Research Notes in Mathematics
B A Barnes, G J Murphy
MRFSmyth&TTWest
Riesz and Fredholm theory in Banach algebras
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Research Notes in Mathematics
B A Barnes, G J Murphy
MRFSmyth&TTWest
Riesz and Fredholm theory in Banach algebras
Pitman Advanced Publishing Program BOSTON LONDON -MELBOURNE
67
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
it Pitman Advanced Publishing Program BOSTON LONDON MELBOURNE
PITMAN BOOKS LIMITED 128 Long Acre, London WC2E 9AN PITMAN PUBLISHING INC 1020 Plain Street, Marshfield, Massachusetts
Associated Companies Pitman Publishing Pty Ltd, Melbourne Pitman Publishing New Zealand Ltd, Wellington Copp Clark Pitman, Toronto
© B A Barnes, G J Murphy, M R F Smyth & T T West 1982 First published 1982 AMS Subject Classifications (main) 47B05, 47B30, 47B40 (subsidiary) 46BXX, 46JXX
British Library Cataloguing in Publication Data
Riesz and Fredholm theory in Banach algebras (Research notes in mathematics, 67) 1. Banach algebras 1. 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 512' 55 82-7550 0A326. R54 AACR2 ISBN 0-273-08563-8 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 form 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
CHAPTER
0.1 0.2 0.3 0.4 0.5 0.6 0.7
OPERATOR
0
THEORY
Notation . . . . . Fredhoim operators
Riesz operators . Range inclusion .
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F.1
Minimal ideals and Barnes idempotents
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F.2
Primitive Banach algebras
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F.3 F.4
General Banach algebras
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Notes
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Action on the commutant The wedge operator . . Notes
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CHAPTER F
CHAPTER
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FREDHOLM THEORY
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RIESZ
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43
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THEORY
R.l
Riesz elements:
algebraic properties
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R.2
Riesz elements:
spectral theory
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R.3
Riesz algebras:
characterisation
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R.4
Riesz algebras:
examples
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R.5
Notes
CHAPTER
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C*
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63
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70
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73 77
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C*-ALGEBRAS
C*.1 The wedge operator . . C*.2 Decomposition theorems .
C*.3 Riesz algebras . C*.4 A representation . C* . 5 Notes . .
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81
APPLICATIONS
A
CHAPTER A.l
Fredholm and Riesz elements in subalgebras
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A.2
Seminormal elements in C*-algebras
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A.3
Operators leaving a fixed subspace invariant .
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A.4
Triangular operators on sequence spaces
A.5
Algebras of quasitriangular operators
A.6
Measures on compact groups
A.7
Notes
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BA.l Spectral theory . . . . . . . BA.2 The structure space . . . . . BA.3 Minimal ideals and the socle
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BA. 4 C*-algebras
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108
BIBLIOGRAPHY . .
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112
INDEX
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CHAPTER
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98
BANACH ALGEBRAS
BA
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NOTATION . . .
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122
Introduction
This monograph aims to highlight the interplay between algebra and spectral theory which emerges in any penetrating analysis of compact, Riesz and Fredholm operators on Banach spaces.
The emphasis on algebra means
that the setting within which most of the work takes place is a complex Banach algebra, though, in certain situations in which topology is dispensable, the setting is simply an algebra over the complex field.
The
choice of spectral theory as our second main theme means that there is little overlap with other extensions of classical results such as the study of Fredholm theory in von-Neumann algebras. We use the monograph 'Calkin Algebras and Algebras of Operators in Banach Spaces' by Caradus, Pfaffenberger and Yood (25) as our take-off point, and it should be familiar, or at least accessible, to the reader. view of the Calkin algebra is given in (40)).
(A modern
The original intention
behind Chapter 0 was to provide a summary of classical operator theory, but, it emerged in the course of the work that a quotient technique developed by Buoni, Harte and Wickstead (17),
(41) led to new results, including a geometric
characterisation of Riesz operators (§0.3) and some range inclusion theorems (§o.4).
Thus Chapter 0 contains an amount of new material as well as a
survey of classical results.
On an infinite dimensional Banach space a Fredholm operator is one which, by Atkinson's characterisation, is invertible modulo the ideal of finite rank operators (the socle of the algebra of all bounded linear operators on the Banach space).
This motivates our concept of a Fredholm element in an
algebra as one that is invertible modulo a particular ideal which, in the semisimple case, may be chosen to be the socle. In §F.l we introduce the left and right Barnes idempotents.
For a
Fredholm element in a semisimple algebra these always exist and lie in the socle.
In the classical theory they are finite rank projections related to
the kernel and range of a Fredholm operator. considered in §F.2.
Primitive Banach algebras are
Smyth has shown how the left regular representation of
the algebra on a Banach space consisting of a minimal left ideal may be used
to connect Fredholm elements in the algebra with Fredholm operators on the space.
With this technique the main results of Fredholm theory in primitive
algebras may be deduced directly from the classical results on Fredholm operators.
This theory is extended in §F.3 to general Banach algebras by
quotienting out the primitive ideals.
It now becomes appropriate to intro-
duce the index function (defined on the space of primitive ideals).
The
validity of both the index and punctured neighbourhood theorems in this general setting (first demonstrated by Smyth (83)) ensures that the full range of classical spectral theory of Fredholm (and of Riesz) operators carries over to Banach algebras.
Riesz theory is developed in Chapter R building on the Fredholm theory of the previous chapter and we follow Smyth's analysis (85) of the important class of Riesz algebras.
Results which are peculiar to Hilbert space and
their extensions to C*-algebras, including the West and Stampfli decomposition theorems are given in Chapter C*.
Chapter A contains applications of our
theory to seminormal elements in C*-algebras, operators leaving a fixed subspace invariant, triangular operators on sequence spaces, quasi-triangular
operators and measures on compact groups. ments are listed in Chapter BA.
The underlying algebraic require-
Each chapter contains a final section of
notes and comments.
A fair proportion of the theory developed here is appearing in print for the first time.
Z.mong the more important new results are the geometric
characterisation of Riesz operators (0.3.5); (§0.4);
the range inclusion theorems
the link between Fredholm theory in primitive algebras anc classical
operator theory (F.2.6);
the punctured neighbourhood theorem (F.2.10);
index function theorem (F.3.11);
the
the characterisation of inessential ideals
(R.2.6) and the Stampfli decomposition in C*-algebras (C*.2.6).
(Some of
these results have, however, been known since the appearance of (83)).
This
has required that full details of proofs be given, except for the theorems listed under the notes at the end of each chapter.
Each author has been involved in the development of the ideas presented in this monograph.
The subject has gone through a period of rapid expansion
and it now seems opportune to offer a unified account of its main results.
O Operator theory
This chapter contains the basic results from operator theory or, Banach spaces
often stated without proof. Pfaffenberger and Yood (25).
The main reference is the monograph of Caradus,
Bonsall (13) gives an algebraic approach to the
spectral theory of compact operators; ful references for Fredholm theory;
ded for Riesz theory;
Schechter (80) and Heuser (43) are use-
Dowson (29) and Heuser (44) are recommen-
while Dunford and Schwartz (30) provides an invaluable
background of general spectral theory. Notation and general information is set out in §1.
Fredholm operators
are considered in §2 which contains a proof of the Atkinson characterisation §3 outlines the theory of Riesz operators and, employing a quotient
(0.2.2).
technique due to Buoni, Harte and Wickstead (17),
(41), contains a proof of
the Ruston characterisation as well as a new geometric characterisation of Riesz operators due to Smyth (0.3.5).
This material is used in §4 to prove
range inclusion theorems for compact, quasinilpotent and Riesz operators several of which are new.
Much simpler proofs of these results are avail-
able in Hilbert space and are given in §C*.5.
In §5 we consider the action
of a compact or Riesz operator on its commutant, and in §6 the properties of the wedge operator. 0.1
Notation
IR
and
and
H
C
will denote the real and complex fields, respectively, and
a Banach and a Hilbert space over
X
We start by listing the var-
C.
ious classes of bounded linear operators which will be discussed and, where necessary, defined subsequently: B(x)
the Banach algebra of bounded linear operators on
Inv(B(X))
the set of invertible operators in
B(X);
F(x)
the ideal of finite rank operators on
K(X)
the closed -ideal of compact operators on
I(X)
the closed ideal of inessential operators on
CX)
the set of Fredholm operators on
Q(X)
the set of quasinilpotent operators on
X;
X;
x; X;
X;
X;
the set of Riesz operators on
R(X)
T a $(X),
If
triction of
T
operator on
X*.
to
on
y -Y a(y)x
T,
f(T)
where
r
1
is a suitable contour in
ted with each spectral set
P(W,T)
where
r
=
is the operator of rank < 1,
6(T). If f 6 Hol(Q(T))
217ri
fr
w
and zero on
is a spectral set for
6(T)
and P
:
T
X2
Let
6(T)\ W.
where
T = T1 ED T2,
then
P (w,T)
:
surrounding
p(T)
z 4-
dim(y)
and
:
reduces (commutes with)
denoted S.
T
is the range
we then write
is a spectral set for
If
X
is
If
T
if
P(a,T) c F(x).
it
X.
will denote the dimension of the space Y
Y
and
U will denote the
is a closed subspace of the Banach space
Ix + Y1] = infl]x + y1l.
tive integers).
w
If
X1
0(T2) = a(T)\W.
0(T1) = w,
denotes the quotient space of cosets
the norm
where
X = x1 (D X2
is then the residue of the resolvent operator
P(A,T)
X.
0(T), and f E Hol(a(T))
the corresponding spectral projection is written
(z-T)at -1 the point
closed unit ball of x/Y
P
P
is called a pole of finite rank of
X
defined by
We use the following notation for pro-
T1 = TjX1, T2 = TjX2.
a(T)
is easy to check that function
If
P.
T = T1 (D T2
an isolated point of P(X,T).
with
P4 = P e 13(X)
the kernel of
P(w,T)
of
w
Associa-
T.
(-T)-ld-
2
jections.
A subset
6(T).
is the spectral projection
w
f(z)
surrounding
p(T)
is a suitable contour in
is one on
the adjoint
T*
f(z) (z-T)-ldz
fr
is
which is open and closed in
Q(T)
denotes the res-
is defined by the Cauchy integral
f(T)
=
TIY and
X
functions which are analytic in some neighbourhood of the operator
will be the kernel of
denote the family of complex valued
Hol(6(T))
Let
a & x
a 6 X*,
x E X,
If
X.
is the dual space of
X*
Y.
ker(T)
X, invariant under
is a subspace of
Y
and if
will denote the resolvent set, spec-
T, respectively.
trum and spectral radius of T
and r(T)
0(T)
P(T),
X.
x + Y; it is a Banach space under
denotes the set of integers (posi-
7e(7f+)
The00sure of a subset
X,
S
of a topological space will be
The termination of a proof will be signified by 0 0.2
Fredholm operators
Let
X
be a Banach space over
U
is a compact operator on
T
dim(T(X)) < -.
is the closed unit ball of
an ideal
is of finite rank if
T E B(X)
C.
if
X
The finite rank operators in
X.
and the compact operators a closed ideal
F(X)
is compact where
T(U)
K(X).
form
B(x)
If
T E K(X) Q(T)
the Riesz theory of compact operators states that each non-zero point of is a pole of finite rank of if
dim(ker(T)) < -,
T(X)
is closed in
set of Fredholm operators is denoted that if
T e K(X)
and
A 34 0
The quotient algebra
is a redhoZm operator if
T E B(X)
T.
and if
X,
dim(X/T(X)) < -.
The
It follows from the Riesz theory
D(X).
then A- T E (D(X).
B(X)/K(X)
whose elements are the cosets
is a Banach algebra under the quotient norm.
T + K(X)
It is called the Catkin algebra
and will play a mayor role in our deliberations.
Our immediate aim is to
characterise Fredholm operators. DEFINITION,
0.2.1
of elements
x
=
II{xn}li
(ii)
m(X)
is the linear space of bounded sequences
{xn}
with the supremum norm
E X
n
Q_(X)
(i)
supllxnII. n
is the linear subspace of
consisting of those sequences
Q.(X)
every subsequence of which contains a convergent subsequence, i.e. those sequences with totally bounded sets of terms. It is elementary to check that subspace of
(X),
and
Z.(X)/m(X),
Further, if
Q (X).
is a Banach space and
T C B(X)
{x } E m(X) _> {Tx } E m(X).
n
n
T E B(X)
and if
T({xn} + m(X))
Clearly
Q0(X)
T C B(X),
and
=
let
T
then
Let
.. X
{x } E k
n
(X) > {Tx } e
T c K(x) <=> T = 0.
n
denote the quotient space
denote the operator on
{Tx } + m(X). n
m(X) a closed
X
defined by
For
(Atkinson characterisation)
THEOREM.
0.2.2
the following
T E B(x)
statements are equivalent T E $(X);
(i)
(ii)
T + F(X) E Inv(B(X)/F(X));
(iii)
T + K(X) E Inv(B(X)/K(X));
(iv)
T E Inv (B (X) ) .
co-dimension, so there exist closed subspaces
X
T
ker(T)
=
and
T E (D(X) _> dim(ker(T)) <
(i) => (ii).
Proof.
=
13 Z
Z
and W
is of finite
T(X)
of
X
such that
T(X) ® W.
can be depicted as the 2x2 operator matrix
ker(T) T
Z
=
T22
the subspaces on the top being domains and those on the left ranges; T22 : Z ± T(X)
unmarked entries are zero. T(X) (
the
is bijective and continuous and
is closed so there exists a continuous linear inverse
S22 : T(X) a Z
If
(30) p.57).
T(X) ker (T)
S
= Z
S22
ker (T)
T (X)
then
TS
=
w
and
I
T (X)
Clearly, IL
TS
and
ST
ST
=
Z
ker (T) Z
I
are projections of finite co-dimension so there exist
projections
P , Q
E F(X)
=
I-P
,
TS
so
S
such that
modulo
T
is the inverse of
I-Q
=
ST
F(X).
(ii) _> (iii) is obvious.
If s + K(x) = (T + K(x))-1, there exist K1, K2 E K(x) I - K1, ST = I - K2. Clearly ST = I = TS. (This
(iii) _> (iv) . such that
TS
=
argument is not reversible as we do not know that closed subalgebra of
(iv) _> W.
=
is a
B(X)).
Let
and choose a sequence
T E Inv(B(X))
{xn}
in the
Then
unit ball of ker(T).
{Tx }
T E B(X)}
{T
0 => T({x } + m(X))
11
O
=
L1
{xn} + m (X)
0'
=
{xn} E M(X), 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
T
and
Z
n
X.
such that
X
z
dim(ker(T)) < -. dim(ker(T)) < co, there
Since
X = ker(T) 9 Z.
and
Tx
Clearly
so it is sufficient to prove that
Suppose not;then there exists
Z.
for each n
{Tx } E m(X)
of
is injective on
is bounded below on
IIxn II = 1
is closed in
{x } C Z
n
with
- 0.
n
T({x } + m(X))
=
{x } + m(X)
o,
n
=
n
0,
_> {xn} E m(X).
Thus there exists a subsequence
}
{x
nk IIYII = 1
and
Tx
-> Ty = 0,
such that
x
-*
y E X.
Then
n]c
but Z- ker(T) = (0)
which is a contra-r1k
diction. 5
is closed, the quotient space
T(X)
Since
dim(X/T(X)) < -.
remains to prove Ilyn + T(X)II < 1
for each n
,
IIY n + Txn
for each n
.
II
< 2
Let
{y )C X n then there exists
be such that {x } C _X
n
such that
is invertible so there exists
T
such that
{w n} e Q.(X)
T({wn} + m(X))
t hus {T(w
is a Banach space, it
X/T(x)
{yn + Txn} + m(X),
=
- x) - y} E m(X),
hence there exists a subsequence such that
T (wn
- xn
- ynk
)
z E X,
k
thus since
IIynk +z+T(X)II 10 as k- 00 is closed.
T(X)
the unit ball of 0.2.3
So
If
T E 13(x)
be the spectrum of
T + K(X)
0.2.4
COROLLARY.
W(T) = Q(T).
0.2.5
DEFINITION.
(i) If
the defect of
n
+ T(X)}
has a convergent subsequence, thus dim(X/T(X)) < 0
is compact and
X/T(X)
DEFINITION.
{y
the essential spectrum
w(T)
is defined to
in the Calkin algebra.
T E 4(X)
T, d(T) = dim(X/T(X))
the nullity of
T, n(T) = dim(ker(T)),
and the index of
T,
i (T) = i (T) = n (T) - d (T) . X
(ii)
If
T c $(X),
X
set of Fredholm points of Clearly group and
T
is a FredhoZm point of T
is denoted by
P(T) C (D(T) = C\&(T),
i(T1T2) = i(T1) + i(T2)
further for
X - T E c(X), and the
if
D(T). (D(X)
is a multiplicative semi-
T1, T2 E D(X)
((25)
3.2.7).
Moreover the set of Fredholm operators is invariant under compact perturbations and
i(T + K) = i(T)
if
T E (D(X),
theory is of crucial importance.
K E K(X)
((25)
4.4.2).
Perturbation
THEOREM.
0.2.6 1 IS1 !
there exists
If T E (D(X)
< 6 => T + S c D(X)
and
6 > O
such that
i(T + S) = i(T).
((25)
s E B(x),
4.4.1).
Consequently,
4(X)
is an open semigroup in
is continuous on
(D(X)
and therefore constant on connected components of
B(X)
and the index function
More detailed information is available if the perturbation is caused
(D(X).
by a multiple of the identity. THEOREM.
0.2.7 for
n(T),
and
< 6
XI
Tf T E (D(X), n(A + T),
d(X + T)
respectively for o <
d(T)
3.2.10).
((25)
there exists
6 > O
such that
A + T E $(X)
are constant and less than or equal to
< 6.
a]
This important result we call the punctured neighbourhood
theorem.
An index-zero Fredhoim operator may be decomposed into the sum of an invertable operator plus a finite rank one.
0.2.$
T E
THEOREM.
T + AF E Inv(B(X))
i(T) = O => there exists
F e F(X)
such that
(% # 0).
As in the proof of 0.2.2 we may write
Proof.
ker T T
W
where
Z
= T(X)
F E:
and
(D (X)
T22
ker T
and
have the same dimension since
F(X) by means of the isomorphism
ker (T) W
F
J
:
i(T) = 0.
Construct
ker(T) -> W.
Z
J
T(X)
If
A # 0,
0.2.9
T + AT E Inv(B(X)) .
THEOREM.
= r _l
If T E B(x)
then
a(T)om{A E )>(T)
and
i(A - T) = 0}
O(T + K) .
KSK(x) 7
The result may be restated as follows:
Proof.
i(A - T) = o} = U
and
{X E O(T)
p(T + K).
KEK(X)
Let
P (T + K); then for some K0 E K (X) ,
AE
KEK (x) A - T - Ko E
hence
(X)
and
A E P (T + Ko) But then
i(A - T - Ko) = 0.
A - T E O(X)
and i (A - T) = 0. Conversely, let generality take
A - T E D(X)
Then
A = 0.
i(A - T) = 0. and
T E I(X)
Without loss of
i(T) = 0
and, by 0.2.8,
0 E p(T + K1) ,
such that
K1 E K(X)
there exists
and
Riesz operators
0.3
The ideal of inessential operators
on a Banach space
I(X)
is defined to
X
be the inverse canonical image of the radical of the Calkin algebra. the radical is closed in the Calkin algebra,
of finite rank of x E X
radius
B(X).
is a Riesz operator if the non-zero spectrum of T consists of poles
T E B(X)
If
is closed in
I(X)
Since
T.
and
d(x,E)
c > 0,
denotes the open ball centred at
x
of
E.
0.3.1
Let
DEFINITION.
compactness
q(B)
of
If BC U L1(xi,E) (The pointsl x
q(B)
Clearly
=
is the infimum of
B
cover by open balls in
X
of radius
we say that
E > 0
such that
B).
is totally bounded in
X,
is a compact subset of
0.3.2
Let B be a bounded subset of x, u
Of X and T C 8 (X) . q(T(U)) <
so if
B}.
B
is closed in
X.
'Then
sup q(T(B)) < 4q(T(U)).
q (B) <1
B.
Thus
q(B) = 0 < > B LEMMA.
has a finite
is a finite E-net for
{xl,... xn}
inf{E > 0 : there exists a finite E-net for
q(B) = 0 <=> B
B
E.
need not necessarily lie in
i
x; the measure of non-
be a bounded subset of
B
the closed unit baZZ
X,
Proof.
The left hand inequality is obvious. q(T(U)) < E
suppose that
and let
B
To prove the right hand one
be a bounded set such that
q(B) < 1.
Then
n
T(U) CU A(yi,E),
(y1,... yn E X)
1
n C U A(Txi,2c),
(xl,,.. xn E U)
1
n
2T(U) CU ©(2Txi,4C).
and
1 m
B CO
Now
, . . .
z zm E X)
1 m
CU A(b3,2) , 1 m
tJ (b, + 2U). 1 m
Thus T(B)CJ (Tb. + 2T(U)), 1
n
m
Cl (Tb +U A(2Txi,4c)), mn CU U (Tb + A(2Tx1,4s)), 1 1
m n
C U U A(Tb. + 2Txi,4s)
11 so
q (T (B))
0.3.3 Proof.
< 4E , If
LEMMA.
Let
{xn} E 9
11xn + m(X)11 < S,
11{xn} - {yn}11 < S. {yn}
(X), q({xn}) = 11{xn} + rn(X)11.
for each
q({xn}) < E + S
Since
C > 0.
then there exists
{yn} E m(X),
This is a finite
and, as
C
is arbitrary,
{yn} E m(X)
such that
there exists a finite E-net for (6+6)-net for
q({xn}) < S.
{x 1.
