Riesz and Fredholm Theory in Banach Algebras (Research Notes Inmathematics Series)

B A Barnes, G J Murphy M R F Smyth & T T West University of Oregon/Dalhousie University! Department of Health and Social Services, Northern Ireland! Trinity College, Dublin

Riesz and Fredholm theory in Banach algebras

Pitman Advanced Publishing Program BOSTON-LONDON MELBOURNE

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Associated Companies Pitman Publishing Pty Ltd. Melbourne Pitman Publishing New Zealand Ltd, Wellington Copp Clark Pitman, Toronto

© B A Barnes, G J Murphy, M R F Smyth & T T West 1982 First published 1982 AMS Subject Classifications (main) 47B05, 47B30, 47B40 (subsidiary) 46BXX, 46JXX British Library Cataloguing in Publication Data Riesz and Fredholm theory in Banach algebras (Research notes in mathematics, 67) 1. Banach algebras I. Barnes, B A. II Series 512' .55 QA326 ISBN 0-273-08563-8 Library of Congress Cataloging in Publication Data Main entry under title. Riesz and Fredholm theory in Banach algebras (Research notes in mathematics, 67) Bibliography: p Includes index 1 Banach algebras 2. Spectral theory (Mathematics) I. Barnes, B A (Bruce A) II. Series QA326.R54 512' 55 82-7550 ISBN 0-273-08563-8 AACR2 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical. photocopying, recording and/or otherwise without the prior written permission of the publishers This book may not be lent, resold, hired out or otherwise disposed of by way of trade in any fonn of binding or cover other than that in which it is published, without the prior consent of the publishers ISBN 0 273 08563 8 Reproduced and printed by photolithography in Great Britain by BiddIes Ltd, Guildford

Contents

CHAPI'ER

0

OPERATOR

.

THEORY

. . . .

0.1

Notat~on

0.2

Fredholm operators

0.3

Rlesz operators

0.4

Range

0.5

Act~on

0.6

The wedge operator

0.7

Notes

CHAPTER

1

. . . .

8

12

lnclus~on

on the commutant

.

F

.

3

15 17

. . . . .

FREDHOLM

19

THEORY

F.l

Mlnlmal ldeals and Barnes

F.2

Prlm~tlve

Banach algebras.

29

F.3

General Banach algebras ••

35

F.4

Notes •

43

CHAPTER

R

RIESZ

ide~potents

•

23

THEORY

R.l

Rlesz elements:

algebraic propertles

53

R.2

Rlesz elements:

spectral theory

54

R.3

Rlesz algebras;

characterisat~on.

60

R.4

Rlesz algebras:

examples

62

R.5

Notes

CHAPTER

C*

C*.l The

w~dge

C*.2

63 C* -ALGEBRAS operator

Decompos~tlon

theorems.

70 73

C*.3 Rlesz algebras

77

C*.4 A representatlon • .

78

C*.5 Notes • . • . • •

81

CHAPTER

A

APPLICATIONS

A.l

Fredholm and Riesz elements

A.2

Sem~normal

A.3

Operators

A.4

Tr~angular

A.S

Algebras of

A.6

Measures on compact groups

96

A.7

Notes • • •

98

CHAPTER

BA

elements leav~ng

a

~n

~n

subalgebras

C*-algebras

f~xed

sLllspace

88 ~nvar~ant

operators on sequence spaces quas~triangular

BANACH

86

operators

90

92 94

ALGEBRAS

BA.l Spectral theory

100

BA.2 The structure space

101

M~n~mal ~deals

105

BA.3

and the socle

BA.4 C*-algebras • • . • . • . • .

108

BIBLIOGRAPHY

112

INDEX

118

NOTATION

122

Introduction

ThlS monograph alms to hlghllght the interplay between algebra and spectral theory whlch emerges In any penetratlng analysls of compact, Riesz and Fredholm operators on Banach spaces. that the

set~ng

The emphasls on algebra means

wlthln whlch most of the work takes place is a complex

Banach algebra, though, In certaln situations in which topology lS dlspensable, the settlng lS slmply an algebra over the complex field.

The

choice of spectral theory as our second maln theme means that there is Ilttle overlap Wlth other extenslons of classlcal results such as the study of Fredholm theory In von-Neumann algebras. We use the monograph 'Calkin Algebras and Algebras of Operators In Banach §paces' by Caradus, Pfaffenberger and Yood (25) as our take-off pOlnt, and (A modern

It should be famlilar, or at least accesslble, to the reader. view of the Calkin algebra lS glven in (40».

The original

lnten~on

behlnd Chapter 0 was to provlde a summary of classical operator theory, but, i t emerged In the course of the work that a quotlent technlque developed by Buonl, Harte and Wlckstead (17),

(41) led

~o

new results/lncluding a geometrlc

characterisatlon of Rlesz operators (§O.3) and some range inclusion theorems (§O.4) •

Thus Chapter 0 contalns an amount of new materlal as well as a

survey of classlcal results. On an lnflnlte dlmensional Banach space a Fredholm operator lS one whlch, by Atklnson's characterlsatlon, lS invertlble modulo the ldeal of finlte rank operators (the socle of the algebra of all bounded linear operators on the Banach space) •

ThlS motlvates our concept of a Fredholm element in an

algebra as one that lS lnvertlble modulo a partlcular ldeal whlch, In the semislmple case,

~ay

be chosen to be the socle.

In §F.l we lntroduce the left and rlght Barnes idempotents.

:For a

Fredholm element In a semislmple algebra these always eXlst and lie In the socle.

In the classical theory they are flnite rank proJections related to

the kernel and range of a Fredholm operator. considered in §F.2.

Prlmitive Banach algebras are

Smyth has shown how the left regular representatlon of

the algebra on a Banach space consistJ..ng of a mlnimal left ideal may be used

to connect Fredholm elements Ln the algebra WLth Fredholm operators on the space.

\hth thLs technLque the main results of Fredholm theory Ln p£LmltLve

algebras may be deduced directly from the classLcal results on Fredholm operators.

This theory LS extended in §F.3 to general Banach algebras by

quotLenting out the primLtLve Ldeals.

It now becomes approprLate to intro-

duce the Lndex function (defLned on the space of primLtLve Ldeals).

The

validity of both the index and punctured neLghbourhood theorems Ln thLS general setting (fLrst demonstrated by Smyth (83»

ensures that the full

range of classLcal spectral theory of Fredholm (and of Riesz) operators carrLes over to Banach algebras. Riesz theory LS developed In Chapter R bULlding on the Fredholm theory of the prevLous chapter and we follow Smyth's analysLs (85) of the iwportant class of RLesz algebras.

Results which are peculLar to HLlberr space and

their extensions to e*-algebras, lncludLng the West and Stampfli decOMposition theorems are gLven in Chapter e*.

Chapter A contaLns applLcatLons of our

theory to semLnormal elements in e*-algebras, operators leavLng a fLxed subspace invarLant, triangular operators on sequence spaces, quasLtrLangular operators and measures on compact groups. ments are listed in Chapter BA.

The lliLderlying algebraLc requLre-

Each chapter contaLns a fLnal sectlon of

notes and comments. A faLr proportLon of the theory developed here is appearlng in prlnt for the first tLme.

~ong

tile more LIDportant new results are the geometrLc

characterisatLon of RLesz operators (0.3.5); (§0.4),

~~e

range inclusLon theorems

the link between Fredholm theory In prLIDltive algebras and classLcal

operator theory (F.2.6);

the punctured neLghbourhood theorem (F.2.10);

index functLon theorem (F.3.ll);

the

the characterLsatLon of Lnessential Ldeals

(R.2.6) and the StampflL decomposltLon In e*-algebras (e*.2.6).

(Some of

these results have, powever, been known since the appearance of (83».

Th~s

has reqULred that full details of proofs be gLven, except for the thecrems listed under the notes at the eno of each chapter. Each author has been involved Ln the development of the ldeas presented in this monograph.

The subject has gone through a perLod of rapid expansion

and Lt now seems opportune to offer a unLfled account of LtS maLn results.

o

Operator theory

This chapter

cont~ns

often stated

w~thout

the

bas~c

proof.

results from operator theory on Banach spaces

The

Pfaffenberger and Yood (25).

ma~n

ful referenceS for Fredholm theory; R~esz

,lh~le

theory;

the monograph of Caradus,

Bonsall (13) gives an

spectral theory of compact operators;

ded for

~s

reference

algebra~c

Schechter (BO)

approach to the

and Heuser (43) are use-

Dawson (29) and Heuser (44) are recommen-

Dunford and Schwartz (30) provides an

~nvaluable

background of general spectral theory. Notdt~on

and general

~nformat~on ~s

set out

~n

§l.

Fredholm operators

are cons~dered ~n §2 wh~ch contains a proof of the Atk~nson character~sat~on §3 outl~nes the theory of R~esz operators and, employ~ng a quotient

(O.2.2) • techn~que

Buon~,

due to

the Ruston

Harte and

character~sat~on

W~ckstead

operators due to Smyth (0.3.5).

range

~nclus~on

theorems for compact, are new.

wh~ch

R~esz

characterisa~on

Th~s mater~al ~s quas~n~lpotent

used R~esz

and

~n

of

§4 to prove

operators

Much simpler proofs of these results are avail-

able ~n H~lbert space and are g~ven in §C*.5. of a compact or

(41), contains a proof of

geometr~c

as well as a new

R~esz

several of

(17),

operator on

~ts

In §S we consider the action

commutant, and

~n

§6 the properties of

the wedge operator. 0.1

Notat~on

lR

and

and

H

€

w~ll

denote the real and complex

a Banach and a

H~lbert

space over

f~elds,

cr.

respect~vely,

We start by

l~sting

and

X

the var-

ious classes of bounded l~near operators wh~ch w~ll be ~scussed and, where necessary, B(X)

def~ned

subsequently:

the Banach algebra of bounded l~near operators on

Inv{B{X» F(x)

the

the set of ~deal

of

~nvert~ble

f~n~te

operators

B(X);

~n

rank operators on

X;

K(x)

the closed

~deal

of compact operators on

I (X)

the closed

~deal

of

~

the set of Fredholm operators on

(X)

Q(X}

the set of

~nessential

quasin~lpotent

X·,

operators on X;

operators on

X;

X;

X;

R(x)

the set of R1esz operators on

If

T

B (X)

S

rXT)

I

(T)

0

I

truro and spectral radius of Y

and if

T

T

operator on

X*.

to

on

Y.

X*

Let

X.

wlll denote the resolvent set, specker(T)

X, 1nvarlant under

is the dual space of

x s X,

If

r (T)

T, respectively.

is a subspace of

triction of

Y -+ a(y)x

and

X.

a s X*,

Hol (0 (T»

a

~

T, X

fiT)

2'1T1

r

where otT)

fr

1 ---.

r

Let

then

peW,T)

: T

=

A

P.

