' E to(E). b r c is trivial. P r o p o s i t i o n 2.3.5.5

II ( 0 )

Let E be an involutive quasiunital Banach

algebra which is not unital, F an associated mvolutive unital Banach algebra, and x'o the element of r(F) which vanishes identically on E . For every x' E r ( E ) , let ~' denote the unique element of r(F) such that "~'IE = x' (Corollary E3.4.12).

~.3 lnvolutive Banach Algebras

309

a) r ( F ) is the convex hull of {~' I*' e r(E)} tO {x{)}. b) ro(F) = {'~' [ x' e r0(E)} tO {x{~}. a) Take y ' e v(F)\{x~)} and set x I :=

1 y'lE II~'lEll "

Then x' E r ( E ) , ~'IE = x' =

1

Ilu'IEII

y' = IlY'IEII~' + (1

y'IE

-IlY'lEII)x~.

Hence y' belongs to the convex hull of {~' I x' r r(E)} 12 {x{)}. The reverse inclusion is trivial. b) follows from a). P r o p o s i t i o n 2.3.5.6

ii

( 0 ) If E is an involutwe quasiunital Banach algebra,

then every state of E which is an algebra homomorphism is a pure ,.tate.

Let x' be a state of E which is an algebra homomorphism. Take y', z' E 7-(E) and c~, f~ E IR+ \{0} such that +f~=l,

x'=ay'+~z'.

Take x E E . Then ly'Cz)l ~ <_ ~'(~*~),

Iz'(~)l ~ _< ~'(~'~)

(Proposition 2.3.4.10 b)). Thus o = x'(x*x) - I x ' ( x ) l 2 = ~y'(x*x) + ~z'(x*x) - I ~ y ' ( x ) + f~z'(x)l ~ >_

>_ ~ ( ~ + 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<~x II = IIz'zll = Ilxll ~

It is obvious that ~ is continuous. Since

I~(x')l = I~'(~)1 _< I1~'11 Ilxll = I1~11, V

x E C(T(E)) and

I1~11= _< Ilxll. The first assertion follows, since the map E

~C(7(E)),

is obviously linear and involutive. Now choose x E E with IIxll-

V

x:

,~x

II~ll. By Proposition 2.3.4.10 b), V

V

I,~(x')l ~ -- I~'(~)1 ~ ~ ~'(x'x) -- x"~ (:~) <_ II~'~:';'~II

12.3 lnvolutive Banach Algebras

311

for every x' E r(E). By the above considerations, V

Ilxll~ = I1~11~ =

sup I~(~')1 ~ _< 11~*'7~11 _< IIx*xll < Ilxll ~ ,

x'~r(6)

so that V

II~'1~ II = Proposition 2.3.5.9 algebra.

( 0 )

IIx'xll = Ilxll ~ 9

m

Let E be an involutive quasiuntial Banach

a) If E is unital, then r(E) is compact. b) If E is separable, then r(E) is metrizable and separable. By Proposition 2.3.4.13 c), E'+# is a compact set of E~ and it is metrizable whenever E is separable. a) follows from the above remark and the fact that r ( E ) = {z' e E '#+ [ x' (1)-- 1}

(Corollary 2.3.4.7). b) follows from a), Proposition 2.3.5.5, and the above remark. Proposition 2.3.5.10 algebra.

II

( 0 ) Let E be an involutwe quaszunital Banach

a) An element of r(E) belongs to T0(E) iff it is an extreme point of El#+. b) The convex hull of vo(E) is dense in r ( E ) . c) If E is unital then every nonempty closed face of 7(E) possesses a point of to(E). d) If E is unital, then, given x E E , the extreme points of ~(r(E)) are ~o~t~i~d ~ ~(~o(E)).

a) Take x ' E to(E)

y~,ztE E ~# and a,f~ El0,1[ such that ~ + ~ =1,

,~y' + Bz' = x'.

Then 1=

[Ix'I[

=

[ray'

+/~z'[[ =

allY'l[

+

f~l[z'[[

31~,

~. Banach Algebras

(Corollary 2.3.4.14) and we deduce successively that Ilu'll = Ilell = 1,

u',z' e r(E), ~t ~

Zt "

Hence x' is an extreme point of E'+# . The converse is trivial. b) By Proposition 2.3.4.13 b), c), E'#+ is a compact convex set of E~. By a), to(E) 13 {0} is the set of its extreme points. Let A' be the convex hull of to(E) U {0} and B' the convex hull of r0(E). Define

~,-A'\{o} ~B',

~'..~~~. 1

!

Take x' E v(E). By the Krein-Milman Theorem (Theorem 1.3.1.10 c)), there is a filter ~ on A' converging to x' in E~. Since the map E~

~-+,

U':

;llu'll

is lower semicontinous (Proposition 1.2.6.6), 1

=

IIx'll _< lim inf IIVII _< lim sup I1~'11 <- X,

lim IIVII = 1. Hence ~(~) is a filter on B' converging to x' in E~ and so B' is dense in r(E) . c) By Proposition 2.3.5.9 a), r(E) is compact. Hence, by the Klein-Milman Theorem (Theorem 1.3.1.10 b)) every closed face of r(E) has an extreme point. -1

d) Let a be an extreme point of ~(r(E)). Then ~ (a) is a closed face of v r(E) (Proposition 1.2.7.10) since x is continuous (Proposition 2.3.5.8). By c), this face contains an element x' E to(E). Thus

P r o p o s i t i o n 2.3.5.11 ( 0 ) If E ,s an involutive quasmnital Banaeh algebra, then the following are equivalent:

~.3 lnvolutive Banach Algebras

313

a) For every x' E Re E' there are y', z' E E'+ such that

IIx'll- I1~'11+ IIz'll.

x'= y ' - z', b) Ilxll = I1~11 Sot every x e R e E .

c) Re E '# is the closure in E~ of the convex hull of r(E) U ( - r ( E ) ) . If these conditions are fulfilled then v

d)

II x ~ II = IIx~ll = Ilxll 2 for every x ~ R e E .

e)

There is a unique order relation wh,ch renders E an ordered Banach space wzth

E+ = {x e R e E l x' e E'+ ==~ x'(x) e IR+}. Moreover,

E~ = {x' e Re E ' I x ' ( E + ) C lR+}. If IK = C, then E+

---

{x e E I x' e E'q . ==~ x'(x) e ll:t+}

E'+ = {x' e E' I x'(E+) C IR+}. If in addztion E is unital then:

f) For every x E E+ and every n G lN there is a y G Re E such that yn " - - X .

g) The following are equivalent for every x E Re E" gx) x G E+. g2) : q y E R e E , g3) 3 y G E , g4)

x=y2. x - - y*y.

c~ >_ ]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<~<1, so, by b), lll - x l l - - I l l

- ~l[ _< 1.

We define (Proposition 2.2.3.13 a)) y := [1 + ( x - 1)]-~. By Corollary 2.3.2.19 b), y 6 Re E and by Proposition 2.2.3.13 c), y" = x.

316

2. Banach Algebras

g)

gl =~ g2 follows from f). g2 ~ g3 is trivial. g3 =~ g4. We have

o < a-

Ilzll_ c, - -

X'

(~ * ~) _< c,

for every z' ~ r ( E ) . By b), v

Ilal - x l l - - I 1 ~ 1 1 -

sup I ~ - x'(~'y)l ___a

z'Er(E)

g4 =~ g5 is trivial.

g5 =~ gl. By b), v

z'(x) _> 0 for every x' E r ( E ) . By e), x E E+. h) By c) and Proposition 2.3.5.9 a) (and Proposition 2.3.1.22, Proposition 2.3.2.7) the absolute convex hull of E~. # is a 0-neighbourhood in E', so the assertion follows from Corollary 1.7.1.8 b). II Remark. a) Let E be the involutive Banach algebra obtained from the vector space ~2 using the pointwise multiplication, supremum norm, and involution

1K2 ___, 1K~,

(~, a) ~ - , (~, ~).

Then 7(E) = 0 but IIx211 = Ilxll 2 for every x E R e E . b) By Corollary 2.3.4.15, the infimum of y' and z' in E~ is 0, where y', z' are the elements from a). L e m m a 2.3.5.12

( 0 ) Let T be a set, ~ a subset of IR T, ~7 the convex

hull of j r , and ~ the closure of ~ m the topology of pointwtse convergence on IR T . Then

sup x(t) = sup x(t) for every t E T .

~.3 lnvolutive Banach Algebras

317

Take a E IR with sup x(t) < zE~"

and let

{x

"u :=

e ~tr I x(t) _< a } .

7/ is a convex set of ]RT containing Jr and it is closed in the topology of pointwise convergence. Hence 7/ contains ~ and so supx(t) <_ a

.

xEff

Since a is arbitrary, sup x(t) < sup x(t) . xE~

xE~"

The reverse inequality is trivial.

I

P r o p o s i t i o n 2.3.5.13 ( 0 ) Let E be a quasiunital Cresp.unital) involutive Banach algebra, A a subset of 7"(E), B the convex hull of A , and B the closure of B m r ( E ) . Then the foUowzng are equivalent: m

a) r0(E) C B. b) T(E) = B .

c) (resp. d)) For every x E Re E , sup x~

( resp.

~'(~)

=

sup x'(~ ~) = x' EA

d)

sup xtEr(E)

sup x'Er(E)

is fulfilled if

112~11= for all x E Re E .

~up

x'EA

x'(~)

~'(~)

~'(~)).

318

~. Banach Algebras

a =~ b follows from Proposition 2.3.5.10 b). b =~ a is trivial. b =~ c follows from Lemma 2.3.5.12. c =~ b (resp. d =~ b). Let B d be the closure of B in E ~ . By Proposition 2.3.4.13 c), B ~t is a compact set of (E'+#)E. Take y' E r ( E ) \ B ~t . By Proposition 2.3.2.25, there is a y E Re E such that

x'(y)

sup

< y'(y).

z'E B ct

We may assume that

{[y[}<

1.

First suppose c) is fulfilled. Then

{1'({/) <

sup x'(y)= sup x'(x) _< sup x'(x) < y'(y) x'ET(E)

z'E A

z'E Bct

which is a contradiction. Now suppose d) is fulfilled (and E is unital). By Proposition 2.2.3.13 and Corollary 2.3.2.19, there is an x E Re E such that x~= 1 +y. Then

y'(x ~) < sup x'(x 2) = sup x'(x ~) = sup x'(x 2) = z'E~(E)

z'E~

z'EA

= 1 + sup x'(y) _< 1 + sup x'(y) < 1 + y'(y) = y'(x ~) 9 '6~

z'EB ct

(Lemma 2.3.5.12), which is a contradiction. Hence ~'(E) c B d , r ( E ) = r ( E ) N B et = B . As for the last assertion, we have that

9 ' ( ~ ) _< ll~'llllx~ll= l{~ll for every x' E z(E) and hence

sup ~'(x2) <__ll~ll- sup x'(~).

z'~(E)

This implies d).

z'EA

II

~,.3 lnvolutive Banach Algebras

319

Proposition 2.3.5.14 ( 0 ) Let E be an involutive unital complex (tea 0 Banach algebra such that x r e E r (x r e Re Er), x r ( 1 ) = [[x'[[ ==~ x r e E + .