Thus
It follows that
q({x}) _< 11{xn} + m(X)11. 9
Conversely, let
so for each n there exists j (1 < ) < Q)
yl,... y
say
IIxn - y3II
q({xn}) < 6; then there exists a finite 6-net for
If
< 6.
z
= y
n
{xn}
such that
for each such n we obtain a sequence
,
7
NOW
{z } E: M(X).
n
II{xn} - {zn}II < 6,
hence II{xn} + m(X)II < 6,
II{xn} + m(X)II < q({xn}) .
so
Recall that T c B(x)
induces an operator
where
T E B(X)
X = kw(X)/m(X),
by virtue of the equation
T({xn} + m(X))
0.3.4
LEMMA.
=
{Txn} + m(X).
q(T(B)) < 2IITIL.
IITII < sup
q(B)
sup{II{Txn} + m(X)II
=
IITII
sup{q({Txn})
II{xn} + m(X)II < i},
=
q({xn}) < 1},
:
(0.3.3)
sup q (T (B)) .
<
q(B)
be a bounded subset of
B
choose an infinite sequence xn+l e B\L)A(x.,6) SO
for
n > 1.
If
e > 0
IITII
=
2IITII >
{Txn}
sup{q({Txn})
sup
q(B)<1 10
in
q(B) > 6 > 0.
IIx - x m n
II
q({Txn}) >
> 6 T(B)
for
m # n
where
q(T(B)) - e.
q({xn}) < 1} > 11 q(T(B)) - e,
q(T(B)) ,
Then we may
inductively so that
B
apply this to the set
such that
:
such that
Clearly
]],,
q({xn }) > 6/2.
to obtain a sequence
thus
{xn}
X
and
q(B) < 1, Now
We have now, somewhat laborously, set up the machinery required for our characterisations of Riesz operators.
0.13.5
For
(Ruston characterisation)
THEOREM.
T E B(X)
the following
statements are equivalent
(i)
(ii)
T c R (x) ; r (T + K (x) ) = 0;
(iii) r(T) = 0; (iv) (v)
lim q(Tn(U))l/n = 0; n for each E > 0 there exists
such that
n E 7+
has a finite
Tn(U)
En -net.
T be a Riesz operator, then if
Let
(i) <_> (ii).
proof.
{X E 0(T)
IAl
:
P E F (X) .
r (T - TP) <6 and
r (T + K (X) )
<
the set
is finite and the corresponding spectral projection
> d}
Now
d > 0
If
P(X,T) E F(x).
the corresponding spectral projection
0 # A E 6(T)
then
TP E F (X) ;
inf r (T + K) < S, KeK(x)
and since
6
T E B(X)
Conversely, let then
r(T + K(x)) = 0.
is arbitrary
satisfy
A - T + K(x) E inv(B(X)/K(X)),
(0.2.2),
A - T e CX).
contain points of
If
each neighbourhood of
for
0 <
Al
< 6,
F
p(T)
and some positive and
But if the non-zero boundary points of 0(T) must be isolated.
isolated, all non-zero points in such and
must
A
thus using the punctured neighbourhood theorem
P(T),
therefore this punctured neighbourhood lies in 0(T).
o J A
so, by the Atkinson characterisation
0 # A e aa(T)
(0.2.7), n(A - T) = 0 = d(A - T)
point of
If
OCT + K(x)) = {o}.
a contour in
p(T)
surrounding
A
A
S;
is an isolated
a(T)
Let
are all A
be one
but no other point of
0(T).
Then
P(A,T + K(X)) since A
z - T - K(X)
=
P(A,T) + K(X)
=
27ri
r
is invertible inside and on
is a pole of finite rank of
(z - T - K(X))-1dz r.
So
=
0,
P(A,T) E K(X)
and
T.
11
This follows at once from 0.2.4.
(iii).
Combining 0.3.2 and 0.3.4 we get
(iv).
IITII < 4q(T(U)) < 81[TI1,
(§)
and the equivalence follows by considering This is now clear since
(iv) <_> (v).
q (Tn (U) )
?1111n.
1
Tn(U)
has a finite En-net <_>
< En is
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
and
S
0.3.6
THEOREM.
(ii)
S E B(x), T E R(x)
'(iii)
Tn E R(X)
T.
and
S, T c R(x)
(i)
[S,T] E K(X) > S + T E R(X);
[s,T] E K(x) _> ST, TS E R(x);
and
(n > 1), T E B(x), I ITn - TI J (n > 1) _> T c R(x) .
-> 0
[Tn,T] E K(x)
and
Another useful consequence involves functions of a Riesz operator.
T E: B (X) 0.3.7 (ii)
Let
f E Hol (a (T)) .
and
THEOREM.
T E R(X)
(1)
If T E B(x)
and
f(z)
and
f(O) =O> f(T) E R(X);
does not vanish on
U(T)\{o}
then
f (T) c R (x) => T E R (X) .
In fact f (0) = 0 => f (T) = Tg(T) , ';here hence 0.3.7(i) follows from 0.3.6(ii).
g E Hot (c1 (T) )
and
[T,g (T)] = 0
Range inclusion
0.4
The machinery developed in §3 allows us to deduce properties of an operator S
from an operator
use S
S-1(U)
T
provided that
In this section we shall
S(X) C T(X).
to denote the inverse image of
U
under
S
whether, or not,
is invertible.
0.4.1
that
THEOREM.
If S,T E B(x)
and
S(X) C T(X)
there exists
n=1
12
such
S (U) C T1 (T (U) ) .
Proof. X= S-1(T(X)) = S-1(T(U n U)) C S-1n T(U)) _ S
n > 0
is continuous hence
S-1
(T(U)) is closed in
n=l
X
n S-1(T(U))= X. n=1
therefore by the Baire
category theorem ((30) p.20), + n £ B'
But
x £ X
and
1
nS
(T(U))
E > 0
nS-1(T(U))
has a non-empty interior for some
is homeomorphic to
such that
S-1(T(U)),
S(L(x,E)) C T(U).
so there exist
Hence there exists
{x
}C U
n Sx = lim Txn, and if llyll < E, there exists {zn} C U such that n S(x + y) = lim Tzn. Thus Sy = lnm T(zn - xn) and {zn - xn} C 2U. such that
n
Finally, if
there exists
fly 11< 1,
{yn} C U
such that
Sy = n lim Ty n
2E-1
where
n =
0.4.2
COROLLARY.
0.4.3
THEOREM.
S(X)C T(X)
and
If S,T E 13(X),
n
T E K(X) _> S E K(X).
ST = TS,
and
S(U) C n(T(U))
then
Sn(U)Cnn(Tn(U)). By hypothesis the result is true for
Proof.
n, and let y E U,
S > 0.
Then there exists
n = 1.
z E U
Suppose it is true for such that
IISny - fnTnzjI < / 61IS11-1
Thus
so
Sn+ly I
- fln STnz 1
1
I
IISn+ly - nnTnSz1I
But there exists
w c U
< ;Z 6,
(t)
< / a.
such that
IISn - nTwII < / 6n-nIITnII-1,
so
IlnnTnSz - nn+lTn+lw11 <
From (Th) and (*) we see that if
(*)
1, S.
y c U,
there exists
w E U
such that
f1Sn+ly - nn+lTn+lw11 < &
that is
Sn+1(U)C-
U
n+1Tn+1(U)
and the proof follows by induction 0
13
Combining 0.4.1 and 0.4.3 we get 0.4.4
COROLLARY.
0.4.5
THEOREM.
If SW C T(X) If S,T E $(X),
and
ST = TS
S(X) C T(X)
then
and
T E Q(X) > S £ Q(X).
CS,TJ £ K(X)
then
T E R(x) _> S c R(x). Let V
Proof.
and
denote the closed unit ball of there exists
{xn} + m(X) E V,
{y} £ m(X)
If
E > 0
such that
1 + E
<
llxn + ynll
X = 2 (X)/m(X).
r xn + y n C U.
then t 1 + £
Now there exists {z } C U n
> 0
rl
such that
(0.4.1)
S (U) C.flT(U)
so there exists
such that
S(xn + yn)
fTz n
-
1 + E
so
E
(n > 1) ,
IIS(x n + yn ) - (1 + £)flTzn I I<E(1 + C)
hence IIS(xn + y
since
<
n
)
{zn} C U.
(n > 1),
- r1Tznll < E(1 + E) + EniITiI
Now
{yn} c m(X), hence
{Syn}
(n > 1),
C
m(X), therefore
(IS({xn} + m(X)) - fT({zn} + m(X))Il < E(1 + E) + En[ITII,
and since
{xn} + m(X), {zn} + m(X) E V we get
S(V) (:::-TIT (V)
and
Sn(V) C pn(Tn(V))
which gives 14
[S,T] £ K(X) => [S,T] = 0, (n > 1),
Ilsnll < rlnllrnll,
so, by 0.4.3,
But
r(T).
r(S) <
thus
r(T) = 0 by 0.3.5, hence
r(S) = 0
and
S £ R(X)
again by 0.3.5 . Action on the commutant
0.5 If
T £ B(X),
of
8(X)
and
Obviously
0.5.1
hence
is a compact (Riesz) operator on
T
'0-1
:
- T) -1 E Z (T)
S -; S(A -
T)-1
,
and then
£ B(Z(T)),
l c p(T).
Conversely, if
3
£ p(T),
I = (A - T)V(I)
=
(A - T)V(I)
The next result states that if is also compact.
0.5.2
If
{T(S )}
n
Sn C Z(T),
_ {S T} 1 n 1
X
ST(U) = TS(U)C! T(U) S
to
is a compact operator on
T
X
then
As we remark (p. 20) the converse statement is false.
E
IIsn11 = 1
(n > 1), we need to show that
has a norm convergent subsequence.
closed unit ball of
of
V(I) (A - T)
=
T £ K(x) > T £ K(z(T)).
THEOREM.
Proof.
Z(T).
such that
Thus
A c p(T) .
since everything commutes, thus
T
V £ B(Z(T))
there exists
is the identity on
(A - T)V = V(A - T)
SIE
Z(T) we
0 (T) = 0 (T) .
l £ P(T) > (A -
If
Z(T).
has a compact (Riesz) action on its commutant.
LEMMA.
Proof.
T on
is the operator of multiplication by
S -} ST
:
11TH = ffTlf T
say that
T
T which is a closed subalgebra
denotes the commutant of
Z(T)
and put
If
E = T(U).
hence by continuity
is contained in
mapping the compact Hausdorff space
Let
S £ Z(T)
S(E) C. E.
U and
be the IISII < 1,
Now the restriction
X(E); the set of continuous functions E
to
X.
Since
IISII < 1 > Ilsx - Sx.II _< IIx - x'If, 15
the set
{SIE
S E Z(T), IiSII < 1}
:
is an equicontinuous subset of
CE)
and is therefore, by the Arzela-Ascoli theorem ((30) p.266), a compact subset Hence if
CA(E).
of {S
Sn E Z(T)
and
which converges uniformly on
}
IISnII < 1
there exists a subsequence
{S
E, i.e.
Tx}
converges uniformly on
nk {S
i.e.
U.
is norm convergent ,
T}
A more complete result is true for Riesz operators. 0.5.3
THEOREM.
Let
Proof.
T E R(X) < = > T E R(Z(T)).
A J 0.
Lemma 0.5.1 shows that
is an isolated point of
C(T) <-> A
6(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
:
then
S E Z(T),
P(X,T)
:
(S E Z(T)).
SP(X,T)
S
commutes with
S
X
=
X1 ® X2,
T
=
T1 9 T2,
S
=
S1 ® S2,
so
(S E Z(T))
P(A,T)S = S1 0 02,
where
P(A,T)
(S E Z(T))
S1 E Z(T1), therefore
P(A,T)(Z(T))
Suppose now that
= {P(A,T)S
:
T E R(X), then
0(T)
and
thus
dim(P(A,T)Z(T)) < - hence
dim(X1) < W.
Conversely, let
point of
0(T)
and
S E Z(T)} _ {S1 0 02
0 # A E 0(T)
It follows that
T E R(Z(T))
:
S1 E Z(T1)}.
is an isolated point of
dim(Z(T1)) < dim($(X1)) <
T E R(Z(T)). and
0 # A E 0(T); then
dim(P(A,T)Z(T)) = d3-m(Z(T1)) < Co.
A
is an isolated
Since the algebra
generated by T1
is contained in
it must also be finite dimensional,
Z(T1)
thus there exists a non-zero polynomial G(T1) _ {a}
thus
{xn}
and
pendent set (A - T1)
But
p(T1) = 0.
for some positive integer
k.
if
= Xxn (for each n), also there exists(0 #)a E x1 n It follows at once that the infinite linearly inde-
T x 1
T a = Xa.
such that
such that
is infinite dimensional it contains an infinite linearly inde-
(} - T1)-1(O)
pendent set
(a - T1)k = 0
p
{a
xn}
of rank one operators lies in
is finite dimensional, hence so is
(0)
Z(T1)_
(A - T1)
Thus (0) = X1,
thus
T c R(x) , The wedge operator
0.6
we define the Wedge operator
T E B(X)
If
TAT : S } TST Clearly 0.6.1
IIT A TII
THEOREM.
=
T /\T
on
B(X)
by
(S E 8(X)) . IITII2.
(ii) compact, (iii) Riesz,
is (i) finite rank,
T E 8(X)
or (iv) quasinilpotent <=> TAT is.
n
Proof.
Let T = E ai Q xi then
(i)
i=1
n
n
t E a 4t xi}S( E a j=1 J i=1 1
TST =
Thus
{TST
a x) = J
S a (Sx.)a, 2 x1, i,J=1 1 J J
S E B(X)} C span {a, Q x
:
n
1 < i, j < n}
which is a finite dimensional subspace of Conversely, suppose that
T # 0
and
B(X).
T AT is finite rank.
contains an infinite linearly independent set that
T a
0
{Txi}1 choose
If
a c X
T(X)
such
and then the set
{TAT(a Q xi) }1
=
{T(a 4d xi)T}1
=
is an infinite linearly independent set in
{T*a Q Txi}1
T n T(B(X)).
17
U be the closed unit ball of
Let
(ii)
B(X).
exist
If
T
is compact,
x1,... xk E U
with
X, so the set
in
{TSTxi
{(TSTxi,..., TSTxk)
x E U
such that for each Fix
{STx1
E > 0
i (1 < 1 < k)
S E B1}
:
there
is bounded
Thus the set
is totally bounded.
which is contained in the product space
s E B1},
:
there exists
i, then the set
S E B1}
:
the closed unit ball of
B1
is totally bounded, hence,if
T(U)
IIT(x - x1)11 < E.
and
X
X(k}
in the product topology, is a subset of a totally bounded set, hence is itself
Therefore there exist Si.." Sn c B1
totally bounded.
of the above set is within
If x e U
of
E
for some
(TSi Txl,... ITSTxk)
3 (l<3
and S E B1
IITSTX - TS3 Tx!I < IITII
+ IITII Thus
such that each point
IIT(x - xi)II + IITSTx1 - TSi Txi1I
IIS11
IITx1 - Txll
IISi 11
IITST - TS,TII < (21ITII + 1)E
bounded, i.e.
T,% T
Conversely, let sequence in
X
and the set
{TAT : A E Bl}
is totally
is compact.
and TAT be compact.
T 34 0
and
< (211T11 + 1)E.
a E X,
0
{a t& x1}
If
{x,}
is a bounded
1s a bounded sequence in
B(X),
thus {T a Q Tx1}
{TA T(a & x1)
is a bounded sequence in
IIT*a 0 Tx
1
-
and thus has a subsequence such that
B(X)
T*a & Tx
n
1
11
=
IITx1
11T*U11
m
Hence
{Tx }
has a convergent subsequence and
(iii)
If
1s a Riesz operator, then for
such that
T
IITn - Kn 11
< En
(n > N).
T
-
n
II
.
Tx1
-> O.
m
is compact.
c > 0
there exists a compact
Without loss of generality take
IITII, 1lKn11 < 1.
T)n (I (T
18
A
- K/\Knll =
sup IITnSTn - KnSKnI! IISf1<1
211TH - KnI
< 2En.
K
is compact by part (ii), so, using the Ruston characterisation (0.3.5),
RnA K
n
TA T
is Riesz.
Conversely, let TAT be Riesz on
$(X);
is a Riesz operator ((25) 3.5.1), i.e. is Riesz on
Theorem 0.5.3, T2
S -
then its restriction TA TIZ(T) ST2
is Riesz on
Z(T)
and by
X, hence, by the Ruston characterisation, so
T.
is
(iv)
I
I (Tn T)nI I =
r (T
I
IT
A TnI I
=
= lim II (TA T)nIIlln
I ITnI
Jim
=
TA T
0.7
IITnII2/n
=
r(T)2.
n
n Clearly
12
is quasinilpotent <_>
T
is .
Notes
Compact or completely continuous operators arose first in the guise of quadratic forms in Hilbert's work (46) in 1906.
Fredholm, three years
earlier, had considered resolvent expansions of certain integral operators (33).
The spectral theory of compact operators was worked out by F. Riesz
(74) in 1918 for the important special case in which the underlying Banach space is a space of continuous functions.
Fredholm theory in its present
form dates from the work of Atkinson (5) and Gohberg (36), Yood (102),1954.
(37) in 1951 and
The characterisation of Fredholm operators as those which
are invertible modulo the compact or finite rank operators (0.2.2) Atkinson.
is due to
Ruston started work on Riesz operators (76) in 1954 while Calkin's
work on ideals of operators in Hilbert space (20) had appeared in 1941 and Kleinecke (53) introduced the important ideal of inessential operators, which is the biggest closed ideal contained in the Riesz operators, in 1963.
The
properties of Riesz operators were examined by Heuser (42) in 1963; in more detail by Caradus (21) and West (94), (95) in 1966.
This class is of interest
on account of the spectral properties which Riesz operators share with compact Operators.
From the algebraic standpoint, if commutativity (or commutativity
modulo the compact operators) is added to a result valid for extends to
K(X), it often
Examples of this phenomenon occur in 0.3.6 and 0.4.5.
R(X).
The technique, set up in §0.2, and used to study Riesz operators in §0.3 is based on work of Buoni, Harte and Wickstead (17),
replacing T
by
Tn
(41).
Observe that by
in inequality (§) on p.12, then taking n-th roots and
using 0.2.4 we get 19
r(T)
=
r(T + K(X))
=
lim q(Tn(U))1/n n
Further information on measures of non-compactness and the essential spectrum is given by Zemanek(l07),(1O8)The representation
algebra on
X
as a subalgebra of
73(X)
T + K(X) - T
of the Calkin
is due to Lebow and Schechter (57).
It is not known if this representation is irreducible, or if its image is closed in
B(X), although Lebow and Schechter have shown that the latter
statement is true if ((57) 3.7).
X
satisfies the Grothiendieck approximation property
The equivalence of (i) 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
T = V + F where
ible, F is of finite rank and rV,F] 2 = 0.
Note that [V,F] = 0 <=> zero
is either in the resolvent set of rank ((25) 1.4.5), and in this case
T, or is an isolated pole of T
V
T
is invert-
of finite
is called a Riesz-Schauder operator.
Ruston proved the equivalence of (i) and (ii) in 0.3.5.
Condition (iv) is
due to G.J. Aurphy and the geometric characterisation (v) of Riesz operators to M.R.F. Smyth.
The range inclusion theorem for compact operators (0.4.2) is attributed to R.S. Phillips; the result for quasinilpotent operators (0.4.4) is due to M.J. Ganly,while the result for Riesz operators (0.4.5) is due to M.R.F. Smyth The fact that a compact operator acts compactly on its commutant (0.5.2) was observed by Bonsall (11),
(12) and used to develop an algebraic approach
to the spectral theory of compact operators.
Theorem 0.5.3 showing that a
Riesz operator has a Riesz action on its cormutant 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 ingenious counter-example of a weighted shift operator on
k. l.
Murphy overcame the main difficulty, which is to determine the commutant, by choosing the weights so that it is the strongly closed algebra generated by the operator.
The wedge operator
T A T
was introduced by Vala (91) and its properties
were studied by Alexander (4), who also considered the generalised wedge operator defined by
S A T : V -> SVT
2
(V E B (X)) .
The following result generalises 0.6.1. 0.7.1
THEOREM.
If S,T E B(x) then
S
and T belong to classes (i)
(ii) or (iii) <=> S AT does (i)
(ii)
(iii)
the non-zero finite rank operators; the non-zero compact operators;
the Riesz operators which are not quasi-nilpotent.
is quasinilpotent <=> either
S
or
T
is quasiniZpotent.
Further
S AT
F Fredholm theory
In 0.2.2 we saw that a bounded linear operator on an infinite dimensional Banach space
X
is Fredholm <_>
invertible modulo
it is invertible modulo
The fact that
K(X).
is a primitive ideal) and that
F(X)
F(X)<=>
it is
is a primitive algebra ((0)
B(X)
is the socle of
B(X)
motivates the
work of this chapter.
Fredholm theory in Banach algebras was pioneered by Barnes (7), semisimple and semiprime algebras.
Banach algebras by Smyth (83).
(8) in
This theory was extended to general
Here we adopt a simpler approach due again
to Smyth and use the case of a primitive Banach algebra,where Fredholm theory is straightforward,with natural analogues of the rank nullity, defect and index of a Fredholm operator for Fredholm elements of the algebra (those invertible modulo the socle), to build up to the general case.
§1 contains information on minimal ideals and minimal idempotents, also in F.l.l0 we exhibit the Barnes idempotents for a Fredholm element. are fundamental to the rest of the chapter. Banach algebra is developed in §2;
These
Fredholm theory in a primitive
the nullity, defect and index of a
Fredholm element in a primitive algebra are defined, and a representation connects these quantities with the nullity, defect and index of a certain Fredholm operator.
This allows us to deduce the important results of
Fredholm theory in primitive Banach algebras directly from their counterparts in operator theory.