If

P

OCT)

at the point

X.

Ilx +

tive integers). denbted

S.

O(T), and f

reduces (commutes wlth)

of

Associadeflned by

S

Hol(O(T»

If

we then wrlte

T

1S a spectral set for

iJ.I

O(T 2 )

1S the range

=

O(T)\'W.

If

A

1S

If

T

1f

P(A,T) S F(x).

It

1S then the resldue of the resolvent operator A.

Y

Y

and

U

wlll denote tile

is a closed subspace of the Banach space

denotes the quotlent space of cosets

the norm

peW,T)

where

w111 denote the dimension of the space

closed un1t ball of

T.

u)

the corresponding spectral projection is wrltten

function

(z_T)-l

A subset

We use the follow1ng notatlon for pro-

Tl

P(A,T)

x/y

surrounillng

w1th

is easy to check that

dim(Y)

peT)

is called a pole of finite rank of

Z -+

O(T).

lS a spectral set for

Tlx l , T2 = Tlx 2 • Tl ED T2 and O(T l ) = w,

where

an isolated pOlnt of P(A,T).

a (T)\.W.

p2 = P S B (X)

the kernel of

surroundlng

f (Z) ( Z-T) -1 dZ

W and zero on

P : T = Tl ED T 2 , T

fr

OCT)

is a sUltable contour 1n

ject10ns. and

27Tl

piT)

W is the spectral projection

ted with each spectral set

is one on

OCT). If f S Hol(O(T»

f(z) (Z-T) -1 dg

whlch is open and closed In

where

the adjolnt

1S the operator of rank < 1,

x

denote the family of complex valued

lS a suitable contour In

P(W,T)

T*

1S defined by the Cauchy 1ntegral

1 ---

f(T)

denotes the res-

TIY and

functions whlch are analytlc in some neighbourhood of the operator

wlll be the kernel of

yll = 1nfllx + yll.

x + Y; It is a Banach space under denotes the set of lntegers (pOS1-

:g(z,+)

The ~I6sure of a subset

X,

S

of a topolog1cal space w111 be

The term~nat~on of a proof w~ll be s~gn~f~ed by • 0.2

Fredholm operators

Let

X be a Banach space over
(x);

(ii)

T + F(x)

EO:

Inv(B(X)/F(x»;

(iii)

T + K(x)

EO:

Inv(B(x)/K(x»;

A

(iv)

T

proof.

EO:

Inv(B (X» •

(i)

=>

(il).

T

EO:

4>(X}

~>

dim{ker(T»

co-dimension, so there exist closed subspaces

x

T

ker(T)

T(X)

ED Z


(iii) ii..s

such that

-

Q

F (X)

modulo

S + K(X)

If

TS

T

I

•

obv~ous.

=> (iv) .

(~ii)

such that

I - P

TS

so

IS F (X)

P, Q

I

- Kl ,

(T + K(X»

-1

there ex~st

,

Clearly

ST= I - K2 ·

=

ST

{T :

I

=

Kl , K2 IS K(X) TS. (This

T IS

B(x)}

and choose a sequence

{x} n

argument is not reversible as we do not know that

is a

A

B(X».

closed subalgebra of (~v)

=> (il.

unit ball of ker{T) •

Then

0=> T{{x } + m(X» n

{TX } n

in the

0,

=> {x } + m{X) n

0,

=> {x } IS m{X) , n

so the unit ball of

ker{T) is compact, hence

Next we show that

T(X)

exists a closed subspace T{X) = T(Z) T

and

T

is

is bounded below on

[ [x [[ = 1 n

for each n

is closed in Z

of

X

inject~ve

Z.

on

Z

Tx

->-

n

diction.

and

Tx

~

->- Ty =


{x } + m(X) n £

00.

so it is sufficient to prove that

{Tx } IS m(X) => T({x } + m(X» n n

I IYII

Since X

Suppose notithen there exists

and

=> {x } n

X.

such that


~

y

IS

x.

Then

which is a contra-

5

Since

is closed, the quotient space

T(X)

remains to prove

Ilyn Ilyn

+ T{X)

+Tx

n

II II

{w } E JLoo(X)

dim(X/T(X»

hence there

T(W

II y

thus

since

~

~

for each n , then there for each n •

{y

n

- x ) - y } n

- x

+

~

y

+ T (x) j I

z

{y

+ T(X)}

X/T (X) If

DEFINITION.

(i) If

the defect of

00

Calk~n

•

weT)

is deflned to

algebra.

= aCT).

A

~s

of

po~nts

peT) B

necessar~ly

Of

and

Bex).


0

{xl' ••. x }

is a compact subset of

8

x

E.

q(B) '" 0 B

X

T consists of poles

T.

DEFINITION.

Compactne8S

(The

B(x).

E.

0.3.1

If

is closed in

(X)

4q(T(U».

the closed unit ball

X,

-

proof.

The left hand inequality is obvious.

s uppose that

< £

q(T(U»

and let

B

To prove the right hand one

be a bounded set such that

q(B) < 1.

Then n

T(U) CU t.(Yi'£)'

(Y 1 " "

1

Y n £ X)

n

Cv t.{Tx.,2£), ~

1

n

2T(U)

and

C.V t.{2TX. ,4£). ~

1

m

NOW

CV

B

t.(z ,1),

1

J

m

Cv

t.(b ,2),

1

J

m

CU

+ 2U) •

(b j

1

m T(B)CV 1

Thus

(Tb.

+ 2T(U»,

(Tb j

+ U t. (2Tx. ,4£» ,

J

m

n

cU

~

1

1

mn (Tb. + t. ( 2Tx ,4£», ~ J

CuU 1 1

m n

CUU t.(Tb. + 2Tx.~ ,4£), J

1 1

so

0.3.3

LEMMA.

Proof.

Let

II{x} - {y

n

{y}

n

If

}I I

q({x }) < £ + Q n

{x} E ~ (X), q({x }) = n m n

Ilxn + m(X)

for each

n

•

< 4£

q(T(B»


O.

II

< 0,

Since

£

+ m(X)

{y} £ m(X)

n

t

is arbitrary,

I I·

{Yn} £ m(X}

then there exists

This is a fin~te

and, as

I ]{xn }

such that

there exists a finite £-net for (£+0) -net

q({x }) < n

-

for

o.

{x}. n

Thus

It follows that

q({x }) < I I{x } + m(X) I I· n n 9

q({x }} < 0; then there exists a finite o-net for n such that Y£I so for each n there eX.l.sts j (l.2 J .2 £)

conversely, let say

Yl ""

Ilx n

-y.!! J



n

SUp{q({TX }) n

[[xm - xn [[

> 0

thus

10

211TII>

n

q({x }) < l} > ~ q(T(B»

sup q(T(B». q (B)'::'l

n

Then we may

induct.l.vely so that

q({x}) > If E > 0 apply this to the set T(B} n to obtain a sequence {TX} such that q({TX}) > ~ q(T(B»

so

o.

-

Ef

for

m of nand

where

q (B) .2 1,

- E.

~ow

We have now, somewhat laborously, set up the machinery required for our Characterisations of Riesz operators.

0)3.5

THEOREM.

For

(Ruston characterisation)

the follouJing

T E B{x)

statements are equivaLent (i)

{ii}

R (x)

T E

(iv)

K(x}) = 0,

reT + A

(iii)

;

reT)

= 0,

n l/n lim q(T (U)} = 0; n

(v)

E > 0

for each

+

e~sts

there

has a finite

n E li'

En-net. (i) (l.i).

~.

Let

T

the correspondlng spectral projection IAI >

{A E aCT) P E

F (x)

reT +

K(x)}




O.

aCT +

therefore this punctured nelghbourhood lies in point of

If

6,

K(x»

satisfy

E Inv(B(x}/K{x}),

contain points of

TP E F(x);

lnf reT + K} < KEK(x)

0 is arbitrary

and since

P(A,T) E F(x).

A E aCT)

is finite and the corresponding spectral projection

reT - TP) < 0

Now

•

o}

0 ~

be a Riesz operator, then if

a contour in

peT)

aCT) must be isolated. surrounding

6:

A is an isolated

a (T) Let

are all

A be one

A but no other point of

aCT}.

'!ben

P (A,T + K(x»

s.ince

z - T -

K{x)

P(A,T) + K(x)

-121Tl

fr

is invertible inside and on

A is a pole of fin.i te rank of

(z - T - K(lC» -1 dz

f.

So

0,

P(A,T} E K(x)

and

T. 1.1

(iii).

(ii)

This follows at once from 0.2.4.

(iii) (i v) •

.5.

IITII

Combining 0.3.2 and 0.3.4 we get

.5.

4q(T(U»

(§ )

aiITII,

I IATnl Il/n.

and the equivalence follows by considering (iv) (v).

This is now clear since

Tn(U)

has a finite En-net

q (Tn (U) ) < En • An easy consequence of the Ruston characterisation and of properties of the spectral radius in the Calkin algebra is the following result. [S,T] = ST- TS is the

commutator of Sand T.

0.3.6

THEOREM.

(ii)

S

'(iii)

T

(i)

B(x), T E

E

and

S, T E R(x) R(x)

and [S,T] E T E B(X), liT

E R(x) (n > 1), n (n > 1) => T E R (X) •

n

E K(x)

[S,T]

=> S + T E R(x};

K(x) => ST, TS E R (X) ;

-

and

Til + 0

[T ,T] n

E

K(X)

Another useful consequence involves functions of a Rlesz operator.

B(x)

T E

0.3.7 (ii)

and

f E Hol(a(T».

THEOREM.

If

T E

Let

(i)

T

E R(x)

and

B(x)

fez)

and

flO) = 0 => fiT)

E R(x);

a (T}'\{O}

does not vanish on

then

f(T) E R(x} => T E R(x). In fact

flO)

0=> f(T) = Tg(T),

Ilhere

g

E

Hol(a(T»

and

[T,g(T)]

o

hence 0.3.7{i) follows from 0.3.6(li). 0.4

Range inclusion

The machinery developed In §3 allows us to deduce properties of an operator S

from an operator

use S

S-l{U)

that

S(X)~

to denote the lnverse image of

THEOREM. S (U)

If S,T

C n (T (U)

x=s

12

provided that

T(X). U

In this section we shall

under

S

whether, or not,

is invertible.

0.4.1

S

T

-1

E

B(x)

and

S(x)C T(X)

there exists n >

0

such

) • 00

(T(X»

is continuous hence

00

00

S-l(T(U n U)le S-l(V n T(U» n=l n=l S-l(T(U»

is closed In

X

I

I

'-../ n n=l

S

-1--

(T (U) )=x.

therefore by the Baire

category theorem ({30) p.20),

+

n £ 'I. X

But

such that

SX =

Sex + y)

l~m

n

-

COROLLARY.

