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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© 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 <
~m(T(X»
U
the closed
~s
an ~deal
~s
ball of
un~t
~s
rank operators
compact where ~n
B(x}
form
s
T
theory of compact operators states that each non-zero
po~nt
of
d~m(ker{T»
f~n~te
<
rank of
~f
00,
T s K(X)
quot~ent
The
T(U)
If
~s
Af
and
algebra
T s B{X)
T. ~s
T(X)
set of Fredholm operators ~f
f~n~te
The
~f
K(x).
a pole of
that
X.
of fiaite rank if
~s
X
and the compact operators a closed ~deal
F(x)
R~esz
the
T s B{X)
is a compact operator on
T
00.
C.
~n
closed
A-
t~en
B(x)/K(X)
is a Banach algebra under the
T S
norm.
aCT)
a Fredholm operator if and
~f
dim{X/T(X»
<
00.
The
It follows from the Riesz theory ~(X).
T + K(x)
whose elements are the cosets
quot~ent
and will playa maJor role in our
X,
~(X).
denoted 0
~s
K(x)
It
del~berat~ons.
~s
called the Calkin algebra
Our
~mme~ate
aim
~s
to
characterise Fredholm operators. 0.2.1
DEFINITION. xn s X
of elements
II {xn } II (i~)
(~)
~s
m(X)
It
~s
w~th
w~th
sup n the
the
l~near
space of bounded sequences
the supremum norm
subspace of
too (X)
cons~sting
of those sequences
a convergent subsequence,
wh~ch conta~ns
Q,oo(X) T
~s
B(X)
a Banach space and then
Q,oo (X) •
too (X) ,
{x } s m(X) => {Tx } s m(X). Let n n and ~f T s B(X) let T denote the operator on
Further,
~f
t
A
{Tx } + m(X}.
T{{x}+m(X}) n
Clearly
T
S
those
B(X),
and
n
T
€
K(X) <=> T
O.
m(X) a closed
{x} s Q, (X) => {Tx } s noon X denote the quot~ent space
subspace of too(X)/m{X},
~.e.
totally bounded sets of terms.
elementary to check that.
and
{x } n
I Ixn II
l~near
every subsequence of sequences
~s
A
X
defined by
0.2.2
For
(Atkinson characterisation)
THEOREM.
T
EO:
the following
B(x)
statements are equivalent (i)
T
EO:
4>(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
<
Z
and
~
and
T(X)
W of
~s
of finite
x such that
ED W.
can be depicted as the 2x2 operator matrix
W
T
T(X)
the subspaces on the top being domains and those on the left ranges; unmarked entries are zero. T(X)
T22 : Z
+
T(X)
~s
biJective and continuous and
is closed so there exists a continuous linear inverse
(30) p.57).
the
8 22 : T(X) + Z
If
ker(T) S
Z
ker(T) then
W
TS
and
T(X)
Clearly, 4.
TS
and
ST
ker(T) Z
ST
Z
EE
are projections of finkte co-dimension so there exist
projections
S
ST
is the inverse of
(ii) => (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 =
<
00, there
Clearly
{x } C Z n
with
0, 0,
m(X) •
Thus there exists a subsequence = 1
dim(ker(T»
= ker(T) e z.
o.
=> {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
<
dim{ker(T»
0,
{x
nk
but
}
such that
x
ker (T) =
(0)
Z (\
->~
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) <:
<
mm(X/T (X»
T E
~(X)
the nullity of
T, neT)
and the index of
mm(ker(T»,
T,
- d(T).
B(x),
set of Fredholm
has a convergent subsequence, thus
the essential spectrum
in the
T, d(T) = mm(X/T{X»
= neT)
T E
n
B(x)
T E
0.2.5
Clearly
ex~sts
00
is compact and
WeT)
If
such that
x,
So
COROLLARY.
(ii)
ex
invertible so there
k +
0.2.4
(T)
~s
as
T + K(X)
X
T
0
+
is closed.
DEFINITION.
i
n
+ Tx } + m(X), n
Z E
+
~
be the spectrum of
~
{x }
ex~sts
A
a subsequence such that
) -
the unit ball of
itT)
be such that
m(X),
E
n
ex~sts
T(X)
0.2.3
n
< 2
n
n
{y }eX
< 1
.r(fw} + m(X»
{T(W
Let
00
is a Banach space, it
such that
n
thus
<
X/T(X)
~(T)
=
a Fredholm point of T
is denoted by
~\W(T),
further
T
if
A-
T E
~(X),
and the
~(T). ~(X)
is a multiplicatlve
se~-
i(T l T 2 ) = i(T l ) + ~(T2) for T l , T2 E ~(X) «25) 3.2.7). Moreover the set of Fredholm operators is invariant under compact perturbations group and
and
itT + K) = itT)
if
T E
~(X),
theQry is of cruciaL importance.
K E K{X)
((25)
4.4.2).
Perturbation
"s
there exists
If T E c[J (X) < 0 => T + S E c[J (X) and THEOREM.
0.2.6
11
leT + S)
=
o>
such that
0
S E B(x),
« 29 4.4.1) •
i (T) •
Consequently,
c[J(X)
lS
is contlnuous on
c[J(X)
and therefore constant on connected components of
c[J(X).
an open semigroup 1n
and the index function
B(X)
More detalled 1nformatlon lS avallable if the perturbation is caused
by a multlple of the ldentlty. 0.2.7
for
Tf
THEOREM.
.5. 0 and
I AI
n(T),
d(T)
(25)
n(A + T),
such that
0
0 < IAI
<
o.
ThlS lmportant result we call the
3.2.10).
A + T E c[J(x)
are constant and less than or equaL to
d(A + T)
respectiveLy for
0 >
there exists
T E c[J(X),
punctured neighbourhood
theorem. An lndex-zero Fredholm operator may be decomposed lnto the sum of an 1nvertlhle operator plus a finlte rank one. 0.2.8
T
THEOREM.
E
T + AF E Inv(B(X}} ~s
Proof.
and
¢(X)
i{T)
o => ther>e exists
we may
wrlte
F E
f
(X)
such that
(;"'i'0).
ln the proof of 0.2.2
w
T
T(X)
where
Wand
ker T
have the same dlmenslor. slnce
F £ F(x} by means of the lsoIIlOrphlsm
rn ]cer(T)
W
F
T(X)
If
A i' 0,
0.2.9
n
T + AF
THEOREM. a(T
+
£
If
O.
Construct
: ker(T) -+ W.
Z
Inv (8 (X) ) T
J
leT)
£
B(x}
,
•
then
a (T)"O.
£
{T)
and
leA - T)
O}
K).
KEK(x) 7
'!'he result may be restated as follows:
{A E ~(T)
i(A - T}
and
=
U
O}
peT + K).
KEK(x)
U
A E
Let
P (T + K) i then for some
K£K(x) A - T - K
hence
E
0
~(X)
and
E K (X) ,
K
0
A E peT + K ) 0
i(A - T - K ) = O.
But then
0
A - T E
~(X)
i(A - T) = O.
and
Conversely, let generality take there exists
A - T E
=
A
O.
~(X)
'!'hen
Kl E K(X}
and
i(A - T) =
T E ~(X)
and
o.
Wi thout loss of
=
i (T)
0
and, by 0.2.8,
0 E peT + Kl } •
such that
Riesz operators
0.3
The ideal of inessential operators
reX}
X
on a Banach space
is defined to Since
be the inverse canonical image of the radical of the Calkin algebra.
I
the radical is closed in the Calkin algebra,
is a Riesz operator if the non-zero spectrum of
T E B(x)
of finite rank of If
x E X
radius
E > 0,
and
A(x,E}
Let
q(B)
of
B
cover by oPfin balls in Be U A (xi ,E)
points
1
Xi
Clearly
X
denotes the open ball centred at
B
be a bounded subset of
~s
the infimum of
of radius
we say that
need not
in£{E > 0
q(B)
of
q(B) = 0 <-'> B
necessar~ly
Of
and
Bex).
<
-
sup
q(B)~l
B).
B.
X,
so if
B}.
B
:LS
closed in
X.
Then ~
has a finite
Thus
be a bounded subset of x, u
q(T(B»
B
is a finite E-net for
is totally bounded in
Let
q(T(U»
such that
there ex~sts a finite E-net for
LEMMA. T E
n
lie in
0.3.2
B
X; the measure of non-
E > 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.
£
> 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
<0.
{z } E m(X}.
for each such n we obta.l.n a sequence
Now
n
hence
zn = Yj
If
[ I{xn }
- {z } n
II
I I{xn }
+ m(X) I I <
< 0,
0,
T E B(x}
Recall that
.l.nduces an operator
TE
A
B(x}
where
X = £oo(X}/m(X},
by virtue of the equation
{Tx } + m{X} .
T({X } + m(X)} n
0.3.4
Proof.
LEMMA.
n
IITII .2 sup q(T(B)}.2 21 [Til. q (B).21
[IT[ I
sup{[I{Tx} +m(x)11 n
sup{q({Tx }) n
<
Let
B
:
[[ix} n
: q(ix }) < n-
11,
+m(X)[[ .21}, (0.3.3)
sup q(T(B». q (B).21
be a bounded subset of
choose an infinite sequence n
Xn+1EB\Ut;(X%}
~/2.
for
{x} n
X .l.n
such that
B
Clearly
n>l.
q(B) > 0 >
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)}
<
then
A - T +
(0.2.2),
K(x)
A - T E
reT +
T E B(X}
Conversely, let
and
~(X).
If
p (T),
0
~
=
•
r
the set
then
K{x»
=
{a}.
If
0 ~ A
each ne.ighbourhood of
A must
thus using the punctured ne.ighbourhood theorem for
0 < IAI < 0, peT)
and some positive and
But if the non-zero boundary points of
isolated, all non-zero points in suCh and
0
so, by the Atkinson characterisation
A E 3a(T)
(0.2.?), n(A - T) = 0 = d(A - T)
a (T)
0 >
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 <
II Sn+lY -
that is
n
T £
K(x} ->
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 +
<
{y}
n
£
If
E > 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
<
£
(n ~ 1),
(n ~ 1) ,
so
(n > 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) <
But
o
reT)
T E: B(X),
Z(T)
of
B(X)
T
S -+- ST
I ITI I
I ITI I.
and
Jbviously say that 0.5.1
T
A
E:
(A -
A
hence
denotes the cornmutant of
and
S E: R(X)
o (T)
-1
If
Conversely,
(A -
is a compact (Riesz) operator on
T
-1
E:
Z(T),
A E:
~f
p(T),
~s
the identity on
T)V (I)
(A -
h(S
also compact.
THEOREM.
~.
)t n 1
If
S
n
E:
co
n
to
tt
is a compact operator on
K (Z (T)
T E:
X
then
) •
(n > 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
<
I! x
E
to
- x'
If
SeE)
S E: Z (T)
C E.
and
II S II
.2.
I,
Now the restriction
the set of continuous functions X.
I [, 15
the
{sIE:
set
11sl1
S E: Z(T),
21}
is an equicontinuous subset of
Ste(E)
and is therefore, by the Arzela-Ascoli theorem «30) p.266), a compact subset of
<;(E).
{S
}
u,
~
Hence i f
Sn E: Z (T)
which converges uniformly on
{S
i.e.
nk
T}
II Sn II
< 1
E, l.e.
{S
and
tt
is norm convergent
there exists a subsequence nk
Tx}
converges unlformly on
A more complete result is true for Riesz operators. 0.5.3
THEOREM.
Proof.
Let
aCT) <=v A
s R(x) <=>
T
A f o.
T S R(Z(T»
•
Lemma 0.5.1 shows that
is an isolated point of
O{T),
A
is an isolated point of
and in this case the associated
spectral projections are connected by the formula
P(A,T)
Now if
5
5 ->- SP(A,T)
Z(T),
S
P(A,T)
where
Sl
then
T
Tl E& T 2 ,
S
51 E& S2'
P(A,T)5
Sl E& O2 ,
S
s
Z (T) ) •
commutes with
Xl @
P(A,T)
so
x2 '
(5 s Z (T) )
(S
S
Z (T) )
Z(T l ), therefore
{P(A,T)S
Suppose now that and
thus
dim(P(A,T) Z (T»
dim (Xl)
Conversely, let OCT)
and
S S Z(T)}
T S R(X), then
OCT)
li.
5
X
P{A,.) (Z (Tl)
point of
(S
<
1
It follows that
00.
<
0
00
hence
T s R(Z(T»
T and
dim{P(A,T)Z(T»
S
A s aCT)
lS an isolated point of
dim(Z(Tl »
2 dim(B(x l »
<
00,
R(Z{T». 0 ~ A S O(T); then = ~m(Z(Tl»
<
00.
A
is an isolated
since the algebra
generated by
Tl
ex~sts
thus there O{T l ) = {A}
(A -
is contained
Z(T l )
i t must also be
a non-zero polynomial (A - Tl)k = 0
thus
Tl)-l{O)
~n
p
such that
for some positive
fin~te
p(T l ) =
~nteger
k.
dimensional,
o.
But If
inf~n~te linearly inde(for each n): also there eXists{O ;6)aE: Xl*
is infinite dimensional i t contains an
pendent set
{x} and * n T a = Aa.
T1Xn = AXn
such that
It follows at once that the infinite linearly inde-
{a ~ x}
pendent set Tl )
(A T
of rank one operators lies in
n
-1
is
(0)
fin~te
dimens~onal,
Thus
hence so is
thus
E: R{x) •
0.6
The wedge operator
B(X)
T E:
If
T 1\ T : S
-+
TST
(S
on
B (X)
by
E: B (X) ) •
IITA Til
Clearly 0.6.1
wedge operator T!\. T
we define the
is (i) finite rank, (ii) compact, (iii) Riesz,
T E: B{x)
THEOREM.
or (iv) quasiniZpotent <=>
T!\
Tis.
n
Proof.
(i)
T
L a. ~ Xi i=l ~
n
TST Thus
Let
S
x.)s( ~
n
L a. ~ j=l J
x.) J
L a (Sx.)a. 6l x . i,J=l ~ J J ~
E: B (X)} C span {a j 6l x~ : 1'::' ~, J < n}
f~nite dimens~onal
which is a
then
n
( L a 6l i=l ~
{TST
I
Conversely, suppose that
subspace of T f.
0
and
S(X). TAT
contains an infinite linearly independent set that
T *a
01
0
lS
fini te rank.
{Txi}~
choose
If
a E: X*
T(X)
such
and then the set
is an infinite linearly independent set in
T f\ T(B (X»
•
1.7
(ii)
Let If
S(X}.
exist
U be the closed unit ball of ~s
T
compact,
{TSTXi
{(TSTxl , ••• , TST~)
S
:
in the product topology,
:
Bl },
£
~s
of the above set is wi~n x £ U
and
~s
wh~ch
£
~.e.
there
conta~ned ~n
is
<
(l
~
the product space
< k)
~s
x(k)
itself
such that each po~nt
Sl' ••• 5 n £ Bl
for some
J (12J2n).
S £ Bl
Ilsll
IIT(x - x.) II + I ITSTx ~ - TS j Tx.ll ~ ~
TI\ T
~s
and the set
{TAT
compact.
If {x} ~s a bounded Tia and T 1\ T be compact. * and aia£x, {a 6a x } ~s a bounded sequence ~n B(X),
Conversely, let sequence in
i
e~sts
there
{STx
(TSjTx l , ••• ,TSJ~)
of
IITST - TSjTl1 ~ (211TII + 1)£
bounded,
£ > 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 <~ R = eA where e E Min(A); -Zf R is a minimaL right ideal of A and x E A then xR is either a minimal right ideaL or is zero. (ii)
R
(iii)
~rOOf.
(ii),
F.1.4
+ enxn
(~)
(in)
By (14) 24.17,
xk E A
(0).
•
If {e l , ... en' f}CMin(A) and (1 < k < n) and ejx j ~ 0 then
LEMMA (Exchange).
where
x E rad(A)
BA.3.1 and BA.3.2
f
23
It suffices to show that
proof.
is contained In the right hand side.
s1.nce
Now
and
e.A
lS a min1.mal right ideal.
J
Thus
e.A ]
F.l.S ef
DEFINITION.
=0 =
fe
for
A set
W of ldempotents of
e, fEW
Observe that If
and
e
{e l , ••• , en}
1.S orthogonal if
A
f f. is an orthogonal set of idempotents 1.n
A,
then
We need two further technlcal lemmas. F.l.6
LEMMA (Orthogonallsation).
orthogonal subset of Ml.n{A) is such that
f E Min (A) {el"'"
Suppose that
and that
fA ¢; R,
{e, , ••• , e}
is an '!hen if
R = elA + .•• + enA.
there exis ts
such that
e n+ 1
is an orthogonal-subset of Min (A)
en +l }
n
~
R+ fA
and
= elA + •••
+ en+1A. Proof.
Wri te
fA ¢: pA
and
(1 - p)fA
p = el
+ ••• + en'
f = pf + {l - plf,
Then
P
2
l t follows that
g I say.
Clearly
pg = 0,
hence if
2
Further e n + l '; 0, for if g e n + l := e n +l . which is false. Now (O)'; e n+ lA egA, hence
=
~en+l
e~en+l
e n +1 }
Now, since
en+l.
24
R
(l -
~
pA.
p) f
Since
,; 0,
hence
is a mlnlmal right ldeal which (F.l.3) contains a minimal
idempotent
{el"'"
and
= P
0 = en+1P~
:=
en+l~
(1
lS an orthogonal subset of (1 - p)fA
g(l - p)
£
=
gA
gA
(F.l.3)
e n + 1 = g (l - p) ,
gp,
then
g
e n+ 1 E Mln (A) •
< k < n),
Min (A) •
I
(1 - p) fACfA + pA,
gpgp = 0 Further
so the set
elA + ••• + en+lACR + fA.
so
Conversely, as
(1 -
e n +l
p)fA
= gel
gA
- p),
by
~n~mality
(F.l.3)
g(l - p)A n+l R + fA C. ~ ekA •
g~ving
Therefore
1
F.l. 7
Let
LEMMA.
be a right ideal of
R
rrrinimal right ideals of A. is finite, further, if n 2 p = L e = p E soc (A)
1
k
Proof.
Us~ng
lying in a finite sum of
A
R"M~n(A)
Then every orthogonal subset of
is a maximal such subset then
{e l , ... , e} n and R pA.
the exchange and
lemmas we can find an
orthogonal~sation
...
+ f A. Suppose {fl ,· .. £ }CMin(A) such that R CflA + m m } that {e l , .. ··, e is an orthogonal subset of R I'\M~n(A). Then n e = f 1a l + for some E A (1 < k ~ m), so f.a. .j o for + f a 1 mm ~ J J some J wh~ch we take to be 1, and then by the exchange lemma
orthogonal set
...
e 2 = elb l + f 2b 2 + ••. + fmbm since e 2 i elb l by orthogonallty,
for some
Thus
Which we take to be stage at which m RC.L e A, and
1
k
(1 ~ k .:. m)
so
m
~
f A = ~ 1 k I
~f
(1 ~
A
€:
~
k
m) ,
and
f b i 0 for some J (2 ~ j < m) J J Repeating the process m times we arrive at a
2.
m
bk
at
th~s
point the process
ter~nates
since
°
for some k then em+lek e m+ l E R" Mln(A) i Hence no {e 1 , " •. , e m+l } ~s not an orthogonal set.
orthogonal subset of
contains more than
R" Min (A)
m
elements.
Suppose now that {e , ••• , e} is a maximal orthogonal subset of 1 n n 2 R "Min (Al and put p = L e k • Clearly p = p E soc (A) and pAC. R, so it only rema~ns to show €hat RCpA. If not, choose w E R\.pA and write y
=
fky
(1 - p)w.
i
0
Then
for some k
y i 0, (1 < k
ideal which must contain an fA
¢
pA
and, by F.l.6,
and
<
f
s~nce
m
IDln~mality
m) •
By
Then
YUfk.j 0
E
M~n
Now
(A) •
there exists
e
so
n+1
r
m
y E Ret fkA, fkyA = yufkA
y =
~A,
fky
is a minlmal right
fAcyAc::.(l - p)A,
E M~n(A)
so
hence
such that
hence {e 1 , •••
,en+t 25
pA + fA =
is orthogonal and e n +1 £ pA + fACR, proves the lemma F.l.8
n+l L ekA.