Then E is strongly symmetric and a(x) C ~ ( r ( E ) ) for every x E E (x E Re E ) . Let a E a(x). By Proposition 2.2.4.30, there is an yr E E r , such that

yr(x) = a.

Iiyri[ = y'(1) = 1, Assume IK = 113. We set X r :__ y r .

Then

~'(~) = ~ ' ( ~ ) = ~ , and, by hypothesis, x' E r(E). For the case IK = lR, set x r := re y' E Re E ' . Then

1

[[x'[I >__x'(1) --

(y'(1) + y"(1)) = ~(y'(1) + y'(l*)) : 1,

IIx'll <_ Ily'II = 1 (Proposition 2.3.2.7), so that [[x'[[ = x'(1) - 1. By hypothesis, x' E r ( E ) . Thus 1

,

1

,

x'(~) = ~(y (~) + ~"(~)) = ~(~ (~) +

y,

(~.)) = ~.

By the above, v

a(y*y) C y'y ( r ( E ) ) C IR+ for every y E E . E is therefore strongly symmetric.

1

8~0

~. Banach Algebras

P r o p o s i t i o n 2.3.5.15 ( 0 ) Let E be an involutive unital complex (real) Banach algebra, such that

Ilxll = r(x) whenever x E No E (x E Re E ) . Then the following are equivalent:

a) E ts strongly symmetric. b) x' E E'+ whenever x' E E' (x' E Re E') and IIx'll = x'(i). If these conditions are fulfilled, then:

c) x E No E (x e Re E) ~ ~(r(E)) is the convex hull of a ( x ) . d)

~: - ( g , x E E =~ a(x) C ~r(r(E)).

e)

If F is an involutive subalgebra of E , then .for every x' E Re F', there are continuous positive linear forms y', z' on F such that

x'-- ~'- z', f)

IIx'll--Ily'll + IIz'll.

There ts a untque order relation on E with respect to whtch E ts an ordered Banach space with

E+ = {x E E (x E ReE) Ix' E E+ ==# x'(x) E R + } .

g)

If x" ts a linear form on E' such that x"(E'+) C I t + , then x" is continuous.

a =~ b. We may assume that x'(1) = 1. Take x E E . Then ax* x + ~l E No E

(ax* x + ~l E Re E) ,

so, by assumption,

[[cex*x + ~1[[- r(cex*x +/71) for all a, ~ E IK. By a), a(x*x) C IR+, so by Corollary 2.2.4.31,

z'(z'z) ~ lrt+. Thus x' E E~ (Proposition 2.3.4.13 d)). b =~ a & c & d follows from Proposition 2.3.5.14 and Corollary 2.2.4.31. e & f & g. Assume first F = E . By c),

1~.3 lnvolutive Banach Algebras

r(x) <

sup

I~'(~)1 _< II+il,

so that

I1~11 =

sup I~'(~)1

x'e~(E)

for every x E Re E . e), f) and g) follow from Proposition 2.3.5.11 b =~ a & e &h. Let now F be arbitrary. By Corollary 2.3.2.23, there is an u' e Re E ' , so that

,,'IF = =',

I1,,'11 = I1~'11.

e so t h a t By the above considerations, there exist v', w t E E +,

u'- v'- w',

I1~/11= IIv'll + IIw'll.

We set

y' := v'lF,

z' : - w'lF.

Then y', z' are continuous positive linear forms on F , so t h a t XI __ yl _ ZI and

I1~'11 ~ I1r + IIz'll ~ I1r + IIw'll- Ilu'll = IIx'll, SO

Iix'll = Ily'll + IIz'll.

1

If T is a Hausdorff space, then the bounded Radon measure ~ on T is positive zff

C o r o l l a r y 2.3.5.16

I1~11 = #(T). Let B be the set of bounded Borel functions on T . It is an involutive unital Banach algebra with respect to the structure induced by s176176(Example 2.3.2.4). Identifying ~u with the continuous linear form

B=

~ lK,

x:

" /xd#,

the assertion follows from Proposition 2.3.5.15 a =~ b.

m

3~

1~. Banach Algebras

2.3.6 Involutive Modules Throughout this subsection E is an involutive (unital) Banach algebra. Definition 2.3.6.1 ( 0 ) An involutive (unital) E - m o d u l e ( E - a l g e b r a ) is a (unital) E-module (E-algebra) F endowed with an involution such that F is an involutive Banach space (algebra) and ( ~ ) " = ~. ~. ,

( ~ ) " = ~. ~.

/or (x, a) e E • F .

If F and G are involutive (unitaO E-modules (E-algebras) then a map u : F --~ G is called a homomorphiam of involutive E - m o d u l e 8 ( o f unitaO E - algebras) i.f :t is an mvolutive homomorphism of E-modules (of (unital) E-algebras}.

E is an involutive (unital) E-algebra. If F is an involutive unital E-algebra then the map E

3, F ,

x:

~xl

is a homomorphism of involutive unital E-algebras. P r o p o s i t i o n 2.3.6.2 ( 0 ) If F is an mvolutive E-module, then F' is also an involutive E-module. In particular, E' zs an involutive E-module. By Proposition 2.2.7.2, F' is an E-module. Take ( x , a ' ) E E x F' and a E F . Then (a, (xa') ~ = (a*, xa') = (a'x, a') = (x ~ a '~ = (a, a'*x*),

(~, (~'~)') = (~., ~,~) = ( ~ . , ~,) = ( ~ ' , ~") = (a, ~'~"). Hence

(x~')" = ~"x',

(~'~)" = ~'~"

which proves that F' is an involutive E-module. P r o p o s i t i o n 2.3.6.3

I

Let F be an involutive E-module and A an involutive

set of F . Then {x ~. E I a E A ==~ xa = ax} is an involutive closed subalgebra of E .