S,T E

and

y £ U,

B(x),

0 > O.

nn T n Sz II


ST = TS,

and

w E U

~s

true for

Then there exists

~

(T(U}),

so there exist

Hence there exists

{x}C U n such that

such that

Sy = n lim Ty

n

n

S £ K(x}.

then

S{u)C: n(T(u»

n

1.

Suppose i t is true for

z £ U such that

(t)

o.

such that

1 - nn+ 1 Tn+ wII < ~

(*)

o. y c U,

From (t) and (*) we see that if

II S n+ 1y

T(U} •

C

-1--

•

By hypothesis the result

[ Innn T Sz

S

{y} C U

there exists

S{X)C T(X)

If

But there exists

so

has a non-empty interior for some

•

THEOREM.

n, and let

so

2£

C nn (Tn (U) )

Proof.

S (f1(x,£»

such that

-1

0.4.2

(U)

(T(U})

is homeomorphic to

.

n

?

-I--

lim Tx , and if Ilyll < £, there exists {z}C.U n n n Tz 'l'hus Sy lim T(z - x ) and {z -x}C2U. n n n n n n

where

0.4.3

(T (U) )

Ily II < 1,

Finally, i f =

-1 - - -

£ > 0

and

£ X

nS

nS

there exists

w £ U

such that

- nn+ 1 T n+ 1WII < 0 ,

and the proof follows by induction.

1.3

Combining 0.4.1 and 0.4.3 we get 0.4.4

COROLLARY.

0.4.5

THEOREM.

If If

S(X)C: T(X)

S,T

B(x),

£

and

then

ST = TS

and

seX) C T(X)

T E Q(X) => S E Q(X).

[S,T] E K(x}

then

T E R(x) => S E R(x) •

Proof.

Let

V

denote the closed unit ball of

{x} + m(X) E V,

and

there exists

n

Ilxn + y n II

1 +


0

so there

ex~sts

X = £oo(X)/m{X).

m(X}

such that

(n ~ 1) ,

£

xn + Yn then {

} CU. 1 + E

Now there exists

{z } n

c. U

n > 0

such that

S(U}

C TjT(U)

(0.4.1)

such that


1),

hence Ils(xn + y n ) - T)Tzn 'I < E(l + E) + EnII T ' I

since

{z} C U. n

Now

A

{y} E m(X}, hence n

{sy} n

£

m{X}, therefore

A

Ils({x } + m(X}} - nT({z } + m(X}) II < £(1 + E) + EnIITII, n

{x } + m(X}, {z } + m(X) E V

and since

S(V}

n

n

C

nT(V)

n

and

[S,T] E K(x} => [S,T] = 0,

(n

which gives

l4

we get

> I),

so, by 0.4.3,

A

thus

reS)
1)

n

ue need to show that

I

has a norm convergent subsequence, X

E

and put

E = T(U) •

hence by continuity

is contained in

ex(E);

mapping the compact Hausdorff space

II s II

T)

Let

U

be the

1

ST(U) = TS (U) c: T(U) S

peT)

such that

Thus

V(I) (A -

T

II s II = 1

Z (T) ,

{S T}

Z(T).

T)V(I)

A E:

V E: B(Z(T»

As we remark (p. 20) the converse statement is false.

K (X) =>

T E:

closed unit ball of

of

Z(T) we

and then

ex~sts

there

The next result states that if

0.5.2

Z(T).

= O(T).

since everything commutes, thus

~s

on

S -+- SeA - T)-1 E: B(Z(T»,

(A - T)V = V(A - T)

T

T

P (T) •

E:

I

which is a closed subalgebra

is the operator of multiplication by

peT) => (A - T)

T)

T

compact (Riesz) action on its commutant.

has a

LEMMA.

Proof.

I

o

reS)

Action on the cornmutant

If

S E

by 0.3.5, hence

tt

again by 0.3.5 0.5

A

r(T).

< 1 =>

II SX

-

SX'

II


a

hence/~f

a subset of a totally bounded set, hence

llTSTX - TS J Txll -< IITII

Thus

Bl }

£

the closed unit ball of

S £ Bl } ~s bounded ~ totally bounded. Thus the set

Therefore there ex~st

totally bounded.

If

S

x £ U

i, then the set

p~x

~

X, so the set

Bl

is totally bounded,

such that for each

xl""~£U

11 T(x - x ) II < £.

with in

T(U)

X and

X

~

~

thus {T * a 6a Tx } ~

is a bounded sequence ~n

and thus has a subsequence such that

B(x)

* Ta6aTx ~

II

- Tx.~

m

Hence

{Tx}

has a convergent subsequence and

(iii)

If

~s

~

T

a

R~esz

operator, then for

< £

11

(T 1\ T) n

-

K

n

1\ K

n

II

n

(n

> N).

£

W~thout

T >

a

~s

II

-+

a.

m

compact.

there

loss of

< 2! I~ -

-

a compact

e~sts

general~ty

K

n

take

11 < 2£n.

K

n

K AK

1S compact by part (ii), so, using the Ruston character1sation (0.3.5),

n n T f\ T is R1esz.

Conversely, let

T 1\ T

B(x);

be Riesz on

then its restriction

I

T 1\ T Z (T)

ST 2

is a Riesz operator «25) 3.5.1), 1.e. S 7 is R1esz on Z(T) and by 2 is Riesz on X, hence, by the Ruston characterisation, so

Theorem 0.5.3, T

T.

is (iv)

lim

r(TI\ T)

'!'hus

II (T 1\ T)nlil/n

r(T)

2

.

n

Clearly 0.7

T 1\ T

is quas1n11potent

T

is

•

Notes

Compact or completely cont1nuous operators arose first in the guise of quadratic forms in H11bert's work (46) 1n 1906.

Fredholm, three years

ear11er, had considered resolvent expansions of certain 1ntegral operators (33) •

The spectral theory of compact operators was worked out by F. Riesz

(74) in 1918 for the 1mportant spec1al case in which the underlying Banach space is a space of cont1nuous funct1ons.

Fredholm theory 1n its present

form dates from the work of Atkinson (5) and Gohberg (36), Yood (102),1954.

(37) 1n 1951 and

The characterisat10n of Fredholm operators as those which

are invert1ble modulo the compact or f1nite rank operators (0.2.2) Atkinson.

1S due to

Ruston started work on R1esz operators (76) in 1954 while Calkin's

work on ideals of operators in H11bert space (20) had appeared 1n 1941 and Kleinecke (53) introduced the 1mportant ideal of 1nessential operators, which is the biggest closed ideal contained in the Riesz operators, in 1963.

The

propert1es of Riesz operators were exam1ned by Heuser (42) in 1963; in more detail by Caradus (21) and West (94),

(95) in 1966.

Th1s class is of interest

On account of the spectral propertles which Riesz operators share Wlth compact Operators.

From the algebralc standpoint, 1f commutat1v1ty (or commutativity

modulo the compact operators) is added to a result valid for

R(x).

extends to

K{x), i t often

Examples of this phenomenon occur in 0.3.6 and 0.4.5.

The technique, set up ln §O.2, and used to study Riesz operators in §O.3 is based on work of Buoni, Harte and W1ckstead (17), replacing

T

by

Tn

(41).

Observe that by

1n inequa11ty (§) on p.12, then taking n-th roots and

using 0.2.4 we get

19

"-

reT)

reT

lim q(Tn(U»l/n •

+ K (X))

n

Further information on measures of non-compactness and the essential spectrum is given by Zemanek(107),(108)'The representation algebra on

X

B(x),

closed in

representat~on

T

of the Calkin

is due to Lebow and Schechter (57). is irreducible, or if ltS image is

although Lebow and Schechter have shown that the latter

statement is true if «57) 3.7).

B(X)

as a subalgebra of

It is not known if this

T + K(x) +

X

satlsfies the Grothlendleck approxlmation property

The equivalence of (l) and (iv) in 0.2.2 is due to Buoni,

Harte and Wickstead «17) Theorem 3).

If

T

is an index-zero Fredholm

operator on a Banach space Murphy and West (62) followed by Laffey and West (55) have strengthened 0.2.8 by showing that ible, F

is of finlte rank and

rV,F] 2

is elther in the resolvent set of

T

rank «25) 1.4.5), and In thls case

I

T

=

O.

T

=

V + F

Note that

where

V

[V,F]

or lS an lsolated pole of

is invert-

0 zero T

of finite

lS called a Riesz-Schauder operator.

Ruston proved the eqUlvalence of (i) and (li) in 0.3.5.

Conditlon (iv) is

due to G.J. :1urphy and the geometrlC characterisatlon (v) of Riesz operators to M.R.F. Smyth. The range inclusion theorem for compact operators (0.4.2) lS attributed to R.S. Phllllps; Lhe result for quasinllpotent operators (0.4.4) lS due to M.J. GanlY,while the result for Rlesz operators (0.4.5) lS due to M.R.F. Smyth The fact that a compact operator acts compactly on ltS commutant (0.5.2) was observed by Bonsall (11),

(12) and used to develop an algebralc approach

to the spectral theory of compact operators.

Theorem 0.5.3 showlng that a

Riesz operator has a Rlesz action on its cornmutant and its converse is due to Smyth (81).

The converse of 0.5.2 was shown to be false by I.S. Murphy (63),

with an lngenlous counter-example of a welghted shift operator on

~l'

Murphy overcame the main difflculty, whlch lS to determlne the commutant, by choosing the welghts so that It is

L~e

strongly closed algebra generated by

the operator. The wedge operator

T 1\ Twas lntroduced by Vala (91) and its propertles

were studied by Alexander (4), \1ho also considered the generallsed wedge operator defined by

V + SVT

(V £

B (X»

•

~

follow~ng

0.7.1 (ii)

or

(~)

(ii) (iii)

result

THEOREM. (ii~)



general~ses

If S 1\ T

S,T £ B(x)

0.6.1.

then

s

and

T

belong to classes

(1)

does

the non-zero finite rank operators; the non-zero compact operators; the Riesz operators Wzich are not quasi--niZpotent.

is quasiniZpotent either

s

or

T

is quasiniZpotent.

Further

s /\ T

F Fredholm theory

In 0.2.2 we saw that a bounded linear operator on an inf~n~te d~mens~onal

Banach space

X

invertible modulo is a primitive



1s Fredholm

B(x)

The fact that

K(X).

~deal)

~nvert~ble

it is

and that

F(X)

modulo a

~s

F(X)

is

algebra «O)

pri~t~ve

mot~vates

B(X)

is the socle of

~t

the

work of this chapter. Fredholm theory semisimple and

~n

algebras.