However th1.S implies that
contradl.cttng the maximality of
{e l , ••• , en}
which
•
DEFINITION.
If
x £ A
the
xa
O}i
right annihilator of x
l.n
A
l.S
defined by
{a
ran (x)
\mile the
left annihilator of
{a
lan(x}
DEFINITION.
F.l.9
A
£
£ A
If
ax
x £ A
x
in
A
is defl.ned by
O}.
we say that
2
p = p
£ A
idempotent for x in A l.f xA = (1 - p)Ai while Barnes idempotent for x in A i f Ax = A(l - q) •
q
is a 2
=q
left Barnes £ A is a right
:"ote that (i)
the Barnes idempotents l l.f they exist,are not normally unique;
(ii)
p
loS
a left Barnes idempotent for
(iil.)
q
is a rl.ght Barnes idempotent for
x in x
A => lan(x} l.n
APi
A => ran (x) = qA.
The next result is fundamental as it connects the eXl.stence of Barnes idempotents in the socle with left or right invertib1.1ity modulo the socle.
Let A be a unital semisimple algebra and x E A. Then x is left (right) invertible modulo soc(A)<=> x has a right (left) Barnes idempotent in soc (A) •
F.l.IO
THEOREM
~.
Let
u
(Barnes idempotents).
be a left inverse of
right invertibility is sl.milar). (1 < 1. < n)
x
modulo
soc (A)
Then there eXl.st
ei
(the proof for E Ml.n (A)
such that
ux - 1
n L e i al.' 1
-=>
n ran(x) C L eiA, 1
==i>
ran(q)
qA,
where
q=q
2
£
soc (A)
(F.l. 7) •
,
a
1.
E: A
Now
hence A Ax
axq + axel - q)
ax
Axe. A(l - q) .
= Ax e e
Aq
Aq
Observe that
implies that
wh~ch
¢
L
ex~sts
g £
L('\ Ag =
(0)
A
S~nce
L.
for, if
~t
which is contained in some maximal and
Ag.
by maximality of
L,
Thus there and
So
z
L,
a £ A,
(0) •
Hence
£
y £ L,
yz + yag,
y
yag
y - yz
£
LA Ag
o
xag
ag £ ran (x)
qag.
=
qA,
so
ag
=
qag
=
which
0
Contradiction which concludes the proof.
g~ves
=
1
z £ L,
a
The reVerse inclusion is obviously
•
F.I.ll
EXAMPLE.
on a Banach space Pictorlally as
of
it follows that
maximal~ty.
z + ag,
hence, for
true
x £ L
contradlcting
L + Ag = A
by minimality of
Suppose not, then
L 6l Ag.
1
Thus
A
1 £ L
Now
M~n(A)\L.
IJe shall show that
A(l - q) .
ux - 1 £ soc (A) were,
In particular, for some
so
AX" Aq = (0).
Ax
is a proper left ideal of
left ideal soc (A)
axel - q) ,
Let
X
X
and
~n
We construct Barnes ldempotents for a Fredholm operator X.
Recall that
F(x) = soc(B(x».
T
T
is examined
the proof of the Atkinson characterisation (0.2.2).
= ker(T) e z = W e T(X) dim(ker(T)}, dim(W) <
where 00
If
Z A
and £
Ware closed subspaces
B(x),
27
z
Jeer (T) ker(T) All TA
ker(T)
Al2
w
1--"='-----1----'==--\
z
Since
T22
is invertible,
A
A22
ran(T)<=>
E
Z
T(X) LT22A21 T22A22
o
A21
So if
A E ran(T) ker(T)
~
w
A
Z
T(X)
and a right Barnes idempotent for
T
is
Q
since
ran(T)
for if
=
Q!(X).
It
easily checked that
B(X)T
= B(X)
(I - Q).
B E B(x},
W
T'X)
ker(T) BII B'r
while if
ker (T)
Bl2
t---+--\ Z
B22
C E B (X)
C{l - Q}
ker(T)
Z
ker(T)
B12T22
Z
B22T22
f
ker(T) Z
Z
T(x~rn
leer (T)
28
LS
Z
ell
Cl2
C21
C22
ker(T'EB .ker(T'rn ker (T)
Z
ker(T)
Z
I
Z
Z
C22
Since
is ~nvertLble the equations
T22
be solved uniquely for that
~s
Z
B12 , B22
arb~trary
an
Barnes ~dempotent for
is
TB(X)
w~ll
A
A
~s,
is a
x, Y
pr~m~tive
A
£
tation of
and
If
X" 0 "y
space
l~near
Now
such that
~{zyln =
X
are two
Y
d~mens~on.
F.2.1
LEMMA.
Let
there exist
(il)
dim (eAf)
(iii)
dim (Re)
~.
Since
whose kernel
Af
~.
C
such
~s
£
TI(A),
primitive if
CO)
representation.
£
A
y
,Jote that,
is zero.
fa~thful irreduc~ble
Then there
ex~st
is a subspace of
~ (A) ,
~(Ay)n =
hence
spaces we
wr~te
d~m(X)
inf~n~te d~mens~onal
and
such that
n £ X
X
which is So
,,0
f
=
such
there exists
so
= dim(Y)
xAy" (0) to mean
or they have the same
be a right ideal of A,
R
represen-
~,
X.
~(XzY)n = ~(x)~
But then
l~near
or
be a
~(Ay)n
e, f £ Min (A) u, v
x
~
X.
=
then
uev;
d:l.m (AfjRf) •
We have observed that ~s
Banach algebra over
= 1; = dim(Rf);
d~m(Ae/Re)
(i)
A
irreduc~ble
and let
that either the spaces are both
(iv)
B(X)
T.
{OJ => 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) <
infinite dimensional so
~s
<
d~m(Re)
So if ~f
d~mension
00
00
I
then
I
Rev
~s
so
~s
un~ty.
~s
dim (Rev)
and
dim (Re) < dim(Rf).
and therefore So is
Re,
and conversely. (iv)
Let
Sf
5
linearly
~s
=>
Suev
=>
Sue
be a subset of
is linearly
~ndependent
ue f 0,
s~nce
Re = Rtue <::. Rue CRe) . dim(AejRe);
modulo
~ndependent
Rf,
linearly independent modulo Ruev, by
~s
(To see this
A, then
~ue
modulo
= Ae
so
dim (AfjRf) <
tue = e
potent of
th~s
section
e
-+
that is
Ae ,
important representation.
which is
=> dim (AfjRf) < d~m(AfjRf).
will denote a
idem-
f~xed m~n~mal
B(Ae)
on the Banach space
=> 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) <
00
The infinite dimens~onal results are clear For the remainder of
Re.
dim (AejRe) <
It follows that
similarly
Rue
(~),
= 0,
invar~ant
~t
~s
under
It
~s
x(y)
Further
for each
X,
xy
for
y
E
Ae.
Banach algebra
left ideal of
A which implies that
representat~on
~s
x S A,
e~ther
L
~f
is a subspace of
it follows that = 0
L
or
L =
A
This is an
norm reducing, hence continuous and
fa~thful. "-
=
prim~tive
Ae.
L
s~nce
Ae ~s
a
Thus the
~rreduc~ble.
Observe that
x(Ae)
and Si~e
xAe,
ker(x) = xA
ran(x)~
and
ran (x)
Ae = ran(x)e. are
r~ght ~deals
of
the rank, nullity and defect of the operator
A,
it follows from F.2.1
xS
B(Ae)
that
are independent of
the particular choice of
e S Min (A) .
As the
follow~ng
example illustrates
we can say even more when dealing with the algebra of bounded
l~near
operators on a Banach space. F.2.2
EXAMPLE.
subalgebra of
Let
~
g
where
F(x).
conta~n~ng
B(X)
m~nimal
algebra and we fLX our y
be a Banach space and let
X
y S X,
A
be any unital closed
is a primitive Banach
Ldempotent to be the rank-one projectLon and
g S X*
Then
A
g(y) = 1.
The representation space is
now
A(y
since
~
A
Ay
g)
conta~ns
.r(x ~ gl
defines the
x
g
~
~
g,
all rank-one operators .
Tx
Then, if
T S A,
g
~
A
correspon~ng
T S B(X ~ g) •
It LS clear that the rank,
A
and defect of
nul1~ty
F.2.3
DEFINITION.
F.2.4
THEOREM.
If
(il
(L)
x
A
soc(A)
we defLne the rank of
~nduct~ve
Note that
n = 0 => x = O.
=
{x
C
exists a non-zero
usA
such that
v
xue
p.2.S
~f
n
xAeCl: f.Ae 1
Since
(1 -
I O.
Then
x
soc (A) •
f S Min(A) • f) v
o
SLDce
xueA
LS a
Now there
Lt follows that
hence, by the induct~on hypothesis/ x - fx
x c soc (A)
Conversely,
fAe cxAe.
C
S
dim (xAe) < n + 1.
Suppose, then, that
ex~sts
dim«l - f)xAe) < n,
S
soc (A) •
which completes the proof. £
soc (A)
n
I
xAC l: f A 1
dim (xAe) < n
and
where
(l < ~ <
nl
l
by F.2.1
l
DEFINITION.
exists a
rank (xl
rank (x)
dim (xAe) < n => x
minimal right ~deal which therefore contains an
hence
by
A : rank (x) < oo}.
hypothesis assume that
is primLtive there
Therefore
x
X.
~s obv~ous.
(ii) As an
A
S
as an operator on
T
x = 0 <=> rank(x) = 0,
(~i)
~.
are equal to those of
T
y S A
x
lS
such that
defined to be a FredhoZm element of xy - 1,
yx - 1 € soc (A) •
A
if there
The set of Fredholm 31
elements of If
~s wr~tten
A
~s
soc(A)
A
invertib~lity
By BA.2.4
invertibil~ty
A
a proper ideal of
then
modulo
modulo soc (A)
k(h(soc(A»). ~(A)
it follows that
i-lext we link Fredholm elements in
Proof.
x s
THEOREM.
~(A)
=> ~
A
ran (x) (\ Ae
x(Ae)
rank (p)
so
00
of
x
DEFINITION.
rank(p) <
x s ~(A)
If
which
k(h(soc(A»).
of
Ae.
Ae.
F.l.9 and F.l.lO.
(F.2.4).
Ae,
and
00
(F.2.4) th~s
•
theorem is false.
we define the nullity~ defect and index
by
n (;;;),
nul (x)
Now if
x s
~
(A),
def (x)
~
q, p
d (i),
ind (x)
is a Fredholm operator on
nullity, defect and index; where
A
pAe,
Example F.4.2 shows that the converse of F.2.7
to
semigroup of
which is closed in
Ae/(l - p)Ae
Ae/x (Ae)
otherw~se
qAe,
qA(\Ae
(1 - p)Ae
xAe
~nvertible
with Fredholm operators on
p, q
rank(q) <
n (5{)
therefore
is
~s equ~valent
is a FredhoLm operator on
\1e use the Barnes idempotents
ker (5{)
x
Since the latter ideal is closed in multiplicat~ve
is an open
is stable under perturbations by elements of
F.2.6
~(A)<=)
x s
(We often implicitly make this assumption,
modulo soc (A) . A =
~(A).
further,
Ae
nul (x) = rank(q)
are right and left Barnes idempotents for
with the same and x
def(x) = rank(pl in
A,
respectivel~
.,.We are now in a position to reap the benefits of the connection between the Fredholm theories in a primitive Banach algebra with minimal ideals and
on a Banach space. F.2.8
THEOREM.
Proof.
x
{x S
Inv(A)
o
nul (x)
Inv (A),
S
<=>
xA = A = Ax,
<=>
x s
and the Barnes ldempotents
<=>
x S
and nul (x)
F.2.9
THEOREM
(Index) .
(l)
The map
x
(ii)
lnd(xy)
-+
ind(x)
:
-+
-f'
ind(x)
lnd(y)
(iv)
lnd(x)
ind(x + u) ,
(i) The map
h'
VJ
x
-+
X
(x
S
rank(q) = def(x)
tt
is con tinuous;
(x, y
S 1>
(A) ) ;
y lie in the same component of
and
X
z
p = q = 0,
o
rank{p)
lnd(x) + lnd(y),
=
(iii)
Proof.
def(x) }.
u
k(h(soc(A»»
S
lS contlnuous.
•
Now use the continuity of the
index for Fredholm operators. ind (xy) = i (xy) = l (x) l (y) = i (x) + l
(E)
(iii)
x
If
x
Y
and
(iv)
x
yare connected by a path In
In the Fredholm operators on and
lies in
F.2.l0
and
u
lnd (x) + ind (y) •
the same is true of
Ae. {x + AU
are connected by the path ;\row use (iii)
•
=
(9)
o < ,\ <
l}
which
tt
THEOREM (Punctured nelghbourhood).
If
x S
there exists
S > 0
such that (l)
nul(x +
(ii)
def(x +
(iii)
lnd(x + \)
Proof. 0.2.7
A) A)
is a constant
~
nul (x) ,
(0 <
is a constant
~
def(x)
(0
is a constant,
(I AI
f
<
I AI I AI
< s);
< s);
< s) •
APply the punctured neighbourhood theorem for Fredholm operators •
If
and
lnd(x) < 0 (> 0)
there exis ts U S soc (A) such that x + AU is left (right) invertible for A # o. We may choose u s pAq where p and q are left and right Barnes idempotents for x &n A (respectively). F.2.ll
THEOREM.
x s
33
~.
f~rst
We consider
the case of
left and right Barnes idempotents for
Because
1 - p.
xz
Since
ind(x)
t(s B(Ae»
while
s (qae) so
the spaces
5 (£ B(Ae»
such that
A
£
A,
in
qAe
~t
then
such that
and
pAe
Write
sqae
psqae;
'G.(qae)
S~milar1y
qAe u
pstpae
pae,
and
"u(ae)
vuae
qtpsqae
qtsqae
qae,
by choice of
and
t. x
representat~on
= 0 = vx,
+ A-1 v)
x + AU
(x
hence
S
+ AU)
x +
AU
(x
UV
So -+
X
~s
hence for
+ AU)
A
p,
f~nite
-1
+ A
ind(x)
AA
A.
= qtp
then
Hence
uv
p,
vu
uz
o
xv
so
Si~larly
q
1,
~s
such that
Inv(A)
for
A"
be a basis for
qAe
and
therefore
{c l ' ••• , c n } where n < m.
let
{d1 , ••• , dm} a basis for pAe theorem to construct St tEA
v
1.
1 - .t? + P
20
A
q.
vu
+ q
has a right inverse and
For the case
onto a basis of pAe
A " 0,
1 - q
v)
ex~st
faithful.
has a left inverse in
(z
(1 - q)A.
€
£\pAe.
"\pAe
psqtpae
Thus
z
psqae,
uvae
(y
be
1 - q,
= psq,
'G."(ae)
yu
yx
have the same
Now
Now
q
Ax = A(l - q) ,
y £ A(l - p),
takes a basis of
reverses the process.
S IqAe.
u\qAe
as the
and
P
So by the Jacobson density theorem «75) 2.4.16) there
dimension. S, t
x
,re may take
px = xq = 0,
= 0,
= o.
y, z € A
(1 - p)A, hence there exist
xA
~nd(x}
~n
o.
Now use the Jacobson density
S (c
1.
ts
Then
)
(1 < i < n),
di
(1 < 1. < n) •
1.S the ident1.ty map restr1.cted to
x + AU
gives
c1.
left-1.nvertible in
with the case of
1.nd(x) > 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) <~ soc(A')
invertib~lity
I (Al •
A
~
Thus
(A)
modulo
then
x £
~(A)
then
A
~
(A) •
=
(0) <=> ~(A)
soc (A)
x
is invertible
At the other
Inv(A)
(BA.2.2).
is equivalent to invertibility
is an open semigroup of
under perturbations by elements of
<~
I(A).
A
which is stable
Note that by BA.2.2 and BA.2.S
we have
Inv(A) ,
If
P
Inv (A I
TI(Al
£
then
) ;
A/P
III (A) ,
~s
~ (A • ) ;
pr~mitive
a
fact enables us to develop Fredholm theory the structure space of
A/r(A)
I (A ') •
I (A) •
unital Banach algebra and this ~n
A.
is isomorphic to
Further, by BA.2.3, h(I(A»
=
h(psoc(A».
Thus using BA.2.2 we get
~(A),
X £ <-?
x
is invertible modulo
I(A),
<=>
x
is invertible modulo
P
F.3.3
LEMMA.
If s'
(P
£
h(soc(A»).
there exists a unique
£ Min(A'}
P £ TI(A)
s ¢ P,
further
Proof.
The first statement follows from BA.2.S and BA.3.S.
s
¢
s + P f 0,
P,
+
(s
But sas -
s'
AS
P) (a
+
~(A')
£
£
M~n(A/P)
s + P £
so if
p) (s
so
rad(A) CP,
+
•
a £ A
p)
sla's'
sas + P.
AS'
~or
A(S
+ p),
so
(s + P) (a + P) (s + P)
such that
some
A £~,
hence
Now since
s + P E: Min (AjP)
that is F.3.4
THEOREM.
o of
TI(A)
(1)
£
£
~(AjP),
+ p
£
Inv(AjP),
If
Min(A')
Proof.
~(A)
there exist
y£
such that if
y + p y
(ii)
If x
•
and
A
a1
,
(1
5i
2
=¢
t'. £ Min(A') ]
1 ~
and a finite subset
> 0
! Ix - y! I
<
then
£
(p £ 0);
(P £ TI (A)" Q) • then
~(A)
= Inv(A)
suppose that this is not the case. Ui
£
b ., t
t);
for each
]
Then if (l .::. j
]
<
m)
and the theorem is tr1v1al so x £
~(A)
in
there exist
elements
such that
A
and
i, j
!.'.,
ux - 1 - L s a 1 l 1
£ rad(A) ,
n xu - 1 - L t b j £ rad(A) • 1 j
By F.3.3 the set in
TI (A) •
h({sl" •• sir t l , ••• t m})
ux + J
x
A
and
1 + J
xu + J,
is lnvertible modulo
yl 1 < £ o => 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 <
P n }, then there ex~sts
NOW if
{Pi"'"
I Ix
Ek => 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 <
(0
<
I AI
I
then there exis
E);
< E);
~.
Choose
£
as in F.3.3 and label It
punctured neighbourhood theorem In
priIDltive algebras (F.2.10) there exist
nul (x + A + P k ) nul{x + P k )
=
£
for
o(x}
mln{£ o , £1"." is similar
0 <
for
IA!
=0
o
=> index}
(1 ~k < n).
< £k
and the result for
£n }
<
which is a constant
vex)
The proof
follows.
tt
We remark that In a general Banach algebra if l(X)
Then by F.3.4 and the
such that
positive numbers
Take
£0.
~(A)
x c
then This
but the converse is not necessarlly true.
fact has important consequences wmch were first observed by Pearlman (69). If
lS an lndex-zero Fredholm operator on a Banach space then we have the
T
following lmportant decomposltlon (0.2.8), ible and
F
T = V + F
V
where
leT) < 0
of finite rank (Wlth analogous results If
is lnvertor > 0).
The converse lS obviously true and by F.3.7 and F.3.8 It extends to general However, as the next example shows, if
Banach algebras. index) = 0 F.3.10
i t does not follow that Let
EXAMPLE.
HI' H2
x
x £
~(A)
and
has a correspond:Lng decomposition.
be infinite dimensional separable Hilbert
A = B(H l ) e B(H 2 ) • Then A is a semislmple Banach algebra soc(A) = F(H l } e F(H 2 >. Consldering Fredholm theory in A relative to
spaces and take and
the socle,if
T £
~(A),
T
Tl
e
T2
and
ind(T) = l
HI
(T l ) + i
H2
{T 2 }.
5 = Ul e V 2 where U l lS the forward unllateral shift on HI and V 2 Suppose that the backward unllateral shift on H2 • Clearly lnd(S) = O.