~.3 Involutive Banach Algebras

3~3

Let x E E such that xa

--

ax

for every a E A. Then 9 "~ = (~'x)" = ( ~ ' ) " = ~ " for every a E A. Hence

{x ~_ E la 6_ A ==~ xa = ax} is an involutive set of E . It is easy to see that this set is a closed subalgebra of E. m P r o p o s i t i o n 2.3.6.4 E t then

( 0 ) Assume E quasiunital. If x, y E E and x', y' E

x* xtx ~_ x* ylx, whenever x' < y', x*xty + y*x'x < x ' x % + y *X t y whenever x' > O. If x' is positive, then (y*~,x*x'~) = (~y'y~', x ' ) = ((y~*)*(ux*), ~') e ~t§ and so x*x~x is positive. Hence o < (x - ~ ) ' X ' ( x - y ) = x * x ' x - ~*x'~ - ~ ' ~ ' ~ + y ' x ' ~ ,

x*x'y + y*x'x < x*x'x + y*xty. If x' <_ y', then

0 < x * ( y ' - x')x = x*y'x - x*x'x, by the above considerations. Thus

x*x'x < x*ytx,

m

3~,4

2. Banach Algebras

Proposition 2.3.6.5

Take x E E and let x' be a posstive linear form on E . If E is quasiunital, then

II~'~'~ll = ~'(~'), and X* X t ---- Xt X = O ~

whenever x*xtx = O.

If ~ is an approximate unit of E, then

IIx'x'xll = l~(y'y,x'x'x)

=

lim(xy'yx" x') = (xx" x')

by Propositions 2.3.6.4 and 2.3.4.10 a) (and Proposition 2.3.4.9). Assume that

X*XIX- -

O.

By Proposition 2.3.4.6 c), I(y,x'x)12= I(xu, x')l 2 < x'(xx*)x'(y*u)= o

for every y E E. Thus

XtX --0 and x'x' = (x'z)" = 0

by Proposition 2.3.6.2. Example 2.3.6.6 Let ( E t ) t E T be a family of (unital) involutive E-modules, p e [1, cr u {0}, and F the (unital} E-module

{~ e I-I Et l (llS(t)U)teT e if(T)} tET

defined in Example 2.2. 7.21. Then F endowed with the involution

F

~ F,

is an involutive (unital} E-module.

~,~ ~, (~(t)'),~r II

I?,.3lnvolutive Banach Algebras

3~5

P r o p o s i t i o n 2.3.6.7 ( 0 ) Let n E IN and assume the involutive vector space En,11 (Proposition 2,.3.1.31 a)) is endowed with a norm such that sup

llm, ll <_ llall _< ~.S 11%11,

i,jE~In

s,$=l

Ilall = Ila'll for all a E E11111. We recursively define

E (~ := E ,

E (p) := (E(n-t)) '

for every p E IN and put 11

"5" E(P-1)11,n --+ ]K,

b:

; Z (b"'a3s ) t j=l

~(P) and endow .L~ ~(P) B111 with the norm for all p E IN and a E ~,,n 111

II

defined inductively for all p E IN. We have for all p E IN O {0} " a)

a e E 11, (p)11=}. sup s~/E~n

II~,~II<_ II~II _< ~ llmjll. t,3=l

...,.11 zs an involutive E-module and for p ~ 0 the map b) ~(p) E(n),11 ---} [~,E(p-1) )' 11,11

~

a

:

.~

~ a

is an isometry of involutive E-modules.

c) If we identify ~(P) .a.~11~11 with (En 111)(n) v:a the isometries in b}, then a E E 111' (p)111' ==~ j~(,)a--(jE(v)a:j),~e~,, ~t-,,n,vt

7t11"1

l

We prove the proposition by induction on p. a) follows from Proposition 2.1.4.25. b) We may assume that p ~ 0 9 Take a E ~(n). L.n,n Then for each b E E~lnn-1) (b,~*) = (b*,~)= ~ z,$=l

((b*),~,a,)=

856

2. Banach Algebras

=


(b,,,.;,) = s~$=l

= ~ (bj,,Ca*).)= (b, a'), :,2=1

so t h a t

and I1,,'11 = I1~"11 = Jl~'ll = JJ~ll =

I1-11.

Let x 6 E and a 6 E (p) Then for all b 6 E~,~n-1) n

n

(b,~) - ~

(b,,,xaa,) = ~

so=l

(bijx,a,,)= (bx,a~ = (b,x'd),

s,$=1

so t h a t x a ~- x a ~

The relations

are proved similarly. Finally, the relations

(.~)" = ~ ' . ' ,

(xa)" = . ' x '

are easy to see. re(P) . , .+l) c) Assume first a 6 ~ . , n . Then for every b 6 E.(p

(b,jE~p~a> = (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 <_ I1~11

for every x' E a(E) (Proposition 2.2.4.19). Thus ~ e C(a(E)) and -- ~ + , 8 ~ ,

x~ -- x~,

I1~11~ <_ I1~11,

proving the first assertion. If E is unital, then "i'(x') = x'(1) = 1

for every x' E a(E) (Proposition 2.1.1.12), proving the second assertion, Theorem 2.4.1.3

m

( O ) Let E be a Gel?and algebra.

a) ao(E) is compact.

b) a(E) is locally compact. c) If E is unital, then a(E) is compact. d) If E zs separable, then ao(E) and a(E) are metrizable. e) If a(E) ss not compact, then ao(E) is its Alexandroff compactification and ~ E Co(a(E)) for every x E E .

a) and d). By the Theorem of Alaoglu-Bourbaki, E#~, is compact and it is metrizable whenever E is separable. By Proposition 2.2.4.19, ao(E) = {x' E E#~, Ix, y E E ~

x'(xy) = x'(x)x'(y) }.

b) and e) follow from a). c) By Proposition 2.1.1.12, a(E) = {x' e ao(E) l x'(1) = 1}.