Banach algebras by Smyth (83).

straightforward/w~th

Th~s

~n

by Barnes (7), (8)

theory was extended to general

Here we adopt a

to Smyth and use the case of a is

p~oneered

Banach algebras was

semipr~me

s~pler

approach due

aga~n

Banach algebra/where Fredholm theory

pr~mit~ve

natural analogues of the rank

null~ty,

defect and

index of a Fredholm operator for Fredholm elements of the algebra (those invertible modulo the socle), to §l contains in F.l.lO we

~nformat~on

exhib~t

bu~ld

up to the general case.

min~mal ~deals

on

the Barnes

~dempotents

are fundamental to the rest of the chapter. Banach algebra

~s

Fredholm element connects these

developed ~n

prim~t~ve

quant~t~es

Fredholm operator. Fredholm theory

Th~s

If

obta~n

defect and

~ndex

~t ~s

of the

Fredholm theory

null~ty,

algebra are

P

Banach algebras ~s

a

Alp

a Fredholm theory

(69) and Smyth (83)

the

idempotents, also

~ndex

defect and and a

def~ned,

~n

a

These pr~m~tive

of a

representat~on

~ndex

of a

certa~n

allows us to deduce the important results of

then the quotient algebra

§3 to

§2;

m~n~mal

w.r.th the nullity/defect and

~n pr~m~~ve

in operator theory. A

a

~n

and

for a Fredholm element.

~n

d~rectly

pr~m~t~ve ~deal

is

of a general Banach algebra

pr~m~t~ve.

the general case.

necessary to replace the

pr~m~t~ve

case by

from their counterparts

func~ons

Th~s

fact

~s

exploited in

.\s observed by Pearlman numer~cal

(of

valued

f~n~te

null~ty,

support)

defined (for each Fredholm element) on the structure space of the algebra. §4 besides containing observat~ons on extensions of the theory also conta~ns results on of

')')

general~sed ~nd~ces

algebra~c

elements) •

and on the algebraic kernel (the largest idea'l

F.l

~deals

Minimal

sect~on

In this

w~ll

A

unital.

necessar~ly

(TOpolog~cal

In such an algebra the m~nimal

of the

sem~s~mple

be a

cons~derat~ons w~ll

~deals

right

sooLe of A, (wh~ch

stated for right

division algebra

theorem (14) 14.2,

A

left ideals

e E A

minimaL

~s

~f

eAe

is a

is a Banach algebra then, by the Gelfand-Mazur

eAe =

~e).

~s

denotes the set of

M~n(A)

the set of rank-one project~ons ~n

compact Hausdorff space and n,

~nimal

m~nimal ~dem-

A.

M~n(B(X»

on

to be the sum

Definitions and theorems are usually

A non-zero ~dempotent

(~f

~s def~ned

soc (A)

corresponding statements may be made for left ideals.

~deals,

DEFINITION.

patents of

which is not

not enter into our

equals the sum of the

(14) 30.10) or (0) if there are none.

F.l.l

~

algebra over

Our results apply equally well to a semiprime algebra over

discuss~ons.

~).

and Barnes idernpotents

~dempotents ~n

the

and closed sets of functions of

n,

C(n)

points of

If

n

is a

denotes the algebra of continuous functions

C(n)

character~st~c funct~ons

are the

m~nimal ~dempotents

,mile the

~solated

B(X).

n.

are the characteristic

M~n(C(n»= ~

Thus

of open

n

possesses no

isolated points. F.l.2

DEFINITION. r~ght ~deal

for any The

l~nk

(i)

LEMMA. xA =

RIC R

I

between minimal

algebras is set out F.1.3

A r~ght ~deal

~n

either

R

of

Rl = (0)

~dempotents

~s

A

and

§BA.3, for reference

I

minimaL

or

R ~ (0)

and ~f

Rl = R.

m~n~al

~t

~f

~deals

~n sem~simple

is restated here.

Let A be semisimple then

0 x

0;

is a minimaL right ideal of A either

0 "W{yln.

finite

(1)

jn

ker(T).

algebra, then

under each element of

and

for

pr~m~tive un~tal

fa~thful

xAy

on the

~(x)~"

invar~ant

Z £ A

A

~dempotent

be a

possesses a

For suppose that

that

so a right

with range

p)B(x),

(1 -

Recall that an algebra

Min (A) " ¢.

A

P

X

Observe

Banach algebras

Pr~~t~ve

In th~s section

if

~n

B(X)

may

C 22

C12 ' B22T22 the equality.

ker(T}

~dempotent

shows that any

and hence is a left Barnes

that

ver~f~es

~s any ~dempotent in

B(x)p,

lan(T}

that

=

sat~sfies

T(X)

F.2

which

closed complement of

T

s~m~lar analys~s

A

B12T22

a minimal left

~deal,

eAf -F Af

(0) •

Av,

Choose a non-zero so

f

~

uv

for some

v

eAf.

E

u

E

A. ?Q

Also

v = ev,

hence

By (i) ,

(ii)

f = uev.

eAf= eAuevC eAev = t!:ev

,

Rf

RuevC.Rev.

dim (Rf) < dim(Re) •

Similarly,

By

(iii)

(~)

Eurther if

Rf

whose

dim (Rf)


Sue

be a subset of

is linearly

~ndependent

ue f 0,

s~nce

Re = Rtue x

Hence

tt

to denote the left regular representation of the

=0

00

t S A.

Oe shall wri te

A.

A

xAe

for some

=> dim (AejRe)
0

A

qAe

and the same argument

A F o.

for

A similar proof deals

•

Note that th1.S method of develop1.ng Fredholm theory 1.n a pr1.mitive Banach algebra

A

requires that

M1.n{A)

F ¢.

and then Fredholm theory 1.S trivial for

Lut

Min (A)

~(A)

=

=¢

Inv(A)

soc (A) = CO)

and obviously the

null1.ty, defect and 1.ndex of any Fredholm element (however these concepts are defined) must be zero.

The Calkin algebra of a separable Hilbert space is

an example of a primitive Banach algebra with zero socle. F.3

General Banach algebras

\e now extend our theory

to a unital Banach algebra

quotient algebras as building blocks. is semis1.mple and we write

s·

write

=

{x' : XES}.

x

A

using its primitive

The quot1.ent algebra

.for the coset x + rad (A)

I

In general the socle of

A

A'

=

and if

S

A/rad(A)

c. A

does not eX1.st so in

its place we use the presocle. F.3.1

The presocle of

DEFINITION.

{x E A

psoc(A)

Clearly

psoc(A)

=

P5oC(A}

I(A}

Clearly F.3.2

I(A}

A,

\~1.le

if

A

1.S semisimple,

The ideal of inessential elements of

A

1.5 defined to be

k{h(psoc{A)}).

1.S a closed ideal of

DEFINITION.

there exists

is def1.ned by

x' E soc (A ') }.

1.S an ideal of

soc (A) •

A

yEA

An element such that

x

A. is called a Fredholm element of

xy - 1, yx - 1 E psoc(A).

A

if

The set of 35

Fredholm elements of If

psoc(A)

modulo

modulo

while if

I

=

psoc(A)

By BA.2.4

~(A).

is written

is a proper ideal of

psoc (A)

extreme

A

psoc (A) = A

=

rad{A) 0

! Ix - y! I


Y is I \x - y II < £ o ~> Y

!!x so

Q

Now

is a closed ldeal of

that is

has a flnite complement

J.

Hence there exists

invertlble modulo

J.

lS lnvertible modulo

JC:P

Now P

E:

o

for

for

>

0

such that

P £ TI (A)'-.Q,

proving

(ii) •

If

P

£

TI(A)

then

u'x' - 1', x'u ' - l' ~

£

soc(A'},

u'x' - l' + P', x'u' - l' + pi £ soc(A'/P'),

(BA.3.4)

37

~

ux - 1 + P, xu - 1 + P E soc (A/P) ,

=>

x + P E ~(A/P).

n yl I
Y + P k E ~(A/Pk)

E

-

= min{Eo,E l , •• •

g~ves (i)

Ek } ~n

Recall that if,

(BA.2.6)

Ek > 0

{l < k < n) .

such that :\ choice of

tt

a prlmitlve algebra

A,

Min (A) =

¢

then the nuillty

and defect of every Fredholm element are defined to be zero. F.3.5

For each

DEFINITION.

index functions

IT (A) -+ ::.t by

v (x)

(p)

nul (x + P) ,

(x) (p)

def (x + P) ,

cS

~nd(x

lex) (P)

x E ~(A)

we def~ne the nullity~

+ P).

By F.3.4 each of these functlons has finlte support In

on

h (psoc (A» If

Obviously

lex)

t-

= vex)

TI{A)

and is zero

- o(x).

is a primitive Banach algebra then, for each minimal idempotent,

is the unique minimal ideal

(0) (0)

A

•

defect and

P E II (A),

1'1in (A) c. P

hence

wh~ch

fails to contain it (F.3.5).

soc (A)C P.

case the support of the nullity, defect and

So

It follows that, In this

~ndex

functions conslsts of

the zero ideal, so

nul (x)

v (x)

The concepts of

(0),

def (x)

/) (x) (0),

DEFINITION.

t (x) (0) •

nullity, defect and index can be extended to a general

Banach algebra as follows.

F.3~6

ind (x)

If

x E

~(A)

we deflne

~f

r.

nul (x)

V (x) (p) ,

P£TI(A)

Since

def(x)

L O(x)(P), pdI(A}

index)

L P£l1{A}

~ 0,

Vex) (p)

=

nul (x) def(x)

l(X)(P).

o(x) (p) > 0

it follows that

0 vex)

-

0,

o(x)

_

o.

0

Now F.2.8 extends to the general case. F.3.7

THEOREM.

Xnv(A)

=

~.

{x

~(A)

£

: nul (x)

Apply BA.2.2(v)

{x

def(x) }

= 0

~(A)

£

vex) ==

0

-

o{x)}.

•

'!he properties of the index of a Fredholm element in a primitive Banach algebra given F.3.8 (1)

'!he map l(xy)

(iii)

1 (x)

(iv)

leX

F.3.9 E

> 0 (1)

(ii) (Hi)

F.2.9 extend easily to the

function.

~ndex

THEOREM (Index).

topology on (ii)

~n

x

~(A)

-+ 1 (x)

-+ :ll1 (A)

is eontinuoUB in the pointuJise

aIT(A). =

=

lex) + ley), 1 (y)

if

+ u) = l(x),

x

(x

(x, y

and

y

£

~(A),

~(A»;

£

lie in the same corrponent of u

£

I{A}).

Fix

THEOREM (Punctured neighbourhood) •

such that for each P £ 11 (A), vex + A) (p) is a constant ~ vex} (p), o(x + A) (p) is a eonstant ~ o(x) (p), leX

+ A) (P)

is a cor.stant,

(! AI

~(A).

< £) •

x E

~(A)

(0

< IAI
x + J

x E R E(X)C T(X) , ker (S) C ker (E) ;

ST

F ='> F(X)CS{X), ker{T)L ker(F);

TF

T -=> ker (F) C ker (T)

ET

T ='> T(X)C E(X);

F = ST, r~ght

~nformat~on.