Let
there exists
F £ soc (A)
FI E
<~>
+ Fl £ Inv
But
Ul
F(H I
P2 £ F(H 2 )
},
where
such that
B(H l
}
and
iH (Ul + F I ) = iH (U I ) 1
and
S + F £ Inv{A) • S + F
(U I
Then
F = FI 9 F2
I
+ F 1) e (V2 + F 2) E Inv(A)
V2 + F2 E Inv{B(H 2 }) • - 1
+ 1
while
1
which is impossible. To overcome this dJ..fflculty we employ the index function. tations Pl<~
SI e 52 + 51
H(Hl ) 9 (O),
contaln
soc(A}.
and
51 9 52 + 52
The represen-
are clearly irreducible therefore
(0) e B(H2 > are prlmitive ideals of A which do not Suppose pc. II (A) , soc (Al ¢::. P, then there eXlsts
P2
=
E E: Min (A)
are non zero.
I
E2
=
P
0,
E ¢ P.
such that If
I
o I
Si.nce
E ¢ P 2 se by Thus we have shown that El
Pl'
t
=
El
PI
and
soc (A) •
e:Lther
E2 ,
6)
=
SA.3.S, P
0,
pr il!ll. t:L ve :Ldeals which do not conta:Ln P
E
P2
P2 ,
or
El
E2
siIllLlarly :Lf
are the only two
So for
T E: ell (A)
and
On the other hand if T = Tl 6) T2 I PI or P 2 , t (T) (P) = O. (T) " 1 (T) (P 2) = iH (T 2 )· It is now easy to see that if (T) (PI) = 1. H I'
1. {T}
and
:: 0
1 then
T2 ,
T = V + F
i
(T)
HI
1
=0
and
where
V E:
2 0 so/by applying 0.2.8 to :LH (T 2 ) 2 Inv(A) and F E: soc (A) "
Tl
This :Ldea can be made precise.
F.3.11
there eX1."sts A
If x c ell(A)
THEOREM.
and
such that
u E: I (A)
leX)
{p}
< 0 (> 0)
for all
P E: TI{A)
is leftrright) invertible for
x + Au
I o.
Proof.
Ive cons:Lder the case
1 (x)
_ 0,
the rema:Ln:Lng cases may be handled
as in F.2.11. Let
x
ell (A)
x,
:Ln
such that
A'
p' E: soc(A')
x'
then
p, q E: A
exist for
£
is invertible modulo
orthogonal subset of
sl"'"
sn
(Sl + ••• + sn)p modulo P
Thus xA
(l -
{PI"'"
(sl + ••• + s }p}A n
{sl' ••• , sn} C P, Pm}
except for deduce that Now for
of P
TI(A} ,
Pk x
such that
p')A'.
{s'l"'"
S:Lnce s'n}
is an
modulo rad(A),
hence p
so if
A
E:
= (I' -
x'A'
M:Ln(A') and
(sl + ••• + sn)p
P
Thus there
q' are left and right Barnes idempotents
p' and
(F.l.lO), :Ln part:Lcular
there eX:Lst
soc(A'}.
(p E: TI(A».
modulo P,
which is true for all but a f:Lni te subset i t follows that
(I < k < m) •
Since
:LS invertible modulo
P
x
is r:Lght invertible modulo
1 (x) (p) "" 0 for
1 .:::. k .:::. m I by F. 2.11 we may choose
P
I Pk ~E A
(P E: TI (A) )
P
we
(1 .:::. k .:::. m) •
such that
41.
with
x + Atk
invertible modulo
Pk
o.
Af
for
Put
Now
S'
€
i
soc(A') => s,
~
E
I(A)
A(
LSi) ( L si)Ptk si¢P k 1
for each
i
I
hence
'\: €
I(A).
n
x + A,\:
Then
x +
x + A~
which is invertible modulo primitive ideal except
x + AUk
x
Pk
modulo
Pk
•
modulo
Pk ,
Pk
for
A
f O.
for
P
"I P k •
Further
~
11e5 in every
So
modulo
P
m lIfrite
U
L
1
'\.
x + AU
Then
modulo
x + A~
which is invertible modulo x + AU = x modulo
Pk
for
P
P
is invertible modulo
P
x + AU
A 1= 0
E:
Inv(A)
for
for
Pk
A "I 0
and
1 < k ..::. m,
(1 < k < m) •
Thus, for
for
f Pk
P € II (A) •
•
while
-
A "I
0,
x + AU
It follows by BA.2.2 that
A final generalisation of our theory remains. F.3.12
DEFINITION.
An
inessentiaL ideaL of A. that
xy - 1, yx - 1
€
K
~deal
An
K of x s A
A
such that
K C I (A)
such that there exists
is called a K-FredhoLm element of
K-Fredholm elements is denoted by
y S A A.
such
The set of
~K(A).
We can develop a Fredholm theory relat:Lve to each SUch (BA.2.4), without loss of generality we can assume to equal k(h(K».
:LS called an
K
K
and, by
to be norm closed or
The statements and proofs all go through with only the
obvious modificatlons.
An inessential ideal of particular importance is
the algebraic kernel which lS considered in §F.4. FA
Notes
Fredholm theory in an algebraic setting was pioneered by Barnes (7), 1968, 9 In the context of a
(8), In
semiprime rlng {one possesslng no non-zero
nilpotent left or right ldeals} .
He used the concept of an ideal of finite
order to replace the flnite dlmenslonallty of the kernel and co-range of a Fredholm operator. F.4.l
DEFINITION.
A right ideal
J
In a semiprlme ring
A
has
finite
order if It is contalned in a finlte sum of minlmal right ideals of A (wlth a corresponding deflnltion on the left). written
ord(J),
whose sum is
The
order of an ideal J,
lS deflned to be the smallest number of minimal ldeals
J.
The connectlon with our work is clear, for if
x E
~(A)
are left and right Barnes ldempotents, then the left ldeal
and
p
Ian (x)
and
q
Ap, and
the right ldeal
ran{x) = qA, both have flnite order so the nullity, defect
and index of
are defined by the formulae
If
x
nul (x)
ord(ran(x) )
ord(qA) ,
def(x)
ord (lan (x) )
ord(Ap) ,
lnd(x)
nul (x) - def(x) •
A
ord(Ap)
lS prlmltlve and rank(p)
x E
~{A}
then
so the deflnltlon
ord(qA) =
rank{~and
of these concepts coincides
w~th
our
own. The index theory which Barnes obtalns is more general than that developed
in Chapter F as it lS purely algebralc In character, but each result must be proved ab lnitlo, and the prelimlllary manipulahons are rather involved. Our approach, developed by Smyth, Vla the left regular representation of a primitive algebra
A
Predholm elements In direct.
on A
Ae
where
e E Min(A),
and Fredholm operators on
and the Ilnk between Ae
(F.2.6) is more
However our theory lS less general than that of Barnes, for F.2.1(ii
43
requires that
A
be a Banach algebra.
representat~on wh~ch
The
correspondence between the
x
and
we have used is well known «75) 2.4.16), the d~mens~ons
of the kernel and the co-range of
are the key to our expos~ tion of Fredholm theory.
example of a pr~m~t~ve Banach algebra A
such that
x
is a Fredholm operator on ~n
F.2.6 is false F.4.2
EXAMPLE.
°
Let
T
~s
0B(X)/K(X)
sh~ft
on a separable
prim~tive
a
show~ng
Ae
ile now
¢
and an
g~ve
x
t
an ~(A)
that the converse of
be an operator on a Banach space
X
such that
(T+K(x»
H~lbert
to be the closed un~tal subalgebra of A
Min (A) #
with
general.
WeT)
(T)
(The bilateral
Then
A
x
space
~s
an example) •
R{x) generated by T
Banach algebra with
Min (A)
and
Take
A
K(x).
i ¢ and, as in F.2.2, A
the rank, But
null~ty
T € Inv(A)
nul(T}
0=
and defect of T in B(X) are those of T ,.. so T € Inv(B(Ae)}. Suppose that T E ~(A)
def(T)
so, by F.2.8,
T E
Inv(A) ,
hence
~n
B(Ae).
then
T E Inv(A) modulo
K(X) • However, the unital Banach algebra T + K{X)
so
0A/K(x} (T + K(X»
A/K{x)
~s
generated by the element
has connected complement
«14) 19.5}.
Further
I}
IAI Therefore
\ AI
{A:
T E Inv(A) modulo Th~s
:5.. llc.0A/K(X)
('1'
'TT:x-+-x
(T + K(X) }c.oA/K(X)
+ K(x»
(T + K(x».
which contramcts the fact that
K{X). a drdwback of the
exh~bits
for a general
0B{x) /K{X}
A -+-
representat~on
B(Ae}
primit~ve
Banach algebra.
Further
lnves~gatlons
into this
case have been carrled out by Alexander «4) §5). If, however, more useful.
44
A In
~s
a prlmltive C*-algebra then the representatlon
the flrst place, as we see in §C*.4,
Ae
'IT
is
can be given the
inner product
<x, y>e
ey*xe
y*x
(x, y E: Ae) ,
under WhlCh l t becomes a Hllbert space in the algebra norm.
'If
lS then a
faithful irreduclble *-representatlon WhlCh is therefore an isometry. Henc.e
(BA.4.2) •
~urther,
the converse of F.2.G, lS valld in this case.
atlon of C*.4.2 and c*.4.3 shows that, Slnce ideal of
A
WhlCh does not contaln
a singleton set, the
THEOREM.
Let
A
lS the only primitive
by BA.3.5,
'\
III
c* .4.3 becomes
ln C*.4.3 lS dlspensable and we can take
TI2
the representation deflned above. F.4.3
soc(A)
(0)
In fact an examin-
TI
to
Thus we have
be a primitive unital C*·algebra lu'ith
e E: Min (A) ,
then (il (ii)
1T(soc(A» 1T (&OC (Al )
F(Ae) ; K{Ae) ;
(iii)
1T(R(A) )
R(Ae) f"\ TI(Al ;
(iv)
TI (
(R (A)
lS the set of Rlesz elements of
A
relatlve to
soc (A)
defined ln
R.l.l) • F.2.3 and F.2.4 contaln a definltlon of rank for elements of a prlmitive Banach algebra as well as a characterlsation of the socle as the set of elements of flnlte rank.
rln alternatlve defin1tlon of finite rank elements
via the wedge operator 1S glven in C*.l.l xl\X E: F(A))
(x
is of f~n1te rank in
A
if
and we show that, ln a C*-algebra, the set of flnite rank
elements is equal to the socle (C*.1.2). this result to semasimple algebras.
Alexander «4) 7.2) has extended
In primitive algebras the two defin-
itions are eqUlvalent. Returning to Fredholm theorY,Barnes'ldeas for semisimple algebras were extended by Smyth (83) to general Banach algebras and th1S approach 1S fOllowed here in §F.3.
Pursuing suggestions of Barnes (8) and Pearlman (69)
45
Smyth introduced the x £ 4l(A}
~ndex func~on
~nd(x)
and
= 0,
then
sum of an lnvertible plus an this in F.3.10
~s
decomposi~on ~n
i (T) < 0 [v,F1 2
v + F
T
(~O),
= 0
Let
lnessen~al
~s
element.
g~ven
V
is left
[V,
F] = 0
by Murphy and West (62) and
(r~ght)
T E 4l(X},
~nvertlble
T.
i{T)
~t
then
accordlng as
and the decomposl tion may be chosen so that
[v, FJ = VF - FV.
= 0,
then;elther
the
on this
This result
~s
best
posslblel~n
i t is not always posslble to choose a decomposition such that for example/lf
~nto
The orlglnal example of
~'urther lnforma~on
be a Banach space Wlth
where
F E F (X)
where
X
problem that/if
is not always decomposable
due to Pearlman (69).
Laffey and West (55). is shown that
x
the operator case
w~th ~~e
(F.3.5) to cope
T
=V
+ F
T E Inv(B(X»
where
V E Inv(B(x»,
that
[v, F] = 0,
~ €
F(x)
and
, or zero is a pole of fin~te rank of
Using the techniques of thlS chapter these results can also be trans-
21anted lnto Banach algebras.
The lndex functlon for Fredholm elements In
a general Banach algebra has also been deflned by Kral]evlc, Suljagic and Veselic (110) maklng use of the concept of degenerate elements dlscussed In
§R.5. If
A
{Th~s
one may adjoln a unlt and proceed as In thls chapter. necessary in Chapter R, Eor setting} •
'¥
=
'¥ {A}
R~esz
ex~sts
We say that
R
y € A
x E A
lS
quasi-invertible
and
I
A
modulo an ldeal
x + y - xy, x + y -yx € F.
such that
Let
k(h(F})
I
all of whose scalar
mul~ples
elements of
F.
The elements of
relatlve to
A
statlng some useful results
~n
F.
'¥
\1e conf~ne
quasi-Fredholm theory.
F
The set
R
and let lle In
are the set of Rlesz and ~nessentlal elements of
(respectlvely) rela~ve to
Fr>edholm
theory must be done In a non-unital
lS the set of all such elements.
F denote the set of elements In Then
wlll be
However as Barnes (8) and Smyth (83) showed,a dlfferent approach
may be adopted. if there
theor~
lS a non-unltal algebra then,ln order to carry out Fredholm
'¥.
A
are called the
quasi-
oursel"es here to The flrst follows
from the fact that a quasl-inver~le ldempotent must be zero.
F.4.4
THEOREM.
Eve~
idempotent
Of
'¥
lies in
F.
In operator theory much lnterest has been focussed lmpllcltly upon the quasi-Fredholm ideals lncludlng the ldeals of flnlte rank, compact, strictly singular and
inessen~al
operators.
In the algebralc context we note the
following very general result starting with any quasl-Fredholm ideal
46
J.
The proof depends on elementary properties of the radical and the fact that we can ~dentify the structure space of F.4.5
THEOREM.
Let
J
A/J
w~th
the hull of
be an ideal of the algebra
J
such that
A
«83) 4.2) Fe JC '1',
then (l) (ii) (~ii)
(iv) {v}
(vi) (vii)
x E 'I' <=> x + J x E I
<='> x + J
x E R <""'> x + J
is quasi-invertible in A/J, is -z,n the radical of A/J, is quasinilpotent in A/J,
FC. J c...I eRe '1';
= h(J) = h(I}; I 1-S the largest left or right ideal lying in A/J is serrtisimp le <-> J = I. h(F)
F.4.6 COROLLARY. Anfj one of the sets each of the others. «83) 4.3}
I,
The results thus far are valid for an index theory we need to restrlct
F
R, '1', h{F)
'1'-,
uniquely determines
arbitrary J..deal
to lJ..e J..n
F
of
A,
for
psoc(A).
The monograph (71), §A gJ..ves an interesting account of Fredholm theory for linear operators on linear spaces with no reference to topology. Mizor~-Oblak
Related work lS due to Kroh (54).
(59) studies elements of a
Banach algebra whose left regular representatJ..ons are Fredholm operators. If one is concentratJ..ng on an ldeal This J..S exhibJ..ted by Yang (98) who
F
the choice of the J..deal is important.
stud~es
operators on a Banach space
invertJ..ble modulo the closed J..deal of weakly compact operators.
If the
space is reflexive then every bounded linear operator is weakly compact and the Fredholm theory becomes trJ..vial. There have been many extensJ..ons of the classJ..cal Fredholm theory of linear operators of wh~ch the most important is the theory of semi-Fredholm Operators. ?4.7
DEFINITION.
and if elther
neT)
T E
B(x)
J..S
semi-Fredholm
or
d(T)
J..S fJ..nlte.
if
T(X)
J..S closed J..n
X
The basLC results for seIDl-Fredholm operators are gJ..ven ln (25), this class of operators has proved of central J..mportcnce J..n modern spectral theory. F.4.8
DEFINITION.
semi-Fredholm
An element
x
in a semlsimple unital algebra
A
is
if l t J..S elther left or right J..nvertible modulo soc (A) • 47
~f
By F.I.IO
~s
X
~dempotents
(left) Barnes
~n
conf~ne
modulo soc (A),
x
has
so we could use the methods of
theory
~n
Banach algebras.
r~ght th~s
In this
ourselves to Fredholm theory.
Another extension of the generalised
~nvertible
soc (A)
sem~-?redholm
chapter to develop a monograph we
(r~ght)
left
~nverses,
class~cal
general~sed
or
theory leads to operators which have Fredholm operators (named relatively
regular operators by Atkinson (5». F.4.9
STS =
T E B(x)
(ii)
and both F.4.l0
T € B(x)
(~)
DEFINITION.
TST = T,
has a gene~alised inverse
S € B(x)
~f
s. a generalised Fredholm operator ~f
~s
ker(T)
and
T(X)
T E B(x)
THEOREM.
T(X)
are complemented subspaces
~n
has a generalised inverse <=> T
~s
closed ~n
X
X.
is a generalised
Fredholm operator. Proof.
Let
general~sed ~nverse
have a
T
it follows that
E
E
2
Barnes idempotents for
?' = F
T
2
~n
and B(X).
S,
then if
I - E, I - F
E = TS,
are left and
IJe collect the following
TS
E => 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
E
-> 1.(x)
E
:
onto a
E
Inv(A) + K.
and that 1.f
llx - y!I <
1.
loS an index if, for
e
i(x + z)
It follows at once from the definition that Z E
and
continuous semi group hOIIlOIIlOr:;>h1.sm
~
DEFINITION.
F.4.l2
Inv(A)C
there eX1.sts
>
E
=
1. (x)
such that
0
(x
y
E
E
ct>, and
ICy).
Jur uniqueness result loS somewhat surpr1.s1.ng, roughly lot states that, for
a fixed
To make th1.S pre Close we need
K, the index is un1.que.
F.4.l3
DEFINITION.
i :
If
equivalent 1.f there
and j
:
a group isomorph1.sm
loS
e
are ind1.ces they are
G -+ H
such that the
following diagram commutes
"'e
J
H
F.4.l4
THEOREM.
Proof.
Let
such that i(xu) •
x, Y E
j(y),
Let
=
1 E K.
I, yu
y
to get
since
(eoi) (x)
be such that
xu = w + k
Thus
the right by =
There is, at Most, one index up to equivalence.
j(w)
j (x)
I
=
i(x) = i{y).
Clearly
for some
w E Inv(A)
x = wy + k'
e.
Now there eX1.sts
1.(y)-l
where
i(u) and
1jJ : A -+ A/K
Mult1.ply on
Thus
j(x) = J(w)J(y}
e
Now we can def1.ne a map
e
be the canon1.cal hOlllOmorphism.
e = 1. (x}1.(u)
k E K.
k' E K.
and it follows 1.mmediately that
so
u E
G -+ H by
is an 1.solllOrphism
•
The eX1.stence theorem
is as follows. F.4.lS
THEOREM.
subgroup of
(1.)
An index exists
<=> 1jJ(Inv(A»
InV(A/K).
If the condition in (i) is satisfied the group is discrete, and an index may be defined by setting (ii)
l.
(x)
is a closed normal
1jJ (x) 1jJ (Inv(A) )
(x
e:
G
Inv(A/K)/1jJ(Inv(A»
~.
e:
(i)
Suppose that
j
~s
Inv(A/K) -+ H : W(x) -+ j(x)
H, with w(Inv(A»
a well
S~nce
ker(8) = W(Inv(A».
then the map def~ned
~s
Inv(A)
~n
open
~s
W(Inv(A»
A,
so ~s
G = Inv(A/K)/W(Inv(A)
example, there is no ne~ghbourhood
g~ves
thus
Inv(A/K).
open in
A/K.
Inv(A/K).
Hence
tt of
poss~bll~ty
which
~s cont~nuous,
a discrete group.
for spectral theory, for
su~table
an analogue of the punctured
obta~n~ng
In a sense, as the next result shows, any
theorem (0.2.7).
~ndex
Fredholm
~s
deflned here is not
~ndex
onto
a closed normal subgroup of
W(Inv(A»
Part (ii) now follows easlly The abstract
8
W is open,
is a closed normal subgroup of
Conversely, suppose
homomorph~sm
group
r~se
sat~sfactory
to a
~s
spectral theory
encompassed
w~th~n
the work of thls chapter.