A
If
P
then the quotient algebra
is a primitive ideal of a general Banach algebra A/P
is primitive.
§3 to obtain a Fredholm theory in the general case.
This fact is exploited in As observed by Pearlman
(69) and Smyth (83) it is necessary to replace the numerical valued nullity, defect and index of the primitive case by functions (of finite support) defined (for each Fredholm element) on the structure space of the algebra. §4 besides containing observations on extensions of the theory also contains results on generalised indices and on the algebraic kernel (the largest ideal of algebraic elements).
Minimal ideals and Barnes idempotents
F.1
A
In this section
will be a semisimple algebra over
(Topological considerations will not enter into our
necessarily unital.
Our results apply equally well to a semiprime algebra over
discussions.
In such an algebra the socle of
1).
which is not
C
A,
is defined to be the sum
soc(A)
of the minimal right ideals (which equals the sum of the minimal left ideals (14) 30.10) or (0)
if there are none.
Definitions and theorems are usually
stated for right ideals, corresponding statements may be made for left ideals. F.1.1
A
division algebra (if theorem (14) 14.2,
potents of
eAe
is a
is a Banach algebra then, by the Gelfand-Mazur Min(A)
eAe = Ce).
denotes the set of minimal adem-
is the set of rank-one projections in
compact Hausdorff space and the idempotents in
Q,
is minimal if
A.
Min(13(X))
on
e E A
A non-zero idempotent
DEFINITION.
and closed sets of
?3(X).
If
(
is a
denotes the algebra of continuous functions
C(Q)
are the characteristic functions of open
C(Q)
while the minimal idempotents are the characteristic
0,
functions of isolated points of
Thus
Q.
Min(C(Q))=
< > Q
possesses no
isolated points. F.1.2
DEFINITION.
A right ideal
for any right ideal
R1C R, either
R
of
is minimal if
A
R1 = (0), or
R # (0)
and if
F1 = R.
The link between minimal idempotents and minimal ideals in semisimple algebras is set out in §BA.3, for reference it is restated here. F.1.3
LEMMA.
(i)
xA = 0
(ii)
(iii)
R
Let A be semisimple then
x = 0;
is a minimal right ideal of A <_> R = eA
if R is a minimal right ideal of A
and
where
x E A
e e Min(A); then
xR
is either
a minimal right ideal or is zero. Proof.
(1)
(iii)
F.1.4 + enx n
By (14) 24.17,
LEMMA (Exchange). where
x c rad(A) = (0).
BA.3.1 and BA.3.2
xk E A
If
{e1,... en, f}GMin(A)
(1 < k < n)
and
e,x, J
54 0
and
f = e1x1 + .,,
then
J
23
E ekA k
fA +
=
ekA.
E
k#j
It suffices to show that
Proof.
A
e
is contained in the right hand side.
j
ejA
F.1.5
3
3
1
3
e,x,
since
e.x.A = e.A
Now
=
and
0
e.A
is a minimal right ideal.
e.x3A C fA +
E
ekA ,
A set
W
of idempotents of A
DEFINITION.
of = 0 = fe for e, f 6 W Observe that if
Thus
3
J
f.
and e
{ell..., en}
is orthogonal if
is an orthogonal set of idempotents in
A,
then
(e1 + ... + en)A
e A + ... + enA
=
=
1
e1A a e2A ... 9 enA.
We need two further technical lemmas. F.l.6
orthogonal subset of Min(A) is such that
f 6 Min(A) { e1
Suppose that
LEMMA (Orthogonalisation).
. . . ,
and that
£A
{e1,..., e
en+l
is an orthogonal subset of Min(A)
en+1}
}
R = e1A + ... + enA.
there exists
R,
n
is an mien if
such that
and
R t fA = elA + .,,
+ en+l A. Proof.
p = e1 + ... + en.
Write
fA ¢` pA
f = pf + (1 - p)f,
and
(1 - p)fA
idempotent
Then
and R = pA.
p2 = p
it follows that
g, say.
Clearly pg = 0,
Further Now
en+l # 0, (0)
hence if
for if
en+1A CgA,
en+l = g(l - p),
g = gp,
hence
Now, since
=
(1 - p)fA = gA
g(1 - p) 6 gA
(F.1.3),
=
then
g = gpgp = 0
en+l £ Min(A).
(1 < k < n), eken+l = ekpen+l = 0 = en+lpek = en+lek is an orthogonal subset of Min(A). {el,..., en+l}
24
hence
(1 - p)f # 0,
is a minimal right ideal which (F.1.3) contains a minimal
en+l - en+12. which is false.
en+1
Since
(1 - p)fA C fA + pA,
so the set
Further
e A + ... + en+lA C R + fA.
so
1
Conversely, as
(1 - p)fA
by minimality (F.1.3)
en+l = g(l - p),
=
=
g A
g(1 - p) A
=
en+l A.
n+l
Therefore
R + fA CIE ekA S
giving
fA C pA + en+1A
1
Let
LEMMA.
F.1.7
R be a right ideal of A
minimal right ideals of A. is finite, further, if n p ek = p2 E soc(A)
orthogonal set
for some
which we take to be
m
ak E A
so
(1 < k < m),
Suppose
Then
is an orthogonal subset of RR Min(A).
{e1,..., en}
J
such that R Cf1A + ... + fmA.
{f1,... fm}C Min(A)
el = fl al + ... + fmam some
is a maximal such subset then
R = pA.
Using the exchange and orthogonalisation lemmas we can find an
Proof.
that
Then every orthogonal subset of RAMin(A)
felt ..., en}
and
lying in a finite sum of
fjaj # O for
and then by the exchange lemma
1,
m
=
E f k A
e1A +
fkA. 2
Thus
e2 = e1b1 + f2b2 + ... + fmbm
since
e2
by orthogonality,
e1b1
J
which we take to be in
m
so
is not an orthogonal set.
{e1,. .., em+l}
orthogonal subset of
and put p = ELL ek.
m
contains more than
R n Min(A)
Suppose now that {el,. en} n -1
R/\Min(A)
(2 < j < m)
J
E fk A = F ekA; at this point the process terminates since 1 and if em+i E R R Min(A) then em+lek # 0 for some k
stage at which RC E e A, 1 k (1 < k < m)
j
and
Repeating the process m times we arrive at a
2.
m
for some
0
f b
(1 < k < m),
bk e A
for some
Hence no
elements.
is a maximal orthogonal subset of
Clearly p2 = p E soc(A)
and pAGR,
so
If not, choose w e R\pA and write m in since y E RC-) fkA, y = E fky so
it only remains to show That RC pA. y = (1 - p)w. fky 34 O
fkyu = fk
Then
for some k for some
y
0,
u c A.
ideal which must contain an fA ¢ pA
and
(1 < k < m).
and, by F.1.6,
By minimality
Then yufk # 0 f c Min(A).
there exists
fkyA = fkA,
is a minimal right
so
Now
hence
fA G yA c(l - p)A,
en+l e Min(A)
such that
hence felt .... en+1}
25
n+l
is orthogonal and pA + fA = en+l
e pA + fAC R,
However this implies that
E ekA.
contradicting the maximality of
{e1,..., en}
which
proves the lemma , DEFINITION.
F.1.8
x E A
If
the right annihilator of
x
in A
is
defined by
ran (x)
{a E A
=
xa = O};
while the left annihilator of
lan(x)
{a E A
=
DEFINITION.
F.1.9
idempotent for
x
in A is defined by
x
ax = O}.
If
x E A
we say that
in A
if
xA = (1 - p)A;
Barnes idempotent for
in A
x
p = p2 E A while
is a left Barnes
q = q2 E A
is a right
Ax = A(1 - q).
if
Note that the Barnes idempotents,if they exist,are not normally unique;
(i)
(ii)
p
is a left Barnes idempotent for
(iii)
q
is a right Barnes idempotent for
x in x
A -> lan(x) = Ap; in
A => ran(x) = qA.
The next result is fundamental as it connects the existence of Barnes idempotents in the socle with left or right invertibility modulo the socle. F.l.lO
THEOREM
algebra and
(Barnes idempotents). Let A
x E A.
Then
x
is left (right) invertible modulo
has a right (left) Barnes idempotent in
soc(A).
Proof.
modulo
Let
u
be a left inverse of
right invertibility is similar). (1 < i < n)
be a unital semisimple
x
soc(A)
Then there exist
e.
(the proof for E Min(A),
such that
n
ux-1 = Eea 1
1,
n ran(x) C E e. A, 1
_>
ran(q)
=
i
qA,
where
q = q
2
E soc(A)
soc(A)< > x
(F.1.7).
a
E A
Now
ax
axq + ax (I - q)
=
ax(1 - q),
=
Observe that Axn Aq = (0). We shall show that A = Ax a Aq which implies that Ax = A(1 - q). Suppose not, then
hence AxC A (1 - q) .
is a proper left ideal of A
Aq
Ax
ux - 1 E soc(A)
left ideal
L.
soc(A) St L
for, if it were,
g E Min(A)\L.
exists L
Since
x 6 L
it follows that
contradicting maximality.
L + Ag = A by maximality of
Now
Ag.
L,
Thus there and
So
z E L,
a c A,
so yag = y - yz E L fl Ag = (0).
Hence
In particular, for some
z + ag,
=
1
and
L e Ag.
=
A
1 C L
by minimality of
Ag = (0)
which is contained in some maximal
hence, for y C L, y
xag
Thus
yz + yag,
=
=
0
=
qag.
ag e ran(x) = qA,
,
so
ag = qag = 0
which gives
F.1.11
EXAMPLE.
on a Banach space
a
The reverse inclusion is obviously
contradiction which concludes the proof.
true
1 = z e L,
We construct Barnes idempotents for a Fredholm operator X.
Recall that
F(X) = soc(B(X)).
T
T
is examined
pictorially as in the proof of the Atkinson characterisation (0.2.2). Let
of
X
X = ker(T) e Z = W ® T(X)
and
where
dim(ker(T)), dim(W) < -.
If
Z
and W
are closed subspaces
A E B(X),
27
ker(T)
Z
ker(T)
W
ker(T)
TA =
T22`
T (X)
Since
z
is invertible,
T22
ker (T)
Z
A
A11
12
A22
A21
w
T (X)
T22A21
A S ran(T)<> A21 = 0 = A22.
2A 22
So if
A E ran(T) ker(T) A
=
W
A11
Z
A12
T (X)
and a right Barnes idempotent for
T
is
ker (T)
ker(T)
I
It is easily checked that B (X) T = 6(x) (I - Q) .
since ran (T) = Q-8 (X) .
for if
B E B(X),
ker(T) BT = z
ker(T)
T ,X)
W
B11
B12
T (X)
B22
B21
ker(T)
Z
ker(T)
W
B12T22
z
222
Z
B22T22
while if C e 6(X), ]cer (T) C(I - Q) = ker(T) z
28
ker(T)
z
C11
C12
C21
C22
ker(T)
ker(T) z
ker(T)
Z
I
z
Z
C12
C22
T22
Since
is invertible the equations
be solved uniquely for that
B12, B22
B12T22
= C12, B22T22 = C22 may which verifies the equality. Observe
is an arbitrary closed complement of
Z
Barnes idempotent for
T
ker(T) B(X)
is any idempotent in
A similar analysis shows that any idempotent
lan(T)
B(X)P,
=
TB(X)
in
ker(T).
whose kernel
B(X)
(1 - P)B(X)t
=
and hence is a left Barnes idempotent for
T.
Primitive Banach algebras
F.2
A will be a primitive unital Banach algebra over
In this section that
Min(A) # 0.
A
and
For suppose that
A
tation of
x
xAy = (0) - either
O 34 4j(y)n.
Now
such that and
X
Y
X.
1(Ay)n
invariant under each element of
x
(0)
such
E 11(A),
.dote that,
i(A),
is zero.
Then there exist
C, n E X
is a subspace of
X which is
hence
But then
11(zy)n = C.
y
or
be a faithful irreducible represen-
1
and let
0 34 y
on the linear space
that (x)C
If
is primitive if
C
is a primitive algebra, then
x, y E A
z E A
A
Recall that an algebra
A possesses a faithful irreducible representation.
that is, if
so a right
X
satisfies
T(X)
is
P
in
with range
(Ay)n = X.
So there exists
t4i(xzy)n = i(x)C # 0
are two linear spaces we write
such
so
dim(X) = dim(Y)
xAy / (0)
to mean
that either the spaces are both infinite dimensional or they have the same finite dimension. F.2.1 (i)
LEMMA.
Let
there exist
e, f E Min(A)
u, v E A
and R
such that
be a right ideal of A,
then
f = uev;
(ii) dim(eAf) = 1; (iii) (iv)
Proof.
Since
dim(Re) = dim(Rf);
dim(Ae/Re) = dim(Af/Rf). (i)
Af
Choose a non-zero v c eAf.
We have observed that eAf 34 (0). is a minimal left ideal,
Af = Av,
so
f = uv
for some
u E A.
Also
hence
v = ev,
f = uev.
(ii)
By (i),
eAf = eAuevC eAev = Cev whose dimension is unity.
(iii)
By (1),
G Rev. Rf = Ruev.
Rf
dim(Re) < -, so is dim(Rev)
dim(Rf) < -, then
Similarly, if
dim(Rf) < dim(Re).
Further if
So if
Rev
is infinite dimensional so is
and
dim(Re) < dim(Rf). Re,
and therefore so is
and conversely.
(iv)
Let S£
=>
be a subset of A, then
S
is linearly independent modulo
Suev Sue
Rf,
is linearly independent modulo Ruev, by (I), is linearly independent modulo
(To see this since
ue # 0,
dim(Ae/Re);
so
:due = Ae
similarly
tue = e
for some
dim(Ae/Re) <
It follows that
Re = Rtue C Rue cRe).
Rue = Re. t E A.
Hence
dim(Af/Rf) <
> dim(Ae/Re) < dim(Af/Rf).
dim(Af/Rf) <
The infinite dimensional results are clear , For the remainder of this section potent of
e
will denote a fixed minimal idem-
Ue shall write
A.
x- x : A-> B(Ae) to denote the left regular representation of the primitive Banach algebra A on the Banach space
Ae,
that is
important representation. xAe = 0 => x = 0,
it is faithful.
which is invariant under left ideal of
x(y) = xy
y E Ae.
for
This is an
It is norm reducing, hence continuous and since
x,
Further if
for each
is a subspace of
it follows that
x E A,
A which implies that either
L
L = 0
or
L = Ae.
L
Ae
is a Thus the
representation is irreducible.
Observe that
x(Ae) = xAe,
and ker (x) = ran (x) n Ae = ran (x) e . Since
xA
and
ran (x)
are right ideals of
the rank, nullity and defect of the operator
A,
it follows from F.2.1
x E F3(Ae)
that
are independent of
As the following example illustrates
e 6 Min(A).
the particular choice of
we can say even more when dealing with the algebra of bounded linear operators on a Banach space. F.2.2
subalgebra of
containing
B(X)
A be any unital closed
be a Banach space and let
X
Let
EXAMPLE.
Then A
F(X).
is a primitive Banach
algebra and we fix our minimal idempotent to be the rank-one projection
y Q g where
and
g 6 X*
X,
y E.
The representation space is
g(y) = 1.
now
A(y 0 g) since
Ay & g
=
Then, if
contains all rank-one operators.
A
T (x 0 g)
F.2.3
DEFINITION.
F.2.4
THEOREM.
T
are equal to those of
x 6 A
If
It is clear that the rank,
T 6 B(X ta g).
nullity and defect of T
as an operator on x
we define the rank of
by
X.
rank(x) = rank(xl
x = 0 <=> rank(x) = 0;
(i)
soc(A) = {x 6 A : rank(x) <
(ii)
(i) is obvious.
(ii) As an inductive hypothesis assume that Note that A
T C A,
Tx 43 g
=
defines the corresponding
Proof.
X 0 g,
=
n = 0 => x = O.
dim(xAe) < n -> x e soc(A). dim(xAe) < n + 1.
Suppose, then, that such that
u 6 A
is primitive there exists
xue # O.
minimal right ideal which therefore contains an exists a non-zero
f 6 Min(A).
(1 - f)v = 0
xueA
is a
Now there
it follows that
hence, by the induction hypothesis, x - fx e soc(A).
dim((l - f)xAe) < n, Therefore
Since
v c fAe c xAe.
Then
Since
x 6 soc(A)
which completes the proof. n
Conversely, if hence
xAe C E f.Ae 1
F.2.5
z
DEFINITION.
exists a
y 6 A
x E soc(A), and x
xA C E f A where
z
i
f,
e Min(A)
(1 < i < - n)
dim(xAe) < n by F.2.1
-
is defined to be a Fredholm element of A if there
such that
xy - 1,
yx - 1 E soc(A).
The set of Fredholm 31
A
elements of If
is written
is a proper ideal of
soc(A)
A
then
it follows that
is an open multiplicative semigroup of
$(A)
is stable under perturbations by elements of Next we link Fredholm elements in THEOREM.
x E (D(A) > X^
ran (x) A Ae
=
=
x(Ae)
xAe
=
p, q
qA() Ae
=
Ae/R(Ae)
=
d(x)
Ae/(l - p)Ae
=
Ae.
Ae.
F.1.9 and F.l.lO.
qAe,
(F.2.4).
Ae,
and
pAe,
=
rank(p) < -
=
rank(p)
of
which is closed in
(1 - p)Ae
=
k(h(soc(A))).
is a Fredholm operator on
n(R) = rank(q) = rank(q) <
therefore
A which
A with Fredholm operators on
We use the Barnes idempotents
ker(x)
so
is invertible
since the latter ideal is closed in
k(h(soc(A))).
invertibility modulo
Proof.
x
By BA.2.4 invertibility modulo soc(A) is equivalent to
A = (D(A)).
F.2.6
x c $(A)<-;
(We often implicitly make this assumption, otherwise
modulo soc(A).
A
D(A).
(F.2.4)
Example F.4.2 shows that the converse of this theorem is false. F.2.7
of
DEFINITION.
If
we define the nuZZity, defect and index
X E O(A)
x by nul (x)
Now if
=
def (x)
n (x) ,
x E (A) ,
R
q, p
d (x) ,
ind (x)
is a Fredholm operator on
nullity, defect and index; where
=
further,
=
i (x). Ae
nul(x) = rank(q)
are right and left Barnes idempotents for
with the same and x
def(x) = rank(p) in
A, respectively
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.
Inv(A) _ {x E O(A)
nul(x) = 0 = def(x)}.
:
x E Inv(A),
Proof. <_>
xA = A = Ax,
<=>
x E (D(A)
and the Barnes idempotents
<=>
x E (D(A)
and nul(x) = rank(p) = 0 = rank(q) = def(x) .
F.2.9
THEOREM
(Index).
(i)
The map
x --- ind(x)
(ii) (iii) (iv)
:
-> Z
(D(A)
ind(xy) _ ind(x) + ind(y), ind(x) = ind(y)
if x
ind(x) = ind(x + u), (i) The map
Proof.
x ± x
p = q = 0,
is continuous;
(x, y E (P(A));
and
y Lie in the same component of
(x E (D(A),
(D (A);
u E k(h(soc(A)))).
is continuous.
Now use the continuity of the
index for Fredholm operators.
(ii) ind (xy) = i (xy) = i (x) i (y) = i (x) + i (9) = ind (x) + ind (y) (iii)
if
and (iv)
y
are connected by a path in
in the Fredholm operators on and
x
lies in F.2.10
and
x
y
the same is true of
{x + Au : 0 < A < 1}
are connected by the path
u
(NA)
Ae.
which
Now use (iii) .
(D(A).
THEOREM (Punctured neighbourhood).
If x c (D(A)
there exists
c > 0
such that (i)
nul(x + A)
is a constant < nul(x),
(0 <
IAI
< E);
(ii)
def(x + A)
is a constant < def(x),
(0 < IAI
< E);
(iii)
ind(x + A)
is a constant,
Proof.
(I X1
< E).
1pply the punctured neighbourhood theorem for Fredholm operators
0.2.7 F.2.11
If x s $(A)
THEOREM.
u e soc(A)
such that
potents for
x + Au
u e pAq where
We may choose x
in
A
and
ind(x) < 0 (> 0)
there exists
is Left (right ) invertible for
p
and q
A
o.
are Left and right Barnes idem-
(respectively).
33
We consider first the case of
proof.
ind(x) = O.
left and right Barnes idempotents for xA = (1 - p)A, hence there exist
1 - p.
xz
Since
while
such that
then Ax = A(1 - q),
A,
such that yx = 1 - q,
and pAe
qAe
have the same finite
t(E S(Ae))
=
takes a basis of
s(c ]3(Ae))
reverses the process.
sqae
uIgAe = slgAe.
=
psqae;
Similarly
Write
u(gae)
=
qAe
vIpAe = t`pAe.
=
uvae
=
psqtpae
=
pstpae
=
pae,
and
vu(ae)
=
vuae
=
qtpsqae
=
qtsqae
=
qae,
s
and
as the representation yu = 0 = vx,
x
x + Au
x + Au
=
Hence
1-q+q
=
ind(x) < 0
{dl,..., d} a basis for theorem to construct
vu = q
Similarly
A.
1-p+p
1.