B

~,.~ Gel]and Algebras

333

C o r o l l a r y 2.4.1.4 ( 0 ) Let E be a Gelfand algebra which is not unital and F an associated unital Banach algebra of E . Then the map

a(F)

} ao(E),

x' : " x'IE

is a homeomorphism. If a(E) is not compact and if we identify a(F) with ao(E) by means of the above homeomorphism, then a(F) is the Alexandroff compactification of a ( E ) . The given map is obviously continuous. It is injective by Proposition 2.1.1.12 and it is surjective by Proposition 2.1.1.7. Hence the assertion follows from Theorem 2.4.1.3 a), c), e). I P r o p o s i t i o n 2.4.1.5 ( 0 ) If E ,s a Gelfand algebra and ~ is the set of regular maximal proper ideals of E , then Ker x' E ~ whenever x' E a(E) and the map

a(E) ---+ ~,

x' ~ ; Kerx'

is bsjective. If E ts unital, then a(E) # 0.

Step 1

x' e a(E) =~ Ker x' e

If E is unital, then the assertion follows from Proposition 2.1.1.12. Hence we may assume E is not unital. Let IK x E be a unital Banach algebra associated to E and 5':IK•

; IK,

(o~,x) ~ ~ a + x'(x).

Then 5' E a(lK x E) (Proposition 2.1.1.7) and KerS' is a maximal proper ideal of IK x E (Proposition 2.1.1.12) distinct from E . Identifying E with {0} • E , Ker x' = E f3 Ker 5'. Hence Ker x' E ~. Step 2

The map is injective

Take x', y' e a(E) with Ker x' -- Ker y'.

33~

~. Banach Algebras

Then there is an a E lK with ye = ax' (Lemma 1.2.6.3). Thus

a~,(~:) = ~ , ( ~ ) = ~,(x)~ = ~2~,(~)~ = ~ , ( ~ ) for every x E E . Hence a = 1 and x ~ = y'. Step 3

The map is surjective

First suppose that E is unital. Take F E ~. Then F is closed (Corollary 2.2.4.16). Let q : E ~ E / F

be the quotient map. E / F

is a unital Banach

algebra (Proposition 2.2.1.14) and a division algebra (Proposition 2.1.2.12 b)). If K = Ft, then by Corollary 2.1.5.19,

o ( ( x , o)) = ~ E / F ( X ) . ElF

By Corollary 2.2.5.5, there is an isometry of unitai normed algebras u : E l F lK. Put

x' :- u o q . Then x~E a(E) and Ker x' = Ker q = F . In particular, a(E) ~ 0 (Proposition 2.1.1.14 c)). Now suppose that E has no unit and let K x E be a unital Banach algebra associated to E . Take F E ~'. Let G be a maximal proper ideal of K x E containing {0} x F and different from {0} x E . By the above, there is a character y~ of IK x E such that K er y' = G. Let x' be the restriction of y' to E (identified with {0} x E ) . Then x' E a(E) and Ker x e = F .

I

2.4.1.6 ( 0 ) If E is a unital aelfand algebra, then x E E is not inverttble iff x'(x) = 0 for some x' E a ( E ) .

Corollary

x is not invertible iff it belongs to a maximal proper ideal of E (Corollary 2.1.2.11), i.e. iff x E Ker x' for some x ~ E a ( E ) (Proposition 2.4.1.5), which is equivalent to x'(x) = O.

1

~.,~ Gel]and Algebras

C o r o l l a r y 2.4.1.7

335

( 0 ) Let E be a Gel.fand algebra and x, y E E .

a) a ( x ) = "~(a(E)) if E is unital, a(x) = ~(a(E)) U {0} if E is not umtal. b)

I1~11~ = ~(=).

c)

o(~u)c

a(x)a(u),

r(xy) < r(x)r(y),

~(~ + y) c o(~) + o(u), ~(~ + u) c r(~) + ~(u).

a) First assume that E is unital. Take a E IK. Then a ~_ a(x) ~=~ a l - x is not invertible.

Hence a belongs to a(x) iff there is an x' ~. a(E) such that 9 '(~1 - ~) = o

(Corollary 2.4.1.6), i.e. such that = ~'(x) = ~ ( ~ ' ) .

Thus o(~) = ~ ( o ( E ) ) .

Now assume that E is not unital. Let F be a unital Banach algebra associated to E. Then a s ( x ) = aF(x) = ~(a(F)) = ~(a(E)) LI {0}

by the above result and Corollary 2.4.1.4. b) and c) follow from a). C o r o l l a r y 2.4.1.8

I

Given A C 9 and a > 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))) <

IIPm(x)- f(x)li

for every m e IN, m _> 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<, n-'0

Since

n'-'O

it follows a([1 + x] ~ = a ( f ( x ) ) = f ( a ( x ) ) C [(1 + a) a , (1 + b) a] C IR+ by Corollary 2.4.1.9 and Lemma 2.2.3.12. Corollary 2.4.1.11

I

Let E be an involutive unital complex Banach algebra.