~nessent~al
~deal,
then the results of the classical spectral theory of
bounded
l~near
that
t.he results of
~f
As we have seen, if
operators extend to Banach algebras.
is an
Now we show (informally)
r'redholm theory extend, then
class~cal
K
K
must be an
inessential ideal. He shall make use of the as those ideals lation
po~nt
valid.
Let
isolated
po~nt
A
S
~ts
of
relat~ve
that,
~n wh~ch
spectrum.
of
i
K, the results of
AS
and
0 I
o(x).
It
x E K
of inessential
Suppose that
to the ideal
Since
dO (x) •
character~sat~on
x S K,
~s
0 < I~
- II
v(~ - x)
< s,
clearly
class~cal
suff~c~ent
~s invert~ble
punctured
ne~ghbourhood conta~ns po~nts
O(~
are both zero for
-
x)
0 < I~
-
o(~
and
-
Ai
of
x)
O(x)
~n
R. 2.6
P(x)
< ~
and
K
~ndex
and
Fredholm theory are A
is an
to do so for each
modulo
ex~sts
hence
s > 0
v(~
such that
But this
- x)
It follows by the
p(x),
A - x S
K, hence
are constant.
theory that th~s punctured neighbourhood lies ~n isolated point of
a generalised
"Je need to show that
o(x) •
A - x
~s
Thus, by the punctured neighbourhood theorem, there for
~deals
each element has zero as the only possible accumu-
hence
and class~cal
A
~s
an
is therefore an lnessential ~deal.
atgebpaic If it satlsfies a polynomial an algebra is atgebpaic if every element therein ~s algebraic.
An element of an algebra ~s
identlty,
wh~le
The algebpaic algebra.
key~el
of an algebrd lS the maximal algebraic ideal of the
Its existence is demonstrated in (48) p.246-7 where it is shown
to contain every
r~ght
or left algebraic ideal.
The original setting for algebraic Fredholm theory was a semisimple Banach 51
algebra and it was in thlS context, and relatlve to the socle, that Barnes(7) developed the theory lD 1968.
In 1969 he extended it to semiprime algebras.
In the general case the socle does not always eXlst
~nd,
ior this reason,
smyth (83) and Vesellc (93) lndependently developed Fredholm theory relatlve to the algebralc kernel.
In fact Smyth has shown «84)§3) that the algebraLc
kernel of a semislmple Banach algebra lS equal to the socle.
A lLttle more
effort extends thlS result to seIDlprime Banach algebras.
A
If
1s a
general Banach algebra and Lf Srrqth's result lS applled to the quotient algebra
A' = A/rad(A)
it foJlows that tCle algebraic kerpc: cf
contained In the presoclc.
A
is
R Riesz theory
In this chapter the Ruston to define
R~esz
character~sation
of Riesz operators (0.3.5) is used
elements of a Banach algebra relative to any closed two-sided
proper ideal, and elementary
algebra~c
developed in §R.l in this general
properties of Riesz elements are
sett~ng.
It transpires, however, that in
order to obtain the deeper spectral theory of Riesz elements the ideal must be an inessential ideal and such a situation is investigated in §R.2. ~s
Finally the theory of Riesz algebras ~n
Riesz algebras are listed
§R.4.
Note that the algebras considered in
this chapter will not
necessar~ly
R.I
Riesz elements:
algebraic
Let
A be a Banach algebra and let
R.l.l
rex
+
DEFINITION. K)
o.
=
x
RK (A)
A
t
be unital.
propert~es
K be a proper closed ideal of
is a Riesz element of
R (when
= R(A)
will denote the set of Riesz elements of This
defini~on
~deal
K
~s
(relative to
A
A. K)
if
unambiguous from the context)
A.
is motivated by the Ruston characterisation of Riesz
operators (0.3.5). inessential
developed in §R.3 and examples of
In the next section, having restricted
K
to be an
we shall demonstrate the familiar spectral properties of
Riesz elements. Let
[x,yJ = xy - yx
follow~ng
R.l.2
denote the commutator of
THEOREM.
(i)
x
£
R, y
£
K
=> 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) <=13(x)
a consequence
a closed subset of
be a unitaL Banach aLgebra and
A -
W(x)
Proof.
(y E K),
is a compact set.
of the next theorem R.2.2.
R.2.2
w(x) = 0A/K(x + K),
Obviously
0A/K{x + K)CO(x + v)
and since Clearly
x E A
o(x).
x E A, then
is not a Fredholm element of index-function zero}.
A
of
complements the result may be restated as follows,
p(x + y}
A -
X E
A-
X
cP (A)
and
1 (A -
x)
a},
yEK and, by
us~ng
U p(x
F.3.ll,as
+
y)
E Inv(A) + K}.
yEK This last statement is true since
A E UP(x + y) <=> A - x - Y E Inv(A) YEK <=> A - x E Inv(A) + K
R.2.3 o(x) Proof. ition
LEMMA. A Riesz point of x and p(A,X} E K. Let
A
A
be a Riesz
is 1so1ated in
p(A,x)
1 211"i
Jr{z
po~nt
o(x},
of
for some
•
which is in
x
y E K,
o{x)
wh~ch l~es
in
is an isolated point of
o{x).
hence the associated spectral
- x) -1 dz e: A,
By the de fin~dempotent
where
r
is a circle ln
Slnce
A
is a Fredholm pOlnt of
p(x)
surrounding
A
but no other point of
A E PA/K(x + K),
x,
a(x).
hence the assoclated
spectral idempotent
rherefore R.2.4
p (A,X) E K
£et
THEOREM.
p (x) •
A - x E
for
(F.3. 9 ),
I~ - AI < E,
V{J-l-x),
Then every
x E A.
is a Riesz point of x.
30(x)
then every nelghbourhood of
If, in additlon,
~ - x ~ ¢
A/K.
be a unital Banach algebra and
lying in
A E 3a(x)
If
ln
• A
Fredholm point of x Proof.
o
p(A,X} + K
p(A,X + K)
there eXlsts
A E
contalns points of
>
such that
0
and, by the punctured nelghbourhood theorem
COl-x}
0 < IJ-l - AI < E.
are constant for
It follows
that
v (J-l
o
- x)
C(J-l - x)
for
0 < IJ-l -
Ai < E,
A
thus, by F.3.7, thlS punctured neighbourhood of A
lS an lsolated pOlnt of
o(x}
lies ln
p(x).
Therefore
WhlCh lS, by definitlon, a Rlesz point
•
The next result corresponds to the Ruston characterlsation of Rlesz operators (0.3.5). R.2.S
COROLLARY.
algebra and a(x)
Of
~.
If
'l'hen
x E A.
and
x E
then
r (x + K)
A - x + K E Inv(A/K) , hence A
is a Rlesz pclnt of
POints of
R <=> each
o(x) •
o(x) •
A - x E
0,
E
so lf
If
a(x)'{o}
is an isolated point
0
0
,;
,;
A E 3a(x},
A E a(x),
then then, by R.2.4,
Hence every non-zero boundary point of
It follows that
o{x)'\{O}
a(x)
{A E a(x)
: rAI > 0 > J}
is a finit~ set
a(x)
is a dlscrete set of Rlesz
x, and, by R.2.3,the assoclated spectral ldempotents lie in
Conversely, lf each non-zero pcint of set
A
be a unital Banach
A
p(A,X) E K.
x E R,
is isolated ln
Let
(Ruston characterlsatlon)
lS a Rlesz pcint of
{Ak}~' say.
idempotent associated Wlth thlS set, p = 1: p(Ak,x) E K
I
and
K. x, the
The spectral rex - px) < O.
I
57
< 0,
r (x + K) .:. r (x - px)
Thus
0 is arbitrar~ly small,
and since
tt
rex + K) = 0
In terms of the Browder spectrum this result states that
x
R <=> B(x}C{o}.
£
and
s B(x)
T
.bte that
then
The next result
~s
B(T)
~f
X
character~ses
K
~deals
the
out Riesz and Fredholm theory, and is Riesz algebras in §R.3.
fin~te dimens~onal l~near
is a
space
empty.
~mportant
relat~ve
wh~ch
to
we can carry
in the characterisation of
The theorem is valid for
~deals wh~ch
ne~t~er
are
two-s~ded.
closed nor
Let A be a unital Banach algebra and J a left or right ideal of A. Then JC:I(A} <=> zero is the only possible accumulation point of o(x) for each x S J. R.2.6 THEOREM.
Proof.
If
J C
I (A),
then
x S J => x S I (A) => r (x + I (A) )
accumulat~on po~nt
is the only
Conversely, let
of
be a left
J
possible accumulation point of may, and do, assume that
projection with unit u
S
pAp Now
= PI
p p
p(A,X)
and
0
such that A
is
show that p
A ~s
hypothes~s,
By
~deal
o(x)
SA.
sem~s~mple,
of
po~nt
of
We show that Thus
hence
q2 F PI Now
P
(so = q
I idempotent P 2....S J.
q2 S J),
+ q' 2 2 P2'
S
and
0
F
we
A S o(x).
so the associated spectral
p S J.
p
¢
pAp
Inv(pAp),
soc (A) •
there ex~sts
~s
a Banach algebra
hence there
ex~sts
and our next task
~s
to
Clearly then
Xl S PlAPl
such that
The hypothesis implies that zero
Set
with s~milar relat~ons holding for
P 2 S plAP l ,
q'2 = PI - Q2'
¢ soc (A) • Label th~s P2 ¢ soc (A) , PIP2 = P2P l = P 2 -F PI'
so, at least one of
then
O(x),
the only
0A(x) = 0A' (x')
x S J
I(A) = k(h(soc(A»),
so, by BA.3.10,
F o.
hence zero
S J.
Suppose not, then
f.l
S~nce
Let
px
~s
such that zero
(x S J).
0p Ap (xl) contains at least two points. I 1 is the only poss~ble accumulat~on point of
an isolated point
A
~s semis~mple.
an isolated
(px) pAp p = upx
pSI (A) •
¢ ~n(A}
A
0,
O(x} by R.2.5.
q2' q'2
and
process we get an infinite
Cont~nuing th~s
{Pn}~
of ~dempotents ~n
PnPm
PmP~
then
{~}l
and
A -+ 0 n
as
p + w S J
n -+
1
an
~s
properly
I An I
sA,
~nf~n~te
< 2-n
J
sequence
ll ~! 1-1
~
(n
w = wp S J.
and
{x + I (A)
~n
1),
Now
elements
necessar~ly ~n
relat~ve
un~tal
R.2.6 so as to
obta~n
= crA(x)lj{O}
~f
Al
o
rex + I(A»
def~ned ~n
§R,l.
x E: A,
A/I (Al
rhus
In R.2.1
def~ned
so
R~esz
tt
JC:I(A) ~s
not
points of an
relative to Fredholm
po~nts
of
and the results of R.2.3, R.2.S and
def~ni~on
the
J) .
(x E:
of a Banach algebra which
~deal
~n
a full Riesz theory
and denote by
x E J.
~s a kernel (BA.2.3),
I(A)
algebra have been
To extend this
and
(BA,2.8) ,
to a closed
the e lemen t.
non-un~tal
elements.
quas~n~lpotent
unital have been a
A E cr(x)'{O}
~s a le ft ~deal of the Banach algebra
E: J}C rad (A/I (A) )
X
(n > 1).
This gives the required
the proof of R.2,S, that
~s se~s~mple s~nce
A/I (A)
cr(p + w) •
of
p(A,X) E I(A), if
of
cons~st~ng ent~rely
0A (x)
= Pn
(n > 1) => (1 + A ) E cr(p + w) n
{x + I (A) : x E J}
Therefore
element
q n~
1
accumulat~on po~nt
It now follows, as
R~esz
~
and
contra~ct~on, so that
but
= PI'
ql
w = L: A
Then
00
1 + A s cr(p + q) n n
Thus
set
= pqn = ~ (n~l). ~jow choose a d~stinct complex numbers !uCh that
of
soc(A), and satisfy~ng
- Pn - l (n ~ 2), ~s a properly ~nf~n~te orthogonal family of ~dempotents ~n m > n.
for
~P
such that
{An}~
= Pn
all lying outs~de
J
family
str~ctly decreas~ng
un~tization
and ~f
a non-unital algebra let
A
~m(A)
=
of
Then,for
A.
~s an ~deal of
J
be 00,
A, ~t ~s also an
1
ideal of
We cons1der the case of se~slmple
AI'
for general
A
fOllows on
leA) = k(h(soc{A») prim1t1ve
~deal
of
factor~ng
out the
and 1t 1S clear that A 1S contalned
a pr1mi t1 ve 1deal of
AI'
~n
further 1 f
a
A
ra~cal.
soc (A)
pr~m1~ve
PIE II (AI)
=
(and Then,
AI)' the result ~f d~m(A}
soc (AI) • ideal of and
=
00,
Now every AI'
PI -:/J A
also
A
lS
then
P InA E II (A) ((48) p.206). It follows that I (A) = I {AI)f\ A and the proofs extend 1mme~ately. The case of f~n~ te d1D·ens Lcr,al A is tr~v1al. 59
follow~ng
We shall need the
consequence of the punctured neighbourhood
theorem in Chapter C*. R.2.7
Let
THEOREM.
element
in a semisimple unital Banach algebra
x
for some
11
E
Also
index.
D
of the function
and R.2.4
~nd(A
hence
A
Inv(A)
E
and
x
-+
ne~ghbourhood
continu~ty
by
discont~n-
except for a set of
nul (A - x)
{A Em,
of the
\lIUch must be countable
The result now follows from F.3.7
theorem.
character~sation
algebras:
No study of
or Fredholm theory
R~esz
inessen~al ~deals ~ntroduced
algebras such that A
x),
(A E nl
•
~esz
where
- xl = 0
def(A
nullA - x) = 0
by the punctured
of the
11 - x
n, then every point of n is a Riesz point of
ind (11 - x) = 0,
Proof.
uities
If
A.
is a countable discrete set.
a(x)('In
R.3
n be a connected open set of Fredholm points of an
~s
A = I{A}.
an algebra over
~s
complete
in §F.3.
Recall that
c.
w~thout
analysis
deta~led
a
To this end we
exa~ne
those
I(A} = k{P E TI{A) : SOC(A')C:P'}
\Ie
start
~s
a Riesz algebra if
w~th
general
algebra~c
cons~der-
ations. R.3.l
DEFINITION.
It follows
~n
~mmediately
the structure spaces of <=>
A'
is a
algebra
R~esz
from the
A
algebra,
and
A'
def~n~tion
A
A = ICA).
and the homeomorphism between
(BA.2.5), that
and that
h(soc(A)} is empty <=> A/soc(A) this case
A
is a
A
~s
~esz
a
se~s~mple
algebra
Riesz algebra <=>
~s a rad~cal algebra (BA.2.3), s~ncet in
A = I(A) = k(h(soc(A)).
Thus, sem~simple Riesz algebras are
'close' to the~r socles in the sense that the socle is cont~ned ~n no primitive ideal.
It follows, and
w~ll
be illustrated in §R.4, that the class
of Riesz algebras is a large one embrac~ng many important spec~al algebras. The characterisation of R~esz Banach algebras lS an lmmed~ate consequence of R.2.6. R.3.2
THEOREM.
A Banach algebra
(Smyth characterisatlon)
A
algebra <=> zero is the only possible aCCUMulation point of
is a Riesz
a(x}
for each
x E A. It follows that, if a Banach algebra
a(x)'{o}
is a discrete set,
a(x}
A
~s
a RLesz algebra, then
LS countable and
da(X}
=
a(x)
(x E A).
A slmple consequence of thlS is R.3.3
Let
COROLLARY.
the Banach algebra
B A
A.
be a closed suhalgebra, and
a closed ideal/of
J
is a Riesz algebra => B and A/J are Riesz
algebras. We have noted the dlscretene$ of the non-zero spectrum for every element For general Rlesz algebras the structure space
In a Rlesz Banach algebra.
is dlscrete In the hull-kernel topology. R.3.4 Proof.
1f A is a Riesz algebra, neAl
THEOREM.
Wl thout loss of generall ty take
of accumulatlon pOlnts of
TI{A}
A
is discrete.
to be semislmple.
is contalned In
h(soc(A}}
Then the set
(BA.3.6}, and lS
tt
therefore empty
The converse of R.3.4 lS false, for the Calkln algebra of an lnflnite dlmenslonal Hllbert space has zero as l ts only pruD.l h ve
j
deal but l t lS not
However, we do get a converse result In the commutative
a Rlesz algebra. case. R.3.S
:f ~ is a commutative Banach algebra then A is a Riesz ~s discrete in the hull-kernel topology <~> TI(A) is dis-
COROLLARY.
algebra <=> IT(A)
crete in the Gelfand topology. Proof.
The eqUlvalence of the discreteness of IT(A)
Gelfand topologles follows from BA.3.7.
The proof lS completed by applYlng
R.3.4, and, conversely, by notlClng that If h(soc(A)} = ~
Semislmple then
(BA.3.B)
In the hull-kernel and
TI(A}
lS dlscrete and
A
is
tt
The lnterestlng examples of Rlesz algebras are non-unltal as the next result lndlcates. R.3.6
THEOREM.
A unital Banach algebra is a Riesz algebra <=> it is finite
dimens1:onal l"1odulo its radical. ~oof.
Take
unless
A = soc(A),
A
to be semlslmple and unltal. It follows that
Whl ch lS imposslble.
e i E Mln(A) (l
2
i, j
Obvious
2 •
(1 < l n)
Now
< n).
If
A
TIlen
(BA.3.3), hence
A/SOC (A)
If
A
lS a Rlesz algebra then,
is a unltal radlcal algebra
soc(A), 1 = e l + where + e n n n L L e Ae and dim(e Ae j ) < 1 i=l 1"'1 l J l A is fiB.l te dlmenslonal. TIle converse lS A
The next result is an immediate consequence of R.2.6 if
A
algebra, but i t is also true for general algebras (a proof of
LS a Banach th~s
LS
indicated in §5), SO i t is stated in full generality.
R.3.7
A left (right) ideal
THEOREM.
<-;,
L
of an algebra A is a Riesz algebra
L CI (A) • An obvious corollary of this is that every left or right ideal of a Riesz
algebra is Ltself a RLesz algebra. R.4
Riesz algebras:
Let
A
examples x E: Ai
be a Banach algebra and
= XY},
{y E: A : yx
z(x)
the commutant of
A.
is a closed sub algebra of
x, Let
x
denote the
bounded linear map
x
y -+ xy
=
Z(X) -+ z(x).
0B(Z) (x),
0A(x)
R (X)
denotes the set of RLesz operators on the Banach space
and R.4.2
we shall be
and we write the latter set
oonsider~ng
Recall that
O(x).
Then
X.
operators on the Banach spaces
In R.4.1 Z(x)
and
A. R.4.1
each If
E:
A
then
A
let
L
x E: A
plica tion by R.4.2
three (i) (ii) (iii)
Proof.
If
COROLLARY.
x
x
on
A is a Banach algebra such that is a Riesz algebra.
and
x
A,
R
E: R{A)
(x
R
€ R(A)
(x E:
X
x /\ x Let
u
x
invariant under
for mult~-
A
is a Riesz algebra if any of the
E: A)i
A),
(x
-+ XUX E: R(A)
E:
R(z (x»
respectively.
A Banach algebra conditions hold,
Lx
E:
denote the operators of left and right
x
COROLLARY.
foll~ing
X
A
and put
T
so
~f
(i)
T = I
€ A) •
Lx
(~~)
or or
Rx (~~i)
or
x l\ x.
hold then
Clearly Z{x) = Z TJZ
E: R(z)
«23)
is 3.5.1)'
x
In cases (i) and (~~) th~s ~mplies that E: R(z) and in case (~~i) that 2 ,2 = (x )~ R(z) " (0 •3• 5) , ~mp 1"~es (x ~ ~ wh~ch, by the Ruston characterisation that . .c ...
x e:
R(Z) •
Thus, in each case, by R.4.1,
A
is a Riesz algebra
..