=
uz = 0 = xv
let
pAe
s, t E A
{c1,..., cn}
where
n < in.
such that
so
1,
has a right inverse and is therefore in
For the case
uv = p,
A # 0,
has a left inverse in
(x + Au) (z + X-1 V)
hence
=
vu = q.
is faithful.
x
hence for
(y + X-1 v) (x + Au)
Thus
uv = p,
So
t.
then
psqae,
uv(ae)
by choice of
onto a basis of pAe
u = psq, v = qtp
Now
Now
q be
and
So by the Jacobson density theorem ((75) 2.4.16) there exist
s(gae)
so
y, z E A
p
px = xq = 0, we may take y E A(1 - p) , z E (1 - q)A.
ind(x) = O, the spaces
dimension.
s, t E A
Because
in
x
-et
Inv(A)
be a basis for
for
A 34 O.
qAe
and
Now use the Jacobson density
Then
t(di)
=
c1
(1 < i < n),
9(ci)
=
d,
i
(1 -< i <- n) .
is the identity map restricted to
is
x + Au
gives
A
left-invertible in
for
qAe A
O.
and the same argument A similar proof deals
ind(x) > 0 0
with the case of
Note that this method of developing Fredholm theory in a primitive Banach
algebra A
requires that
Lut
Min(A) # 0.
and then Fredholm theory is trivial for
Min(A) = 0 <> soc(A) = (0)
'(A) = Inv(A)
and obviously the
nullity, defect and index 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
We now extend our theory to a unital Banach algebra A
The quotient algebra
quotient algebras as building blocks. x'
In general the socle of A
:
x E S}.
At = A/rad(A)
and if
for the coset x + rad(A)
is semisimple and we write write
S' = {x'
using its primitive
S CA
does not exist so in
its place we use the presocle. F.3.1
The presocle of A
DEFINITION.
psoc(A)
Clearly
{x C A
=
psoc(A)
Clearly F.3.2
=
I(A)
A,
while if
A
is semisimple,
The ideal of inessential elements of
A
is defined to be
k(h(psoc(A))).
is a closed ideal of
DEFINITION.
there exists
x' 6 soc(A')}.
is an ideal of
psoc(A) = soc(A).
I(A)
:
is defined by
y C A
An element
x
A.
is called a Fredholm element of
such that xy - 1, yx - 1 C psoc(A).
A
if
The set of 35
Fredholm elements of
while if
modulo psoc(A), extreme
(D(A).
A
psoc(A) = A
then
x £ D(A) <> x
then A = (D(A).
invertibility modulo Thus
I(A).
(D(A)
soc(A)
is invertible
At the other
psoc(A) = rad(A) <> soc(A') = (0) <_> D(A) = Inv(A)
By BA.2.4 modulo
is written
is a proper ideal of
psoc(A)
If
A
(BA.2.2).
is equivalent to invertibility
is an open semigroup of A which is stable
under perturbations by elements of
I(A).
Note that by BA.2.2 and BA.2.5
we have
Inv(A)'
If
Inv(A');
=
then
P £ II (A)
A/P
_
(D(A)'
$(A');
=
I(A').
is a primitive unital Banach algebra and this
fact enables us to develop Fredholm theory in the structure space of
I(A)'
A.
is isomorphic to
A/I(A)
Further, by BA.2.3,
h(I(A)) = h(psoc(A)).
Thus using BA.2.2 we get
x £ $(A), x
is invertible modulo
I (A),
<> x
is invertible modulo
P
<_>
F.3.3 s
LEMMA.
P,
If
further
(P £ h(soc(A))).
there exists a unique
s' £ Min(A')
P E 11(A)
Proof.
The first statement follows from BA.2.5 and BA.3.5.
s 0 P,
s+ P # 0, so if a c A (s + P) (a + P) (s + P)
But
S' £ Min(A')
such that
s + P E Min(A/P).
so
sas - Xs £ rad(A)C P,
=
sas + P.
s'a's' = Xs'
for some
so
(S + P) (a + P) (s + P)
=
3 (s + P) ,
X £ C,
hence
Now since
s + P C Min(A/P) .
that is
of
0
If x £ )(A)
THEOREM.
F.3.4
such that if y e A and
II(A)
y + P E 4)(A/P),
(1)
(P
Proof.
then
Min(A') _ 0
E:
a1, si
(D(A) = Inv(A)
bj, tI
(1 < i < k);
t'j e Min(A')
llx - yl1 < e
then
IT (A)\
suppose that this is not the case. u;
and a finite subset
(P e 0);
(ii) y + P e Inv(A/P), If
e > 0
there exist
for each
i,
j
Then if (1 < j < m)
and the theorem is trivial so x e (D(A)
A
in
there exist
such that
elements
s'i,
and
ux - 1 - E sIa1 £ rad(A), 1
M xu - 1 - ) t b, e rad(A).
j
1
7
By F.3.3 the set
in
II (A).
J
=
k(h({sl,... sQ; t1,... tm}))
ux + J x
1+J
=
0
> y
lix - yll < £
(ii). If
and
=
xu + J,
is invertible modulo
Ilx - ylI < e so
Q
NOW
is a closed ideal of A
that is
has a finite complement
h({sl,... sQ; t1,... tm})
P £ II(A)
0
J.
Hence there exists
is invertible modulo
=> y
Now
J.
is invertible modulo
P
J c P
for
E
0
for
> 0
such that
P e II(A)\52,
P £ II(A)\11
proving
then
u'X' - 1', X'U' - 1' £ soc(A'),
u'x' - 1' + P', X'U' - 1' + P' £ soc(A'/P'),
(BA.3.4)
37
-->
ux - 1 + P, xu - 1 + P E soc(A/P),
_>
x + P E (D(A/P).
(BA.2.6)
Q = {Pi,..., Pn}, then there exists
Now if
lix - YII < 6k > Y + Pk C (D(A/Pk)
Ek > 0
(1 < k < n).
such that :i choice of
gives (i) ,
E = min{EO,El,... Ek}
Recall that if, in a primitive algebra
A,
Min(A)
then the nullity
and defect of every Fredholm element are defined to be zero. F.3.5
For each
DEFINITION.
index functions
II(A)
-
x E (D(A)
we define the nullity, defect and
2 by
v (x) (P)
=
nul (x + P) ,
S (x) (P)
=
def (x + P) ,
i (x) (P)
=
ind(x + P) .
By F.3.4 each of these functions has finite support in
on h(psoc(A)).
If A
1(A)
and is zero
i(x) - v(x) - S(x).
is a primitive Banach algebra then, for each minimal idempotent,
is the unique minimal ideal which fails to contain it (F.3.5).
(0) (0)
Obviously
30 P C 11(A),
Min(A)C P
hence
soc(A)G P.
So if
It follows that, in this
case the support of the nullity, defect and index functions consists of the zero ideal, so
nul(x)
=
\)(x) (O) ,
def(x) = S(x) (O) , ind(x) = i (x) (0) .
The concepts of nullity, defect and index can be extended to a general Banach algebra as follows.
F.3,,.6
3a.
DEFINITION.
If
X E b(A)
we define
nul (x)
I
=
v (x) (p) ,
PEII (A)
def (x)
=
S (x) (P)
E
P£II (A)
ind (x)
=
1 (x) (P) .
E
P£II(A)
Since v (x) (P) > 0,
S (x) (P) > 0
nul(x)
=
0 <_> v(x)
def(x)
=
0 <--> S(x)
it follows that
0,
-
O.
Now F.2.8 extends to the general case. F.3.7
THEOREM.
Inv(A) = {x £ D(A) Proof.
:
nul(x) = 0 = def(x)} = {x £ (D(A)
:
v(x) = O = S(x)},
Apply BA.2.2(v) 0
The properties of the index of a Fredholm element in a primitive Banach algebra given in F.2.9 extend easily to the index function. F.3.8
THEOREM (Index).
(i) the map x -* i(x) topology on 72II (A)
(ii) (iii)
(iv)
:
(D(A) -
if x
1 (x + u) = 1 (x) ,
is continuous in the pointWise
(x, y £ O(A));
1 (xy) = I (X) + 1 (y) , 1(x) = 1(y)
ZII(A)
and y
(x £
Zie in the same component of 4(A).
(A) , u £ I (A) )
F.3.9
THEOREM (Punctured neighbourhood).
£ > 0
such that for each
.
Fix
x £ f(A), then there exis
P £ 11(A),
(i)
v(x + A)(P)
is a constant < v(x)(P),
(0 < 1AI < £);
(ii)
S (x + A) (P)
is a constant < S (x) (P) ,
(O <
(iii)
1(x + A)(P)
is a constant,
(I X1
< £)
a
< £) ;
Choose
proof.
E
as in F.3.3 and label it
punctured neighbourhood theorem in
v(x + X)(Pk)
=
\)(x)(Pk)
<
nul(x + P
k
for
k
for
)
which is a constant
)
0 < IX' < Ek
<
(1 < k < n).
and the result for
E = minfc0, El,..., cn}
Take
primitive algebras (F.2.10) there exist
nul(x + X + P
=
Then by F.3.4 and the
such that
E1,..., En
positive numbers
Eo.
v(x)
The proof
follows.
is similar 0
S(x)
We remark that in a general Banach algebra if t(x) E 0 => ind(x) = 0
x E @(A)
then
but the converse is not necessarily true.
This
fact has important consequences which were first observed by Pearlman (69). T
If
is an index-zero Fredholm operator on a Banach space then we have the
following important decomposition (0.2.8),
ible and
F
T = V + F
where
of finite rank (with analogous results if
V
is invert-
or > 0).
i (T) < 0
The converse is obviously true and by F.3.7 and F.3.8 it extends to general Banach algebras.
ind(x) = 0
it does not follow that
EXAMPLE.
F.3.10
spaces and take and
A = B(H1) S 13(H2).
has a corresponding decomposition.
x
Then A
is a semisimple Banach algebra
A
Considering Fredholm theory in
T = T1 (D T2
T E (D(A),
S = Ul a) V2
where
U1
there exists
where
F2 E F(H2)
<_> U1 + Fl c Inv B(H1)
Clearly
H2.
such that
F E soc(A)
F1 £ F(H1),
2
is the forward unilateral shift on
the backward unilateral shift on
and
relative to
and ind (T) = i H (T1) +i H (T2) . 1
Let
and
be infinite dimensional separable Hilbert
H1, H2
Let
soc(A) = F(H1) a)F(H2).
the socle,if
x 6 (A)
However, as the next example shows, if
ind(S) = 0.
5 + F E Inv(A). S + F = (U1 + Fl)
Then
Hl
and V2
Suppose that F = F1 a F2 ,
(V2 + F2) £ Inv(A)
and V2 + F2 E Inv(B(H2)).
But iH (U1 + F1) = iH (U1) = - 1 while iH (V2 + F2) = iH (V2) _ + 1 1
which is impossible.
2
1
2
To overcome this difficulty we employ the index function. tations
Sl ® S2 } 51
P1_= B(H1)
contain
(0),
soc(A).
and
S1 ® S2 -* S2
P2 = (0) (D B(H2)
Suppose
P C.II(A),
The represen-
are clearly irreducible therefore
are primitive ideals of
A which do not
soc(A) 5L P, then there exists
such that
E E Min(A)
If
are non zero. E2
O,
E V P2
# 0,
E
P = P1.
0
Since
E V P.
P # P1 or P
P1
and P2
soc(A).
So for
Thus we have shown that
On the other hand if
1(T)(P) = 0.
21
BA.3.5, P = P2,
sc by
primitive ideals which do not contain
either
E = E1 (D E2,
or
El
E2
similarly if
are the only two and
T E c1(A)
T = T1 e T2
1(T)(P1) = iH (T1), 1(T)(P2) = iH (T2). It is now easy to see that if 2 1 1(T) E 0 then iH (T1) = 0 and 1H (T2) = 0 so,by applying 0.2.8 to T1 1
and
T = V + F
T2,
2
where
and
V E Inv(A)
F e soc(A).
This idea can be made precise.
If X E D(A)
THEOREM.
F.3.11
u E 1(A)
there exists
l(x)(P) < 0 (> O)
and
such that
for all
P E 11(A)
is Zeft(right) invertible for
x + Au
A / 0. We consider the case
Proof.
the remaining cases may be handled
i(x) E 0,
as in F.2.11.
Let
p, q E A
exist for
then
x E (D(A)
x'
in
such that
p' and
there exist
orthogonal subset of
x'A' = (1' - p')A'. such that
sl,..., sn E A
=
(sI + ... + sn)p
hence p
=
(sI
Thus xA
=
(1 - (s1 + ... + sn)p)A modulo P,
of
11(A),
Now for
Since P
x
is right invertible modulo
1 (x) (P) = 0 for
1 < k < mi by F.2.11 we may choose
tk E pA
=
is an
(P E 11(A)).
it follows that
is invertible modulo
x
{s'1,..., s'n}
which is true for all but a finite subset
except for P = Pk (1 < k < m). deduce that
Since
modulo rad(A),
+ ... + s )p modulo P
{s1,..., s}C P,
{P1,..., Pm}
Thus there
Min(A') and
p
So if
soc(A').
q' are left and right Barnes idempotents
(F.1.10), in particular
A'
p' E soc(A')
is invertible modulo
x'
P ¢ Pk tk e A
(P E 11 (A))
P
we
(I < k < m).
such that
(s1 + ... + sn)pA modulo
41
with
x + Atk
invertible modulo
Put
uk
E
=(
si0k
Pk
A # O.
(1 < k < m) .
sil tk,
i, hence
for each
s'i E soc(A') _> Si e I(A)
Now
for
uk e I(A).
n
X + Auk
Then
si)ptk
x + A( S E¢P s .)(
=
k
r.
modulo
Pk,
1
x + A(E si)ptk modulo
Pk,
sl4pk
Pk
which is invertible modulo primitive ideal except
x
x + Auk
=
modulo
x + Xtk
=
Pk.
Pk
O.
for
A
for
P # Pk.
Further
uk lies in every
So
modulo
P
m
write u = E uk.
Then
1
x + Au
=
x + Auk
Pk
which is invertible modulo x + Au = x modulo
P
is invertible modulo
x + Au E Inv(A)
for P
Pk
modulo
P
for
and
1 < k < m,
(1 < k < m).
Thus, for
for
Pk
A 34 0
P e ]I(A).
while A 34 0,
x + Au
It follows by BA.2.2 that
for XD O 0
A final generalisation of our theory remains. F.3.12
DEFINITION.
inessential ideal of that
An ideal A.
xy - 1, yx - 1 e K
An
K of A x e A
such that
such that there exists
is called a K-Fredholm element of
K-Fredholm elements is denoted by
y e A A.
such
The set of
AK (A).
We can develop a Fredholm theory relative to each such (BA.2.4), without loss of generality we can assume to equal k(h(K)).
is called an
K CI(A)
K
K
and, by
to be norm closed or
The statements and proofs all go through with only the
obvious modifications.
An inessential ideal of particular importance is
the algebraic kernel which is considered in §F.4. Notes
F.4
Fredholm theory in an algebraic setting was pioneered by Barnes (7),
(8), in
1968, 9 in the context of a semiprime ring (one possessing no non-zero nilpotent left or right ideals).
He used the concept of an ideal of finite
order to replace the finite dimensionality of the kernel and co-range of a Fredholm operator.
A right ideal
DEFINITION.
F.4.1
J
in a semiprime ring
A has finite
order if it is contained in a finite sum of minimal right ideals of A (with a corresponding definition on the left). written
ord(J),
whose sum is
The order of an ideal
J,
is defined to be the smallest number of minimal ideals
J.
The connection with our work is clear, for if
x e $(A)
are left and right Barnes idempotents, then the left ideal
and p
and q
lan(x) = Ap, and
the right ideal
ran(x) = qA, both have finite order so the nullity, defect
and index of
x
are defined by the formulae
nul(x)
=
ord (ran (x))
=
ord (qA) ,
def(x)
=
ord(lan(x))
=
ord (Ap) ,
ind(x)
=
nul(x) - def (x) .
If
A
is primitive and
Ord(Ap) = rank(p)
x E (D(A)
then
so the definition
ord(qA) = rank(q)and
of these concepts coincides with our
own.
The index theory which Barnes obtains is more general than that developed in Chapter F as it is purely algebraic in character, but each result must be proved ab initio, and the preliminary manipulations are rather involved. Our approach, developed by Smyth, via the left regular representation of a
primitive algebra A
on
Fredholm elements in
A
direct.
Ae
where
e E Min(A),
and Fredholm operators on
and the link between Ae
(F.2.6) is more
However our theory is less general than that of Barnes, for F.2.1(ii
43
requires that A
be a Banach algebra.
The representation which we have used is well known ((75) 2.4.16), the correspondence between the dimensions of the kernel and the co-range of are the key to our exposition of Fredholm theory.
x
and
example of a primitive Banach algebra A with such that
is a Fredholm operator on
x
Ae
Min(A)
x
Gee now give an
and an
/- 0
x
(D(A)
showing that the converse of
F.2.6 is false in general. Let
EXAMPLE.
F.4.2
W(T)
=
a (T)
T
=
be an operator on a Banach space
GB(x)/K(X)
(T + K(X))
{}.
_
:
such that
X
1XI = 1}.
(The bilateral shift on a separable Hilbert space is an example). to be the closed unital subalgebra of
Then A
R(X) generated by T
Min(A) / 0
is a primitive Banach algebra with
the rank, nullity and defect of
But T E Inv(A)
so
nul(T) = 0 = def(T)
T
in
T E Inv(B(Ae)). so, by F.2.8,
Suppose that T E Inv(A),
T
T E (P(A)
hence
A
K(X).
and, as in F.2.2,
are those of
B(X)
and
Take
in
B(Ae).
then
T E Inv(A) modulo
KW. However, the unital Banach algebra T + K(X)
so
aA/K(X)(T + K(X))
A/K(X)
is generated by the element
has connected complement
((14) 19.5).
Further
{X :
Therefore
1}
IxI = {A
:
=
6B(X)/K(X)
(T + K(X))C_6A/K(X) (T + K(X)).
IXI < 1}C OA/K(X)(T + K(X))
T E Inv(A) modulo
which contradicts the fact that
K(X).
This exhibits a drawback of the representation
TT
:
x ->x : A; B(Ae)
for a general primitive Banach algebra.
Further investigations into this
case have been carried out by Alexander((4) 95). If, however,
more useful. 44
A
is a primitive C*-algebra then the representation
In the first place, as we see in 5c*.4,
Ae
v
is
can be given the
inner product
<x, y>e
ey*xe
=
y*x
=
(x, y E Ae),
7
under which it becomes a Hilbert space in the algebra norm.
is then a
faithful irreducible *-representation which is therefore an isometry. Hence
aA(x)
=
(Y
Tr(A)
(x)
=
(BA.4.2) .
aB(Ae) (x)
Further, the converse of F.2.6, is valid in this case. ation of C*.4.2 and C*.4.3 shows that, since ideal of
A which does not contain
a singleton set, the
THEOREM. Let
A
is the only primitive
by BA.3.5,
A.
in C*.4.3 becomes
in C*.4.3 is dispensable and we can take
72
the representation defined above. F.4.3
soc(A)
(0)
In fact an examin-
to
Tr
Thus we have
be a primitive unital C*-algebra with
e c Min(A),
then (i)
(ii)
'rr(soc(A)) = F(Ae);
Tr(soc(A)) =
K(Ae);
(iii)
Tr(R(A))
=
R(Ae)
Tr(A) ;
(iv)
T;((D (A) )
_
(D (Ae)
Tr(A)
(R(A)
A
is the set of Riesz elements of
relative to
soc(A)
defined in
R.1.1). F.2.3 and F.2.4 contain a definition of rank for elements of a primitive Banach algebra as well as a characterisation of the cocle as the set of elements of finite rank.
An alternative definition of finite rank elements
via the wedge operator is given in C*.1.1 X A x e F(A))
(x
is of finite rank in
A
if
and we show that, in a C*-algebra, the set of finite rank
elements is equal to the socle (C*.1.2). this result to semisimple algebras.
Alexander ((4) 7.2) has extended
In primitive algebras the two defin-
itions are equivalent.
Returning to Fredholm theory,Barnes'ideas for semisimple algebras were extended by Smyth (83) to general Banach algebras and this approach is followed here in §F.3.
Pursuing suggestions of Barnes (8) and Pearlman (69)
45
Smyth introduced the index function (F.3.5) to cope with the problem that,if ind(x) = 0,
and
x c (D(A)
is not always decomposable into the
x
then
sum of an invertible plus an inessential element. this in F.3.lO is due to Pearlman (69).
The original example of
further information on this
decomposition in the operator case is given by Murphy and West (62) and Let
Laffey and West (55).
T = V + F
is shown that
[VIF1 2
=
0
where
T E D(X),
and the decomposition may be chosen so that
This result is best possible, in that
[V, F] = VF - FV.
it is not always possible to choose a decomposition such that
[V, F] = 0
T = V + F
i(T) = 0,
for example,if
then,.either
then it
is left (right) invertible according as
V
where
F E F(X)
i(T) < 0 (> 0),
be a Banach space with
X
where
V E Inv(B(x)),
LV,
F] = 0,
and
F E F(X)
T E Inv(B(X)), or zero is a pole of finite rank of
Using the techniques of this chapter these results can also be trans-
T.
planted into Banach algebras.
The index function for Fredholm elements in
a general Banach algebra has also been defined by Kraljevic, Suljagic and Veselic (110) making use of the concept of degenerate elements discussed in §R.5.
A
If
is a non-unital algebra then, in order to carry out Fredholm theory,
one may adjoin a unit and proceed as in this chapter.
(This will be
necessary in Chapter R, for Riesz theory must be done in a non-unital setting).
However as Barnes (8) and Smyth (83) showed,a different approach
may be adopted. if there exists `Y = VF(A)
We say that
y E A
R
x + y - xy, x + y - yx E F.
such that
and
I
A
I = k(h(F))
Let
The set
FredhoZm elements of
A
F.
all of whose scalar multiples lie in
The elements of
relative to
F.
T
;7e confine
stating some useful results in quasi-Fredholm theory.
R
and let
are the set of Riesz and inessential elements of
(respectively) relative to
F
is quasi-invertible modulo an ideal
is the set of all such elements.
denote the set of elements in Then
x E A
T.
A
are called the quasiourselves here to The first follows
from the fact that a quasi-invertible idempotent must be zero. F.4.4
THEOREM.
Every idempotent of
`Y
Zies in
F.