;

SE = S => ker (E)C ker (S) ; FS

S ='> S(X)C F(X) .

Collat~ng

T{X)

so both

S

these results we see that

E

(X),

and

T

Conversely, let

ker (T)

are T

ker(F) , and

general~sed

SeX)

F (X),

ker (S)

KerCE) ,

Fredholm operators.

be a generalised Fredholm operator/then the pictorial

part of the proof of Atkinson's theorem (0.2.2) shows how to construct a generalised ~nverse 48

S

and ~t follows at once that

TST = T, STS = S

tt

Generalised Fredholm theory for operators has been studied by Caradus (22), (23),

(24), Yang (97), Treese and Kelly (90), among others. conta~ns

generalised Fredholm operators on a Banach space

SeX)

in

The class of

all the projections

so one cannot expect such a tightly organised theory as in the

class~cal

Fredholm case, for example/this class is not, in general, open,or

closed under compact perturbations, but we do have results of the following type «22) Corollary 1). pA.ll

Let T be a generalised Fredholm operator on

THEOREM.

and lel;

X

satisfy I Iv! I < lis II-I, where s is a generalised inverse of T and ei ther ker (V) ::> ker (T) or V (X) C T (X) , then T - V is a V £ B(x)

generalised Fredholm operator. If

in

T

general~sed

is

~lbert

space,there

project~ons

and

E

Fredholm

ex~sts

~ts

generalised inverse is not unique butr

a unique generalised inverse

Fare hermitean.

S

such that the

Such an inverse is called a Moore-

Penrose inverse in the matrix case (of course every matrix has a MoorePenrose inverse) applications. situat~on

(A

concept has recently proved to have many important

b~bl~ography w~th algebraic~sed

has been

inverse semi group

tll~S

and

I

~tems

as follows:

~f each element

xyx = x, yxy = y.

1700

x £ S

The structure of these

is a

conta~ned

sem~group

S

in (64». ~s

has a unique ~nverse sem~groups

This

called an y

such that

is somewhat tractable

and they have been objects of considerable study. The Fredholm theory outstanding

wh~ch

characterist~c

we have developed

an

~ntimate

~n

this monograph has as its

connection with spectral theory.

It

has l~ttle oonnect~on with the Fredholm theory of Breuer (18),

(19) extended

by Olsen (68) , based on the concept of a dimension function

von-Neumann

algebras

( (25)

Chapter 6).

Harte (106) has

invest~gated

~n

Fredholm theory

relative to a general Banach algebra homomorphism. Coburn and Lebow semigroup of a group

wh~ch

( (25)

topolog~cal

Chapter 6)

def~ne

a

generalised index on an open

algebra to be any homomorphism to another

se~­

is constant on connected components of the first semigroup.

Of course, our theory f~ts ~nto th~s very general framework and by spec~al­ iSing a l~ttle we obtain results (due to G.J. Murphy) on the ex~stence and uniqueness of an Let let

~

A

~ndex

denote a

defined in a Banach algebra.

un~tal

Banach algebra with proper closed ideal

denote the set of elements of

A

invertible modulo K.

K

Then

and ~

is

49

an open mult.l.plicative semigroup,

discrete group i (x)

=

with unit element

G

e x

E

K),

x

1.(x)

E

:

x + Y

E:

•

(i)

Apply the

£

R

bas~c

Let x and f(O)

THEOREM. x

y.

He have the

R·, R·,

x E: R, y E: A and [x,yJ £ K => xy, yx E: (ih) x,y £ R and [x,y] £ K-> x + y £ R; {iv} x £ R (n > 1), x .... x in A and [x ,x] n n n

R.1.3

and

analogues of 0.3.6 and 0.3.7.

{li}

~.

x

£ K

(n > 1) => x

£

R.

properties of the spectral radius to elements in £

A

a

f

E:

HoI (O' (x) ) ,

....-:> f(x)

£

R;

and

A/K

then

53

(ii)

(iii)

R and

I':

(if

A is unita~)

cr(x)'-{o} -> f(x) Proof.

~s

(i)

xg(x)

cr(x + K)C:cr(x}

~ntegral

(iii)

x

f(x + K) = f(x) + K.

f(x}

I':

f

I':

f(x)

one

o => f(x)

f(O)

~mmed~ately ver~f~es

Hol(cr(x + K»

since

{oJ.

f(cr(x + K»

{oJ,

cr(x + K) =

so, by hypothesis,

i cr (x + K) •

cr(x + K),

cr (x + K) C cr (x),

How

hence

x

f

cr(f(x + K»

cr(f(x) + K»

character~sations

two

,

of the radical of a

characterisat~on involv~ng

lnv(A)

~s

un~tal

well known (BA.2.8)

involving the set of quas~n~lpotent elements We recall that if 1/I(k(h(K}» R.1. 4

THEOREM.

rad(A) = {x ~.1.5

1~

is the

rad(A/K)

=

I':

Let

Banach

COROLLARY.

54

wn~le

that

~s due to Zemanek (104).

A -+ A/K

then

A be a unital Banach algebra~ then

Let

A

£

A : x + Q(A)C:Q(A)}.

be unital then I':

A

x + RCRJ.

Riesz elements: spectral theory

Recall that if l(A}

I

K.

(BA.2.3).

x + Inv(A)C:lnv(A)} = {x

A

Q(A)

canon~cal quot~ent homomorph~sm

k(h(K» = {x e: A : x + ~K(A)C$K(A)} = {x R.2

does

so

algebra which lead to chardcterisaticns of the kernel of the hull of The

R.

£

therefore

~K(A) •

g~ve

Next we

then

cr(f(x + K»

K)

o 1. f(cr(x + K»

thus

R;

R,

£

K

not vanish on

Observing that

representation of

and

4> (A) => 0

I':

I':

does not vanish on

f

(~~),

f £ Hol(cr(x»,

cr(x + K)Ccr(x),

Now

and

and

f(x}

cr (f(x) +

cr(x)'-{o} => x

g £ Hol(cr(x».

x £ A

Since

x £ ~K(A)

a consequence of R.l.2

where

that if

f

~K(A) .

£

Using the Cauchy

(ii)

does not vanish on

f(x)

A

is a Banach algebra then

of inessential elements of

A

A' = A/rad(A)

is defined by

and the

~deal

(\ {p

I (A)

p r::J soc (A r) } •

€ Il(A)

We, henceforth, lnsist that

K

lS a closed inessentlal ideal of

that

A

and

lS closed ideal of

K

KC

I (A) •

carried out relative to this fixed ideal from

~K

K,

A, that is,

Our Riesz theory will be so we shall drop the subscript

~.

and

We are gOlng to deduce the spectral properties of Rlesz elements from the

A

Fredholm theory of Chapter F whereln i t is assumed that

lS unltal.

A

Thus, from R.2.l to R.2.6, when we use results from Chapter F,

wlll always

be unital and, at the end of the section, we shall show how these results may be extended to non-unltal algebras. R.2.l

DEFINITION.

A

plex number

Let

A

be a unltal Banach algebra.

lS called a Fpedholm point of

Fredholm or essential spectpum

A

w(x)

The Weyl spectpum of

W(x)

x

( \ CJ (x

of

x

In

A

x

If

If

A-

x € A,

x €~.

a com-

The

lS deflned to be the set

lS not a Fredholm pOlnt of

x}.

lS deflned to be the set

+ y) •

y€K The complex number invertlble, or If of

CJ

(x) .

A lS called a Riesz point of

A

lS a Fredholm pOlnt of

x

x

If either

A- x

lS

whlch is an lsolated point

The Riesz spectpum or Bpowdep essential spectpum of

x

in

A

lS

deflned to be the set

A

6 (x)

We note that

w(x),

lS not a Riesz pOlnt of

W(x)

and

Sex)

x}.

are all compact subsets of

~

anu

the incluslon

55

w{x)C. W(x)C 13 (x)C a (x) ,

is valid for

W{x)

Let

THEOREM.

A

Tak~ng

u

The

Clearly,

~t

follows that

inclus~on

Sex)

whenever

~s

X

K

is proper,

w{x}CW(x). ~s

W(x) 2).

it follows,from (BA.4.5), that

{ak}~'

p (H)

Then

O(f}=O(f n

Relatlve to the decomposltion

n-

H =

s

n

l)V{aJ,

(H)

n

ED

(1-5

n

)tw

we have

where

h

n

x j (1 - SI: ) (H) •

n

By (BA.4.5),

a(x + y ) n

Now if

A S a(x + Yn ) for each n, hence, Slnce A S o(x + y} . It follows that 0 (x)"w Co (x + y) •

A S a(x),w,

is open in

A,

then

To prove the reverse inclusion, suppose that A S p{x)uw,

n':::' m, 74

so we can choose

m > 1

such that

A ¢ o(x)'w,

Inv(A)

then

A ¢ O(x),,{\}~.

Then, for

h

=

n

h

m

I (l - s

n

) (H),

"lhere the inverses

ex~st

the decomposition

H =

W

Then

:1 -

Also

h

Ilwnll.2 llwmll

Now, s~nce

n

by virtue of the

n

CH}

s

(1 -

$

n

)

cho~ce

(H) ,

(A - h ) -11 (1 - s

m

of

m.

n

) (H)

I

Then, relative to

'Trite

~ )-~

(A

n

s

0. - h )-1

and so

n

for

n > m.

Fix

n > m

! lY

so that

l

- Yn !.::.llwmll:l

(Y - Y ) Is (H) = 0, n n

:J

[:

~

(A - x -

-1

y)

n

hence

W

n

[(A:

(y - Yn )

[:

y ) n

(y

(A _Ohn ) -l z

f )-1 n

(A : h ) -lJ [: n

J.

:]~ u-: )1']· n

Now,by (BA.4.5),

r{

(A -

Therefore

x - Yn )

1 -

(A -

-1

r{ W (Y - Y )}

(Y - Yn ) }

n

n'




correspond1ng to the spectral set

I \1 ~ €2}. Then p E H(A), and p commutes wlth x*x, I I (x - xp) * (x - xp) II = II x*x - px*x II r(x*x - px*x)
- ex*xe)e

def~ne

E

y*x.

ey*xe



second.

so

= e*

by

<x, y>e

thus

e

(If

e

I

H) e

is a *-representation of

A

79

TI e

(i)

(span AeA)

ker TI e e (BA.3.5).

p

(iii)

Proof.

the unique ppimitive ideal of A which does not contain

e

def~nition

It follows at once from the on

A

H : e

e

TI (A)::::> K (H ); e e

(ii)

of

F(H );

He'

If

z + y (z E H ). e

TIe

a

~s

*-representat~on

denote the rank-one operator on

x III Y

Then

yex*ze

yx*z

TI (yx*)z e

let

x, Y E He'

that

(x III y)z,

y

Now every element of AeA is of the form yx* where TI (yx*) = x III y. e From this we conclude that 'IT ~s irreducible x, Y E He' hence (i) follows. e thus

on

H thus ker(TI) e e ker('IT ) = p .

e

pri~tive

is a

of

A

and

e ¢ ker(TI ), e

s~nce

e

(ii) follows from

(~)

•

s~nce

because,

B(H ) (BA.4.1) e In our main theorem

and

Let

THEOREM.