R.4.3
EXAMPLE.
operator on
A
The Banach algebra (x E A) •
is compact
A
~f
~s
x "x
a compact
compact Banach algebras have been studied by
Alexander (4), and, by R.4.2, are Riesz algebras. R.4.4
EXAMPLE.
The Banach algebra
continuous L.C.C. (resp. R.C.C.) A
ex E A).
R.4.2, are R.4.5
~s
A
~f
Lx (resp. Rx) stud~ed
These algebras have been
~et
EXAMPLE.
topology.
1
is a
R.4.6
EXAMPLE.
"-
Now
R~esz
G
Let
R.4.7
EXAMPLE.
Let
~s
Ln
R(x),
then
B (X)
algebra.
vanLshLng at the orLgLn then
R.4.B
EXAMPLE.
a
in the
compact, so by R.3.5,
~s
A
A
a closed subalgebra of
LS a Riesz algebra by R.3.2. T S R(X)
generated by
To see
nom~al
The Gelfand
compact.
X be a Banach space, let
R~esz
group.
~s the dual group
Ll(G)
X be a Banach space and
uni formly closed subalgebra of is, therefore, a
abel~an
is discrete <=> G
algebra <=> G
wh~ch ~s conta~ned
B(X)
by Kaplansky (SO), and, by
G be a locally compact
space of the commutative Banach algebra ~nduced
is a compact operator on
algebras.
R~esz
~-(G)
left (resp. right) completely
peT)
th~s,note E
R(x)
T,
and let
A C R (X)
then
that
~f
A
p(z)
be the and
LS a poly-
(0.3.7) •
Let
T
be a non-nLlpotent quasLnilpotent operator on a
Banach space and let
P
be the algebra generated by
of polynomials Ln I, and
I
T).
Then
Inv(P)
LS the only non-zero Ldempotent Ln soc(P) = (0).
SO
and
I
(the algebra
consists of the non-zero multLples of
of the radLcal as Ln «14) 24.17) we see that Min(P) =
T
P. P
Using a characterLsatLon is semisLmple and since
I(P) = k(h(soc(P») = (0)
I
and
P
is not
a Riesz algebra. Now let R.3.6,
A
A
be the closure of
is a Riesz algebra.
With a dense subalgebra R.S
wh~ch
P
in
B(x),
then
A =
~ ~
rad(A)
so, by
Thus we have constructed a Riesz algebra LS not a Riesz algebra.
Notes
Attempts to develop a theory of compact or weakly compact elements algebra
A
pre-date the development of a
R~esz
theory.
~n
a Banach
I:e list some of the
early defLnLtions.
63
M. Freundllch (34) 1949:
a
~
ta,
a
at
~
lS a compact element of
are compact on
T. ogasawara (65) 1954: A
t
t
A
if both operators
A. lS a weakly completely continuous element of
if both the above operators are weakly compact on
A.
K. Vala (92) 1968:
If the wedge operator
t" t
: a
-+
tat
t
is a compact element of
lS compact on
A.
B. J. Tomluk and P. K. \ong (77) 1972: continuous element of
A
A
t
is a weakly seml-completely
If the wedge operator lS weakly compact on
A.
These concepts have also been studled by Alexander (4), Bonsall (12), Kaplansky (50), ogasawara and Yoshinaga (66), Yllnen (99) the deflnitlons is entlrely satlsfactory.
I
(101).
None of
Thls lS lilustrated In the case
of the wedge operator of Vala by an example due to Smyth (86) of a semlslmple Banach algebra
A
In which
x" x
is compact (x
E
A)
but
x 1\ y
need not be.
However, Ylinen (100) has shown that some of these deflnltlons cOlnclde Wlth the reasonable deflnltlon of the set of compact elements In a C*-algebra glven J..n C*.l.l.
The followlng artlflclal deflnltlon has been proposed by
Smyth for a general Banach algebra
A.
It has the merlt that the compact
elements form a closed two-slded ldeal Wlth many of the expected propertles. Let
S
{u
A
uj. U
lS a flnlte rank operator}
and
set
F
{x E A
x" u
is of flnite rank for each
u E S}.
E
semlprlme then
F
=
S) •
If
J
= k(h(F»
(If
A
lS
the compact elements are deflned
to be the set
r
tx E J
x" Y
lS a compact operator for each
y
E
F}.
Rlesz theory as presented here dates back to 1968 when Barnes (7) derlved the spectral properties of inessentlal elements of a semlsJ..mple Banach alqebra using the concept of ldeals of flnlte order. of the Ruston characterisatlon. Smyth (82),
{84}
HlS paper contalns an analogue
A more general approach was ddopted by
who developed RJ..esz theory relative to a flxed ldeal
algebralc elements (those satlsfyJ..ng a non-trlvlal polynomlal ldentlty).
F
of He
showed that in a semislmple Banach algebra the socle lS the largest such ldeal (termed the algebraic kernel) and obtalned a Rlesz theory (lncluding the Ruston characterisatlon) for elements
x
such that
rex +
F)
=
O.
The
algebralc kernel lS the largest left or right ldeal of algebralc elements and the proof of ltS eXlst.ence In a complex algebra lS due to AmltSur «48}p.246-7). h4
Veseli~ (93) has demonstrated the existence of the algebraic kernel In a complex Banach algebra uSlng functlon theoretic methods. be degenerate If each element In ideal of algebra:Lc elements of
A.
Ax
He deflnes
lS algebralc.
Thus
Ax
x
A
€
to
lS a left
A I hence l t lies in the algebralc kernel of
Indeed, the set of degenerate elements of
A
lS actually equal to the
algebraic kernel and from thls It follows that Veselic's set of compact In Problem 8, Veselic
elements lS preclsely our set of inessentlal elements. asked If a ted by
X
lS a Banach space and
ST
is yes, for
T
lS such that the algebra gener-
B(x)
€
lS fin:!. te dlmenslonal for each ST
is algebralc for each
S
B (X),lS S
€
S, hence
B{X)T
€
F (X) ?
The answer
lS a left ideal of
algebralc elements WlllCh lS contalned In the algebralc kernel which in turn
F (X)
is equal to the socle
Puhl (72) has deflned a trace functlonal in a
•
sUltable subalgebra of a Banach algebra. We record two further results In Riesz theory, both due to Smyth «83), 6.3, 6.4).
R.S.l
THEOREM.
The sets of Fredholm and Riesz points of an element in a
commutative, unital Banach algebra coincide. Proof.
LeL
Slnce
pix)
A
be a
co~~utatlve,
unltal Banach algebra and let
x
€
A.
lS a subset of both sets, we conslder only the Fredholm and
Riesz pOlnts In
0(x}.
If zero is a Riesz point of
x
in
x(l - p) + P
spectral ldempotent, then
hence zero lS a Fredholm pOlnt of
0(x}
and if
Inv(A}
€
1
p
K
€
x + K
so
lS the associated €
Inv(A/K),
x.
Conversely, let zero be a Fredholm pOlnt of
in
0(x),
It sufflces to show that zero is an lsolated point of
0(X}.
eXlsts
Since zero is the only
y
€
A
such that
xy = 1 + k,
possible accumulation pOlnt of
an isolated paint of which lS false.
Now
o Ap (xyp) = {oJ
Thus
0
€
k
€
K.
then,either,
a
Now there
p(xy),
€
NOW, by commutativlty/ a
then, by R.2.4,
€
or zero is
pixy) => a
Hence there eXlsts a non-zero spectral idempotent
pia, xy)
p
0(xy).
0(k),
where
x
p(O, 1 + k)
and
PA(l-p) (xy(l - p»
€
p(x},
p
and
p(-l,k)SK.
GA(l -p ) (xy(l-D» ~
=
GA(x),{o}.
and again, by commutativity,
0
€
PA(l-p) (x(l-p» 65
Now
~s
Ap
a Riesz algebra with
dimensional modulo tains zero.
un~t
~ts ra~cal.
p, hence, by R.3.6,
Thus
So zero is I.so1ated 1.n
~s
0Ap(xP)
~t ~s
f~n~te
a
set
finite wh~ch
con-
0A (x) = 0Ap (x) UOA(l_P) (x(l-p)) •
The second result is a spectral mapping theorem for the Fredholm and Browder spectra. R.S.2 f
e
THEOREM.
If A i8 a unital Banach algebra., then
Hol(O(x»
(i)
f(W(x»
w(f(x»;
(ii)
f(S(x»
8 (f (x) )
proof.
(i)
S~nce
w(x)
mapping theorem 1.n
A/K.
(ii)
If
and
x s A
•
= 0A/K(x
th~s
+ K) ,
x £ A, lotS bicommutant
result loS s1.mply the spectral
Z2(x)
Z2
loS the commutative Banach
algebra
o
{z £ A
for each
Note that every spectral idempotent of
y
such that
x, and of
f(x)
I
consider Fredholm theory in the commutat1.ve Banach algebra to the ideal
K f"\ Z2
(If
spectral mapping theorem). set, thus so is by R.2.6,
0z (z)'{O}.
Kf'\ Z2 C.I1Z 2 ).
Browder spectra in
Further,
If
Z2'
SA(x) ~ Sz (x)
(0)
=
I
then ~s
Hence
From the
lies 1.n ~n
We
Z2
relative
the result reduces to the ordinary
z £ Kf\Z2'
Now let
O}.
[x,yJ
is a d~screte
a closed ideal of
8z
and
a A (z)'{O}
Z2'
and,
denote the Fredholm and
2 R.2.l, and by R.S.l,
def~nition
since all the spectral idempotents of
X
lie ~n
2
Z2'
The result now follows by applying (~)
•
Next we give the proof of R.3.7 in the sett1.ng of a general algebra. Recall (F.3.l) that the presocle ~oc(A) = {x £ A : x' £ soc(A')}.
of an algebra
A
~s
def1.ned to be the ideal
This 1deal 1S of considerable interest 1n the study of Fredholm theory, i t was introduced by Smyth and 1ts elementary properties are outl1ned 1n (83). Clearly =
»,
I (A) = k (h (psoc CA)
¢ <=>
A/psoc(A)
and
1S a Riesz algebra <=> h (pSOC (A) )
A
1S a rad1cal algebra.
reqUlred for the proof connects the set ideal of
R.5.3 x·
A,
Wlth the set
Let
LEMMA.
I
be a left ideal of the algebra.
L
lhe re
L is a left
M1n(A/rad(A».
x + rad(L)
then
M1n(A')
E
is of the form
Min(L/rad(L»
The f1rst of the two lemmas M1n(L/rad(L}},
Conversely~
M1n(L/rad(L».
E
x + rad(L)
If
A.
for some
x
every element of such that
L
E
x ELand
x' E M1n(A'}. Suppose
Proof.
and
x E L
i t follows that
x + rad(L}
u E xLx\rad(L)
then
I
algebra W1th un1t Wr1te
X·
Conversely, 4
suppose
E ~nn(A').
y
2
x·
L "rad(A}C rad(L) ,
E
Choose
x'A'x'
1S a divls10n
xAx CL, such that w'u',
=
Xl
v'u'.
uw - x, wu - x E
so
1S a d1v1sion
x + rad(L)
Let
t
E
2
Y AYf\rad(L),
y + rad(L} E M1n{L/rad(L}). =
y + rad(L} • say
t
=
We shall show
2
Y zy
yzy E rad(L)
I
z E A.
for
and hence
(Ay)yzyCrad(L) .
2
U
E
- y E radeLl, 1t follows that
it follows that
x
=
V
is such that
Y E L
and note that
E L
Y
Then Slnce At
u'w'
hence, since
I
- x
L/rad(L).
x + rad(L) E Min(L/rad (L}).
algebra so
x·
Then
2
x
It follows that xLx + rad(L)/rad(L)
rad(A) (\ LCrad L.
that
there exists
I
Since
is an 1dempotent of
u E xAx\rad(A}
w = xvx E xLx.
x
x· E Min(A'j.
US1ng a character1sat1on of the radical ({14) 24.l6}, 2 t E rad(A) , hence y Ay 1\ rad(L)Crad(A} • Now
2 - x e: y Ay f\ rad(L)c.. rad(A) , xAx\.rad (A) •
E
algebra and
Then
hence
u E yLy\rad(L} •
x'
1S an idempotent of Now Slnce
y modulo L, 1t follows that there exists
x
A' •
yLy/rad(L} v E A
Choose
is a dlv1sion such that
UV - x, vu - x E rad(L). Wr1 te w = xvx. Then uw - x I WU - X E 2 It follows that x'~x' ~s a divisl0n algebra, hence y AYn rad(L) C rad(A} •
x'
E
Min(A')
•
The second lemma connects the presocles of R.5.4
LEMMA.
pSoc (A) t\ L C
If L
L
and
is a left ideal of the algebra
A. A
then
p!oc (L) •
67
proof.
Choose
x E p soc (A) f'\ L.
Then
x
r
E soc (A')
and, because the socle
is the sum of the minimal left 1deals, there eX1st
{x·} nC A • ..{ r k 1
n
x' = L x'x' k'
such that
(\
r11n (A r
{xk}~c..AXC.L.
Clearly we may assume that
)
Then,
1
{xk + rad(L)}~C:M1n(L/rad(L»,
by R.S.3,
and
n
x - L
~ E
rad(A)() LC.rad(L).
1
x + rad(L)
Therefore
soc (L/rad (L) ),
E
hence
We may now complete the proof of R. 3.7.
LCl{A) = k(h(psoc(A»).
psoc{L)
E
as required •
20nsider the standard lsomorphism
(L +
L/ (L f'I psoc (A) )
x
Suppose first that
psoc{A»/ psoc(A).
The algebra on the r1ght of th1S lsomorph1sm lS a left ideal of
l(A)/ psoc(A)
which lS a radlcal algebra, hence 1t also lS a radical algebra, therefore so also is the algebra on the left. algebra, so
L
L
which conta1ns
idempotent of
L/rad(L)
L
1S a R1esz algebra. psoc (A) f'\ L.
lles in
P/rad(L)
idempotents, soc(L/rad(L»C:P/rad(L). structure spaces,
P/rad(L)
fore cannot contain primitive ideal of L/ (L" psoc (A) ) (L +
~et
is a rad1cal
can conta1n
P
be a pr1mit1ve
Then, by R. S .3, every m1n1mal thus, the ideal generated by these
However, by the homeomorph1sm of
1S a pr1m1tive 1deal of
soc(L/rad(L», L
Slnce
L
L/rad(L)
Wh1Ch there-
lS a R1esz algebra.
psoc (A) f'\ L.
Thus no
It follows that
is a rad1cal algebra, hence by the above 1somorph1sm
psoc(A»/ psoc(A)
1S a radical left 1deal of
therefore, lles 1n the rad1cal of Therefore
L /psoc (Ll
is a R1esz alqebra.
Conversely, suppose that ideal of
Then, by R.S.4,
L C I (A)
We conclude W1th
A/ psoc(A)
A/ psoc(A), wh1ch 1S
which,
I(A)/ psoc(A).
• some
remarks on R1esz algebras.
here in §R.3 lS due to Smyth (85).
'i'he theory presented
R1esz algebras are related to the class
of modular annih1lator algebras def1ned by Yood (103).
(An
algebra lS
semiprime if it has no non-zero 1deals W1th square zero). A semlprime algebra is deflned to be moduZar annihiZator If every modular maximal left (r{ght) ideal has a non-zero r1ght (left) annihilator.
If we restrlct to
the semisimple case,the classes of R1esz and modular annihilator algebras
coincide.
A summary of the propert1es and a llSt of examples of modular
annih1lator algebras 1S glven by Barnes (9).
The more 1mportant examples
* H-
in th1S class (W1th the addlt10nal assumpt10n of sem1s1mpl1c1ty) are
algebras (Ambrose (1», dual algebras (Kaplansky (50), (51», annihilator algebras (Bonsall and Gold1e (15»
and Banach algebras with a dense socle.
Th1S class 1ncludes most of the 1mportant Banach algebras possess1ng m1nimal 1deals. ~lthough
a semipr1me modular annihilator algebra 1S a Riesz algebra, a
sem1pr1me R1esz algebra need not be a modular ann1hilator algebra. example 1f B $
1S
{oJ.
B
is a rad1cal Banach algebra w1th
'rhen
B
Then
ran(B) = (0)
1S a max1mal modular left 1deal of
A
A
I
For
let
but
A
ran (B)
be in
1S not a modular ann1hilator algebra, but, by R.3.6,
is a R1esz algebra.
Such a
B
can be constructed as the closure 1n
A
A
B(X)
of the algebra generated by a non-n1lpotent quas1nilpotent operator. Jbserve that the result of R.4.5 holds also for compact non-abel1an In fact, 1f
dy
(f*g) (x)
«45) 20.10). L f : g -+ f*g
denotes a f1xed Haar
f
G f(xy
Slnce (g S L
1
G (G»
-1
measure on
G,
then, 1f
G.
f, g S Ll(G)
)g(y)dy
1S compact 1t 1S easy to ver1fy that the operator 1S compact for each
f
S
Ll(G).
Hence
Ll(G)
is
LCC. Kraljevic and Vese11c (109) def1ne
spectrally finite Banach algebras as
those for Wh1Ch the spectrum of every element 1S a finite set. algebras are prec1sely those wh1ch are equal to their presocles.
In fact these (1c9)
conta1ns an approach to the d1mens1on concept in Banach algebras somewhat d1fferent from that of Barnes (8),
(9).
69
C* C* -algebras
B(x)
The wedge operator whlch has been deflned on the algebra proves to be partlcularly effective ln studied in §l.
C*-algebras.
in Chapter 0
ThlS operator lS
§2 contains the decompositlon theorems of West and Stampfll
for operators in Hilbert space and their analogues ln C*-algebras.
A modl-
fication of a conjecture of Pelczynski leads, ln §3, to a characterisation of A representatl0n is constructed in §4
Riesz algebras among C*-algebras.
which provides an isometric map of finite rank, compact, and Rlesz elements of a C*-algebra onto the corresponding operators in Hilbert space.
In §5
short proofs are provlded for the range inclusion results of §0.4,which are available in Hllbert space. C*.l
The wedge operator
The wedge operator defined ln
§O.6
has been used by Vala (91)
f
(92).
Alexander (4), and Ylinen (99) to deflne classes of finite rank and compact elements in Banach algebras,
(in C*-algebras,Deflnition C*.l.l).
In the case
of the algebra of bounded linear operators on a Banach space these definitions are justified by Theorem 0.6.1, whlch shows that the flnite rank and compact elements of this algebra are, precisely, the finite rank and compact operators on the Banach space. algebra,
The deflnlt10ns are unsatisfactory ln a general Banach
an example due to Vala (92)
may fail to be closed under addltlon.
showlng that the set of compact elements The deflnltions are satisfactory,
however, in the case of a C*-algebra,and in thlS section we demonstrate thlS fact. C*.l.l
DEFINITION.
rank or compact in
An element
A
C*.1.2
THEOREM.
of a C*-algebra
if the wedge operator
rank or compact operator on
A.
x
soc(A)
A
A
x" x : a
lS sald to be finite -+
xax
lS a flnite
respectively.
is the set of finite rank elements in a C*-algebra
Proof.
Let
x E soc (A) ,
1 < L < k.
then
x
where
Then k
xAx
L a.e.Aa.e., L L ]] L,]= 1 k
C
L a e.Ae., L,]=l L L ] k
L
a.a:b .. ,
i/]=l L where each
b
is a fixed element of
L]
Conversely, suppose that dLm(xAx*x) < soc(A) Z
L
(BA.3 .3) •
]
dim (x*xAx*x) <
SLnce
00
thus
so LS
x
in
A.