Then for every x E Re E with a(x) C ]0, ~ [ and for every n E IN, there is a y E Re E such that a(y) C ]0, cx~[ and yn = x .

~. Banach Algebras

338

By Corollary 2.2.5.9, there is an c~ > 0, with r(1 - crx) = limoo[l(1 - ax)"ll ~ < 1 and

Put y := a-~[1 - (1 - ax)]~. By Corollary 2.3.2.19, y G Re E . Since

~,(~-

1) c ~ ( o ) n (a,,(x)- 1)c l - 1,1[

it follows, by Corollary 2.4.1.10, that I

~(~) c 10, oo[. C o r o l l a r y 2.4.1.12

Let E be a unital Gelfand algebra. Take x E E and put

f := { P ( x ) Q ( x ) - ' l P , Q q Kit], Q(x) inverttble}. If F is dense m E , then the map ~(E)

, ~(x),

~' :

; ~'(~)

ts a homeomorphism.

By Corollary 2.4.1.7 a), the map in question is surjective. Since it is continuous and a(E) is compact (Theorem 2.4.1.3 c)), it is sufficient to show that this map is injective. Take x', y' E a(E) with

9 '(~) = r Then x'lF = y'lF,

II

so that x' = y' (Proposition 2.2.4.19). T h e o r e m 2.4.1.13

( 0 ) If E is a Gelfand algebra, then [1~[Ioo = r ( x ) = lim I[xnll~ < Hx[[ n--~oo

for every x e E and given &stinct x', y' E a ( E ) , there is an x e E with

~(x') = o # ~(r

~,.4 Gel.land Algebras

339

That

II~ll = r ( x ) =

lira II~-II ~ _< II~ll

n---~oo

:or every x E E follows from Corollary 2.4.1.7 b) and Theorem 2.2.5.4. By ~roposition 2.4.1.5, there is an x E E such that x E Ker x'\Kery'. Jence ~(~') = ~'(~) = o # y'(~) = ~(~').

~orollary 2.4.1.14

m

( 0 ) If E is a Gelfand algebra, then

radical of E = {x e E I x is quasinilpotent} = {x e E I s = 0}. If E is semi-simple, then there is a unique order relation on E with respect to which E is an ordered Banach space, such that E+ = {x e E la(x) c ~ + } = {x e E I s e C0(a(E))+}. Moreover, the Gelfand transform of E is injective iff E is semi-simple.

By Proposition 2.4.1.5 and Theorem 2.4.1.13, :adical of E =

f'~ g e r x ' = {x e E I ~ = O} = {x e E I x is quasinilpotent}. x'Ca(E)

Now suppose that E is semi-simple. By the above, the Gelfand transform is injective and so we may identify E with its image under it in go(a(E)). By 3orollary 2.4.1.7 a), {x e E la(x) c IR+}= {x E E [ s e C0(a(E))+}

and the last assertion now follows.

1

Remark. If E is the Gelfand algebra of Example 2.2.4.23, then it follows from the above corollary that a(E) is empty.

Proposition 2.4.1.15

( 0 ) Let S be a (unital) Gelfand algebra and u its Gelfand transform. Then the following assertzons are equivalent: a)

There is an ~ > O, such that

r(x) >_ ,~llxll for every x E E .

340

b)

tL Banach Algebras

E Is semi-simple and Im u is closed.

If these conditions are fulfilled, then the following assertions also hold.

c)

The map

E~lmu,

x:

,~ux

is an isomorphism of (unitaO Banach algebras and its mverse has norm smaller than !Q .

d)

There is a unique order relation on E , wtth respect to which E is an ordered Banach space such that E+ = { x s E l a ( x ) C IR+} = {x E E I~ E Co(a(E))+}.

e)

Given x E E+ and n E ~ , there is a unique y E E+ such that y" = x . This y is in the closed real subalgebra of E generated by x and 1

t

Ilull ___ ~llxll ~ 9 f) For every x E E with a(x) C I t , there extst uniquely x +, x - E E+ such that X-----X+ - x - ,

X+X - = 0 ,

x +, x - belong to the closed real subalgebra of E generated by x and iix+ll < l~llxll,

ilx_[i _< ~]lxl1.1

a =~ b & c. By Theorem 2.4.1.13,

allxll ~< Iluxll ~< Ilxll for every x and so b & c follows from Proposition 1.2.1.18 b),d), Proposition 2.4.1.2, and Corollary 2.4.1.14. b =~ a. By Corollary 2.4.1.14, u is injective and the assertion follows from the Principle of Inverse Operator. d) follows from b) and Corollary 2.4.1.14. e) By Corollary 2.4.1.7 a), ~ is positive. Put f :a(E)

" IK,

x' : ~ ~(x')~.

~.4 Gel/and Algebras

341

By b) and Corollary 1.3.5.17 (used for lK = IR), f E I m u and the assertion follows from a), c), and d). f) By Corollary 2.4.1.7 a), ~ is real. Define g" a(E) - - r IK,

h" o(E)

-~ IK,

;sup{~'(x'),O},

x"

x' : ; s u p ( - ~ ( x ' ) , O } .

By Corollary 1.3.5.17 (used for IK = IR), g, h E Im u and the assertion follows from a), c), and d). I Corollary 2.4.1.16 ( 4 ) If E is a finite-dzmensional Gel/and algebra, then the foUowmg assertions are equivalent: a)

There is an c~ > 0 such that

r(x) ~

~llxll

for every x ~. E .

b)

E is semi-ssmple.