In operator theory much interest has been focussed implicitly upon the quasi-Fredholm ideals including the ideals of finite rank, compact, strictly singular and inessential operators.
In the algebraic context we note the
following very general result starting with any quasi-Fredholm ideal
46
J.
The proof depends on elementary properties of the radical and the fact that we can identify the structure space of F.4.5
THEOREM.
Let
A/J with the hull of A
be an ideal of the algebra
J
J
such that
((83) 4.2)
F c J C'Y,
then (1)
x c Y' <_> x + J
is quasi-invertible in
(ii)
x c I < > x + J
is in the radical of A/J;
(iii)
x c R <=-> x + J
is quasiniZpotent in
(iv) (v) (vi)
A/J;
A/J;
FL J GI CRC5;
h(F) = h(J) = h(I) I
is the largest left or right ideal lying in
(vii)
A/J
F.4.6
COROLLARY. Any one of the sets
Y';
is semisimple <=> J = I.
each of the others.
I, R, 5', h(F)
uniquely determines
((83) 4.3)
The results thus far are valid for an arbitrary ideal index theory we need to restrict
F
to lie in
F
of
for
A,
psoc(A).
The monograph (71), §A gives an interesting account of Fredholm theory for linear operators on linear spaces with no reference to topology. Related work is due to Kroh (54).
P3izori-Oblak (59) studies elements of a
Banach algebra whose left regular representations are Fredholm operators. If one is concentrating on an ideal
F
the choice of the ideal is important.
This is exhibited by Yang (98) who studies operators on a Banach space invertible modulo the closed ideal of weakly compact operators.
If the
space is reflexive then every bounded linear operator is weakly compact and the Fredholm theory becomes trivial.
There have been many extensions of the classical Fredholm theory of linear operators of which the most important is the theory of semi-Fredholm operators. F.4.7
DEFINITION.
and if either
n(T)
T E T3(X)
is semi-Fredholm if
or
is finite.
d(T)
T(X)
is closed in
X
The basic results for semi-Fredholm operators are given in (25), this class of operators has proved of central importance in modern spectral
theory. F.4.8
DEFINITION.
An element
x
in a semisimple unital algebra
A
is
Semi-Fredholm if it is either left or right invertible modulo soc(A). 47
By F.l.lO if
x
is left (right) invertible modulo soc(A), x
(left) Barnes idempotents in
soc(A)
has right
so we could use the methods of this
chapter to develop a semi-rredholm theory in Banach algebras.
In this
monograph we confine ourselves to Fredholm theory. Another extension of the classical theory leads to operators which have generalised inverses, or generalised Fredholm operators (named relatively regular operators by Atkinson (5)). DEFINITION.
F.4.9
T E B(x)
(ii)
and both
has a generalised inverse
is a generalised FredhoZm operator if and
ker(T)
THEOREM.
F.4.lO
T E B(X)
(i)
s E B(x)
if
STS = S.
TST = T,
are complemented subspaces in
T(X)
T c B(X)
T(X)
has a generalised inverse <_> T
is closed in
X
x.
is a generalised
FredhoZm operator.
Let T
Proof.
it follows that
have a generalised inverse E = E2, F = F2
Barnes idempotents for
T
in
and B(X).
S,
then if
I - E, I - F
E = TS,
F = ST,
are left and right
We collect the following information.
TS = E > E(X)CT(X), ker(S)Cker(E); ST = F > F(X)CS(X), ker(T)C_ker(F);
TF = T > ker(F)C ker(T); ET = T > T(X)C E(X); SE = S > ker(E)C ker(S);
FS = S > S(X)C F(X) . Collating these results we see that
T(X) = E(X), so both
S
and
T
Conversely, let
ker(T) = ker(F), and S(X) = F(X), ker(S) = ker(E), are generalised Fredholm operators. T
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 inverse
48
S
and it follows at once that TST = T, STS = S .
Generalised Fredholm theory for operators has been studied by Caradus (22), (24), Yang (97), Treese and Kelly (90), among others.
(23),
The class of
generalised Fredholm operators on a Banach space contains all the projections in
so one cannot expect such a tightly organised theory as in the
8(x)
classical 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).
satisfy
v c 8(x) T
Let The a generalised FredhoZm operator on
THEOREM.
F.4.11
I
I V I
I
<
IIsI1-1,
ker(V)D ker(T)
and either
or
where
s
x
and let
is a generalised inverse of
V(x)C T(X),
then
T - V
is a
generalised FredhoZm operator. If
is generalised Fredholm its generalised inverse is not unique but,
T
in Hilbert space,there exists a unique generalised inverse projections
E
and
F
are 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), and this concept has recently proved to have many important applications.
(A bibliography with 1700 items is contained in (64)).
situation has been algebraicised as follows:
inverse semigroup if each element xyx = x, yxy = y.
x e S
a semigroup
This
S is called an
has a unique inverse
y
such that
The structure of these semigroups is somewhat tractable
and they have been objects of considerable study.
The Fredholm theory which we have developed in this monograph has as its outstanding characteristic an intimate connection with spectral theory. has little connection with the Fredholm theory of Breuer (18),
It
(19) extended
by Olsen (68), based on the concept of a dimension function in von-Neumann algebras ((25) Chapter 6).
Harte (106) has investigated Fredholm theory
relative to a general Banach algebra homomorphism. Coburn and Lebow
((25) Chapter 6) define a generalised index on an open
semigroup of a topological algebra to be any homomorphism to another semigroup which is constant on connected components of the first semigroup.
Of course, our theory fits into this very general framework and by specialising a little we obtain results (due to G.J. Murphy) on the existence and uniqueness of an index defined in a Banach algebra.
Let A let
(D
denote a unital Banach algebra with proper closed ideal
denote the set of elements of A invertible modulo K.
K
Then
and $
is
49
and
an open multiplicative semigroup,
A continuous semigroup homomorphism
DEFINITION.
F.4.12
G with unit element
discrete group
(D + K C 1.
} G
i
is an index if, for
e
onto a
x E (D,
i(x) = e <=> x e Inv(A) + K. It follows at once from the definition that zt
and that if
c K),
there exists
x E (D,
i(x + z) = i(x)
such that
E > 0
(x E
y c
and
< E => i(x) = i(Y). x - Y fur uniqueness result is somewhat surprising, roughly it states that, for
DEFINITION.
F.4.13
To make this precise we need
K, the index is unique.
a fixed
If
i
:
(D -> G
and j
equivalent if there is a group isomorphism
are indices they are
H
:
such that the
G --r H
6
following diagram commutes
F.4.14
THEOREM.
Proof.
Let
such that i(xu).
x, y c
the right by
be such that
xu = w + k y
to get
tit
i(x) = i(y).
w c Inv(A)
x = wy + k'
Now there exists
i(y)-1 = i(u)
Clearly
for some
i(w) = e.
since
(Ooi)(x) = j(x),
Let
(D
uy - 1, yu - 1 E K. Thus
= j (y) ,
There is, at most, one index up to equivalence.
where
and
k' E K.
Now we can define a map
and it follows immediately that
6 8
: A -- A/K be the canonical homomorphism.
so
u E $
e = i(x)i(u) =
k c K.
Multiply on
Thus
j(x) = 3(w)3(y)
:
G --* H
by
is an isomorphism . The existence theorem
is as follows. F.4.15
THEOREM.
subgroup of (ii)
(1)
An index exists
<=> (Inv(A)) is a closed normal
Inv(A/K).
If the condition in (i) is satisfied the group
is discrete, and an index may be defined by setting
>_ (x)
=
iU (x) 4 (Inv(A))
(x E D) .
G =
Proof. 8
Suppose that
(i)
Inv(A/K) -* H
:
H, with
ta(x)
:
is a well defined group homomorphism onto
j(x)
-
then the map
(D -> H,
:
ker(b) _ 1(Inv(A)).
Since
is open,
tU
is a closed normal subgroup of
tp(Inv(A))
Conversely, suppose Inv(A)
j
is open in
w(Inv(A)) so
A,
8
is continuous, thus
Inv(A/K).
is a closed normal subgroup of is open in
4'(Inv(A))
Inv(A/K).
Hence
A/K.
is a discrete group.
G=
Part (ii) now follows easily . The abstract index defined here is not suitable for spectral theory, for example, there is no possibility of obtaining an analogue of the punctured neighbourhood theorem (0.2.7).
in a sense, as the next result shows, any
Fredholm index which gives rise to a satisfactory spectral theory is As we have seen, if
encompassed within the work of this chapter.
K
is an
inessential ideal, then the results of the classical spectral theory of bounded linear operators extend to Banach algebras.
Now we show (informally)
that if the results of classical Fredholm theory extend, then
K must be an
inessential ideal. lie shall make use of the characterisation of inessential ideals in R.2.6
as those ideals in which each element has zero as the only possible accumulation point of its spectrum.
Suppose that
that, relative to the ideal valid.
Let
x E K
isolated point of A E aa(x).
Since
and
is a generalised index and
i
K, the results of classical Fredholm theory are
0 34
A c a(x).
'7e need to show that
x c K,
0 < lu - a1 < e,
A - x
is invertible modulo
v(p - x)
and
K, hence
are both zero for
p(x)
0 < lu - Al < g
a(x)
and
K
hence
A - x c D.
such that
But this
v(p - x)
and
It follows by the classical
theory that this punctured neighbourhood lies in isolated point of
E > 0
are constant.
8(Tt - x)
Punctured neighbourhood contains points of 641 - x)
is an
It is clearly sufficient to do so for each
a(x).
Thus, by the punctured neighbourhood theorem, there exists for
A
p(x),
hence
A
is an
is therefore an inessential ideal.
An element of an algebra is algebraic if it satisfies a polynomial identity, while an algebra is algebraic if every element therein is algebraic.
The algebraic kernel of an algebra is the maximal algebraic ideal of the algebra.
Its existence is demonstrated in (48) p.246-7 where it is shown
to contain every right or left algebraic ideal.
The original setting for algebraic Fredholm theory was a semisimple Banach 51
algebra and it was in this context, and relative to the socle, that Barnes(7) developed the theory in 1968.
In 1969 he extended it to semiprime algebras.
In the general case the socle does not always exist and, for this reason, Smyth (83) and Veselic (93) independently developed Fredholm theory relative In fact Smyth has shown ((84)§3) that the algebraic
to the algebraic kernel.
kernel of a semisimple Banach algebra is equal to the socle.
A little more
effort extends this result to semiprime Banach algebras.
A
If
is a
general Banach algebra and if Smyth's result is applied to the quotient algebra
A' = A/rad(A)
it follows that the algebraic kernel of
contained in the presocle.
cn
A
is
R Riesz theory
In this chapter the Ruston characterisation of Riesz operators (0.3.5) is used to define Riesz elements of a Banach algebra relative to any closed two-sided proper ideal, and elementary algebraic properties of Riesz elements are It transpires, however, that in
developed in §R.1 in this general setting.
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. Finally the theory of Riesz algebras is developed in §R.3 and examples of
Note that the algebras considered in
Riesz algebras are listed in §R.4.
this chapter will not necessarily be unital. R.1
Riesz elements;
Let A
algebraic properties
be a Banach algebra and let
R.l.l
DEFINITION.
r(x + K) = 0.
x E A
K be a proper closed ideal of
is a Riesz element of
RK(A) = R(A) = R
(when
will denote the set of Riesz elements of
K
A
(relative to
A.
K)
if
is unambiguous from the context)
A.
This definition is motivated by the Ruston characterisation of Riesz operators (0.3.5).
K
In the next section, having restricted
to be an
inessential ideal we shall demonstrate the familiar spectral properties of Riesz elements.
Let
[x,y'] = xy - Y.
denote the commutator of
x
and
y.
We have the
following analogues of 0.3.6 and 0.3.7. R.1.2
(ii)
THEOREM.
(i)
x E R, y E A
(iii) x,y E R and
(iv) Proof.
0
R.1.3
(i)
x
n
x c R,
y E K => x+ y c R;
and [x,y] E K => xy, yx e R; [x,y] E K -> x + y c R;
c R (n > 1) , xn-* x in A
-
-
and [xn,x] E K (n > 1) _> x E R.
Apply the basic properties of the spectral radius to elements in
THEOREM. Let
x E A and
f E Hol(a(x)),
A/K
then
xER and f(O) =0-f(x) ER; 53
f does not vanish on
f(x) s R and
(ii)
(iii)
(if
A is unital)
f does not vanish on
and
x £ (K(A)
(I (x)\{O} > x £ R;
6(x)\{O} _> f(x) £ AK(A) . Proof.
(i) is a consequence of R.1.2 (ii), observing that
= xg(x)
where
Using the Cauchy integral representation of that if
x £ A
f(O) = 0 => f(x)
g e Hol ((S (x)) . f £ Hol(0(x)),
and
one immediately verifies
f(x)
f £ Hol(0(x + K))
then
since
0(x+K)CQ(x) and f(x+K) = f(x) +K. (ii) Since f (x) £ R, 0(f(x) + K) NOW (iii)
6(x + K) C Q(x),
Q(f(x + K))
0
=
=
{O}.
U(x + K) = {O},
so, by hypothesis,
0(x + K),
f(0(x + K))
f((J (x + K))
=
x £ mK(A) _> 0 j o(x + K).
not vanish on
thus
=
Now
0(x + K) G 0(x),
hence
x E R.
therefore
f
does
so
0(f(x + K))
G(f(x) + K)),
=
f(x) £ A(A) 9
Next we give two characterisations of the radical of a unital Banach algebra which lead to characterisaticns of the kernel of the hull of The characterisation involving
Inv(A)
is well known (BA.2.8), wnile that
involving the set of quasinilpotent elements
We recall that if
tU
(k(h(K))) = rad(A/K) R.1.4
THEOREM.
K.
Q(A)
is due to Zemanek (104).
is the canonical quotient homomorphism A - A/K then (BA.2.3).
Let A be a unitaZ Banach algebra, then
rad(A) _ {x £ A : x + Inv(A) C Inv(A)} = {x £ A : x + Q(A) C Q(A)}. '2.1.5
COROLLARY,
Let A be unital then
k(h(K)) = {x £ A : x + CD R.2
Riesz elements: spectral theory
Recall that if A I(A)
54
K(A)} = {x E A : x + RCR}.
is a Banach algebra then
of inessential elements of A
A' = A/rad(A)
is defined by
and the ideal
I(A)
= (\{P E 11(A) : P`DsoC(A`)}.
K
We, henceforth, insist that that
K
is a closed inessential ideal of
is closed ideal of A and
Our Riesz theory will be
K C I(A).
carried out relative to this fixed ideal
K,
A, that is,
so we shall drop the subscript
and RK. K We are going to deduce the spectral properties of Riesz elements from the
from
(D
Fredholm theory of Chapter F wherein it is assumed that
A
is unital.
A will always
Thus, from R.2.1 to R.2.6, when we use results from Chapter F,
be unital and, at the end of the section, we shall show how these results may be extended to non-unital algebras. R.2.1
Let A
DEFINITION.
plex number
A
be a unital Banach algebra.
is called a FredhoZm point of
Frednolm or essential spectrum
w(x)
=
{A E C
The Weyl spectrum of
W(x)
=
:
of
x
in
A
x
if
x E A,
a com-
The
is defined to be the set
is not a Fredholm point of
A
If
A - x c (D.
x}.
is defined to be the set
x
na(x + y). yCK
The complex number invertible, or if of
0(x).
A A
is called a Riesz point of is a Fredholm point of
x
x
if either
A - x
is
which is an isolated point
The Riesz spectrum or Browder essential spectrum of
x
in
A
is
defined to be the set
a (x)
=
We note that
{A E C : A
w(x),
is not a Riesz point of
W(x)
and
a(x)
x}.
are all compact subsets of
C
and
the inclusion
55
w(x)C W(x)C S(x)Ca(x),
x E A
is valid for
w(x) = aA/K(x + K),
Obviously
(y E K),
and since OA/K(x + K)CO(x + y)
Clearly,
of the next theorem R.2.2.
Let
THEOREM.
R.2.2
is a consequence
is a closed subset of
be a unital Banach algebra and
a(x).
x E A, then
A - x is not a Fredholm element of A of index-function zero}.
{l E C
_
W(x)
A
(3(x)
is proper?
it follows that w(x)CW(x).
The inclusion W(x)C(x)
is a compact set.
W(x)
Clearly
K
whenever
Taking complements the result may be restated as follows,
Proof.
U
p(x + y)
C :
A-xE
(D (A)
and
I (A - x) = O},
yEK and, by using F.3.ll,as
U
p (X + y)
_
{}EC
l - x E Inv(A) + K} .
yEK
This last statement is true since
E
3
U p (x + y) <> A- x - y E Inv(A) for some y c K, yEK
<> X - x E Inv (A) + K . R.2.3
a(x) Proof. ition
A Riesz point of x
LEMMA.
which is in
a(x)
is an isolated point of
and p(A,x) E K. Let A
be a Riesz point of
A
is isolated in
p(l,x)
=
2Ii
f(z r
a(x),
x
which lies in
a(x).
By the defin-
hence the associated spectral idempotent
- x)-ldz E A,
where
F
is a circle in
Since
A
is a Fredholm point of
surrounding
p(x)
but no other point of
A
hence the associated
A E pA/K(x + K),
x,
6(x).
spectral idempotent
p(A,x + K)
p(A,x) + K
=
in A/K.
0
=
therefore p(A,x) E K 0 THEOREM. Let A
R.2.4
Fredholm point of x Proof.
lying in
A E 20(x)
If
If, in addition,
p - x E
(D
for
A - x E (D,
there exists
Then every
x.
contains points of
A
E > 0
such that
and, by the punctured neighbourhood theorem
< E,
lp - aj
S(p - x)
)(p - x),
(F.3.9),
is a Riesz point of
26(x)
then every neighbourhood of
p(x).
x E A.
be a unital Banach algebra and
0 < lp - AI < E.
are constant for
It follows
that
V(1 - x)
0
=
S(p - x)
=
for 0 < p - Al < E,
thus, by F.3.7, this punctured neighbourhood of is an isolated point of
3
0(x)
A
lies in
p(x).
Therefore
which is, by definition, a Riesz point .
The next result corresponds to the Ruston characterisation of Riesz operators (0.3.5), R.2.5
COROLLARY.
algebra and
of 0(x) Proof.
and If
x e R <=> each
Then
x E R,
then
is a Riesz point of
is isolated in
points of
r(x + K) = 0,
0(x).
0(x).
so if If
A - x E (D.
is an isolated point
0 / A E 6(x),
0 / l E 3
(x),
then
then, by R.2.4,
Hence every non-zero boundary point of
It follows that
0(x)\{O}
{A E 0(x)
FAl
> S > J}
0(x)
is a finite set
0(x)
is a discrete set of Riesz
x, and,by R.2.3,the associated spectral idempotents lie in
Conversely, if each non-zero point of set
A e o(x)\{O}
p(A,x) E K.
A - x + K E Inv(A/K), hence 3
Let A be a unitaZ Banach
(Ruston characterisation)
x E A.
is a Riesz point of
n {Ak}l,
say.
K.
x, the
The spectral
n
idempotent associated with this set, p = Z p(}k,x) E K, and
r(x - px) < S.
1
57
r (x + K) < r (x - px) < S,
Thus
and since
r(x + K) = 0 0
is arbitrarily small,
6
In terms of the Browder spectrum this result states that
x c R <=> SW C {O}. and
dote that if
then
T 6 P(X)
is a finite dimensional linear space
X
is empty.
(3(T)
K relative to which we can carry
The next result characterises the ideals
out Riesz and Fredholm theory, and is important in the characterisation of The theorem is valid for ideals which are neither
Riesz algebras in §R.3. closed nor two-sided.
R.2.6 THEOREM. Let A ideal of A. of
Proof.
If
x e J.
then
J G I(A),
x 6 J => x E I(A) => r(x + I(A)) = 0,
Conversely, let
J
be a left ideal of
possible accumulation point of
may, and do, assume that A
projection
with unit p u 6 pAp
6
and
such that
Now A
6 A.
We show that
(px) = {A}.
pAp p = upx 6 J.
show that p c I(A). P = pI V Min(A)
q2 9t pi
(so
q2 e J),
Now pl = q2 + q'2 idempotent
p., C J. cn
p2,
pAp
0 # .l E c3(x).
is a Banach algebra hence there exists
and our next task is to
Clearly then
x1 6 p1Ap1
such that
The hypothesis implies that zero ap Ap (x1), hence this set contains
then g2p1 = plg2
with similar relations holding for
p2 6 p1Ap1,
we
so the associated spectral
p e J.
q2 = p(1.1,x1) e piAp1,
so, at least one of
then
and
x 6 J
p ¢ soc(A).
islthe only possible accumulation point of Set
a(x),
there exists
contains at least two points.
O.
Let
px E Inv(pAp),
Thus
Suppose not, then
u
GA(x) = aA(x')
Since
I(A) = k(h(soc(A))),
so, by BA.3.lO,
an isolated point
such that zero is the only
(x c J).
is semisimple.
is semisimple, hence
ap Ap (x1)
a(x)
A
A is an isolated point of
p = p(X,x)
hence zero
0(x) by R.2.5.
is the only accumulation point of
By hypothesis,
a left or right
J
J G I(A) <=> zero is the only possible accumulation point
Then
for each
a(x)
be a unitaZ Banach algebra and
q2, q'2 L soc(A).
p2 ¢ soc(A),
q'2 = pl - q2' Label this
p1p2 - p2p1 = p2 # pl,
and
Continuing this process we get an infinite strictly decreasing family of idempotents in J all lying outside soc(A), and satisfying {pn}i pnpm = pmp = Pn for m > n. (n > 2), Set q1 = pl, q = pn - pn-l is a properly infinite orthogonal family of idempotents in J {qn}1 such that qnp = pq = qn(n > 1). Now choose a properly infinite sequence then
n
co
of distinct complex numbers such that
{an}1
and
A
n -> 0
n > . Then
as
IAnl
< 2-njlgnll-1
(n > 1),
and w = wp E J.
w=E A qn E A, 1 n
Now
p + w c J and 1 + An E a(p + qn) (n > 1) -> (1 + an) E Thus
1
is an accumulation point of
A E O(x)\{O}
It now follows, as in the proof of R.2.5, that Therefore
{x + I(A)
consisting entirely of quasinilpotent elements.