1
n n
(m i' n) •
00
An 1
e
0,
NOw, for each
00
so
x E soc (A) <=> xx* E
x
Then
CJA(x).
where
dim(xAx) <
be self-adjoint. Let 2 CJ (x AX I Z) = CJz (x ) is a finite
Now
k
x = L Ae,
Thus
xAx*x CxAx,
~,ow
(BA.4.4), Lt suffices to assume that
be the commutant of
set,
e Ae
dLm(xAx) <
therefore
00,
L]
e*
n
e
n
2
(l
n
and
o
e e m n
nl
> dLm(xAx) > dim(e xAxe ) n
dLm(e Ae ). n
n
From thLs Lt LS easy to conclude that each For suppose that dLm(eAe) < 00, where e 2
n
e
(and consequently
n
e
E
A,
then
eAe
x)
E
soc (A)
LS a Riesz
algebra which, by the Smyth characterLsation (R.3.2), LS equal to its own socle.
But if
pxp
P
AP
A,
hence
C*.1.3
THEOREM.
~.
Let
II x n
fun (eAe)
pexep
for some scalar
and
E
soc (A)
x E
- x I[
soc (A) •
-+
O.
p
and
E
x
E
MLn(A),
A
so
eAe = soc(eAe)C.soc(A)
•
is the set of compact elements in a C*-algebra
Choose Then
x
n
E
sOC(A)
such that
A.
Ilx II < I [xl I + 1, n -
71.
I!x AX -x"xll n n
By
C*.1.2,
Hence
x"x
xA x
<
(21lxll +l)IIx
Conversely I let
x
->-
o.
A.
A.
be a compact element of
Now the operator
A.
-xiI
is the uniform linu.t of operators of finite rank on
is a compact operator on
operator on
n
x*x 1\ x*x
A
1
then
x" x
is a compact
being the composl.tion of the
operators
~
a
ax* ->- xax*x ->- x*xax*x
is compact on
(a E: A)
Thus, Sl.nce
A.
is sufficient to assume that
x E soc (A)
x
(BA.4.4), l. t
l.S self-adjol.nt.
By considering the commutant of
a(x)
<:=> x*x £: soc (Al
x
and argUl.ng as l.n C*.1.2,we see that
has zero as l.ts only possible accumulation pOl.nt.
Thus, if
E
> 0,
the set
a (x) (\ {A:
is finite
1 AI > e:}
{Al, ••• ,A n },
corresponding to
A. l.
xe. l. so
\
-2
hence ~
If
say.
l.S the el.genprojectl.on of
x
then
(l < i
A.e. l. l.
2
n),
(e1 " e i ) (xl\x) = eil\ e i , ei~ e i
C*.1.2,
is a compact projection which is therefore of finite rank.
e i ~ SOC(A)
(1
2
l.
2
n), hence
n
p = Ee. E soc(A). 1 l.
Using self-
adjolntness
IIx(l - p) II hence C*.1.4 (l)
72
x E
soc (A)
THEOREM.
xl\. x
r(x(l - p»
<
E,
•
The foUObJing statements are equivaLent in a C*-aZgebzoa
is a Riesz opezoatozo on
A;
A:
(i~)
is a Riesz element of A relative to the ideal
x
rex + soc (A) ) =
(~ii)
Proof.
soc(A};
o.
(ii) and (iii) are
eq~valent
definit~on,
by the
while the equivalence
tt
of (~) and (~i) follows exactly as in 0.6.1 C*.2
Decomposi~on
F~rst
we state our two decomposition theorems in a Hilbert space
decompos~t~on
theorems H, the West
of a Riesz operator into the sum of a compact operator plus a
quasin~lpotent
and
Stampfl~'s
Then using the
mach~nery
generalisation of it using the Weyl spectrum.
already constructed these results are proved in a
C*-algebra setting. C*.2.1
THEOREM,
K E K(H),
(West decomposition)
is normal,
K
OtT) = O(K),
T E R(H)
and
=>
where
K + Q
T
Q E Q(H) •
« 25) 3 .5) • Recall that.if
T
is a Riesz operator,every non-zero point of
index-zero Fredholm and that the Weyl spectrum of
O.
WeT)
T
Thus i f
£
£
R(B),
OCT)
T
OCT}
is
is defined by
A - T i (p°(H)}.
WeT) C {O}.
Stampfl~
(88) has generalised C*.2.1 as
follows, C*.2.2
THEOREM,
K E K{H),
and
(Stampfli decomposition)
a(s)
T E B(H)
=>
K + S
T
where
= WeT).
( (88) Theorem 4).
The extensions of these two results appear as corollaries of the next
theorem.
A
is a unital C*-algebra and
K
a closed
~deal
of
A.
Let x E A and suppose that w is a subset of a (x) suah that every point of w is isolated in o(x), and that the aorresp~ng speatral idempotent Ues in K, then there exists a normal element y E K 8uah that a(x + y) a(x)'w. C*.2.3
THEOREM.
~.
We shall make use of the Gelfand-Naimark embedding of
for a suitable Hilbert space
A
into
B(H)
H «14) 38.10). 73
If
w
is a finite set the proof is trivial, thus suppose not.
the points of
are isolated in the compact set
w
{Ak}~'
able set
where the
ness to
a (x)"w.
that
- ~ -;. O.
\
ponding in
to
Ak ,
Let
Pk
and put ~(H)
a(x + y)
Urite
f
fPk'
Yn -;. y S K,
Since
n
and
n
(n
n
o(g) = {a}, n
and thus
n
a(fn )
Now setting
q
= 0
o
'
Yn
and
is normal.
y
It remalns to
a(x)\w.
and relative to the decompositlon
f
x
be the self-adjoint idempotent
Let (BA.4.3) •
(x + y ) j s (H)
n
such corres-
is a set of disjoint self-adjoint ldempotents,
is normal for each n, hence show that
{ak}~C a(x)"w,
be the spectral idempotent for
n
Sn(H)
=
{qk - qk-l}~
and since
S K
Sn
w must be a count-
are labelled in decreasing order of near-
k
Thus we may choose a countable set
such that
K
A 's
O(x),
Because
gn = (x + an - An) jPn(H). s (H) = s n
n-
1
(H)
ED
we have
n
> 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'
<
Ilwnlllly-ynll,
<
Ilwmll
Ily - Yn l
x - y )-l(y - y ) E Inv(A).
n
n
A-
x - Y E Inv(A) shows that n therefore a(x + y)C.O'(x)'W •
A-
!
< 1.
Multiplying on the left by
x - Y E Inv(A) ,
Now we apply this result to Riesz and Fredholm theory on fixed closed ideal
LEMMA.
~.
A/sac(A)
=
But first, we identify
In a C*-aZgebra
C*.2.4
sacCA)
KCI (A) •
A,
A¢
that is
A
a(x + y) ,
relative to a
I (A) •
I(A) = soc(A).
is a C*-algebra which is semi simple so, by BA.2.3 ,
k(h(soc(A»)
=
k(h(sac(A») = I(A)
•
75
C*.2.5
COROLLARY.
C*-algebra
Let
(West decompos1tion)
A) then there exists a normal
Proof.
l\pply C*.2.3. with
C*.2.6
COROLLARY. A~
C*-algebra
w=
O(x)"{O}
y
y
€
K such that
€
o(x + y) = {o}.
•
Let
(Stampfli decomposit10n)
then there exists
be a Riesz element of a
x
such that
K
be an element of a
x
o(x + y)
= W(x)
the
Weyl spectrum of x. W(x) = {A € o(x)
Proof.
zero Fredholm p01nts of
: A - x
¢
¢o(A)},
so we must remove all tile index-
O(x) by the add1t10n of a single
y E K.
Th1s is
done in two stages, first the 'blobs' of index-zero Fredholm p01nts are removed one by one, then when th1s has been completed, C*.2.3 1S app11ed to remove the 1so1ated index-zero Fredholm p01nts on spectral idempotents 1n
0
(x)
(1rlhich have assoc1ated
K (R.2.4».
A 'blob' is a connected component of 1ndex-zero Fredholm p01nts of which is not a one-p01nt set.
The blobs are countable, say,
construct sequences Choose
A
n
€ V
n
,
{v }oo.
nl {An} 1C ([:, {un} 1 C K, 'lnd co
then
u
00
n
E K
such that
n
x + L
~ -
An
E
Inv(A).
1
(This is poss1ble by F. 3 J.l) •
F1nally, choose
En < ~ €n-l
so that
n
x + L ~ - An + y E Inv(A) 1
Then co
00
< €
n
€
n
I
00
thus
L
~
converges to
u E K.
Now,by (t) we get, for each n ,
1
x + u - An E Inv(A) <=> An E p(x + u).
o(x) Take
An € Vn , i t follows from R.2.7, that O(x + u)A Vn 1S an at most countable set of R1esz points of x + u. Thus we have removed the countable
Since
set of blobs
V
O(x),
of index-zero Fredholm points of
n
by an at most countable set of Riesz points of
We are left W1th the
X.
O(x + u) •
task of removing a countable set of Riesz p01nts of C*.2.3, there exists C*.3
v € K
replacing each one
tt
O(x + u + v) = W(X)
such that
So, by
Riesz algebras
pelczynski conjectured that 1f the spectrum of every herm1tean element in a C*-algebra is countable, then the spectrum of every element in the algebra is countable.
Th1S conjecture has been conf1rmed by Huruya (47).
An obvious
modlf1cation leads to a character1sation of Riesz algebras among C*-algebras, \"hi ch is or1ginally due to Wong « 96) Theorem 3.1) • algebra and C*.3.1
then
Proof.
If O(h)
is a Riesz algebra.
A
x € A,
A.
has no non-zero accumulation point for each
By virtue of C*.2.4 lt is sufflcient to prove that
If p
wlll denote a C*-
the set of hermitean or self-adjolnt elements of
H (A)
THEOREM.
h € H(A)
A
o(x*x)
has no non-zero accumulat10n point.
be the spectral ldempotent of
x*x
A = soc(A). For
°
€ >
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) <
{A € o(x*x)
Ilx
_ xp
I 12
So
I Ix
- xp I I <
Suppose that
€
p
and i t sufflces to show that
I
~
soc(A)
and put
Pi = p.
n, p
~
n
soc (A) ,
hence
2
€ .
p € soc(A).
Then, as in the proof of
R.2.6, we construct a strictly decreasing sequence of 1dempotents such that/for each
let
{p}7 n
~
and, by BA.4.3, each of these idem-
potents may be chosen self-adJoint. p
¢
soC(A) ,
with unit pr1orl,
P
~
Mln(A)
so there eX1sts
p), such that y
need not be ln
o(y)
y
€
pAp
(wh1ch 1S a C*-algebra
conslsts of at least two pOints.
H(A).
If either
a (y*y)
or
o(yy*)
But, a contain
two pOlnts then uSlng the hypothesls we can construct
P2 strictly less than ¢ soc (A) as ln R.2.6. So suppose that for each y € pAp, Pl and P2 are singleton sets. O(y*y) and a (yy*) If y oj 0,
r(y*y)
r (yy*)
[ [y*yll
[ lyl12
oj
0,
.77
o A (yy*) are singleton sets, neither of which consist pp It follows that y*y and yy* E Inv(pAp), ilence
a~.
so
(y*y) and pnp of the zero point. Thus
Y E Inv(pAp).
P E Min(A)
pAp
a
~s
d~V1S1on
which is a contrad1ct10n.
always construct an 1dempotent
P2
algebra, therefore
Thus start1ng with
sa~sfying
our
pAp PI = P
req~rements
by induction, an 1nfinite strictly decreasing sequence
a:p, and
{Pn}~
we can
and hence, such that,
n, Pn ¢ soc (A) •
for each
Now the sequence
{~}~
then
Put is an lnfin1 te orthogonal fam11y of 1dempotents
00
in
H(A).
Now
u = L 2
-k
~
E H{A),
hence
p + U E H(A),
and
1S an
1
1
o(p + u)
accumulation point of p ¢ soc (A) C*.4 \~
as required
which contradlcts the hypotheS1s.
Therefore
•
A representation
have defined flnite rank and compact elements of a C*-algebra (C*.l.l).
Riesz and Fredholm elements are cons1dered relative to the closure of the soc1e.
In th1S section we construct a faithful *-representation of the C*-
algebra onto a closed subalgebra of the operators on a Hllbert space which maps the finite rank (respect1vely, compact, Rlesz, Fredholm) elements onto the finite rank (respect1vely, compact, R1esz, Fredholm)
operator~
in the
subalgebra. Recall that an element of an algebra is algebraic if 1t trivial polynomial
ident~ ty .
21early
f~nite
satisf~es
rank operators on a
a non-
l~near
space or finite rank elements in a C*-algebra are algebra1c.
If A is a C*-'llgebra, aZgebraic elements of A. C*.4.1
THEOREM.
~.
x £ soc (A) ,
<-> ~>
dim (xAx) < "", x
is
soc(A}
is the largest ideal of
(C*.1.2)
algebra~c.
Conversely, let
J
be an ideal of algebraic elements of
A.
By
R.2.6,
JCI(A) = soc(A).
Suppose that x £ J\SOC(A), then, by BA.4.4, n 2 x*x e: J\soc(A} • But x*x = L ;\Pi where Ai sIR and Pi = Pi = pi 1 (1 < i < n). Clearly some Pi (say p) e: J\soc(A} • But p £ soc(A),
7Et
=
p
is a compact element of wh~ch
A P
(C* .1.3)
is idempotent, so
soc (A)
E
A
wh~ch
(C*.l.2),
fin~te
a
representat~on
~s
p 1\ P
a compact operator on
rank operator on
•
contradict~on
is a
The construction of our produce a natural
~s
P/\P
that is
I
is done in stages.
hence
First we
of Hilbert spaces associated with the
fam~ly
A,
ideals
~nimal
of a C*-algebra. Let
be a C*-algebra with
A
corresponding <x,y>
m~nimal
Clearly
left ideal of
is
Now if
l~near
A.
in the first
<x, x>
and
H
is clear that
x
e
e
Thus
H
e
representat~on
lfe(a) x
l~near
in the
so
is
Ilxell
<x, x> > O.
Further
2
ident~cal w~th
the inner-product norm.
A,
E H
for
~f
x
e
n
It
and
then
X,
n
x E H.
He
(xe) * xe,
jlex*xe II
closed in
~s
x e -+ xe
n
a
H
and conjugate
var~able
ex*xe
II <x, x>ell
thus the algebra norm on
a Hilbert space under
~s
TI
e
of
A
on
H e
th~s
inner-product.
We
now
as follows,
ax
~e
representation
C*.4.2
LEMMA.
on
with the foZZowing properties:
e
x, y E H e
If
= Ae be the e define the scalar
and let
0,
<x, x> E cr(ex*xe),
H
Min(A) ,
x E He'
«x, 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)<:.T(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'
<
K
E K(H)
and
R
has a West
00.
Olsen (67) showed that Lf where
Qn = O.
T E B (H)
and
Tn £
K(H),
then
T = K + Q
This result has been extended to C*-algebras
by Akemann and Pedersen (2).
An LntrLguLng property of the ideal of compact operators on a HLlbert space, origLnally due to Salinas (77), LS the followLng. Let
T
£
B{H)
then
reT + K(H»
inf K£I(H)
R
where the eigen-
reT + K) •
Sal~nas'
In fact
algebraic~sed
proof
~s
valid in Banach spaces.
by Smyth and West (87), who showed that for a large class of
commutative Banach algebras,
~nclu~ng
the C*-algebras, the above property
holds for every element and for every closed that
th~s ~s
~deal.
Pedersen (70) proved
true for all C*-algebras, and Murphy and West (60) gave an They also showed that the class of
elementary proof.
~s
algebras in which
compr~sed,
ideal is
This property was
algebra
1..S
C~lfand
space.
dense
~n
commutat~ve
Ba.'lach
property holds for each element and for each closed
roughly, of those algebras whose Gelfand transform the sup-norm algebra of conttnuous functions on the
Further
~lgebra~c
information on the spectral
ra~us
may
be found in the eleg&1t monograph of Aupetit (6).
The mo~fied Pelczynski conJecture wh~ch character~ses C*-R~esz algebras
is due to Huruya (47) and Wong (96).
The following result
~s
stated
~n
(27)
4.7.20, see also (10).
C*.5.6
If A is a
THEOREM.
C*-~lgebra
foll~»ing
the
statements are
equivalent: is a Riesz algebra;
(i)
A
(li)
A
socCA},
if
J
(iii)
is a closed left (resp. right) ideal Of
lan(ran{J}} = J (iv)
(re8p. ran(lan{J»
some Hilbert space (v)
(vi)
(vii)
then of
K(H)
for
H,
'.JJze Gelfand space of every maximal corrzmutar;ive C*-subalgebra of is discrete; left(resp. right) multiplication by on A for each x E A, every non-zero point of a(x) x
=
x
Pelczynski's Kirchberg (105).
A
is a weakly compact operator
is isolated in
a(x)
for each
x* E A.
Such algebras are also called
84
C*-.~ubalgebra
is *-isornorphic and isometric to a
A
A
= J),
dual algebras.
conJecture has been ver~fied ~n Banach *-algebras by
A Applications
In
th~s
chapter our general
L~eory
is applied to a number of
examples, particularly to algebras of operators. often use operator be the
~ndex
We recall
notat~on
function f~rst
~ndex
but the
assoc~ated w~th
(unless
As a consequence we shall otherw~se
part~cular
the
spec~fic
specified) will
algebra.
the definitions of the various spectra in R.2.l.
be a unital Banach algebra and Fredholm spectrum of
~n
x
A
K
a
f~xed inessent~al
ideal of
Let
A.
A
The
is
W(x)
the Weyl spectrum is
(\o(x +
W(x)
k);
kEK while the Browder spectrum is
A
S(x)
Jur I
appl~cations
can be
Spectral mapping
~s not a Riesz point of
class~f~ed
propert~es.
x}.
under three main headings.
The spectral
mapp~ng
theorem holds for the
Fredholm and Browder spectra (R.S.2) but not, ~n general, for the Weyl spectrum.
It does, however, hold for the Weyl spectrum for triangular
algebras of operators on sequence spaces and for certa~n quasi diagonal operators on
H~lbert
Lif~ng theorems.
II and
satisf~es
S
Inv (B (X) )
E
some
space. Suppose that
addi~onal
T
algebra~c
E
B(x) or
~s invertible modulo
analyt~c cond~tion.
satisfying the same confu tion and such that
T -
K(x)
Can one find S E
K (X) ?
85
III
Compact perturbations.
B(x)
satJ..sfJ..es an algebraJ..c
or analytic condJ..tJ..on, can one describe (\a(T + K),
where the J..ntersection
is taken over all A.I
K
Suppose that
E:
satisfYJ..ng the same condJ..tJ..on?
K(X)
£
T
Fredholm and RJ..esz elements J..n subalgebras
We fix some notatJ..on whJ..ch shall remain in effect throughout the chapter. A
denotes a unJ.. tal Banach algebra and
B
a closed subalgebra with 1
Be A.
E
KA J..S a fixed closed J..deal of A contaJ..ned J..n I (Al, and ~ = KAn B , Ue inves tJ.. gate the relatJ..onshJ..p between the hence by R.2.6, ~c I(B). Fredholm elements
qJ (A)
Clearly
qJ (B)
A
in C ct>
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)
\
crA (T)
E crA(T),
crB(T)
be any closed unital *-subalgebra of operators on
Proof.
hence
wlll denote the sets of
and if T E R(B),
K B T E R(A) f) B.
Further, If
A/KA , and
R(A), R(B)
T E R(B), then since
O'B (T) = crA(T).
is invertible In
respectlvely.
R(A)"B,
Conversely, suppose
T + KA
(BA.4.l).
T + ~ E Inv(B/~),
Thus
Next we conslder Riesz elements. Riesz elements in
=
A/KA
is a compact operator in
H B~
H
and let
containing T. and Q a quasi-
ThlS lS a Corollary of A.l.4 and the West decompoSition in the
algebra
B
(C*.2.S)
..
We have the followlng lnformatlon on the Browder spectrum.
~.
Simllar to A.1.4
..
An lnterestlng consequence of thlS is THEOREM. "& et then, for each x
A.L 7 A,
sex) ~.
B = {y E A : xy
Set
If
SA (x)
K
be a closed inessential ideal of
= ~{G(x + y) : y E K and xy = yx}.
spectrum of that
be unital and
A E A,
x =
in BB (x)
B,
(A.L6).