A

TI

(A)

Fredholm elements of a C*-algebra C*.4.3

~deal

~s

e

TI

e

(A)

~s

closed

will denote the set of Riesz and

relative to the closure of the socle.

A

be a C*-algebra.

thepefope isometnc) peppesentatian

cont~nuous,

(TI, H)

'lhepe exists a fCJ:':thful *-(and of A with the following

properties: (i)

'IT(soc(A})

(ii)

TI(soc(A) )

F(R)

n 'IT (A) ;

K(H)n

TI(A);

(iii)

'IT(R(A»

R (H) fI TI (A) ;

(iv)

'If((A»

(H) fI TI (A)

Proof.

Let

A

not contain

1\

of

80

be a set which

soc (A) •

p~

if

= A

p

,

eA on

A

i8 unital.

~ndexes

20r each

A E A,

the

pr~~t~ve ~deals

we can choose

eA=

and then, by C*.4.2, there exists a Define

of

A

e~ £

wluch do Min(A)

*-representat~on

~n

Then

~s

TIl

a *-representation of

on the

A

H~lbert

HI'

space

Now

tl ker(TI ) = f1 {p € TI(A) PA::j> soC(A!, by C*.4.2. As A€A A A€h A have a non-zero kernel ~t is necessary to add another representation

ker TI

=

1

order to ensure that the sum

TI

be

fa~th:ful.

theorem «14) 38.10) on the C*-a1gebra

Use the

A/soc (A)

then

representat~on

=

ker(TI)

Now ~f

TIl (x) € F(H I )· A

such that

therefore

'If

~s

and s~nce

TI

so

TI

fa~thful

is a

'If

'If \ (x) € F(H A), ~t follows that

12

2 n},

j

TIl (x) € F(H l ). But ro verify (~), observe that

TI(soc(A»C F(H) •

cs an ~dea1 of algebraic elements of

F(H){\ TI(Al in

soc (A)

*-representat~on

B (H),

closed ~n

(~).

~t ~s

hence

F (H) ()

so

obta~n equal~ty

let

and

p2

=

p*

~

~

~

iT

T = T* € K(H){) 'If(Al, € K(H)~ 'If (A)

is of fin~te rank, so operator

isometr~c

«75) 4.8.6), and

(Al C'If (soc (A) ) c K (H) () 'If (A) • T

LAP.

~

1

~

A.

where

~

S = TI + ~T2

where

TI' T2

are

TI(soc(lGI)~K(H)A 'If(A),

(~~).

The proofs of (~~~) and (~vl are now stIa~ghtforward (see A.I.3) C*.S

€ ffi,

S~nce every

T € F(H){) 'If(A).

~t follows that

K(H){) TI(Al,

~

To

But each compac~ proJect~on

i.

thus

may be written

self-adJo~nt members of Whence we have equallty

then

for each

Pc € F(H)O TI(A),

S C K(H){) 'If(Al

A,

(C*.4.1), therefore

\lhence we have equah ty

(A),

{Al",An }

so

00

P

*-

hence if

there ex~sts a £in~te subset

x € soc(A),

conta~ned

fa~thfu1

cs a

'If (soc (A) )

~n

to construct a *-represen-

for

= 0,

therefore

the ~nverse ~mage of 'If (soc (Al ) ::> F (H) n

'If •

x € span {AeJA :

ker('lf 2 ) = soc (A) , \m~ch ~s

'If~(x)

then

€ span (AeAA) ,

of

(0),

TI2

•

Let us examine the range of X

=

ker TIl{)ker TI2

may

Ge1fand-Na~mark

Put

HI ED H2 ,

'IT1

tt

Notes

Very neat proofs of the range H~lbert

space

H

C*.5.1

LEMMA.

follow~ng

Vla the

(The footnote In (28) S, T €

~ncluslon

announc~ng a~

B(H),

theorems of §0.4 can be

factor~sat~on

g~ven ~n

a

Lemma due to Douglas (28).

extension to Banach spaces is

S(H)CT(H) => there exists

C € B(H)

~ncorrect).

such that

S = TC.

Proof.

SJ..nce

y € ker(T)l.

S (H)C T

such that

(H)

I

then for each

Sx = Ty.

Put

Cx

X €

y.

H

there

ex~sts

C

lJ..near and we prove C

J..S

a unique

81

Let

is co~tinuous by means of the closed graph theorem « 30) p. 5 7) • be a sequence in

H

ker(T).L.

since

=

SU

Tv,

C*.S.2

~s

LEMMA.

=

thus the graph of

v,

S, T E B(H},

S

By induction

n

n n

T C

=

and s

TS

ST

C

S E B (H)

COROLLARY.

Proof.

Apply C*.S.l

C*.S.4

COROLLARY.

Proof.

Apply C*.S.l and C*.S.2

C*.S.S

COROLLARY.

S E

~.

R (H)

•

S E B(H),

S E B (H)

S = TC

by C*.S.l.

=

0

Let Then

1/!

S(H) 8 E Q(H).

ST - TS E K(H)

be the canonical

1jf.S}

hence, by C*.S.2,

Erdos (31) defined an element 0 => either

ax

=

0

or

I

1/! (T)

and

S(H}C.T(H}

r(ljJ (8»

xb

=

Erdos pOlnts out that

simple Banach algebras. prove that an element

The

of a

semis~mple

rank one operator in some faithful single and the operator

h~s

In fact, In (32) x

x 1\ x

0, that ~s

of an algebra

~s

A

s~ngle

to be

slm~lar

single

elements of use of

•

Banach algebra.

th~s

~f

B(X)

are

concept Erdos

to that in §4, see also

work does not extend even to semlI

Erdos,

G~otopoulos

and Lambrou

Banach algebra has an image as a

r~presentat~on

compact.

B (H)

of

8 E R(B)

sem~simple

a

Mak~ng

constructs a representation of a C*-algebra Ylinen (100).

=

x

O.

homomorph~sm

commute and 1/! (S) = $ (T}lj! (C) •

~s val~d ~n

C*.1.2

easily seen to be the rank one operators.

82

K(H}.

•

r(ljJ (T»

=

•

T E R(H) ,

I

and

TS

ST

T E Q(H),

Alexander (4) showed that

axb

S (H)C T(H) => S E

•

into the Calkin algebra. Now

I

•

II sn II 2. I ITn II I Ic n II ,

thus

and

T E K (B)

C*.S.3

So

TC => reS) < r(T}r(C).

=

n,

for each

closed

~s

and the result follows from the spectral radius formula

='>

and,

such that Sxn = Tyn for each n, Yn E ker(T) J.. a closed subspace of H, lAm Yn = v E ker(T}

Cu

hence

v.

lim Cx n n

.....

Then there exists

n

such that

u,

lim x n n

{x}

The

of the algebra x representat~on

is

in §4 may be

used to transfer Lnformation on finite-rank, compact or Riesz operators on Hilbert space to finite-rank, compact or Riesz elements of C*-algebras.

It

could, for example, be used to deduce the West and Stampfli decompositions in C*-algebras (C*.2.5, C*.2.6) from their counterpart theorems for operators (C*.2.1, c*.2.2).

Legg (58) has gLven the C*-algebra counter part of the

Chui, Smith and Ward result (26) that the commutator Ln the West decompoSL tion is quasLnilpotent.

In fact, the more de taL led informatLon on the

West decomposition provided by Murphy and West (61), (see below),is all valLd LIl a C*-algebra.

Akemalln and WrLght (3) have further results on the wedge

operator, and on the left and rLght regular representations in a C*-algebra. For example, they show that Lf operator either GLllespie «35),

S

or

S, T £ B (H)

T £ K(H).

then

R

R = K + Q

[K,

R

on a HLlbert

Lnto the sum of a compact plus

a quasLnLlpotent dLd these two operators commute.

then the commutator

is a weakly compact

(25) p.58) constructed a Riesz operator

space such that for no decomposLtLon of

showed that if

S AT

See also the rema~ks in §F.4.

Chui, Smith and Ward (26)

LS a West decomposition of a Riesz operator

Q]

LS quasinLlpotent.

R

Murphy and West (61) gave

a complete structure theory for the closed subalgebra (called the decornpo-

sition algebra) generated by

K

and

Q.

It emerges that the set of quasi-

nilpotents forms an Ldeal which LS equal to the radLcal, and that the algebra LS the spatLal dLrect sum of the radical plus the closed subalgebra generated

K.

by

The problem of decomposing Riesz operators on Banach spaces has been open it may even characterLse HLlbert spaces up to isomorphism.

for some time.

Some recent progress LS due to RadJavL and LaurLe (73) who showed that if is a RLesz operator on a Banach space and

0 (R) =O\n}~

I

values are repeated accordLng to algebraLc multiplLcLty(then decomposition Lf

f nlAn'




relative to

(Al f\ B ~(T)

a necessary condJ..tion 1f

J..S semJ..sJ..mple.

:.\.1.1 THEOREM. (T £ B).

Proof. then in

£

qJ (A) r) B

qJ (B) •

=

T

in

B.

EXAMPLE.

ix(T) -I- O. T

£

qJ (B) ,

Take

Let

T; then

where

relatJ..ve

=

WA(T)

for each

T

E:

FJ..rst we gJ..ve

B.

then a

Inv{A)nBCInv(B).

If

T

£

B

(J (T) A

=

(T)

Inv(A) () B,

Now the left and rJ..ght annJ..hJ..lator ideals of

T

By F.l.lO,the left and rJ..ght Barnes lde:.mpotents of T In

B are both zero. hence

ing

B

are zero, hence the same is true of the left and rJ..ght annJ..hilator

ideals of

A.1.2

J..n

be semisi7Tlf?le.

B

It suffJ..ces to show that T

A

Let

qJ(B)

and

but the converse does not hold J..n general.

If we do have equality then B

KA ,

B

aB (S)

1B (T)

0

A

E:

Inv (B)

•

KA = K(X) and choose T £ qJ (X) wJ..th be the maxJ..mal commutative subalgebra of B(X) contaln=

B{X},

(JA (S)

(S £

sJ..nce

B

B)

(BA.1.4) , but

T ¢ qJ (B) •

and

For, If

J..s commutatJ..ve, and we can wrJ..te

KB , by F.3 • 11, J..mplYJ..ng that T Fredholm operators of J..ndex zero J..n reX}) whJ..ch is false. V E

Inv(B)

=

T

K £

of Theorem A.l.l J..S not suffJ..cJ..ent

£

T

qJ0 (X)

V

+

K

(the

So the condJ.. hon

for general B.