Since E is finite dimensional, {~ I x E E} is finite dimensional too, and hence it is closed. The assertion follows from Proposition 2.4.1.15 a r b. I P r o p o s i t i o n 2.4.1.17 ( 0 ) Let E, F be unital Gel./and algebras and u" E --~ F a unital algebra homomorphism. a) y' o u e a(E) for every y' ~. a ( F ) . Define ~'a(F) F

b) u"~ =

~a(E),

y':

E

~oqo

.for all x E E .

c) qo is continuous.

d) If u(E) is dense, then qo is injective. e) If F is semi-simple, then u ss continuous.

~y'ou.

~. Banach Algebras

S4e

a)

is trivial. b) Given y ' E a ( F ) , F

E

~(y')

= ~'(~)

= y' o ~ ( x ) = ( ~ ( y ' ) ) ( ~ )

E

= ~ ( ~ ( ~ ' ) ) = ~ o ~(~').

Hence F E A A ltX - - - - X o r

c) Let x e E . Then ( ~ ( u ' ) ) ( ~ ) = v' o ~ ( ~ ) = u ' ( ~ )

for every y' ~. a ( F ) . ~ is thus continuous. d) Let x ' , y ~ ~. a(F) such that ~a(x')= ~a(y'). Then x ' = y' on u ( E ) . Since u(E) is dense and x', yl are continuous (Proposition 2.2.4.19) we get x' = y' Hence ~a is injective. e) By the Closed Graph Theorem, it sut~ces to show that the graph G of u is closed. Take (x, y) E G. There is a sequence ((x,, Yn)),e~ in G converging to (x. y). By a) and Proposition 2.2.4.19, y ' ( ~ - u ~ ) = y ' ( y ) - ~' o u(~:) = 2 i m ( y ' ( y . )

- y' o ~(~n)) =

= lim y'(y. - ux.) = 0 for every y' E a ( F ) . Since F is semi-simple, y-ux=O

(Corollary 2.4.1.14). Thus (x,y) E G and G is closed.

II

Corollary 2.4.1.18 Let E be a Gelfand semi-simple umtal algebra. Any two norms on E , with respect to which E is a unital Banach algebra, are equivalent. II

~.4 Gelfand Algebras

343'

2.4.2 Involutive Gelfand Algebras

Definition 2.4.2.1

( 0 ) An involutive (unital) Gelyand algebra is an

involutive Banach algebra which is a (unital) Gelfand algebra such that when IK = IR, every element is selfadjoint. Hence an involutive complex Gelfand algebra is nothing else but an involutive commutative complex Banach algebra. An involutive real Gelfand algebra is an involutive commutative real Banach algebra E such that x e E ~

a ~ ( ( x , 0)) = o ~ ( x ) , z* = x .

lR and q~ are involutive unital real and complex Gelfand algebras, respectively. Proposition 2.4.2.2

( 0 ) Let E be an involutive real Gelfand algebra.

a) E and E are strongly symmetric. b) Every character of E and the Gelfand transform of E are involutive. c) a(E)

C

E'+ .

d) If E is quasmnital then a(E) C vo(E). a) Take x E E. By Corollary 2.4.1.7 a), a(x*x) = a(x 2) C ~ ( a ( E ) ) U {0} = ~2(a(E)) U {0} C JR+. Hence E is strongly symmetric. Let (x, y) e E. Then o

o~((x, ~)'(x, y)) = o ~ ( ( x , - ~ ) ( x , ~)) = o~((~ ~ + u~, 0)) =

= aE(x2 + y2) C x ~ ~ ( a ( E ) )

U {0} =

= (~-2 + ~2) (a(E)) U {0} C IR+ (Corollary 2.4.1.7 a)). Hence /~ is strongly symmetric. b) is trivial. c) Take x' E a(E) and x E E . Then

~'(~'~) = ~ ' ( ~ ) = ~'(~)~ e ~ + , so that x' E E~. d) follows from Proposition 2.2.4.19 and Proposition 2.3.5.6.

1

3~4

~. Banach Algebras

Proposition 2.4.2.3 ( 0 )

Let E be an involutive commutative complex Banach algebra. Then the following assertions are equivalent: a)

E is symmetr;c.

b)

a(x) C IR ]or every x e Re E .

c)

The Gelfand transform of E is involutive.

d)

Every character of E is involutive.

e) a(E) C E'+ . If E is quasiunital, then the above conditions are equivalent to each of the following ones: f)

a(E) C T(E).

g)

a(E) C to(E).

Let F be an involutwe real Banach algebra such that E zs isomorphic to the complexification of F and that every element of F is selfadjoint. If E is symmetrzc then F ts a symmetrsc involutwe Gelfand algebra. a =~ b follows from Proposition 2.3.1.29. b =~ c. Define

u" E

~C(a(E)),

x~

~.

Then re"~(a(E)) C a ( r e x ) C IR,

r~(a(E)) C a(imx) C (Corollary 2.4.1.7 a)), so that

ux* = u(re x - i i m x) = r ~ - i im x = r ~ + i i m x = ~~ for every x E E , that is to say, u is involutive. c => a follows from Corollary 2.3.1.7 a). c r

d and g ==~ e =~ d are trivial.

d =r e follows from Proposition 2.3.4.5. e ~ f follows from Proposition 2.2.4.19.