~o(B) = {x E ~(B)
{:>.. E
: 1 (x)
a:
yx} :
WB(x).
then the right hand set lS the Weyl
Slnce
crA(x) = crB(x)
(BA.l.4), It follows
Thus It sufflces to prove that O},
A- x ¢
SB(x)
=
WB(x).
then, from R.2.2,
ilIo(B)}:
87
and
S
B
{A
(x)
E
a:
A - x rt R(B)
}.
Since Rlesz pOlnts are automatically index-zero Fredholm points, and Slnce isolated index-zero Fredholm pOlnts are Rlesz pOlnts, l t sufflces to show that an index-zero Fredholm pOlnt of
B.
definition of
B
of
v E Inv(B)
where
Let
generated by
v, v
lS isolated In
A
Without loss of generallty take x = v + f
0B(x)
By F .3.11., f' E soc (B I);
and
0B(x).
also
vf = fv
by the
be the closed unltal (commutatlve) subalgebra
D
-1
and
f
11.
wlth Gelfand space
The set
A
{" (W)
:
WEll}
is bounded away from zero, whl1e
most a finlte number of
wEll.
is an isolated pOlnt of
0D (x),
A.2
f (W)
It follows, since and therefore of
O. for all but at
v + f,
x 0B (x)
that zero
•
Seminormal elements in C*-algebras
In this sectlon
hyponoPma l.
A
If
T*
wl11 be a C*-algebra. lS hyponormal,
is called seminormal.
T
A.2.1 (i)
T
is hyponormal =>
(ii)
T
is co-hyponorma7.. =>
Proof.
Since
1
(T*) = -
TEA
and T*T
lS co-hyponormal.
~
In
TT*,
eit~er
T
lS
case
We conslder the Fredholm theory of these operators
Suppose that
LEMMA.
T
If
T
E
¢(A) :
1 (T) .::. 0;
1
1 (T)
(T),
> O.
(11) follows from (1).
TO prove (1), recalling the deflnltl0n of the lndex functlon F.3.5, lt
suffices to consider the case of prlmltlve hyponormal.
A.
Then there eXlst Barnes ldempot.ents
'take to be self-adjoint (BA.4.3)), WhlCh satisfy
Assume that P, Q E KA
=
lan(T)
Now T*T > TT* => 0 = QT*TQ ~ QTT*Q ~ 0 => Q'I'T*Q = 0 => QT = 0 => Q E lan(T) => QP = Q => rank(Ql < rank(P).
Thus
88.
'I. (T)
nul(T) - def{T)
rank(Q) - rank{P) < 0
tt
AP
T E ¢(A)
is
(which we may and ran{T)
= r;;p...
A.2.2
COROLLARY.
A.2.3
THEOREM.
If T is normal. leT) = O. If T S
is hyponormal (resp. ao-hyponormal) there
~(A)
exists K S KA such that for eaah non-zero right) invertible. Proof. B
A consequence of A.2.1 and F.3.11 • denotes a closed *-subalgebra of
A.2.4
T
~(B)
S
e~ther
l B (T)
by A.I.3.
= lB(T
T + K
hence
=0
+ K)
B,
~nverse ~n
~s ~nvertible
Let A:; subalgebra of A. If T (i) (ii)
B () ~o (A) K S
then ~
and thus in in
T
T + K
such that
A.
But
and hence in
A,
~o (B) •
S
B.
Thus
•
Next we consider the Weyl spectrum of a A.2.5
S
ex~sts
By A.2.3 there
a left or a right
lA(T + K) = 0,
A.
If T is seminormal and T
THEOREM.
Proof. has
is left (resp.
A S C, T + AK
THEOREM.
B (H),
S B
KA
se~normal
= K(H) and let
operator.
be a alosed
B
*-
is seminormal:
WB(T) = WA{T),
there exists (~)
Proof.
sueh that wA (T)
K £ K A (\ B
Clearly
A- T
= a(T + K).
is seminormal for each
A S C.
By R.2.2,
A - T ¢ q,°{A)}, A - T ¢ ~o(B)},
(ii)
Th~s
algebra
follows from
B (C* .2.6)
operators
H
wh~ch
and the
Stampfl~ decompos~t~on
th~s
and
result
BL
~s
~s
that
~f
L
BL'
there ex~sts
{"\ a
(T
~s
a
T
collec~on
the *-closed subalgebra of
are reduced by the subspaces of
normal operator ~n
of
~n
the
•
A consequence of subspaces of
(~)
(A.2.4)
+
K)
Ko
LI
S K(H)A
BL
of closed consisting of
B(H)
then if
T
~s
a semi-
such that
WeT).
KsK(H)
89
Next we see that Fredholm p01nts of 1ndex zero of a seminormal operator are Riesz points. A.2.6
If
THEOREM.
Proof.
T
is a seminormal operator in
liithout loss of generality take
i t suffices to show that either If
~
T*T
adjoint Barnes idempotents (T) = 0 => P = Q,
1
~nd
T
TT*,
A
A,
wA (T)
then
to be pr1m1tive.
Let
T S
o
(Al ~
is 1nvertible or that zero lS an lsolated as in the proof of A.2.1, there eX1st self-
P, Q S KA
such thi'it
But then
QP = Q.
an exam1nation of F.I.ll shows that the underlying
A
Hilbert space sat1sfies H = ker(T} ffi T(H). is a pole of T A.2.7 THEOREM.
If
(S E B). Proof.
of f1n1te rank A and B
Let
T E B
Thus,e1ther T S Inv(A), or zero
tt
have the property that
is seminormal, then
0B(S)
0A(S)
WB(T) = WA(T).
By A.2.6
( \ 0B (T + K) , KE~
~
°A (T
f\
+ K)
by hypothesis
I
KEKA
A.3
Operators leaving a fixed subspace invar1ant
Let
X
be a Banach space and
A = B(x)
and let
Y
which leave soc(B'), of
T
to
and
B
a fixed closed subspace of
be the closed subalgebra of
invariant. I(B).
Y
A
Recall that if
T E B,
Tty
rad(B),
denotes the restriction
Y.
as follows:
Ty 90
Put
cons1st1ng of operators
We need preliminary information on
Define the restr1ction and quotient representat10ns of X/Y
x.
(T E B, Y E y)
I
B
on
Y
and
7T (T) (x + Y)
~s s~mple
It
these
are
representat~ons
B, x
X) •
F (X/y)C 7T (B), thus both q Hence the ~deals P r ; ker(7Tr ),
~rreduc~ble.
of
E:
and that
r
pr~mitive ~deals
are
E:
F (Y)C 7T (B),
to check that
P ; ker(7T) q q
(T
Tx + Y
q
B.
(~)
(iL)
rad(B) = P ,...P = {T E: B r" q {p , p } = {p E: TI(B) : soc(B') ¢P'};
(~ii)
soc(B') = (F(X)nB) ';
(iv)
F(X)'1 BC1 (B) •
THEOREM.
A.3.l
(~i)
J = {T E: B : T(Y) = (0)
(i)
so
B,
JC.rad(B).
But
Assume that
ideals
E'
is
P' ~n
thus (iii)
r
Thus
(0).
an element of
B
P'
~f
P E: 11 (B)
(~v)
straightforward
q
A.3.2
and
If
LEMMA.
iy (7T r (T» ,
1 (T) (P )
i x / y (7Tq (T»
~bserve that we may have
So,
~f
A.3.3 (i)
(ii) ~.
B
•
~s
and
I
P
f=
M~n(B·).
E' E:
of
rad(B)CJ.
B'
The
and, by BA.3.5,
cannot be in both, since P
r or P q I
then
E' E: P',
rout~ne.
•
T E: Inv(B(X»
is also semisimple then
THEOREM.
E'
hence
or E:
1 (T) (P r)
q
such that
Moreover,
•
The proof of the next result
P , P E: 11(B), r q
pr~mit~ve ~deals
or
is a nilpotent ideal of
T(X)CY}
and
soc(B')C P'. I
and
P 1\ P = J, r q
are fushnct
q
P'
either
~s
E
and P'
r
=
P' A P' r q
and T(X)C y};
(0)
q
r
Proof.
T(Y)
Let
T E: B
and ~
T(Y)C Y
but
T\Y
i
Inv(B(y)}
(A.I.I).
then
and Tjy E: q,(y); <po (X) and Tjy E: q,0(y}.
T E: T E: T E: (~)
such that
Suppose that TS - I
and
T E:
and
ST - I = F E: F(x}.
a finite fumensional subspace
Zl
of
Y
T[Y E: q,(y). Since such that
Choose
T[Y E:
= Zl
S E: B{x) there e~sts
e T{Y} •
Choose 91.
a project~on
Pl
B(X)
£
P l (X) = Z
then a projection that
ver~fy
Zl
£
Zl
and
Y1
£
Y,
Y 1
Again
Y = Z2 Gl (Y l'\ F (y) ) , 3.nd Z2 of Y P 2 s B(X) such that P 2 (X) = F(Y) and ker(P 2 )::>z2' (! - P2)S(I - P l ) £ B. If Y E Y, Y = zl + TY 1 where
choose a closed subspace
We
ker(P l ) .JT(Y).
and 1 such that
w~th
STyl = Yl + Fy l · Yl £ Y, therefore S(I - Pl)Y w Y fl F(Y) • Then where z2 S Z2 and W E = z2 +
S~nce
Thus
(I - P2)S(I - P l ) S B and as P l ' P 2 S F(Xl fl B, (I - P 2 lS(I - P l ) is an ~nverse for T modulo F(X)A B. Thus T S ~(B). The converse ~s
obvious.
(H)
then
If
But,(0.2.8), there
ex~sts
iX(T) = ix(T + F)
0,
1
(the case F
(T) (P ) = l r
T + K
is left
T E
~s
0,
T + F
q,0 (B)
})Y
S~nce
B(X). of
Suppose
hypothes~s.
has a left
~nverse
then, by (~),
TIY E q,0(y);
s~mllar).
~nvertlble ~n
T1e
q,°(X).
E
and
(TI Y)
q such that
T + F E Inv(B(X». hence
Y
l (T) (P ) > 0
F (X) II B
E
T
g~vlng
T E ~o(X)
Conversely, let ?urther
0= l(T) (P ) = ~ (Tiy), so that Tiy S ~o(y). r Y such that T + F E Inv(B). Thus F E F(x) fl B
But
l(T)
s~nce
~X(T
must be
q
there
Thus = 0,
T + F
thus
0
ex~sts
~X(T)
+ F) =
S
<
1 (T) (P )
S E B (F.3.11).
~nverse
T + F
~O,
T E ~(B).
Inv(B)
E
•
The next result lS a Corollary of F.3.ll and A.3.3. A.3.4
V
Inv(B(x}),
£
Let
THEOREM.
and
T S B(x) Y
and
-1
V, V
and
TIY E q,0(y).
A.4
Triangular operators on sequence spaces
In
th~s
sectlon
X
w~ll
1hen,
T(Y)Cy.
is invariant under
~ =
and
V + F
where 0
F <=> T E ~ (x)
denote one of the sequence spaces
c
or
9p < co) wlll and be Schauder the usual basls for X. I f {en}~ ~P 00 x S X, CI. £ X· .Jut <x, a> = a(x) and a = ate ) Then <x, a> = LX a n n 1 n n where x = Z a e If T s B(X) the correspondlng matrlx [tij ] is n n 0
(l
.
.
1
defined by t.
~l
92
.
t
. lJ
=
T s B(x)
J
e > ~
(l < i,
j
<
00)
I
lS upper-triangular If
and for convenlence we t
. = 0 lJ
for
l > J.
wr~te
In thLS section algebra of
A
A = G(X),
KA = K(x),
Qnd
B
of upper-triangular operators.
Inv(B)
= Inv(A) 0 B.
A.4.1
LEMMA.
It LS easy to check that
The fLrst lemma is elementary.
Suppose that
and that
T S B
~n
such that
L: A t
L
Suppose that
LEMMA.
x,
is dense in Proof.
If
o
a(T(X»
for some
t"
and -chat
T S B
and, if T r ¢o(X),
(L
0
l
A.l
{A }'"' 1
l
= 0
> 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\
II sil
°
of the matrlx
By A.4.2,
are non-zero.
LS Lnvertible, but ltS
diagonal entrles are not bounded away from zero whlch gives a contradLctlontt If
'1' E B
those of
h.4.4
let
denote the
diagonal operator whose dlagonal entrLes are
T.
THEOREM.
By A.4.3,
Suppose that
s si
1
(i E '.v)
T
and
s
l
=0
(l
¢
W) •
If
also the diagonal entries of
LS invertLble Ln
Tl = T + S, Tl
B(x), and hence in
O}
= L
to be the dlagonal operator wlth dLagonal entrles
Tl E B () ¢o (X) , A.4.2,
W = {L : t
is flnite.
{Sl}~
Slnce
S
where E
B"K(X),
are all non-zero so, by B. 93
so
Bf\ip°{X)Cip°{B). Now suppose that
rhen there eXlsts
ip(B).
TS = I + L.
ST = I + K,
such that
This implies that ~
T E
lto{T~)
T to
and so
=V
R = T - M,
(where
= 0
1to
+ M where
=
R~
then
X
relatlve to the ideal
=
l(S)
>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
<
II x
<
i lxii-lIp n y
GIl
(P
n
y) - x GIl y
II
yll '
+
II x
GIl
y -
(P
- yll + Ilx - p xl!·1 !y! n
n
I
x) III y -+
II ,
0
(n -+ 00) •
Hence
x It y £ B,
C*-subalgebra of
and l t follows that
A,
B
K (H) C B.
is inverse closed
S~nce
B
is a
un~tal
(BA.4.2).
95
A.5.3
THEOREM.
(ii)
ep(B} = epo(B) = epa (H) f\ B.
(i)
T E B => WB(T} = WeT) = WB(T) = w(T). (~)
Proof.
If
T E ep(B},
TS - I E K ~K(H) •
Hence
B
but
iH{S} = - ~H(T)
so
V E Inv{B(H»
and
where
implies that
V E Inv(B},
p
A.S.4
II TP
II
If T
COROLLARY.
T E epo{H}. But the
(0.2.8) •
therefore
such that
ST - I,
and, by A.S.l, Thus
IB{T) = 0, (~),
This proves
- PT
S E B
ep (H)
K(H)
K E
quasidiagonal
~s
lim ~nf
S, T E
iH(T) = O.
~ (B}C B () cI>o (H) C cI> a (B).
T E B(H)
ex~sts
there
Q.Ild
T
Hence
V + K,
=
on
hypothes~s
B
and we have shown that
(~i)
an easy consequence •
~s
~f
0 .
~s
quasidiagonal and if f
£
then
Hol(a(T)}
f(W(T}} =W(f(T». Proof.
If
T
is quasiruagonal
C*-subalgebra A.5.3 and R.5.2 Note that
conta~n~ng
B
~t
K(H)
quasi tr~angular, hence there exists a
~s
and
T.
Then the result follows from
•
T
normal,
K
compact
T + K
~
~n
result applies to a large class of operators A.6
th~s
B{H}.
Measures on compact groups
The background for group and on
G.
M{G}
E(G)
con~nuous
th~s
For
sect~on
let
=
~nimal ~deal ~n
M(G}.
is the
the set of measures in to Haar measure on
G.
G
tr(a(x»
Ll(G)
LEMMA.
G
be a compact
and
T(G}
the set of all
tr~gono
G. Xa(x}
is a central function ~n
A.6.l
Let
algebra of complex regular Borel measures
unitary representations of
that
(45).
denotes the set of eq~valence classes of irreduc~ble strongly
a E L(G}
dimensional
~n
may be found
convolut~on
the
metric polynomials on
96
quasiruagonal, so
~s
ident~ty
M{G)
of
be the corresponding Character; then
and
There Ma'
=
Ma
Xa
ex~sts
AS
T(G} = soc(M{G)}.
i(G)
lS
a constant
= Ll{G) •
cont~nuous
f~n~te
a d
usual we identify
which are absolutely
Note that
1
* L (G)
a
> 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
<~
has flnite lS a set Yk
t
< m),
]1*L l (G) then
O.
Thus
{Y I' •.•
a E: 2: (G) ,
I
Ym}
lS a llnearly lndependent set.
It follows that, for
wlth the posslble exception of a flnlte subset
Ma C jl*L1 (G) •
Eel: (G) ,
there eXlsts y E: L 1 (G) x E: Ma' such that ].l*y = x. *x Hence ]1* (ea*y) x. Thus )l*Mo (a s l: (G)\ E) • e a Put X span{M : a E: l:(G)\E}, X = X and e = l: e E =
a E: l: (G)
\E
and
o
Tjl(X) = X.
Let
Y = {x S Ll(G)
: ]1*x = 0).
since
Me
is always finite
97
Then
Y*M CYf\M
o
Therefore
= MO
~*MO
dimensional and
~
= (0)
(Y (\ X)
¢
(0
if
¢ E
0
~t follows that
E)I so
Y
A MO
Thus
YClan(X).
=
YClan(X).
0
=
We have proved that
Y f\ X
hence
(0),
=
ker(T}n X ~
Ix
(0)
T (X)
and that
~s semis~mple.
Ll(G)
s~nce
(0),
~ 1
(0 ¢ E) •
(0)
= x.
Thus
T £ Inv(B{X», hence S = T + (I - T)T E Inv(B(L (G»). 11 1 e e \.l Thus K = T (T - I) E: f{L (G», and T S + K w~th SK == KS. e \.l ~ is a RLesz-Schauder operator
TfJ
4t
COROLLARY.
A.6.4
Il*Ll(G) has
(i) (~i)
If \.l E: M(G) the following are equivalent: finite ao-dimension in Ll(G);
11 E:
(iii)
(iv)
]J
= \)
(M(G» ;
+
K
UJhere
\)
wher'e
(v)
~.
(i) => (~v)
\.l E
I
T
then
)..l
E:
there exists $1' $2
and
11
A.7 I
hence
(lL)
such that
~
\.l
=> (iLl clearly.
=>
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:
'
then
A
is a RLesz point
Notes Spectral mappLng propertLes.
theorems for essentLal spectra Ln a general context. open semLgroup in a unLtal Banach algebra for
x E A
Napp~ng
Gramsch and Lay (38) prove spectral
A
Assume that
whLch contains
Inv(A)
S
~s
and
define
A - x ¢ S}.
Then Gramsch and Lay say that the spectral mapping theorem holds in thLS context if for each
98
f E Hol(Os(X»
and
x E A
an
a
s
(f(x»
f (a
s
(x».
They show that the spectral mapping theorem holds for a number of essential spectra of Lnterest in operator theory includLng those of Bowder (16)
(S(x)
in our notatLonl, Kato (52) and Schechter (79) and an example is given to show that Lt faLls for the Weyl spectrum II
LiftLng theorems.
W(x).
The decomposLtion theorems of West and StampflL,
proved Ln §C*.2, extend the corresponding liftLng theorems from the Calkin ThLS observatLon applies to the results
algebra to a C*-algebra settLng.
of Olsen and Pedersen mentioned in §C*.5.
~o(X)
extension of the lLftLng theorem
and Smyth has been dLscussed in §F.3.
~o(X)
T
For a Banach space
+ K(X)
Inv(B(X»
X
the
due to Pearlman
Lay (56) pointed out that Lf
has a commutLng decomposLtion Lnto the sum of an invertible and
a finLte rank operator then zero is a Riesz pOLnt of
T
(the Riesz-Schauder
case) • III
{A
Compact perturbations. E
~
:
A - T
¢ ~o(X)}
If
T S B(X), Schechter (79) shows that
is the largest subset of T-,
under all compact perturbatLons of
n
aCT)
which LS Lnvariant
eqULvalently 1
arT
+
K).
Kc:K(x)
The generalLsatLon of this result to Banach algebras LS given in R.2.2. Lay (56) proved that
SeT)
0. {a (T
+ K)
K
E
K(x)
and
KT
The verificatLon of this result in Banach algebras LS contained Ln the proof of R.5.2.
9q
BA Banach algebras
This chapter Ilsts the lnformatlon required from algebra theory, and In particular deals Wlth Banach alqebras over the complex fleld.