{x + I(A)
but A/I(A)
:
and
x c J.
r(x + I(A)) = 0
(x E J).
is a left ideal of the Banach algebra
x E J}
:
(n > 1) .
This gives the required
a(p + w).
p(X,x) E I(A), if
contradiction, so that
CT (P + w)
x c J}C rad(A/I(A))
is semisimple since
A/I(A)
Thus
(BA.2.8),
is a kernel (BA.2.3),
I(A)
J CI(A) .
so
Riesz elements relative to a closed ideal of a Banach algebra which is not necessarily unital have been defined in §R.l.
In R.2.1 Riesz points of an
element in a unital algebra have been defined relative to Fredholm points of the element.
To extend this definition and the results of R.2.3, R.2.5 and
R.2.6 so as to obtain a full Riesz theory in a non-unital algebra let A
non-unital and denote by A
A (x) = CA(x)lJ{O}
if
1
the unitization of
x E A,
and if
J
be
Then, for dim(A) = W,
A.
is an ideal of
A, it is also an
1
ideal of
We consider the case of semisimple
A1.
for general
A
follows on factoring out the radical.
I(A) = k(h(soc(A)))
primitive ideal of a primitive ideal of
P1l1 A c 11(A)
A
and it is clear that A
(and
Then, if dim(A) = -,
soc(A) = soc(A1).
is contained in a primitive ideal of Al,
((48) p.206).
further if
A1), the result
P1 E 1I(A1)
and
Now every All
P1 SSA
also
A
is
then
It follows that I (A) = I (A1)n A and the proofs
extend immediately. The case of finite
A
is trivial.
59
We shall need the following consequence of the punctured neighbourhood theorem in Chapter C*.
Let
THEOREM.
R.2.7
x
element
be a connected open set of Fredholm points of an
in a semisimple unital Banach algebra
then every point of
u e 2,
for some
Q
If
A.
p - x E Inv(A)
is a Riesz point of x
S2
and
is a countable discrete set.
a(x)r(2
ind(p - x) = 0,
Proof.
Also
index.
D
uities
ind(X - x) = 0 (A c O by continuity of the
hence
except for a set of discontin-
nul(X - x) = 0 = def(A - x),
of the function
A ; nul(X - x)
which must be countable
(A 6 2),
The result now follows from F.3.7
by the punctured neighbourhood theorem.
and R.2.4 0 Riesz algebras:
R.3
characterisation
No study of Riesz or Fredholm theory is complete without a detailed analysis of the inessential ideals introduced in §F.3. algebras such that where
A
Recall that
A = I(A).
is an algebra over
To this end we examine those I(A) = k{P E TI(A)
:
soc(A') CP'}
?]e start with general algebraic consider-
C.
ations.
R.3.1
DEFINITION.
An algebra A
is a Riesz algebra if
A = I(A).
It follows immediately from the definition and the homeomorphism between
the structure spaces of A <_>
A'
is a Riesz algebra;
and
(BA.2.5),that
and that
h(soc(A)) is empty <_> A/soc(A) this case
A'
A
A
is a Riesz algebra
is a semisimple Riesz algebra <=>
is a radical algebra (BA.2.3), since, in
A = I(A) = k(h(soc(A))).
Thus, semisimple Riesz algebras are
'close' to their socles in the sense that the cocle is contained in no primitive ideal.
It follows, and will be illustrated in §R.4, that the class
of Riesz algebras is a large one embracing many important special algebras. The characterisation of Riesz Banach algebras is an immediate consequence of R.2.6.
R.3.2
THEOREM.
(Smyth characterisation)
.4 Banach algebra
algebra <=> zero is the only possible accumulation point of
A a(x)
is a Riesz for each
x C A.
It follows that, if a Banach algebra A is a Riesz algebra, then a(x)\{O}
is a discrete set,
a(x)
is countable and 3U(x) = a(x) (x e A).
A simple consequence of this is Let
COROLLARY.
R.3.3
the Banach algebra
be a closed subalgebra,and J
B
A
A.
a closed ideal,of
and A/i
is a Riesz algebra => B
are Riesz
algebras.
We have noted the discreteness of the non-zero spectrum for every element
For general Riesz algebras the structure space
in a Riesz Banach algebra.
is discrete in the hull-kernel topology.
If A
THEOREM,
R.3.4 Proof.
Without loss of generality take
of accumulation points of
is discrete.
Ls a Riesz algebra, TI(A)
11(A)
A
to be semisimple.
is contained in
Then the set
h(soc(A)) (BA.3.6), and is
therefore empty 0 The converse of R.3.4 is false, for the Calkin algebra of an infinite dimensional Hilbert space has zero as its only primitive ideal but it is not a Riesz algebra.
However, we do get a converse result in the commutative
case.
R.3.5
COROLLARY.
:1
is a commutative Banach algebra then
A
is a Riesz
is discrete in the hull-kernel topology <_> 1I(A)
algebra <=> 11(A)
crete
If
is dis-
in the Ge Z farad topology.
Proof.
The equivalence of the discreteness of 11(A)
Gelfand topologies follows from BA.3.7.
The proof is completed by applying
R.3.4, and, conversely, by noticing that if semisimple then
h(soc(A)) _
in the hull-kernel and
11(A)
is discrete and
A
is
(BA.3.8)
ID The interesting examples of Riesz algebras are non-unital as the next
result indicates. R.3.6
A unitaZ Banach algebra is a Riesz algebra <_> it is finite
THEOREM.
dimensional modulo its radical. Proof.
Take
unless
A = soc(A),
A
to be semisimple and unital. it follows that
Which is impossible.
A/soc(A)
NOW if A = soc(A), n
E Min(A)
(1 < i < n).
1
(i < i,
j
< n)
-
-
Then
(BA.3.3)r hence
A =
A
If
A
is a Riesz algebra then,
is a unital radical algebra
1 = e1 + ... + en where n
e Ae E and dim(e Ae.) < 1 E 1 J i=1 is finite dimensional. The converse is
1 J-
obvious 61
The next result is an immediate consequence of R.2.6 if A
is a Banach
algebra, but it is also true for general algebras (a proof of this is indicated in §5), so it is stated in full generality. L
A left (right) ideal
THEOREM.
R.3.7
of an algebra A is a Riesz algebra
Riesz algebras:
R.4
Let A be a Banach algebra and
the commutant of
x E A;
Z(x) = {y e A : yx = xy}I is a closed subalgebra of
A.
x,
Let
x
denote the
bounded linear map
x.y-
xy
Z (x) -* Z (x) .
.
and we write the latter set
(Z) (x),
Then
OA(x) = O
R(X)
denotes the set of Riesz operators on the Banach space
0(x).
Recall that X.
and R.4.2 we shall be considering operators on the Banach spaces
In R.4.1 Z(x)
and
A.
R.4.1 each If
x E A
plication by R.4.2
If A
COROLLARY.
x E A
then
A
let
L
x
on
COROLLARY.
is a Banach algebra such that
X E R(z(x)) for
is a Riesz algebra. and
X A,
R
denote the operators of left and right multi-
X
respectively.
A Banach algebra A
is a Riesz algebra if any of the
three following conditions hold,
(i) Lx c R(A)
(x E A) ;
Rx E R(A)
(x E A);
(ii)
(iii) x A x: u - xux c R (A) Proof.
Let
X E A
invariant under
T
and put
(x E A). T = L
so if (i),
x
or
that G.1
= (x2)-E R(z)
x E R(Z).
X
or
x A x.
(ii) or (iii) hold then
In cases (i) and (ii) this implies that 6 -CV
R
Clearly Z(x) = Z Ti z E R(Z)
and in case (iii) that
x e R(Z)
which, by the Ruston characterisation (0.3.5), implies Thus, in each case, by R.4.1,
A
is
((23) 3.5.1)
is a Riesz algebra .
operator on
A
A
The Banach algebra
EXAMPLE.
R.4.3
is compact if
is a compact
x ,,,,x
Compact Banach algebras have been studied by
(x E A).
Alexander (4), and, by R.4.2, are Riesz algebras. EXAMPLE.
R.4.4
Lx (resp. R
continuous L.C.C. (resp. R.C.C.) if
A
(x C A).
is left (resp. right) completely
The Banach algebra A
x
)
is a compact operator on
These algebras have been studied by Kaplansky (50), and, by
R.4.2, are Riesz algebras. Let
EXAMPLE.
R.4.5
Now
a
is the dual group
LI(G)
space of the commutative Banach algebra induced topology.
The Gelfand
be a locally compact abelian group.
G
G
in the
is compact, so by R.3.5,
is discrete <=> G
i
is compact.
is a Riesz algebra <_> G
(G)
EXAMPLE.
R.4.6
Let
X
which is contained in
$(X)
EXAMPLE.
R.4.7
Let
X
a closed subalgebra of
be a Banach space and A then
R(X),
A
is a Riesz algebra by R.3.2.
be a Banach space, let
uniformly closed subalgebra of
B(x)
generated by
then AC R(X)
T,
To see this, note that if
is, therefore, a Riesz algebra.
nomial vanishing at the origin then
A be the
and let
T E R(X)
p(z)
and
is a poly-
(0.3.7).
p(T) E R(X)
Let
T
be a non-nilpotent quasinilpotent operator on a
Banach space and let
P
be the algebra generated by
R.4.8
EXAMPLE.
of polynomials in I, and
I
T).
Then
of the radical as in ((14) 24.17) we see that So
and
I
(the algebra
consists of the non-zero multiples of
Inv(P)
is the only non-zero idempotent in
Min(P) = , soc(P) = (0).
T
Using a characterisation
P.
P
is semisimple and since
I(P) = k(h(soc(P))) = (0),
and P
is not
a Riesz algebra.
Now let A be the closure of R.3.6,
A
is a Riesz algebra.
P
in
B(X),
then
A = E e rad(A)
so, by
Thus we have constructed a Riesz algebra
with a dense subalgebra which is not a Riesz algebra. R.5
Notes
Attempts to develop a theory of compact or weakly compact elements in a Banach
algebra A pre-date the development of a RieSz theory.
We list some of the
early definitions.
63
M. Freundlich (34) 1949:
a -> ta,
t
is a compact element of
if both operators
a -> at are compact on A.
T. Ogasawara (65) 1954:
is a weakly completely continuous element of
t
if both the above operators are weakly compact on
A
A
K. Vala (92) 1968: t n t : a -- tat
is compact on
A
if the wedge operator
A.
B. J. Tomiuk and P. K. TTong (77) 1972:
continuous element of
A.
is a compact element of A
t
t
is a weakly semi-completely
if the wedge operator is weakly compact on
A.
These concepts have also been studied by Alexander (4), Bonsall (12), Kaplansky (50), Ogasawara and Yoshinaga (66), Ylinen (99), (101). the definitions is entirely satisfactory.
None of
This is illustrated in the case
of the wedge operator of Vala by an example due to Smyth (86) of a semisimple Banach algebra
in which
A
x1\x
is compact (x C A)
but
x,\y
need not be.
However, Ylinen (100) has shown that some of these definitions coincide with the reasonable definition of the set of compact elements in a C*-algebra given in C*.l.l.
The following artificial definition has been proposed by
Smyth for a general Banach algebra
It has the merit that the compact
A.
elements form a closed two-sided ideal with many of the expected properties. Let
S = {u E A : u k u
is a finite rank operator}
and
set
F = {x E A : X& u
is of finite rank for each
u E S}.
semiprime then
F = S).
If
J = k(h(F))
(If
A
is
the compact elements are defined
to be the set
ix E J: x 1 y
is a compact operator for each
y E F}.
Riesz theory as presented here dates back to 1968 when Barnes (7) derived the spectral properties of inessential elements of a semisimple Banach algebra using the concept of ideals of finite order. of the Ruston characterisation. Smyth (82),
(84)
His paper contains an analogue
A more general approach was adopted by
who developed Riesz theory relative to a fixed ideal
algebraic elements (those satisfying a non-trivial polynomial identity).
F
of He
showed that in a semisimple Banach algebra the socle is the largest such ideal (termed the algebraic kernel) and obtained a Riesz theory (including the Ruston characterisation) for elements
x
such that
r(x + F) = 0.
The
algebraic kernel is the largest left or right ideal of algebraic elements and the proof of its existence in a complex algebra is due to Amitsur ((48)p.246-7). F4
Veselic (93) has demonstrated the existence of the algebraic kernel in a comHe defines
plex Banach algebra using function theoretic methods. Ax
be degenerate if each element in ideal of algebraic elements of
is algebraic.
Ax
is a left
A, hence it lies in the algebraic kernel of
Indeed, the set of degenerate elements of A
A.
Thus
to
x c A
is actually equal to the
algebraic kernel and from this it follows that Veselic's set of compact In Problem 8,Veseli6
elements is precisely our set of inessential elements. asked if ated by
is a Banach space and
X
ST
is yes, for
T E 6(X)
is finite dimensional for each
ST
is such that the algebra generS E 6(X),is S c F(X)?
S, hence
is algebraic for each
The answer
is a left ideal of
6(X)T
algebraic elements which is contained in the algebraic kernel which in turn is equal to the cocle
F(X).
Puhl (72) has defined a trace functional in a
suitable subalgebra of a Banach algebra. We record two further results in Riesz theory, both due to Smyth ((83),
6.3, 6.4). R.5.1
THEOREM.
The sets of Fredholm and Riesz points of an element in a
commutative, unital Banach algebra coincide. Proof.
Since
Let A be a commutative, unital Banach algebra and let
x E A.
is a subset of both sets, we consider only the Fredholm and
p(x)
Riesz points in
a(x).
If zero is a Riesz point of spectral idempotent, then
x
in
a(x)
and if p C K
x(l - p) + p C Inv(A), so
hence zero is a Fredholm point of
is the associated
x + K E Inv(A/K),
x.
in
a(x),
it suffices to show that zero is an isolated point of
a(x).
exists
Since zero is the only
Conversely, let zero be a Fredholm point of
y E A
such that
possible accumulation point of an isolated point of which is false.
p =
where
xy = 1 + k,
a(xy).
x
k E K.
then,either,
a(k),
then, by R.2.4,
Now there
0 E p(xy),
or zero is
Now, by commutativity, 0 E p(xy) => 0 C p(x),
Hence there exists a non-zero spectral idempotent p
p (O, xy)
=
p (O, 1 + k)
and
Now
aAp(xyp) = {o}
Thus
0 E 1A(1-p)(xy(1 - p))
=
and
p (-l, k) e K ,
aA(1-p)(xy(1-p))
= aA(x)\{O}.
and again, by commutativity,
0 E
pA(1-p)(x(1-p))
65
p, hence, by R.3.6, it is finite
is a Riesz algebra with unit
Ap
Now
So zero is isolated in
tains zero.
OAp(xp)
Thus
dimensional modulo its radical.
is a finite set which con-
OA(x) = 0Ap(x) v a
A(l-p)
(x (1-p)) .
The second result is a spectral mapping theorem for the Fredholm and Browder spectra. R.5.2
THEOREM.
is a unitaZ Banach algebra,
x E A and
then
f c Hol(a(x))
(i)
If A
f (W (x)) = W (f (x)) ;
(ii) f(R(x)) = S(f(x)). Proof.
W(x) = aA/K(x + K), this result is simply the spectral
Since
(1)
A/K.
mapping theorem in (ii)
If
Z2 = Z2(x)
x E A, its bicommutant
is the commutative Banach
algebra
Z2 (x)
{z a A :
=
for each y such that
0
Note that every spectral idempotent of
x, and of
f(x),
[x,y1 = O}.
lies in
if
spectral mapping theorem). set, thus so is by R.2.6,
aZ??(z)\{O}.
KA Z
WZ (z) 2
Further,
=
z E K n Z2,
Hence
Now let
Browder spectra in
Z2.
SZ (z)
the result reduces to the ordinary
K.-Z2 = (0),
(If
K n Z2.
We
relative
Z2
consider Fredholm theory in the commutative Banach algebra in to the ideal
Z2.
then
aA(z)'{O}
is a discrete
is a closed ideal of
K(-\Z
Z2,
and,
2
WZ
and 2
QZ
denote the Fredholm and 2
From the definition R.2.1, and by R.5.1,
(z e Z2).
2
SA(x) = tZ
(x)
since all the spectral idempotents of
x
lie in
2
Z2.
The result now follows by applying (a.)
Next we give the proof of R.3.7 in the setting of a general algebra.
Recall (F.3.1) that the presocle of an algebra A psoc(A) = {x E A : x' E soc(A')}.
eL
is defined to be the ideal
This ideal is of considerable interest in the study of Fredholm theory, it was introduced by Smyth and its elementary properties are outlined in (83). Clearly
and A
I(A) = k(h(psoc(A))),
= 4) <_> A/psoc(A)
is a Riesz algebra <=> h(psoc(A))
is a radical algebra.
Min(L/rad(L)),
required for the proof connects the set ideal of R.5.3
with the set
A,
Let
LEMMA.
then
x' E Min(A')
where
be a Zeft ideal of the algebra
L
L is a left
Min(A/rad(A)).
x + rad(L) E Min(L/rad(L)).
is of the form
Min(L/rad(L))
The first of the two lemmas
and
Conversely, every element of x £ L
for some
x + rad(L)
If x £ L
A.
such that
x' E Min (A') . Proof.
x £ L
Suppose
and
it follows that
x + rad(L)
u £ xLx\rad(L),
then
algebra with unit Write
Then
2
- x E LT\rad(A)T rad(L),
L/rad(L).
hence, since
Choose
v £ xAx CL, such that
u'w' = x' = w'u',
is a division
x'A'x'
u'v' = x' = v'u'.
uw - x, wu - x £
so
is a division
x + rad(L) E Min(L/rad (L)). suppose
Conversely,
is such that
y E L
and note that
x = y4 £ L
that x' £ P1in (A') . Then since
x
It follows that xLx + rad(L)/rad(L)
rad(A)A L Crad L.
Write
there exists
w = xvx E xLx.
algebra so
is an idempotent of
u £ xAx\rad(A),
x',
Since
x' E Min(A';.
Let
t£
y + rad(L) E Min(L/rad(L)).
y2Ay n
rad(L), say t = y2zy for z E A. yzy E rad(L), and hence
y2 - y C rad(L), it follows that
At = (Ay)yzyCrad(L).
We shall show
x + rad(L) = y + rad(L).
Using a characterisation of the radical ((14) 24.16),
it follows that t £ rad(A), hence y2Ay rad(L)Crad(A). x2 - x £ y2Ay. rad(L)C rad(A),
Then
u £ xAx\rad(A).
algebra and
is an idempotent of
x'
Now since
u £ yLy\rad(L).
2
Y Ay{ rad(L)C rad(A).
Write
w = xvx.
It follows that
Then
v £ A
Choose
is a division such that
uw - x, wu - x £
:s a division algebra, hence
x'Ax'
40
The second lemma connects the presocles of
R.5.4 LEMMA.
A'.
yLy/rad(L)
x = y modulo L, it follows that there exists
uv - x, vu - x £ rad(L).
x' £ Min(A')
hence
Now
L
and
A.
If L is a Zeft ideal of the algebra A
then
psoc(A) PLC fsoc(L) .
67
Proof.
Choose
x E p soc(A)r\ L.
X' E soc(A')
Then
and, because the socle
is the sum of the minimal left ideals, there exist
1[ A'.<', t'Sin (A' )
{x'k}n
Clearly we may assume that
x'x'k.
such that
x'
by R.5.3,
{xk + rad(L) In C Min(L/rad(L)),
Then,
{xk}i C..Ax C.L.
and
n
x - E xxk E
L
1
x + rad(L) E soc(L/rad(L)),
Therefore
hence
Consider the standard isomorphism
L CI(A) = k(h(psoc(A))).
"
as required .
psoc(L)
Suppose first that
We may now complete the proof of R.3.7.
L/(L A psoc(A))
x E
psoc(A))/ psoc(A).
(L +
The algebra on the right of this isomorphism is a left ideal of
I(A)/ p soc(A)
which is a radical algebra, hence it also is a radical algebra, therefore so also is the algebra on the left. algebra, so
L
L
which contains
idempotent of
L/rad(L)
structure spaces,
psoc(A)n L.
lies in
P/rad(L)
fore cannot contain
primitive ideal of
P/rad(L)
(L +
can contain
P
be a primitive
Then, by R.5.3, every minimal thus,the ideal generated by these
However, by the homeomorphism of
is a primitive ideal of
soc(L/rad(L)), L
Let
is a Riesz algebra.
L
idempotents, soc(L/rad(L))C P/rad(L).
L/(L, psoc(A))
is a radical
L/psoc(L)
is a Riesz algebra.
Conversely, suppose that ideal of
Then, by R.5.4,
since
L
L/rad(L)
which there-
is a Riesz algebra.
psoc(A)n L.
Thus no
It follows that
is a radical algebra, hence by the above isomorphism
psoc(A))/ psoc(A)
is a radical left ideal of A/ psoc(A)
therefore, lies in the radical of
A/ psoc(A), which is
which,
I(A)/ psoc(A).
Therefore L C I (A) , We conclude with some
remarks on Riesz algebras.
here in §R.3 is due to Smyth (85).
The theory presented
Riesz algebras are related to the class
of modular annihilator algebras defined by Yood (103). semiprime if it has no non-zero ideals with square zero).
(An algebra is
A semiprime
algebra is defined to be modular annihilator if every modular maximal left (right) ideal has a non-zero right (left) annihilator.