•.4 Gelfand Algebras

345

f =~ g follows from Proposition 2.3.5.6. Now we prove the last assertion. By a =~ b,

os((~, o)) c for every x E F and F is a symmetric involutive Gelfand algebra. C o r o l l a r y 2.4.2.4

( 0 )

1

( S h i r a l i - F o r d ) L e t E be an involutive complex

Banach algebra. Then the following are equivalent: a)

E is symmetric.

b) o(x) C IR for every x ~. Re E . c)

Any involutive unital Banach algebra associated to E is symmetric. a =~ b follows from Proposition 2.3.1.29. b =~ a. Let F be a closed involutive commutative subalgebra of E and let

x E Re F . By Corollary 2.2.4.27,

,,F(~) = ,,s(x) c ~ , and so, by Proposition 2.4.2.3 b =~ a, F is symmetric. By Proposition 2.3.2.21 c =~ a, E is symmetric. b =,, c. We may assume that E is not unital. Let q~ • E be an involutive unital Banach algebra associated to E and let ((~, x) be a selfadjoint element of q~ x E . Then (~ E lR and x E Re E . Thus

or215

~ ) ) = o~(~) c rt

and

oc•

~)) = ~ + ar215

~)) c ft.

By b =, a, q~ x E is symmetric. c =~ a follows from Proposition 2.3.1.28. C o r o l l a r y 2.4.2.5

I

( 0 ) Let E be a symmetric involutive complex Banach

algebra such that x E R e E , r(x) = 0 ==~ x = O. Take z E No E . Then the following are equivalent:

346

~,. Banach Algebras

a) x E R e E . b) a(x) C Ft. a =~ b follows from Proposition 2.3.1.29. b =~ a. By Corollary 2.3.2.16, E(x) is commutative. Since the Gelfand transform of E(x) is involutive (Proposition 2.4.2.3 a =:~ c), it follows rE(ira x) = rE(x)(im x) = 0 (Corollary 2.2.4.27, Corollary 2.4.1.7 a)). By hypothesis imx =0.

Hence x E R e E .

Corollary ~.4.2.S

m

( 0 ) I / E ~, a ,ymma,~ i,~ot~ti~ Cala,d atg~a~a.

then { ~ l x E E} ts a dense involutive subalgebra of C0(a(E)).

By Proposition 2.4.2.2 b) and Proposition 2.4.2.3 a =~ c, the Gelfand transform of E is involutive. Hence {~ I x E E} is an involutive subalgebra of C0(a(E)) (Proposition 2.3.1.16, Theorem 2.4.1.3 e)). The assertion now follows from Corollary 1.3.5.15 and Theorem 2.4.1.13. II C o r o l l a r y 2.4.2.7 ( 0 ) Let E be asymmetrtc involuttve Gelfand algebra. Then the following are equivalent: a)

There is an a > 0 such that

r(~) >_ allxll for every x E E .

b)

The Gelfand transform is an isomorphism of involutive Banach algebras.

c)

The Gelfand transform is boective.

In particular, E is unital iff a(E) is compact.

By Propositions 2.4.2.2 b) and 2.4.2.3 a =~ c, the Gelfand transform is involutive and by Corollary 2.4.2.6 its image is dense in Co(a(E)). a =~ b follows from the above and Proposition 2.4.1.15 a =~b & c. b =~ a & e is trivial. c ~ b follows from the above and the Principle of Inverse Operators. II

I~.4 Gelfand Algebras

Corollary 2.4.2.8

347

( 4 ) Let E be a finite-dimensional symmetric involutive

semi-szmple Gelfand algebra.

a)

The Gelfand transform of E is an isomorphism of involutive Banach algebras.

b) E has a unit. c) There is a unique order relatwn on E with respect to which E zs an ordered Banach space and E+ = {x e E Ix' e E+ ~

x'(x) e ]R+}.

d) E+ C n e E . e)

Given x E E , the following are equivalent: el)

x E E+.

e2) x = y*y :for some y ~. E . e3) x - y2 for some y ~_ E+ .

e4) a(z) c n+. a) By Proposition 2.4.2.2 b) and Proposition 2.4.2.3 a => c the Gelfand transform of E is in.Jolutive. By Corollary 2.4.1.16 b => a, there is an a > 0 such that

r(~) > ~11~11 for every x E E and the assertion follows from Corollary 2.4.2.7 a =~ b. b), c), d), and e) follow from a). Proposition 2.4.2.9

I

( 0 ) Let E be a symmetric involutive complex Banach

algebra for which there is an a > 0 with

r(~) >_ "11~11 for every x E Re E .

a) For every x E No E such that a(x) C lit+ and for every n E IN, there is a umque y E ReE such that a(y) c IR+ and ya = x . y belongs to the closed real subalgebra of E generated by x and

Ilyll _< lllxll~ 9

348

b)

2. Banach Algebras

For every x E Re E there are uniquely determined x + , x - E Re E(x) such that a ( z +) c IR+,

a(z-)

c IR+ ,

x = x + - z-,

x+x - = x-x + = o.

x + and x - belong to the closed real subalgebra of E generated by x and

iix+ll ___ _IIlxll ,

IIx-II _< ~llxll.

c) Suppose a = 1 and E is unital. Given x E Re E , the following are equivalent:

c~) a(x) c ~.+. c~)

There is a y E R e E such that a(y) C It+ and y~ = x .

ca)

There is a y E E such that y*y = x .

c4) x'(x) E It+ for every positive continuous linear form x' on E . If these conditions are fulfilled then E is strongly symmetric.

a) Let F be a closed involutive commutative subalgebra of E containing x (e.g. E(x) by Corollary 2.3.2.16). Then rF(Z) = r ~ ( z ) >

~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 . S