For C*-algebras we do get equalJ..ty. A.I.3

THEOREM.

ct>(B) = qJ(A) (\ B. Proof.

The map

Let

A

be a C"'-algebra and

B

a *-subalgebra of

A;

then

~(B/KB)

is a *-lsomorphism so Thus if

~(A)~B,

T E

hence, in

W(B/KB )

is a *-closed subalgebra of

~(T +~)

then

(BA.4.2).

A.l.4

THEOREM.

Proof.

If

A

and

=

R(B)

B,

It follows from R.2.S that A.l.S

B

THEOREM.

Let

0 ~ :>..

T E R(B)

B~KA'

.

Now

2hen T = K + Q ~here K nilpotent operator in B.

by deflnition

is countable, hence

then

0 ~ P(A,T) E KBC KA •

be a Ries2 operator on a Hilbert space

T

T E ~(B) . .

= GA(T).

T E R(A)

\ P = Q,

1

~nd

T

TT*,

A

A,

wA (T)

then

to be pr1m1tive.

Let

T S l).

(L

then

> 1),

(i

If

> 1).

then

(J ~ 1),

= 0

l=l L LJ

A.4.2

,,0

t

00

is a sequence

denotes the closed sub-

T(X)

is invertible.

T

a S X',

then

00

o

Thus

n

al

~

(L

= 0

If, in additlon,

o

= lX(T),

A.4.3

a



so

t

(n

L Ln

> 1).

1), by A.4.1, hence

T S ¢o(X) niT)

= 0,

then

a = 0,

T(X) = T(X} = X

hence, by F.2.8,

o

LEMMA.

T

for at

T(X)

and

so

d(T} = O.

But

tt

LS LnvertLble ~ost

X.

lS dense in

a finite number of indices

i. Proof.

Suppose that the set

such that

S S B(x)

and

h

W

!I S II

:

t

l

= O}

lS LnfLnite.

< £ => T + S £ ¢o (X) •

Choose

'rake

S

£ > 0

to be the

.J

operator corresponding to the dlagonal matrlx [s where s .. = 0 J) , (L " lJ LJ -1 (l ¢ W) = El Then S £ B and and s (L E W) • < £, sa = 0 lL thus Tl = T + S E B 1\

0.

B

1 (R

=

+ M)

V E Inv{to),

T to - M

=

and

V E Inv{B)

f'\ ¢o (X)C ip0 (B)

A.4.S

ip{B)

then

f{T) E B,

=9

for

(R.S.2)

A.S If

Kto ,

toCB.

By A.4.2,

thus

1 (T)

So

R(X)

to. to)

If we put

R to E

¢(B)

R{X) = X. S

and

Hence

glves

Thus we have

= O.

is an upper triangular operator on

T

W (f{T»

f(W(T»

X

and

• by A.4.4.

Also

and the result follows from the spectral mapplng theorem

•

H

P

is a Hilbert space let ordered by

B~H)

2

P

Q

if

denote the set of hermltean projectlons In QP =- P

PQ

( P(H)CQ{H»

Note that

F

~

LEMMA.

~.

is not an algebra.

B(H)

Q

E

T

E

~(\

{H) => lH(T)

~

F(H)

such that

such that

peT + F) = Let

T + F

o. Q

?ut £

and

P.

R

KA

These

K (H)C ~ K(H) •

O. lH(T) < O.

By F.3.1l ,

has a left inverse

P with

~.

lS denoted by

A = B(H)

Let

Suppose, on the contrary, that £

P,

0,

operators were first studied by Halmos (39) who showed that

A.S.I

for

is quasi triangular If

and the set of quasitriangular operators In

Q.4

= K(x)~

by F.3.11.

The same argument applled to

O.

lim lnf IlpTP - TPII P

an

Kto

Algebras of quasltrlangular operators

F{H) T £

M E

Slnce

(T E B), T £ B

KB

E

•

If

COROLLARY.

f e Hol(a(T»

>

1 (R)

l{S)=-l(T),

But

K, L

denotes the index function In the algebra

all its diagonal entrles are non-zero. 1 (T)

and

TtoS to = I + r,to' lS a Fredholm element of the commutatlve Banach algebra

Tto

of all diagonal operators on

Hence

S E B

Hence

T + F, Q > p;

then then

S, R

E

there eXlsts

and aPE p,

Q6'

PQRQ = 0,

since and since

0

i P

~s f~n~te ~mensional

QB{H)Q

QRQQo = O.

such that that

P (QRQ) = O. s~nce

and,

there~n.

~s

QRQ

So

II RQ

II RQ -

QRQ

I)

2:..

II ·11 Qo II

II s

A = B{H},

C*-subalgebra of

dimensional,

QRQ

Qo £ QB (H) Q

~s

I ! -1

> I! s (RQ - QRQ) Q

0

(for

Q

P

£

T £ B

~

K(H)

KA =

(T, T

-1

and let

£ B(H)

has the property that

at the zero ~deal

B

such algebras

such that

Pn

{o},

-+

I

where

H

and

B

£

H,

! IP n (x

and

Now

1.

be any

T

Q

2:..

contradicting

P),

un~ta1

B => T

£

o

({a})

be separable and

B

Define

lipn T

-

Routine computatlon shows that

x, y

QRQQo = 0,

II

such that

tB (T) (P)

lB(T)

strongly.

{T £ B(H)

B

QB(H)Q,

-1

for

inverse-closed

B)

£

which contains

B, the index function

P £ IT (B) except perhaps F~rst

iH(T) •

let us see that

ex~st.

Let

EXAMPLE.

A.5.2

and

not right invertible

such that

Note that in such an algebra of

0

P £ QB(H)Q,

R £ ~ •

the fact that Now let

t

0

Q < Q

P,

£

to be a projection which is therefore < Q) •

Qo

- QRQ

Q o

not left invertible in the algebra

fin~te

this algebra is

lIs II-

there eXlsts a

(To verify this observe that

So there is a non-zero

we can choose

Thus

{C*.1.2},

TP

B

n

II

-+ 0

f~x

an

~ncreasing

sequence

P

n

P

£

by

(n -+ (0) }.

lS a closed *-subalgebra of

B (H) •

Let

then

GIl y)

-

(x ill y) p

n

II

(P x) GIl

n


0

such

1

L (G)

w~th

with respect

proof.

Since

MO

T(G) = span{MeJ If

]1

form of

a

M(G),

E:

]1.

~(a)

let

Let

e*Xa

e

T (G) .

E:

I

L (G)

Thus algebra.

(x E: L

A.6.2

I

Hence

¢(M(G»)

Let

LEMMA.

jl

S

Suppose

eJ, hence

o

"1(G)

a

then there exists

I

0

2: (G)

E:

such that

«45) 28.39), thus

soc(M(G»CTcG)

lS a closed ideal of

S

M{G)

deflne

Tjl

E:

M(G)

a(T )

1.1

I

0

which lS a Rlesz

B(LI(G))

be the ldentity measure on

=>

e*Xa

•

M(G)

denote the set of Fredholm elements In

If]1

0

be a flxed Fourler-Stlelt]es trans-

(e*v )' (0) AO

It follows that

(G».

Proof.

L: (G»

Mln(M(G»,

E:

Ll(G).

relatlve to

(0 E:

= soc(M(G»

Let

for each

SOC(M(G»

L:(G)}Csoc(M(G».

E:

((45) 28.36). and

~E:

lS flnlte dlmensional,

by

M(G).

= a(jl).

1

E: Inv(B(L (G»), then there eXlsts such jl 1 that T S = T", = ST. If x, Y S L (G), then T «Sx)*y) = jl*(Sx)*y jl Uo jl 1.1 1 = (TjlSX)*y = x*y = T (S(x*y». Thus (SX) *y = S(x*y) (x,y E: L (G) >':'1 By Wendel's Theorem ((4g) 35.5), S = Tv' .Eor some \! c M(G), thus \! = jl in M(G),

A.6.3

T

I

THEOREM.

I

has finite co-dimension in

jl*L (G)

L (G)

T

]1

is a

Riesz-Schauder operator. I

Proof.

I

I

]1*L (G) = T (L (G» hence, by (25) 3.2.5, Slnce ]1*L (G) ]1 co-dlmenslon It lS closed in LI(G). Suppose that {Ol"'" a~}

of dlstlnct elements of

(1 < k < m).

If

L:(G),

and that there exist

AIY l + ••• + AmYm = 0

where

~

S

Ma'

Yk

S

~

(1



A.6.4 Lmplies that Lf ((25) 1.4.5)

4t

=

W(fj)

AO

o

-

W()..l)

=

If

=> (i).

)..l E ¢o (M(G)}

then, by F. 3 .11,

+ K E Inv(M(G»,

LS a factor~sation of E ~HG)

\.l*K = K*~;

¢ = ¢2 E: T(G) •

are idempotents in soc(M(G)} = T(G)

COROLLARY.

Proof.

(~~i)

Conversely, if

K E ¢1*M(G)*¢2

)..l = (00 - $l)*(\.l + K) A.6.5

and

=>

(~v)

and

K E: T(G)

Inv(M(G»

\! £

by A.6.3,and

Obviously (v) => (ni) •

of

Inv(M(G»,

E:

\.l*¢l

(A.6.ll.

)..l

=0 =

¢2*1J,

Now

4t

as Ln (v)

S()..l).

11 E: C

If there exists a maximal modular left ldeal

L

of

A

such that

{x £ A : xAeL}.

BA.2.1

(i)

A is ObV10US, so assume that there exist (BA.2.2),

y so

E A,

x

Z E

rad(A)

x'

such that

has a left inverse in

(iii) follows at once

has a left lnverse in yx = 1 + z.

But

A'. 1 +

{x

£

£

Inv(A)

• A.

Let

P E

there eXLsts a maximal modular (and therefore closed) left ldeal p

z

A.

Now specialise to the case of a Banach algebra such that

Then

A : xA C.L} •

It follows that

P

1S

11(A) L

closed in

then of

A

A. 103

Further, by BA.2.l,

P

is the kernel of the

representation on the quotient space th~s representat~on

the image of operators on

AIL,

Thus,

~t w~ll

A'

A

=

~s

erA' (x')

primitive

If

BA.2.6

algebras

B(A/L) ,

Now

the bounded linear

suff~c~ent

of

to consider the

rad(A)

algebra

A

cont~nuous irreduc~ble

is a closed

~deal

of

A,

and

It follows from BA.2.5 that

Banach algebra.

~n

deal~ng w~th

X.

for Banach spaces

A).