If we restrict to
the semisimple case, the classes of Riesz and modular annihilator algebras
8
A summary of the properties and a list of examples of modular
coincide.
annihilator algebras is given by Barnes (9).
The more important examples *
in this class (with the additional assumption of semisimplicity) are
H -
algebras (Ambrose (1)), dual algebras (Kaplansky (50), (51)), annihilator algebras (Bonsall and Goldie (15)) and Banach algebras with a dense socle. This class includes most of the important Banach algebras possessing minimal ideals.
Although a semiprime modular annihilator algebra is a Riesz algebra, a For
semiprime Riesz algebra need not be a modular annihilator algebra. example if B (D Cl.
{O}.
is
is a radical Banach algebra with
B
Then
is a maximal modular left ideal of
B
A, but ran(B)
is a Riesz algebra.
Such a
B
can be constructed as the closure in
A
in
A
is not a modular annihilator algebra, but, by R.3.6,
A
Then
let A be
ran(B) = (0)
B(X)
of the algebra generated by a non-nilpotent quasinilpotent operator. observe that the result of R.4.5 holds also for compact non-abelian In fact, if
denotes a fixed Haar
dy
(f*g) (x)
=
(xy-1)
f
J
measure on
G,
G.
f, g c L1(G)
then, if
g (Y) dy
G
((4S) 20.10). Lf : g - f*g
Since
G
(g E L1(G))
is compact it is easy to verify that the operator is compact for each
f E LI(G).
Hence
L1(G)
is
LCC.
Kraljevic and Vese h c (109) define
spectrally finite Banach algebras as
those for which the spectrum of every element is a finite set. algebras are precisely those which are equal to their presocles.
In fact these (109)
contains an approach to the dimension concept in Banach algebras somewhat different from that of Barnes (8),
(9).
69
C* C*-algebras
The wedge operator which has been defined on the algebra proves to be particularly effective in studied in §1.
C*-algebras.
in Chapter 0
$(X)
This operator is
§2 contains the decomposition theorems of West and Stampfli
for operators in Hilbert space and their analogues in C*-algebras.
A modi-
fication of a conjecture of Pelczynski leads, in §3, to a characterisation of
A representation is constructed in §4
Riesz algebras among C*-algebras.
which provides an isometric map of finite rank, compact, and Riesz elements of a C*-algebra onto the corresponding operators in Hilbert space.
In §5
short proofs are provided for the range inclusion results of §0.4,which are available in Hilbert space.
C*.1
The wedge operator
The wedge operator defined in
has been used by Vala (91), (92),
§0.6
Alexander (4), and Ylinen (99) to define classes of finite rank and compact elements in Banach algebras, (in C*-algebras, Definition C*.1.1).
In the case
of the algebra of bounded linear operators on a Banach space these definitions
are justified by Theorem 0.6.1, which shows that the finite rank and compact elements of this algebra are, precisely, the finite rank and compact operators on the Banach space. algebral
The definitions are unsatisfactory in a general Banach
an example due to Vala (92) showing that the set of compact elements
may fail to be closed under addition.
The definitions are satisfactory,
however, in the case of a C*-algebra, and in this section we demonstrate this fact.
C*.1.1
DEFINITION.
rank or compact in
An element
A
C*.1.2
THEOREM.
of a C*-algebra A
if the wedge operator
rank or compact operator on
A.
x
soc(A)
A
x n x :
is said to be finite
a -- xax
is a finite
respectively.
is the set of finite rank elements in a C*-algebra
k Proof.
Let
x E soc(A),
1 < 1 < k.
Then k
k
E a 1e1)A(E a.e 3 3
=
where
a
CA,
e
E Min(A)
1
k
xAx
x = E ae
then
=
1
1
a,e.Aa,e,,
E
J
3
1rJ=1 1 1 k
C
a1 e,Ae
E
1
1r3=1
3
k E
ai(Ebij,
i,j=1
b
where each
13
is a fixed element of
Conversely, suppose that dim(xAx*x) < -,
dim(xAx) < -.
therefore
be the commutant of
set,
thus k x = E anen, 1 (m
(BA.3.3).
1
Since
dim(x*xAx*x) <
n).
in
x
aA (x).
so is where
X
e
= e* n
n
nxIZ)
Let
= az(x2) is a finite
=
(1 -< n < - k),
e2 n
and
eme n
=O
n,
-
- > dim(xAx) > dim(e xAxe
n
n
)
=
dim(e Ae ).
n
From this it is easy to conclude that each For suppose that
x C soc(A) <_> xx* E
be self-adjoint.
x
O'(x
so
Then
# O,
Now, for each
Now
A.
dim(xAx) < co.
Thus
xAx*x C xAx,
.low
(BA.4,4), it suffices to assume that
soc(A) Z
e Ae
where
dim(eAe) < -,
n
(and consequently x) c soc(A) n e2 = e E A, then eAe is a Riesz e
algebra which, by the Smyth characterisation (R.3.2), is equal to its own socle.
But if p c Min(eAe)
pxp
=
pexep
for some scalar C*.1.3
THEOREM.
Proof.
Let
and
X,
=
Ap
hence
soc(A)
p c Min(A),
O.
so
eAe = soc(eAe)C soc(A) 0
is the set of compact elements in a C'4-algebra
x E soc(A).
I]xn - xjI -
and x E A
Choose
x
n E soc(A)
such that
A.
IIx nII < IIxII + 1,
Then
71
Iixn A xn - x n xFI
is a compact operator on
xA x
Hence
Now the operator
A.
A.
A.
be a compact element of A , then
x
Conversely, let operator on
, O.
is the uniform limit of operators of finite rank on
x n x
c*.1.2,
By
< (21Ixli + 1)1lxn - x1I
x*x A x*x
x A x
is a compact
being the composition of the
operators
(a c A)
a + ax* - xax*x '' x*xax*x is compact on
Thus, since
A.
is sufficient to assume that
x
x C soc(A)
<<> x*x s soc(A)
is self-adjoint.
By considering the commutant of
and arguing as in C*.1.2,we see that
x
has zero as its only possible accumulation point.
a(x)
(BA.4.4), it
Thus, if
E > 0,
the set
a(x) c {a; Ixi > s} {x1,.... xn},
is finite
corresponding to
eix
so
=
xei
If
say.
ei
is the eigenprojection of
then
a
i
Aiei
=
(1 < i < n),
Xi (eiA ei) (X I\ x) = ei A ei
hence
eih ei
is a compact projection which is therefore of finite rank. n
By C*.1.2,
ei o soc(A)
(1 < i < n), hence
p = Ee. s soc(A).
Using self-
1 1
adjointness Ilx(i - p)II
hence C*.1.4 (i)
72
=
r(x(l - p)) < s,
x s soc(A)
THEOREM. X A x
The following statements are equivalent in a C*-algebra
is a Riesz operator on
A;
A:
is a Riesz element of A relative to the ideal
x
(ii)
soc(A);
r(x + soc(A)) = 0.
(iii)
(ii) and (iii) are equivalent by the definition, while the equivalence
Proof.
of (i) and (ii) follows exactly as in 0.6.1 9 C*.2
Decomposition theorems
First we state our two decomposition theorems in a Hilbert space
H, the West
decomposition of a Riesz operator into the sum of a compact operator plus a quasinilpotent and Stampflii's generalisation of it using the Weyl spectrum.
Then using the machinery already constructed these results are proved in a C*-algebra setting.
K c K (H) ,
is normal,
K
T S R(H) _> T = K + Q where
(West decomposition)
THEOREM,
C*.2.1
and Q c Q(H).
U (T) -G(K),
((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
W(T)
Thus if
{A C 0(T)
=
T C R(H),
:
T
is
0(T)
is defined by
A - T V (Do(H) I.
Stampfli (88) has generalised C*.2.1 as
W(T)C{0}.
follows,
C*.2.2
THEOREM,
K c K(H), (
and
T e B(H) _> T = K + S
(Stampfli decomposition)
where
Q(S) = W(T).
(88) Theorem 4).
The extensions of these two results appear as corollaries of the next theorem.
C*.2.3
A
is a unital C*-algebra and
a closed ideal of
THEOREM. Let x C A and suppose that
that every point of w is isolated in spectral idempotent Zies in such that Proof.
K
K,
a(x),
w
A.
is a subset of 0(x)
such
and that the corresponding
then there exists a normal element y C K
a(x + y) = a(x)e w.
We shall make use of the Gelfand-Naimark embedding of A into
for a suitable Hilbert space
B(H)
H ((14) 38.10). 73
w
the points of
{ ?}l,
able set
where the
a(x),
Let
w must be a count-
are labelled in decreasing order of near-
Ak's
Thus we may choose a countable set
Xk - ak -* 0.
such
{ak}1C Q(),,w,
pk e K be the spectral idempotent for
corres-
x
n
to
ponding in
are isolated in the compact set
y(x)\W.
ness to that
Because
is a finite set the proof is trivial, thus suppose not.
w
If
and put sn =
ak,
(BA.4.3).
qn(H) = On (H)
such that
K
Let
kpk.
qn
be the self-adjoint idempotent
Now setting
qo = 0,
n
yn
)
1
Ak) (qk
(ak
- qk-1) C K,
is a set of disjoint self-adjoint idempotents, yn {qk - qk-1}1 It remains to is normal for each n, hence yn } y C K, and y is normal. and since
show that 6(x + y) = 6(x)\ w. fn = (x + yn)Isl(H)
Write
gn = (x + an
and
and relative to the decomposition
An)lpn(H).
-
fl =
Then
gl
we have
n(H) = n-1 (H) E) pn(H)
fn-1
fn Since
LO
gnu
it follows, from (BA.4.5), that
a(gn) = {an},
and thus
0(fn) _
{ak}i.
a(fn) = 6(fn-1)
Relative to the decomposition
v
{an},
H = s n(H) 9 (1-s n)(1
we have f
n
x + y
=
n
0
hnJ
where
h
n
= xl (1 -
n
) (H) .
By (BA.4.5),
a(x + yn) = a(fn) va(hn) = (a (W )\{Xk}i)L {txk}i = Q(x)\{lk}iNow if
A e a(x)\w,
is open in
A,
then
for each n, hence, since n It follows that G(x)\w Ga(x + y).
A e 6(x + y
A e 6(x + y).
)
To prove the reverse inclusion, suppose that c p(x)u w,
n>m, 74
so we can choose
in > 1
such that
A ¢ 6(x)\w, 1 ¢
Inv(A)
then
Then, for
h =h (1-s ) m n n I
(H)
( A - hn )-1 = (A - h )m- 1 l ( 1 -s n) (H),
and so
,
where the inverses exist by virtue of the choice of
the decomposition H = s n (H) O
wn
Then
(1 - s n ) (H) , write
(D
O
(l - h n )-1]
O
Ilwnll < Ilwmll
Now, since
Then, relative to
m.
Fix
n > m.
for
n > m
so that
lly - ynll
(y - yn)Is n(H) = 0,
0
1 - Yn
Also
=
z
O
' hence wn (y - yn)
(l - f
(A - x - n} -1(y - yn)
O
O
0 (A - hn)-1 z *
*
}-1 n
=
1][O
(J - hn}
O
z
O (A-h) lz
Now, by (BA . 4.5) , yn)-i(y
r{(A - x -
- yn)}
=
r(wn(Y - yn)},
<
I Iwnl I
< 11w! Therefore
1 - (A - x - yn)-1(y - y
A - x - y
c Inv(A)
n
shows that
)
l ly - Ynl I, l lY - ynl l < 1.
c Inv(A).
Multiplying on the left by
A - x - y c Inv(A),
that is
A
O(x + y),
therefore C (x + y) C 5(x)-w 49 Now we apply this result to Riesz and Fredholm theory on A fixed closed ideal C*.2.4
LEMMA.
Proof.
A/soc(A)
K c I(A).
In a C*-algebra
But first,we identify A,
relative to a
I(A).
I(A) = soc(A).
is a C*-algebra which is semisimple so, by BA.2.3,
soc(A) = k(h(soc(A))) = k(h(soc(A))) = I(A)
, 75
Let
(West decomposition)
COROLLARY.
C*.2.5
Proof.
?apply C*.2.3. with
C*.2.6
COROLLARY. A,
C*-algebra
6(x + y) = {o},
,
W = 6(x)\{O}
Let
(Stampfli decomposition)
y E K
then there exists
be a Riesz element of a
y c K such that
A, then there exists a normal
C *-algebra
x
such that
be an element of a
x
o(x + y) = W(x)
the
Weal spectrum of x. W(x) _ {A E 6(x)
Proof.
zero Fredhoim points of
:
0
A - x V
so we must remove all the index-
(A)},
y E K.
6(x) by the addition of a single
This is
done in two stages, first the 'blobs' of index-zero Fredholm points are removed one by one, then when this has been completed, C*.2.3 is applied to remove the isolated index-zero Fredholm points on spectral idempotents in
0(x)
(which have associated
K (R.2.4)).
A 'blob' is a connected component of index-zero Fredholm points of which is not a one-point set.
The blobs are countable, say,
and inductively construct sequences
E0 = 1
+
00
as follows.
{En}1Ct llunll
En-1'
Choose
an c vn,
{)n}1.
{an}1G C, {un}1 C _K,
then
un c K
and
such that
so that
n
x + E uk - an E Inv(A), 1
(This is possible by F.311).
Finally, choose
En < / En-1
so that
n
x + E uk - an + y E Inv (A) 1 for
Ilyll < En.
Then
I E ukll < E llUkil < En E n+l
n+l
2-k
= E
1
n
,
0O
thus
E uk
converges to
u E K.
Now,by (t) we get, for each n,
I
x + u - X n E Inv(A) < > A n E p(x + u).
0(x)
Take
Since
S V , it follows from R.2.7, that U(x + u) (\ Vn is an at most n n countable set of Riesz points of x + u. Thus we have removed the countable X
set of blobs
V
of index-zero Fredholm points of
n
by an at most countable set of Riesz points of
6(x),
6(x + u).
task of removing a countable set of Riesz points of v C K
C*.2.3, there exists
such that
replacing each one
We are left with the
x.
So, by
.
(7(x + u + v) = W(x)
Riesz algebras
C*.3
Pelczynski conjectured that if the spectrum of every hermitean element in a C*-algebra is countable, then the spectrum of every element in the algebra is This conjecture has been confirmed by Huruya (47).
countable.
An obvious
modification leads to a characterisation of Riesz algebras among C*-algebras,
A will denote a C*-
which is originally due to Wong ((96) Theorem 3.1). algebra and
the set of hermitean or self-adjoint elements of
H(A)
If a(h)
THEOREM.
C*. 3.1
A
then
h C H(A)
has no non-zero accumulation point for each
is a Riesz algebra.
By virtue of C*.2.4 it is sufficient to prove that A = soc(A).
Proof.
x E A,
If
has no non-zero accumulation point.
6(x*x)
be the spectral idempotent of
p
C O(x*x) Ix
So
A.
112
- xp
=
I
I (x - xp) * (x - xp) I
IIx - xpll < c,
Suppose that
p 6 H(A), and p
Then
> £2}.
IaI
I
=
I
I x*x - px*x I
and it suffices to show that
p g soc(A)
and put
For
let
c > 0
corresponding to the spectral set
x*x
pi = p.
commutes with I
x*x, hence
= r(x*x - px*x) <
62
p e soc(A).
Then, as in the proof of
{p IT n and, by BA.4.3, each of these idem-
R.2.6, we construct a strictly decreasing sequence of idempotents such that, for each
n, p
soc(A),
n
Since
potents may be chosen self-adjoint. p ¢ soc(A),
with unit priori,
p V Min(A)
so there exists
p), such that y
0(y)
need not be in
y E pAp
(which is a C*-algebra
consists of at least two points.
H(A).
If either
Q(y*y)
or
two points then using the hypothesis we can construct p2 p1
and
a(y*y)
p2
and
r(y*y)
soc(A)
a(yy*)
=
as in R.2.6.
=
IIY*YII
=
If
I'yll2
contain
strictly less than
So suppose that for each
are singleton sets.
r(yy*)
(T(yy*)
But, a
y c pAp,
y # 0,
O,
77
apAp(yy*)
and
PAp(y*y)
so
It follows that y*y
of the zero point.
pAp
Thus
y c Inv(pAp).
are singleton sets, neither of which consist yy* 6 Inv(pAp),
is a division algebra, therefore
which is a contradiction.
p c Min(A)
and
always construct an idempotent
p2
hence
pAp = Cp, and
Thus starting with p1 = p
we can
satisfying our requirements and hence,
by induction, an infinite strictly decreasing sequence
{pn}i
such that,
for each n, pn ¢ soc (A) . Now the sequence uk = pk - pk-l'
in H(A).
Now
then
{uk}l
lies in H(A). Put {pn}1 is an infinite orthogonal family of idempotents
u = Z 2-k uk E H(A), hence p + u E H(A),
and
1
is an
1
a(p + u)
accumulation point of
p
Therefore
as required 0
soc(A)
C*.4
which contradicts the hypothesis.
A representation
the have defined finite rank and compact elements of a C*-algebra (C*.1.1).
Riesz and Fredholm elements are considered relative to the closure of the In this section we construct a faithful *-representation of the C*-
socle.
algebra onto a closed subalgebra of the operators on a Hilbert space which maps the finite rank (respectively, compact, Riesz, Fredholm) elements onto the finite rank (respectively, compact, Riesz, Fredholm) operators in the subalgebra.
Recall that an element of an algebra is algebraic if it satisfies a nontrivial polynomial identity.
Clearly finite rank operators on a linear
space or finite rank elements in a C*-algebra are algebraic.
C*.4.1 THEOREM.
If A
is a C*-2Zgebra,
soc(A)
is the largest ideal of
algebraic elements of A. Proof. <_>
x 6 soc(A), dim(xAx) < -,
> x
is algebraic.
Conversely, let Jr-I(A) = soc(A).
x*x a J\soc(A). (1 < i < n).
78
(C*.1.2)
J
be an ideal of algebraic elements of
Suppose that
x e J\soc(A),
But x*x = E aipi
Clearly some
pi
where
X.
A.
By R.2.6,
then, by BA.4.4, c R
(say p) c J\soc(A).
and pi = pi = pi But p C soc(A), co
p
is a compact element of A
pn p
(C*.1.3), that is
is a compact operator on
A which is idempotent, so PAP is a finite rank operator on
hence
A,
(C*.1.2), which is a contradiction 0
p E soc(A)
The construction of our representation is done in stages.
First we
produce a natural family of Hilbert spaces associated with the minimal ideals of a C*-algebra. Let
A
be a C*-algebra with
and let
e = e* E Min(A),
corresponding minimal left ideal of
x, y C He
If
A.
H
e
= Ae
be the
define the scalar
by
<x,y>
<x, y>e
Clearly
<
,
ey*xe
=
is linear in the first variable and conjugate linear in the
>
Now if
second.
y*x.
=
x E H
e,
(<x, x> - ex*xe)e
=
0,
*
thus
<x, x> E a(ex*xe),
<x, x>
=
fl<x, x>ell
thus the algebra norm on is clear that
x
so
H
Ijex*xeJJ
1Ix112,
I[xel12
=
Further
=
is identical with the inner-product norm.
is closed in
e
e
.
Thus
H
e
define a representation
Tre (a) x
on
He
=
<x, x> > 0.
so
A,
for if
x
n
e H
e
and
x
n
It
then
--- x E A,
= xe;xe=x, n
n
x E H
C*.4.2
ex*xe = (xe) xe,
and
LEMMA.
=
ax
is a Hilbert space under this inner-product. Tr
e
of
A
on
H
e
We now
as follows,
(a E A, x E He),
The representation
(Tre, He)
is a 'f-representation of
A
He with the following properties: 79
(i)
Tr
(ii)
(span AeA) = F(He
e
TTe(A)'-> K(He); the unique primitive ideal of A which does not contain
ker Tre = Pe
(iii)
e (BA.3.5) . It follows at once from the definition that
Proof.
A
of
He:
He.
on z
x, y C H
ker(Tr
e
e
)
Now every element of AeA
is a primitive ideal of
ker(Tre)
is of the form
A
Tr
and since
yx*
where
e is irreducible e ¢ ker(Tre),
= Pe Tr
.
(BA.4.1)
In our main theorem
and
R(A)
Let
THEOREM.
A
is continuous,
e
c(A)
Fredholm elements of a C*-algebra A C*.4.3
is a *-representation
From this we conclude that
(ii) follows from (i) because, since
B(H)
Tre
denote the rank-one operator on
Then
hence (i) follows.
,
thus
He
x & y
(yx*)z = yx*z = yex*ze = y = (x a y)z,
e
Tre(yx*) = x 5II y.
thus
let
x, y C He,
y (z C He).
Tr
on
If
e
(A)
is closed in
will denote the set of Riesz and
relative to the closure of the socle.
be a C*-algebra.
therefore isometric) representation
Tr
(Tr,
H)
There exists a faithful *-(and
of A with the following
properties:
(i) TT(soc(A)} = R(H)n Tr(A) (ii) Tr(SOC(A)) = K(H)( Tr(A);
(iii)
Tr(R(A)) = R(H)
Tr(A);
(iv)
Tr(c(A)) _ (D (H)
Tv(A)
Let A
Proof.
not contain
= 1
80
A is unitaZ.
be a set which indexes the primitive ideals of
soc(A).
such that P = Pe (irk, H X) of A on
Tr
if
9 ACA
Tr
.'or each
X E A,
we can choose
A
which do
eX = e* C Min(A)
and then, by C*.4.2, there exists a *-representation
Define
HX.
on
=
H
1
a XEA
H
.
Then
is a *-representation of A
Trl
ker 7rl = A ker(TrX) = nfPl 6 11(A)
on the Hilbert space
soc(Aj
:
by C*.4.2.