'!here the
results are known they will be referenced In one of the standard texts (14), (75)
Otherwlse proofs are glven.
(48).
I
The algebras will always be
unital and complex, the non-unltal case may be dealt Wlth by adjolning an identity. §l deals with baslc spectral theory and §2 Wlth the space of prlmQtlve ideals in the hull-kernel topology, In the commutative case with the space §3 gathers lnformatlon on minlmal ldeals and the socle,
of maximal ideals.
While basic results on C*-algebras are Ilsted In §4. BA.l Let
§Pectral theory A
be a unl tal Banach algebra and let
invertible elements In A
A.
If
x £ A,
Inv (Al
denote the set of
the resolvent set
p(x)
of
x
is the set
A-
p (x)
Whlle the spectrum
cr(x) = aA(x) =
x £ Inv(A)},
~'p(x).
The subscrlpts may be omltted If
the algebra in questlon is unambiguous. BA.l.l
Inv(A) is an open subset of
empty compact subset of
A
and i f x
£
A,
crA (x)
is a non-
~.
(14) 2.12 and 5.8) • Ue use BA.1.2
a
Let
a B (x) ~ aA (x) (
In
to denote the topologlcal boundary of a set. B
,
(14)
5.12).
:S:or
x £ A
be a dosed 8UbaLgebra of liJhile daB (x)c. acrA (x) •
A
containing
the spectral radlUS is deflned by
1, x.
Then
rex}
supi!
AI
A
E 0 (x}}.
rex) = lLml [xnl [lin. n->oo «14) 2.8). It follows that the spectral radius is Lndependent of the
EA.l.3
containlng algebra.
If
EA.1.4
7-S a maximal eommutative subalgebra of
B
will be unital and
A
containing
it
x
oBex) = 0A(X}'
( (14) 15.4).
If
0
LS a non-empty compact subset of
~,
denote by
Hol(o)
the class
of complex valued functLons whLch are defLned and holomorphic on some neighbourhood of
o.
Hol(O)
may be regarded as an algebra if we restrLct a
combLnation of functLons to the intersection of their domains. and
E Hol(O(x)},
f
then an element
f(x) E A
If
x E A
LS defLned by means of an
A-valued Cauchy Lntegral «14) §7). BA.l.5
If
Hoi (0 (x»
x
E
into
polynomials in
A the map
is an algebra homomorphism of
f + f(x)
A J mapping complex polynomials into the corresponding
x.
«14) 7.4).
BA.l.6
(Spectral mapplng theorem).
O(f(x»
=f{O(x}).
!f x E A and
then
f E Hol(O(x}),
((14) 7.4).
Note that If
x E A
satlsfles
III -
xl
I
< I,
then we can deflne
log x
by means of an A-valued Cauchy lntegral. BA.2
The structure space
From BA.2.1 LO BA.2.5 we shall be dealing with a purely algebralc situation. Banach algebras are introduced after 2.5.
The term ldeal Wlthout either
adjectLve right, or left, means a two-slded Ldeal. Let on
X
A
be a unltal algebra and
is an algebra homomorphlsm of
Operators on
X.
A representatLon
ally) irredueible If under
X
~(x)
for each
~
f 0, x E A
a llnear space, a representation of A
~:
into the algebra A
+
L(X)
L(X)
A
of linear
lS (strictly or algebraic-
and If the only subspaces which are invarlant are
(o)
and
X.
An
ideal
P
of
A
is 101
primitive P
>C
If there exists a maximal modular left ldeal
L
of
A
such that
{x £ A : xAeL}.
BA.2.1
(i)
A <~ p
is a primitive ideal of
P
is the kernel of an
irreducible representation of Ai is the kernel of the (irreducible) left regular representation on P (ii) the quotient space AIL; (iii) If P is a primitive ideal of 01' with L l L 2 CP then either
lS made up of elements
Note that
({l4) 24.12).
are left ideals of A
and
A
Yi £ L2 ) • The set of all prlmi b.. ve ideals of
k
(n
f\ {p
=
hull
£ II (A)
V
of
fen (A)
P E: f}
{p
h(V)
is
r
closed If
If
An ideal.
if if
(iil) (iv)
rad(A) = such that
J
fC II (Al
radical
lS empty,
of
A
the
lS deflned to be
A,
A
rad(A) = A.
lS
collect some lnformation on the radlcal.
(0)
In
lS
nilpotent.
lS
is semisimple;
A/rad(A) J
~'le
(0).
deflned In terms of a
f
A set
The
II (A)
and has a
is an ideal of A, rad(J) J ('! rad(A) is a nilpotent left or right ideal of
; A
then
J
<=- rad CAl;
z £ rad(A) =>l+zE: Inv(A);
(v)
x E: Inv(A) <=> x + P E: Inv(A/p)
Proof. (v)
J (i)
BA.2.2
(ii)
if
If
of
II (Al
lS a non-empty subset of
V
: P::>V}.
«14) p.132).
= h{k{f)l
rad(A) = (\{p : P E: YeA)}.
semiBimple
kernel
the
while If II (Al
E:
lS denoted by
hull-kernel topology
standard topology known as the closure operatlon.
A
For (i) -
=> is
show that
(iv)
see (14) 24.16, 24.20, 24.21.
If not then
a maximal left ideal
L.
Then
hypothesis, there eXlsts
u E: A
hence
I £ L
inverse
y
Ax
Thus Alp
x + P
is lnvertlble for
P E: II(A) •
Q = {a E: A : aAC.L} E: II(A) , such that
has
(P £ II (A) ) •
has a left lnverse in
We
lS a left ideal which lS contalned In
which is a contradiction.
y E: A.
as its lnverse In aa- above,
x + P
So suppose
obVlOUS.
Ax = A.
(P E: II (A) ) •
y + P i~w
A.
ux - 1 E: QCL. So
Ax
A
and
and, by But
-I.
ux E: AxeL,
x has a left
as a left lnverse, and therefore the hypothesis holds for Therefore
Y =
X
y
so,
Let
J
be an ideal of
A
~
and let
A + A/J
be the canonical homo-
morphism. (i)
BA.2.3
P
~
k(h(J»
(:Ll)
(ili)
~(P)
h{J)<=>
£
-1
£
TI(A/J);
(rad(A/J)};
is semisimple <=> J
A/J
k(h(J)} •
«48) p.20S}. The next result lS frequently used ln the text.
Let
BA.2.4
such that
~ith
be a unital algebra
A
xy - 1, yx - 1
£
ideal
J <=> there exists
then there exists
J, y
such that
A
£
y £ A
xy - 1,
yx - 1 E k(h(J}). proof.
q, :
A
=>
lS ObV10US, so let
lS invertible modulo
=>
there eXlsts
=>
q,(x)~{y}
=>
q,(x)~(y),
=>
~(x)
If
BA.2.S
y £ A
such that
q,(y)q,(x) E Inv(A/J}
set f~'
set
k(h(J)}
E
q, (k(h(J»} = rad(A/J)
(BA.2.3),
(BA.2.2),
• x + rad(A)
x' = {x I
£
A'
A/rad(A} •
x E W}.
The stpucture spaces
(1)
xy - 1, yx - 1 E k(h(J}},
- q,{l), q,(y)q,{x} - q,U}
Inv(A/J)
E
x E A
WCA
map
denote the canonlcal homomorphlsm
A/J.
+
X
If
~
n(A), TI(A')
are homeomor-phic under the
p + pi;
(ii)
(iil)
~.
has a left (right) inverse in A<=>x' has a left(rightl inverse in A!.
x
Inv(A)'
Inv(A' }
«14) 26.6).
(i)
(ii) => 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 <1:).
E
and
Inv(A)
x + Q (A)CQ (A)
therefore
•
and the socle
(0), such that
~
theorem «(14) 24.10)
rad(A).
E
Now
«::,
be an algebra over
J.
x E
Inv(A),
and ~t follows that
«::),
rr(x)~
and
x + Inv (A)C Inv (A) .
q E Q(A) <=> 1 + Aq E Inv(A) (A E
dens~ty
0,
~
x E Q(A),
o.
~
rr(x)
)f,
x + Inv(A)C:lnv(A).
Let
and that
is an
)rr(ux - xu)t;"
hence
x c radiAl •
X
rr
vx)t;"
hypothes~s,wh~ch
TI(A}),
(P E
Let u + x
by
Suppose
u.
But
Thus
space
then, since
rr(u}~ =
= A-
v
l~near
on a
By the Jacobson
such that
x)~
xv -
A
n(x)~ ~ 0,
such that
~ndependent.
are linearly
rr(v
0 E Q(A) => 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
<
givlng
e2
If
Therefore
p
P
e
e = e l + '"
and
e
¢
P,
~
qAC.Q
therefore
On the other hand
e or
P CQ.
But
e
and the result follows re~uired
+ en
then
l
Informatlon is also TI(A)
then
P E: TI(A)
If
Aec..Q,
q E: Q,
Q f\Ae
(0).
Q.
e E: soc (A) ,
< n) •
l
Pe
Qc...p
P e ,hence
Pe0l!;.e = (0), and hence/by BA.2.1, elther Hence
Now if
•
where
¢
ei
P
el
E: Min (A)
i.
for some
on the set of accumulation points
IT*(A)
of
in the hull-kernel topology.
BA.3.6
If A is a
Proof.
Let n
P
semisi~~e a~gebra
IT (A)
E:
then
P ¢ h(soc(A».
and
IT*(A)C.h(soc(A». x E soc (A),P.
Then there eXlsts
(1 < l < n) • where a. E: A, e Hence at least x = L: a e E: Mln (A) 1. 1. 1 l 1. IT (A) (e say) So, by BA.3.5, is the disone e does not lle In P. l 1.S closed In TI (A) , joint un1.0n {p}Vh({e}). Now h({e}) so {p} 1.S Thus
•
P ¢ TI*(A)
open, therefore
The Gelfand topology on the structure space of a commutatlve Banach algebra 1.S, in general, stronger than the hull-kernel topology ((14) 23.4). BA.3.7
the
If
~
is a commutative Banach
Ge~fand topo~ogy <=>
Proof.
a~gebra
then
Without loss of general1.ty we may assume
TI(A)
is discrete in
TI(A)
is discrete in the hull-kernel
IT (A)
A
topo~ogy.
to be semis1.mple.
If
1.S d1.screte 1.n the hull-kernel topology, then 1.t 1.S clearly d1.screte
in the Gelfand topology.
Conversely, suppose that
TI(A)
is discrete in the
By the 51.10V 1.dempotent theorem (13) 21.5), if p E: TI(A) 2 p = p E A such that I, p(Q) = 0 (Q E: IT(A) , Q i p)
Gelfand topology.
pep)
there exists
p
where thus
1.S the Gelfand transform of
TI(A)
is the disjoint un1.on
hull-kernel closed, so this topology
{p}
p.
Then
{p}u h({p})
p E Min(A)
by BA.3.5.
1.S hull-kernel open, hence
and Now
IT (A)
p ¢ P, h({p})
1.S d1.screte in
•
If A is a semisimple commutative Banach algebra such that is disC2'ete then h(soc(A» = cpo
TI(A)
~.
such
BA.3.8
that
From the above proof 1.f p ¢ P,
is
so
soc(A) ¢p
P E IT (A),
there exists
p E fun (A)
• ~07
1
Let
BA.3.9
be a unital semisimple Banach algebra such that
A
singleton set for each proof.
Let
w~th
x E A
x E Inv(A)
hence
x E radiAl ,
A oJ
idempotent which is not pAp
Proof.
pAp
op Ap (x)
is a
~s
p E Min(A)
•
y = Al
minimaZ~
then
o (y) =
OJ
where
is a non-zero
x c pAp
Banach algebra with unit
sem~sLmple
p
such that
(BA.2.7), so
x E pAp, by BA.3.9,
~p
pAp =
~f
and
•
I Ix*x[ I
=
A
LS a C*-algebra
I Ixl 12
Gelfand-Na~mark
(x
E
A).
Lt possesses an LnvolutLon * such
~f
(The terminology B*-algebra
theorem states that every C*-algebra
*-isomorphic to a closed * subalgebra of «14) 38.10).
as follows.
Let
A
Let
BA.4.1. (i)
~
A
A
is compact <=> A
also used).
for some HLlbert space
H
A
be a
LS
funct~onals)
on
A
~
of non-
LS locally compact
~sometrLcally-*-~somorphic
to
Co(~)
i
is unital «14) 17.4, 17.5).
C*-algebra~
then
is semisimpZe;
if I is a closed ideal of A, quotient norm is a C*-algebra; (ii)
(iii)
~s
isometrLcally
be a commutatLve C*-algebra then the space
zero characters (multiplLcatLve lLnear
further,
3(H)
~s
The commutative verSLon of the theorem, due to Gelfand, LS
in the weak * topology and
then
r*
I
if ¢ 1:S a continuous *-homomorphism of is closed in B.
¢(A)
«75) 4.1.19, 4.9.2, 4.8.5).
108
rex) = 0 =>
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}) =
shift on
where a(V}
o
U
H = 12 ED 12
and if
T
is the bilateral
H,
D*
T
Take
and
V
V
I
aIe the forward and backward shifts on
are the unit disk,
wh~le
a(T)
is the unit circle.
12,
a(U}
and
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117
Index
algebraic element
51
algebraic kernel
51
annihilator
F.1.8
Atkinson characterisation
0.2.2
Barnes idempotent
F.1.9
bicommutant
66
Browder spectrum
85
Calkin algebra
3
co-hyponorma1 operator
88
commutant
15, 62
commutator
12, 53
compact action
15
compact Banach algebra
R.4.3
compact element
C* .1.1
compact operator
3
Qecomposition algebra
83
defect of element
F.2.7, F.3.6
defect function
F.3.5
defect of operator
0.2.5
degenerate element
65
diagonal operator
93
dual algebra
84
essential spectrum of element
0.2.3
essential spectrum of operator
R.2.1
Fredholm element
F.2.5, F.3.2
Fredholm operator
3
Fredholm point of element
R.2.1
Fredholm point of operator
0.2.5
Fredholm spectrum of element
R.2.1
finite E-net
8
finite rank element
C* .1.1
finite rank operator
3
general~sed
Fredholm operator
F.4.9
generalised inverse
F.4.9
generalised index
49
hermitean element
77
hull
102
hull-kernel topology
102
hyponormal operator
88
~deal
of finite order
F.4.1
index in algebra
F.4.12
index of element
F.2.7, F.3.6
indeJ< equivalence
F.4.13
index function
F.3.S
index of operator
0.2.5
index theorem
F.2.9, F.3.8
~nessential
element-
inessential ideal ~nessential
operator
inverse semi group ~rreducible
representation
F.3.1 F.3.12 8 49 101
kernel
102
K-Fredholm element
F.3.12
left completely continuous Banach algebra R.4.4 minimal
~deal
F.l.2
min~mal
idempotent
F .1.1
modular annihilator algebra
68
null.ity of element
F.2.7, F.3.6
nullity function
F.3.S
nullity of operator
0.2.5
order of ideal
F.4.l
orthogonal
F.l.S
~dempotents
Pearlman example
F.3.10
pelczynski conJecture
77
pole of finite rank
2
presocle
F.3.1
primitive algebra
104
primitive ideal
102
punctured neighbourhood theorem
0.2.7, F.2.10, F
quasidiagonal operator
96
quasi-Fredholm element
46
quasi-invertible element
46
quasi triangular operator
94
radical
102
rank of element
F.2.3
relatively regular operator
48
representation
101
Riesz algebra
R.3.1
Riesz element
R.l.l
Riesz operator
8
Riesz point of spectrum
R.2.1
Riesz-Schauder operator
20
right completely continuous Banach algebra
R.4.4
Ruston characterisation
0.3.5, R.2.5
semi-Fredholm element
F.4.8
semi-Fredholm operator
F.4.7
seminormal operator
88
semiprime algebra
68
semisimple algebra
102
single element
77
Smyth characterisation
R.3.2
socle
106
spectral projection
2
spectral set
2
spectrally finite Banach algebra
69
Stampfli decomposition
C*.2.l, C*.2.5
upper triangular operator
92
wedge operator
17, 70
West decomposition
C*.2.1, C*
Weyl spectrum
R.2.1
Notation
A
22
ker(T}
2
A'
35
k(f}
102
B(X)
1
too (X)
0.2.1
B1
18
Ian (x)
F.l.8
L
62
13 (x)
R.2.1
C
1
ex(E)
15
dim (X)
29
d(T)
0.2.5
def (x)
F.2.7, }.3.6
15 (x)
F.3.5
a(n>
100
t;(x, E)
x
L.C.C.
R.4.4
m(X}
0.2.1
Min (A)
F.l.l
M(J
96
neT)
0.2.5
nul (x)
F.2.7, F.3.E
Vex)
F.3.5
8
ord(J)
F.4.1
H
1
P{W,T), P(A,T)
2
H(A)
77
psoc{A)
F.3.1
801 (a)
2
P
94
h(V)
102
l(T), iX(T)
0.2.5
index)
F.2.7, F.3.6
leX)
F.3.5
It;(T)
94
I (X)
1
I (A)
F.3.1
II (Al
102
TI*(A)
107
Q.{X)
1
q(B)
0.3.1
Q.t;
94
reT}
2
peT)
2
rex)
101
int(n>
110
Inv(B(X»
1
pix), PA (x)
100
Inv(A)
100
R(x)
2
R(A) , RK(Al
R.l.l
radiAl
102
K(x)
1
K
F.3.1
~
86
1.22
[S, 'l~ [X, y]
12
F.l.8
TI\ T
17
rank (x)
F.2.3
S" 'I
20
soc(A)
23
x"x
70
cr (T)
2
a8x
2
cr(x) , crA (x)
100
Q.lA)
104
L: (C)
96
'I*
2
TIY
2
Tf}
93
T(G)
96
'I
97
X
62
R.C.C.
64
ran (x)
R
f.l
"[ (T)
15
u
8
W(x)
R.2.1
WeT)
0.2.4
u.' (x)
R.2.J
X
1
X*
2
x,
35
~,
~+
2
Z(T}
15
Z(x)
62
Z2 (x)
66
~(X)
1
~o(X)
4
~(T)
0.2.5
~(A)
F.2.5, F.3.2
~K(A)
F ..... 12
'F(A)
46
53
ABOUT 'l'HIS VOLUME In this Research Note the authors exploit the relationship between algebra and spectral theory, in order to set into the context of Banach Riesz and Fredholm theory of bounded and C·-algebras the operators on Banach and Hilbert spaces. Fredholm theory is developed in primitive Banach algebras, and then while Riesz theory follows as a extended to the general consequence. Hilbert space are extended to the appropriate setting of C· -algebras, and numerous results are given. The book will be of particular interest to graduate students with a background in Banach algebras and spectral theory. Many of the results are new, and many are familiar results put into a new setting, the emphasis being on the underlying algebraic nature of classical spectral theory. RESEARCH
(lIN
The aim of this series is to disseminate important new material of a specialist nature in economic fOiOl. It ranges over the whole spectrum of mathematics and also reflects the changing momentum of dialogue between hitherto distinct areas of pure and applied parts of the discipline. . The editorial board has been accordingly and will from time to time be recomposed to the full diversity of mathematics as covered by This is a rapid means of publication for current material style of exposition is that of a developing subject. Work that is in most respects final and definitive, but not yet refined into a fOlblal monograph, will also be considered for a place in the series. homogeneous material is required, even if written by more than one author, thus multi-author works will be included provided that there is a strong linking theme or editorial pattern. Proposals and manuscripts: See inside back cover.
A. Jeffrey, University of Newcastle-upon-Tyne R. G. Douglas, State University of New York at Stony Brook •
Board
F. F. University of Edinburgb H. Brezis, Universit6 de Paris G. Fichera, Universitl di Roma R. P. Gilbert, University of Delaware K. , Universitit Stuttgart R. E. Meyer, University of J. Universitit Freiburg L. E. Payne, Cornell University I. N. Stewart, University of Warwick S. J. Taylor, University of Liverpool ISBN 0 273.s63 8
.~
•