S

loss of generality, uhen

\l~thout

se~simple

a (x

~deal

~n

is contained

is a Banach algebra, then

= A/rad(A)

erA (x)

be

B(X)

representations into If

is a Banach space.

hence,by Johnson's theorem «14) 25.7), this represen-

tation is continuous. Banach algebras

wh~ch

AIL

left regular

~rreduc~ble

py.i~tive ~f

is

~s

zero

a

A.

is a Banach algebra and P s TI(A) the primitive Banach and A'/p' are isometrically isomorphic under the map

A

Alp

x+P-+x' +P'. Proof.

The map

~s

~somorphism s~nce

an

rad(A)C:P

(P S

TICA».

A

straightforward computation shows that the mapp~ng ~s an ~sometry

closed subalgebra Proof.

B

P -+ P(\B rad(B)

=

eAe

is a

eAe

~s

closed

B =

Banach algebra and e 2

(0)

~n

A of

since

BA.2.8

Let

(i)

rad(A)

~

e

TI (A)"'h (B)

then the

~s

~dempotent.

onto

IICB)

The map

((14) 26.14), so

..

quasin~lpotent character~sation

due to Zemanek (104).

e C A,

is sewisimp le .

homeomorph~sm

rad(A) f\

The

se~isimple

If A is a

BA.2.7

..

Q(A)

of the

ra~cal

in the next theorem

~s

denotes the set of quas~n~lpotent elemenrs of A.

be a unital Banach algebra, then contains any right or left ideal al! of whose elenents are

quasini lpoten t; (ii)

rad(A)

(iii)

rad(A)

~.

(ii),

{x s A

x + Q(A)CQ(A)}; x + Inv (A) C Inv (A) } •

(i) follows from (14) 24.18. (iii)

We show that

x + Q(A)CQ(A) => x S rad(A) => x + Inv(A)C.lnv(A) ==> x + QCA)CQ(A).

x + ~(A)c:Q(A) .

Let

~rreducible

representation of

~ E X

Choose

there exists

u E A

o 1 A E p(u}

and put

-1

rr(v rr(v rr(v

-1

-1

-1

x - v

x

-1

xv E Q(A)

E P

)rr(xv

Inv(A).

E

If

Thus

u

x E

BA.3

MLn~mal

Let

A

~deal

in

~deals

~

J

minimal

n(u)rr(x}~ =

~.

Choose

Q (A) => v

-1

xv c Q (A) ,

hence

contra~ct~on.

is a

It follows that

(Ll

+ x)-l

u

-1

(1 + xu

-1 -1

)

,

hence

CA

AX + Inv(A)C- Inv(A)

q + x E

Q (A),

«::)!

E

1 + A(q + x}

Thus

(A E x E Q.(A).

Then

a

(0)

and

ide~otent ~s

is a dLvlsion algebra.

(If

of mlnimal Ldempotents in

A

~~nimal

A

J

right ideal of

are the only

a non-zero

lS denoted by

r~ght

~dempotent

lS a Banach algebra Mln(A).

A

e

is a rLght

Ldeals contaLned such that

eAe =

~e)

•

eAe

The set

There are sLmilar

statements for left ldeals. BA.3.1

If

A

"is a semis?:mp le algebl'a, then

CLl

R-

is a min'imal right 1:deal of A

(il)

L

is a '7Iinimal left ideal of

(ui)

(l -

e}A, (A(l - f»

30.6, 30.11).

A L

eA

where

Af where

e

E

Min (A) ;

f s Min (A) ;

1.-8 a m=imal modulm' right (left) ideal of A

if, and only if, e, f S Mln(A) • ((14)

R

be a minimal right ideal 0-1' A and let u E A. 'lhen " either uJ = (0) , or uJ is a minimal right ideal of A. is a minimal right (H) If x E A , e E Mln(A) and xe of- 0 then xeA ideal of A. (l)

BA.3.2

«14) 30.7, If

A

Let

J

(75) 2.1.8).

has minlmal rlght ldeals the smallest rlght ldeal contalnlng them

all is called the

right socle of

A.

If

A

has both mlnlmal rlght and left

ideals, and if the rlght and left socles of

socle of A eXlsts and denote It by exists, is a non-zero ldeal of ideals we put BA.3 • 3 (l) (H)

Let

A.

A

Clearly the socle, If It

soc (A) . If

A

are equal, 'Ie say that the

has no mlnlmal left or rlght

soc (A) = (0).

be a semisimp le algebra 1Ji th idea l soc(A), soc (J) exist; A

Then

J.

Min (J) = J (\ Mln (A) ;

(Hi)

soc(J)

=

Jf\SOC(A);

if A is a Banach algebra and

(iv) Proof. (ii)

then

e, f E Mln(A)

dlm(eAf) < 1.

«14) 30.10, 24.20).

(i)

straightforward.

(iii)

follows from (li) and BA.3.1.

(iv)

«14) 31.6).

Let A be a semisimple algebra, P E canonical quotient homomorphism ¢ : A -+- AjP. BA.3.4

~nd

TI(A),

Then

let

.""l.jP

¢ denote the is semisimp le and

¢(soc(A) )C.sOC(¢(A). Proof.

¢(Min(A»C-Min(¢(A»)

and the result follows from BA.3.1

tt

The relationshlp between mlnlma1 ldempotents and prlmltlve ldeals is important. BA.3.5

Let A be a semisimple algebra. there exists a unique P e E II (A) If e £ Mln{A) 2 If e = e E soc (A) I the set {p E II (Al : e ¢ p} 1

Proof. (BA.3.l) Clearly 106

(il

If

e

E

Min(A)

therefore

1

e

¢

Pe'

P

If

e

'"

{x

then

A(l - el

E A

xACA(l - el}

Q E TI{A)

and

e ¢ Q,

such that e ¢ Pe' is finite.

is a maxlmal modular left ideal

then

lS a prlmitlve ldeal. Q f\Ae =

(O),

Slnce

Ae

is a minlmal left ideal. qAe = (0).

Thus

It follows that

Qe = (0). ~

q

P C Q,

e

(il) (1




C*-algebras

that

then

A

then there exists

a singleton set for each

A Banach algebra

The

E

hence

contains two distinct points.

(x)

BA.4

hence

oJ Y

0

(y E A), hypothes~s

yx, xy E Inv(A) ,

Thus, by (14) 24.16,

•

:'Iow, i f

Thus

r(xy) = 0 then by the

If A is a semisimple Banach algebra and if p

BA.3.10

o

r(xy) > 0,

is not zero.

= o.

x

such that

contrad~ct~on

r (y - Al) = 0

so

0

yEA

0(xy) = 0(yx) which is a

f

Then we claim that

rex) = O.

for suppose there exists and (14) 5.3,

A = ~l.

then

x E A,

is a

0(x)

A

and A/I

in the

into a C*-aZgebra

B

Let

BA.4.2

be a unital C*-algebra and let

A

$ubalgebra of A

then

GB(x) = crA(x)

B

be a closed unital *-

(x E B).

( ( 75) 4.8.2).

Let

BA.4.3

If

(i)

and

be a C*-algebra.

A

f

there exists

f2 E A,

e

= e2

e* E A

such that

fe

=e

ef = f.

If

(n)

such that

contains a right ideal

A

there exists Proof.

there eX";'sts

e = e* E Min(A)

R = eA.

If

(ill)

is a rtrinima3 right 1:deal of A,

F

(1)

e = e

2

~

= e*

R e f f A (fl' E Min (A) , 1 < i < n) 1

1

ouch that

soc (A)

R = eA.

USlng the Gelfand-Nalmark representation this

the elementary assertlon that If a H~lbert

operators on a

proJect~on

is contalned

~s

eqUlvalent to

~n

a C*-algebra of

self-adjo~nt

space then the C*-algebra contains a

proJection Wlth the same range «84) 6.1). If

Then

R

lS a mlnlmal rlght ldeal there eXlsts £2 = f E Min(A) 2 By (1) flnd e = e e* E A such that fe = e, ef fA.

R

fA = efACeA

(lli)

such

R

feA c. fA.

Slmilar argument

Thus

R = eA I hence

e

E

f.

fun (A) •

..

It lS elementary to check: the unlqueness of the self-adjoint idempotents in BA.4.3.

BA.4.4

Let

(1)

soc(A)

Slnce a

C*-~lgebra

be a C*-algebra, then

A

=

(soc(A»*;

x E soc (A) x*x

(ii)

socCA);

E

x E socCA) x*x s soc (A) •

(li1) ~.

(1)

If

x S soC(A) ,

then x E

and each fiEMin(A}.By BA.4.3,

x = ex, (ii)

lS semls1mple lts socle eXlsts.

hence

x*

x*e

=> is clear.

there eX1sts

e

=e

x*x(1 - e) = 0, Ilx - xel1 2

Let 2

=

R = eA

RC~f.A 1 1 where

where R is a right ideal of A 2 = e* E soc (A) • So

e = e

AeCsoc(A).

E

x

S

A

and suppose that

e* E soc (A)

such that

x*x

E

socCA).

Then

x*x E Ae (BA.4.3).

Thus

and

II x

(1 -

e)

112

11(1 - e)x*x(l - e)11

0,

109

so

x = xe

(iii) A/I

soc (A) •

£

Let

I

be a closed ideal of the C*-algebra

A.

Then

I

I*

and

is a C*-algebra (BA.4.l), hence II (x* + I)

Ilx*x + I II

so x*x



£ I

I

X £

(x

+

I)

I~

IIx

+

III 2 .

•

Finally we need a result on the spectrum of an operator matrix.

Q, and

denotes the interior of the set

If

BA.4.5

T

int(a{U)n a{v»

D

=

o

a (T) = a (U) u

then

= ~

intW)

U, V E B{H) •

and

* V

0 (V) •

This follows immediately from the following lemma. BA.4.6

(a(u) va(V) )'dnt{o{O)" a(V»C a(T)Ca(U) v o{V) •

Proof.

Elementary matrix computation shows that (a (U)

u a (V) )' (o (U)" a (V) ) C

Now choose

A

E

a (a {O}"

0

(V) )

a (T) C.a (U) va (V) •

then

A

E

aa (U)

A - V is a two-sided topological diviSOr all bounded linear operators. IIAnll

=1

for each

(l-T)

So

each

l~O

B

n

e~ ther

In the first case there exist

A -

U

A

n

with

(A - U)A + O. n

D (A - V) ..... 0,

so

U)A

n

o .....

o

hence

or

of zero in the Banach algebra of

In the other case, there e~st

A E aCT) • n, and

n, and

ao (V)

\J

Bn

with

0,

IIBnl1

I

for

D

o

o

+

o

B

n

AE

again

a(T) •

o

o (A-T)

B

n

C\ -

0,

V)

d (a (U) () a (V) )C a (T) •

Thus

It is easy to see that the result of BA.4.S fails if we drop the condition that

int(a(U) tl a(V}) =