,
Now
H1.
r1 may
As
have a non-zero kernel it is necessary to add another representation order to ensure that the sum
be faithful.
Ti
theorem ((14) 38.10) on the C*-algebra
tation
A/soc(A)
then ker (Tr) = ker 711 o ker 7T2 = (0) ,
Hl ® H2,
to construct a *-represen-
of A such that ker(Tr2) = soc(A).
(7r2, H2)
in
712
Use the Gelfand-Naimark
so
Put 7 = 7r1 9 7T2 on
Tr
is a faithful *-
representation. Let us examine the range of x c span(AeAA),
then
7r1(x) e F(H1).
Now if
of
such that
A
ker(7r2) = soc(A) ,
x E soc(A), :
F(H),\T1(A)
F (H) tl T1 (A) ,
X
for 7r)(x)
).t,
hence if
6 F(Hx), it follows that
there exists a finite subset
1 < j < n},
therefore 7r(soc(A) )C F(H) .
which is therefore contained in
TI
and since
Tru(x) = 0,
x C span {Ae A
the inverse image of
Tr (soc (A))
eX C Pu
W.
so
7 1(x) E F(H1).
n}
{)1 "
But
t~o verify (i), observe that
is an ideal of algebraic elements of
A,
(C*.4.1), therefore
soc(A)
whence we have equality (i.) .
is a faithful *-representation hence it is isometric ((75) 4.8.6), and
is closed in 13(H) , so F(H) r) 7r (A) CTr (soc (A)) C K (H) n 7T (A) . To 00 obtain equality let T = T* 6 K(H)n 1r(A), then T = E A P. where A. 6IR, i T (soc (A) )
P1 = P*i = P2i E K(H)n T1(A) for each i. But each compact pro3ection is of finite rank, so Pi E F(H) () Tr(A) , thus T E F(H)n 7r (A) . Since every operator s E K(H)1 1T(A) may be written S = T1 + iT2 where T1, T2 are and
self-adjoint members of K(H) (I TI (A) , whence we have equality (ii).
it follows that
T' (soc (A)) J K (H) j 7r (A) ,
The proofs of (iii) and (iv) are now straightforward (see A.1.3) . C*.5
Notes
Very neat proofs of the range inclusion theorems of §0.4 can be given in a Hilbert space
H
via the following factorisation Lemma due to Douglas (28).
(The footnote in (28) announcing an extension to Banach spaces is incorrect). C*.5.1
LEMMA.
S, T 6 B(H),
S(H)C T(H) _> there exists
C E B(H)
such that
S = TC. Proof.
Since
y E ker(T}
L.
S(H)G T(H),
such that
then for each
Sx = Ty.
Put
x E H
Cx = Y.
there exists a unique C
is linear and we prove C
81
H
be a sequence in
lim xn
=
such that
lim Cxn
u,
yn
Then there exists ker(T)1
Su = Tv,
=
v.
n
n
since
{xn}
Let
is continuous by means of the closed graph theorem ((30) p.57).
c
Cu = v,
thus the graph of
S, T E B(H),
C*.5.2
LEMMA.
Proof.
By induction
=
Tyn
for each
lim y = v S ker(T) n n C is closed
H,
is a closed subspace of
hence
Sxn
such that
ker(T)y
n, .
ST =TS and S = TC => r (S) < r(T)r(C) for each
Sn = T n C n
n,
thus
JISnjl < IITnil
and, So
. IlCnll,
and the result follows from the spectral radius formula S s B(H),
T c K(H)
S E B(H),
T E (}(H),
C*.5.3
COROLLARY.
Proof.
Apply C*.5.1
C*.5.4
COROLLARY.
Proof.
Apply C*.5.1 and C*.5.2
C*.5.5
COROLLARY.
and
S(H)C--T(H) _> S E K(H),
ST = TS
and S(H)C.T(H) => S e 2(H).
.
Sc B(H),
ST - TS c K(H)
T £ R(H),
and
S(H) G T(H)
> S E R (H) . Proof.
S = TC by C*.5.1.
into the Calkin algebra. Now
Let j be the canonical homomorphism of B(H) Then IL(S), I (T) commute and I (S) = (T)i (C).
r((P(T)) = 0 hence, by C*.5.2,
Alexander (4) showed that
rP(S)) = O, that is
Erdos (31) defined an element
x
axb = 0 > either
xb = O.
ax = 0
or
S S R(H)
.
C*.1.2 is valid in a semisimple Banach algebra.
of an algebra A
to be single if
The single elements of
easily seen to be the rank one operators.
B(X)
are
Making use of this concept Erdos
cQnstructs a representation of a C*-algebra similar to that in §4, see also Ylinen (100).
Erdos points out that his work does not extend even to semi-
simple Banach algebras.
prove that an element
In fact, in (32), Erdos, Giotopoulos and Lambrou x
of a semisimple Banach algebra has an image as a
rank one operator in some faithful representation of the algebra <=> x
single and the operator xAx is compact. 82
is
The representation in §4 may be
used to transfer information 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 given the C*-algebra counter part of the
Chui, Smith and Ward result (26) that the commutator in the West decomposition is quasinilpotent.
In fact, the more detailed information on the
West decomposition provided by Murphy and West (61),(see below), is all valid in a C*-algebra.
Akemann and Wright (3) have further results on the wedge
operator, and on the left and right regular representations in a C*-algebra. For example, they show that if operator <_> either
S
or
S, T E 8(H)
T E K(H),
then
(25) p.58) constructed a Riesz operator R
Gillespie ((35),
space such that for no decomposition of
R
R = K + Q
then the commutator
on a Hilbert
into the sum of a compact plus
a quasinilpotent did these two operators commute.
showed that if
S AT is a weakly compact
See also the remarks in §F.4.
Chui, Smith and Ward (26)
is a West decomposition of a Riesz operator is quasinilpotent.
[K, Q]
R
Murphy and West (61) gave
a complete structure theory for the closed subalgebra (called the decomposition algebra) generated by
K
and
It emerges that the set of quasi-
Q.
nilpotents forms an ideal which is equal to the radical, and that the algebra is the spatial direct sum of the radical plus the closed subalgebra generated by
K.
The problem of decomposing Riesz operators on Banach spaces has been open It may even characterise Hilbert spaces up to isomorphism.
for some time.
Some recent progress is due to Radjavi and Laurie (73) who showed that if R is a Riesz operator on a Banach space and
0(R) ={xn}1 where the eigen-
values are repeated according to algebraic multiplicity ,then
Olsen (67) showed that if and
K E K(H)
has a West
nI}nI <
decomposition if
where
R
T E 8(H)
Qn = 0.
and Tn c K(H),
then
T = K + Q
This result has been extended to C*-algebras
by Akemann and Pedersen (2).
An intriguing property of the ideal of compact operators on a Hilbert space, originally due to Salinas (77), is the following. Let
T E 8(H)
then
r(T + K(H))
inf KEK (H)
r(T + K),
In fact Salinas' proof is valid in Banach spaces.
This property was
algebraicised by Smyth and West (87), who showed that for a large class of commutative Banach algebras, including the C*-algebras, the above property holds for every element and for every closed ideal.
Pedersen (70) proved
that this is true for all C*-algebras, and Murphy and West (60) gave an
They also showed that the class of commutative Banach
elementary proof.
algebras in which this property holds for each element and for each closed ideal is comprised, roughly, of those algebras whose Gelfand transform algebra is dense in the sup-norm algebra of continuous functions on the
Further algebraic information on the spectral radius may
Gelfand space.
be found in the elegant monograph of Aupetit (6).
The modified Pelczynski conjecture which characterises C*-Riesz algebras is due to Huruya (47) and Wong (96).
The following result is stated in (27)
4.7.20, see also (10). C*.5.6
If A
THEOREM.
is a C*-algebra the following statements are
equivalent: (i)
(ii)
A
is a Riesz algebra;
A = soc(A);
if J is a closed left (resp. right) ideal of A
then
lan(ran(J)) = J {reap. ran(lan(J)) = J); (iv)
A
is *-isomorphic and isometric to a C*-subaZgebra of K(H)
some Hilbert Space
(v)
for
H;
The Gelfand space of every maximal commutative C*-subaZgebra of A
is discrete; (vi)
Zeft(resp. right) multiplication by on
(vii)
A
for each
x
is a weakly compact operator
x e A;
every non-zero point of Q(x)
is isolated in
6(x)
for each
x = x* E A. Such algebras are also called dual algebras. Pelezynski's Kirchberg (105).
84
conjecture has been verified in Banach *-algebras by
A Applications
In this chapter our general theory is applied to a number of specific examples, particularly to algebras of operators.
As a consequence we shall
often use operator notation but the index (unless otherwise specified) will be the index function associated with the particular algebra. We recall first the definitions of the various spectra in R.2.1.
Let
a fixed inessential ideal of A.
The
be a unital Banach algebra and Fredholm spectrum of
W(x)
=
{Ae C
:
A
in
x
A - x¢
K
A
is
};
the Weyl spectrum is
W(x)
=
(\Q (x + k); kEK
while the Browder spectrum is
(3(x)
=
{A C C
:
A
is not a Riesz point of
x}.
Our applications can be classified under three main headings. I
Spectral mapping properties.
The spectral mapping theorem holds for the
Fredholm and Browder spectra (R.5.2) but not, in general, for the Weyl spectrum.
It does, however, hold for the Weyl spectrum for triangular
algebras of operators on sequence spaces and for certain quasidiagonal operators on Hilbert space. II
Lifting theorems.
Suppose that T E B(X)
is invertible modulo
and satisfies some additional algebraic or analytic condition. S 6 Inv(B(X))
satisfying the same condition and such that
K(x)
Can one find
T - S E K(x)?
T E B(X)
satisfies an algebraic
or analytic condition, can one describe () G(T + K),
where the intersection
Suppose that
Compact perturbations.
III
satisfying the same condition?
K E K(X)
is taken over all
Fredholm and Riesz elements in subalgebras
A.1
We fix some notation which shall remain in effect throughout the chapter.
A denotes a unital Banach algebra and A
is a fixed closed ideal of
KA
hence by R.2.6,
Clearly
KB.
to
in A
(D(A)
relative to
(D(B)C $(A)tA B
a necessary condition if
is semisimple.
B
If
be semisimpZe.
B
KB = KA O B,
and
KA,
in
(D(B)
relative
B
but the converse does not hold in general.
wB(T) = wA(T)
THEOREM. Let
I(A), and
contained in
If we do have equality then
A.l.1
a closed subalgebra with 1 E BC A.
We investigate the relationship between the
KB C I(B).
Fredholm elements
B
for each
First we give
T E B.
(D (B) = (D (F)n B
then GB(T) = GA(T)
(T E B),
Proof. then
It suffices to show that Inv(A) c\ B`Inv(B) . T E
in A
ing
T
in
EXAMPLE.
Let
O.
T; then
T E (D(B),
where
Now the left and right annihilator ideals of
cI(B).
B.
T
By F.1.lO,the left and right Barnes idempotents of T in
T E Inv(B) .
B are both zero, hence
iX(T)
T E Inv(A) (l B,
are zero, hence the same is true of the left and right annihilator
ideals of
A.1.2
If
Take
A = B(X),
and choose
IB(T) = 0
since
and
B
K C KB,
B(X)
(BA.1.4), but T V (D(B).
(S E B)
is commutative, and we can write
by F.3.ll,implying that
Fredholm operators of index zero in
with
T E ((X)
be the maximal commutative subalgebra of
B
GB(S) = GA(S)
V E Inv(B)
KA = K(X)
B(X)) which is false.
contain-
For, if T = V + K
T E (D°(X)
(the
So the condition
of Theorem A.l.l is not sufficient for general B. For C*-algebras we do get equality. A.1.3
THEOREM.
Let
A
be a C'"-algebra and
ID(B) = D(A)n B. Proof.
The map
K
86
K
of
B/KB -)- A/KA,
B
a *-subalgebra of A; then
is a *-isomorphism so
t(B/KB)
Thus if
then
T C (P(A)n B,
hence, in
1(B/KB)
is a *-closed subalgebra of
flT + KB) = T + KA
(BA.4.2).
A
THEOREM.
A.1.4
If
Proof.
and
T E R(B), then since
CB(T) = CA(T).
THEOREM.
Further, if
B
T
Let
A/KA, and
R(A), R(B)
KB = B
A KA, Now
0 # A E CA(T),
CB(T) = CA(T).
we have
T = K + Q
then
0
K
Proof.
by definition
P(A,T) 6 KB C KA.
T C R(B) ,
be a Riesz operator on a Hilbert space
where
nilpotent operator in
T E R(A)
is countable, hence
CA(T)
be any closed unitaZ *-subalgebra of operators on
Then
T E D(B) .
hence
will denote the sets of
and if T E R(B),
T E R(A), B.
It follows from R.2.5 that A.1.5
is invertible in
respectively.
B,
R(B) = R(A)n B,
Conversely, suppose
(BA.4.1).
T + KB E Inv(B/KB),
Thus
Next we consider Riesz elements. Riesz elements in
A/KA
is a compact operator in
H B,
H
containing
and let T.
and Q a quasi-
B.
This is a Corollary of A.1.4 and the West decomposition in the
algebra
B
(C*.2.5) 0
We have the following information on the Browder spectrum.
A.1.6 Proof.
and CB(T) = CA(T) _> (3B(T) = aA(T) .
TEB
LEMMA.
Similar to A.1.4 e
An interesting consequence of this is A.1.7 A,
THEOREM. Let
S(x)
Proof.
Set
spectrum of
that If
A be unital and
K be a closed inessential ideal of
then, for each x E A,
= r\{C(x + y)
y c K
B = {y E A : xy = yx}; x
in
QA (x) _ 8B (x)
=
{A E
W
B,
B
(x).
(A.1.6).
°(B) = {x C (D(B)
WB(x)
:
:
xy = yx}.
then the right hand set is the Weyl
Since
CA (x) = C
B
(x)
(BA.1.4), it follows
Thus it suffices to prove that
1(x) = O},
:
and
(3B (x) = WB (x) .
then, from R.2.2,
A - x ! $°(B)}; 87
and
{A E C
=
$B (x)
:
X - x V R(B)}.
Since Riesz points are automatically index-zero Fredholm points, and since isolated index-zero Fredholm points are Riesz points, it suffices to show that an index-zero Fredholm point of
definition of of
Let
B.
generated by
B
(OM :
and
v E Inv(B)
where
D
and
X = 0
Without loss of generality take x = v + f
aB(x)
is isolated in $0(B).
x E
By F.3.11,
of = fv
also
f' E soc(B'),
OB(x).
by the
be the closed unital (commutative) subalgebra
v, v-1
and
f
with Gelfand space
0.
The set
is bounded away from zero, while f (w) = 0, for all but at
W c Q}
most a finite number of
w E Q.
is an isolated point of
aD(x),
x = v + f,
It follows, since
and therefore of
that zero
aB(x) .
Seminormal elements in C*-algebras
A.2
In this section A will be a C*-algebra. hyponormaZ.
T
If
T*
is hyponormal,
is called seminorrnaZ.
A.2,1
LEMMA.
T E A
and T*T > TT*,
is co-hyponormaZ.
T
is
In either case
We consider the Fredholm theory of these operators
Suppose that
T E $(A):
(i)
T
is hyponormaZ => 1(T) < 0;
(ii)
T
is co-hyponormaZ => i(T) > 0.
Proof. Since
T
If
1 (T*) = - t (T) ,
(u) follows from (i) .
To prove (i), recalling the definition of the index function F.3.5, it suffices to consider the case of primitive hyponormal.
A.
Then there exist Barnes idempotents
'take to be self-adjoint (BA.4.3)),which satisfy
Assume that T E $(A) P, Q E KA lan(T) = AP
Now T*T > TT* _> 0 = QT*TQ > QTT*Q > 0 => QTT*Q = 0 => QT = 0 => Q E lan(T) > QP = Q -=> rank(Q) < rank(P).
Thus
as,
1 (T) = nul (T) - def (T) = Yank (Q) - rank (P)
is
(which we may and ran(T) = Qk
A.2.2
COROLLARY.
A.2.3
THEOREM.
If T
is normal, 1(T) = O.
If T 6 f(A)
is hyponormaZ (resp. co-hyponormal) xhere
K F_ A such that for each non-zero
exists
A s C, T + AK
is left (resp.
right) invertible.
A consequence of A.2.1 and F.3.11
Proof. B
denotes a closed *-subalgebra of
If T
THEOREM.
A.2.4
is seminormaZ and
has either a left or a right inverse in T + K
hence
IA(T + K) = 0,
then
T 6 B n e (A)
By A.2.3 there exists
by A.1.3.
T 6 D(B)
Proof.
A.
is invertible in
A,
T + K
such that
K E KB
and thus in
B,
T 6 e (B).
But
A.
and hence in
B.
Thus
iB (T) = 1B (T + K) = 0 , Next we consider the Weyl spectrum of a seminormal operator. A.2.5
Let
THEOREM.
subaZgebra of A.
(i)
and let
If T 6 B
is seminormal:
K 6 KAf1 B
such that
B
be a closed *-
WB (T) = WA(T) ,
(ii) there exists Proof.
(i)
Clearly
WA(T)
(ii)
A = U(H), KA = K(H)
A - T
WA(T) = CT (T + K) .
is seminormal for each
=
{A C C
:
A - T V (Do(A) },
=
{A 6 C
:
A - T V e(B) },
=
WB (T) .
l c C.
By R.2.2,
(A.2.4)
This follows from (i) and the Stampfli decomposition of
algebra
T
in the
B (C*.2.6) 0
A consequence of this result is that if subspaces of
H
and
L
is a collection of closed
is the *-closed subalgebra of
131
operators which are reduced by the subspaces of normal operator in
U (T + KO)
=
$L,
there exists
(_l a (T + K)
=
L,
K0 E K(H) A $L
consisting of
B(H)
then if
T
is a semi-
such that
W (T) .
KEK(H) 89
Next we see that Fredholm points of index zero of a seminormal operator are Riesz points.
If T is a seminormaZ operator in A, then
THEOREM.
A.2.6
A
Uithout loss of generality take
Proof.
it suffices to show that either
point of
OA(T).
T*T > TT*,
If
adjoint Barnes idempotents
1(T) = 0 => P = Q,
Let
If T E B
(S E B).
T E
0 (A),
as in the proof of A.2.1, there exist self-
P, Q E KA
such that
But then
QP = Q.
and an examination of F.1.11 shows that the underlying Thus,either T 6 Inv(A),or zero
of finite rank ,
is a pole of T THEOREM.
Let
is invertible or that zero is an isolated
T
Hilbert space satisfies H = ker(T) ® T(H). A.2.7
to be primitive.
WA(T) = SA(T).
A
and
B
have the property that
is seminormal, then
OB(S) = OA(S)
WB(T) = WA(T).
By A.2.6
Proof.
WA(T)
=
SA(T)
=
SB(T),
(A.1.6)
WB (T) ,
OB (T + K) KEKB
OA(T + K)
by hypothesis,
KEK A
A.3
Operators leaving a fixed subspace invariant
Let
X be a Banach space and Y
A = B(X)
and let
which leave soc(B'),
of T to
Y
B
be the closed subalgebra of A
invariant. I(B).
and
a fixed closed subspace of
We need preliminary information on
Recall that if
T 6 B,
TIY
Y.
as follows-
7T
90
rad(B),
denotes the restriction
Define the restriction and quotient representations of X/Y
Put
X.
consisting of operators
r (T) y
=
Ty
(T E 13, y E Y) ,
B
on
Y
and
7rq (T) (x + Y)
(T E B, X E X) .
Tx + Y
=
It is simple to check that
and that
F(Y)C 7rr (B),
these representations are irreducible.
A.3.1
(ii)
{P
Pq} _ {P E 11(B)
r'
:
thus both
Pr = ker(7r),
B.
rad(B) = PrrN Pq = {T E B
(i)
THEOREM.
q
Hence the ideals
are primitive ideals of
Pq = ker(7rq)
F(X/Y)C 7r (B),
:
T(Y) = (O) and T(X)CY};
soc(B') etp'};
(iii) soc(B') = (F(X), B)';
F(X)1) BC.I(B).
(iv) Proof.
J = {T E B
(i)
Assume that
(ii)
ideals
P'
P'rf P'q = (O).
and
such that
B
or
r
P'
Thus if
q
Moreover,
E'
hence rad(B)C J.
E' E Min(B').
are distinct primitive ideals of
q
P'
is a nilpotent ideal of
T(X) C Y}
and Pr, Pq E 11(B),
is an element of
E
and P'
r
is in either
E'
T(Y) = (0)
But Prn Pq = J,
so J Crad(B).
B,
:
The
and, by BA.3.5,
B'
cannot be in both, since
'
P # Pr or Pq,
and
P E 11(B),
then
E' E P',
thus soc(B')CP', (iii),
(iv) straightforward
The proof of the next result is routine.
A.3.2
and
If
LEMMA.
2' E O(B) then
1 (T) (Pr)
=
iY (7rr (T)) ,
1 (T) (P
=
iX/Y (7rq (T) ) .
q
)
observe that we may have So, if A.3.3
(i)
B
T E Inv(B(X))
is also semisimple then
THEOREM. Let
T E B
(i)
such that
Suppose that TS - I
and
(D(A)n B
but TlY
Inv(B(Y))
(A.l.l).
then
and T!Y E I(Y);
T E >(B) <=> T E tt(X)
(ii) T E to(B) <_> T E to(X) and Proof.
t(B)
and T(Y)C Y
T I Y E 'Do (Y) and
T E t(X)
ST - I = F E F(x).
a finite dimensional subspace
Z1
of
Y
T[Y E (D(Y).
Since
Choose
T[Y E t(Y),
S E B(x)
there exists
such that Y = Z1 ED T(Y).
Choose 91
of
Z2
choose a closed subspace
P2 E B(x)
then a projection
such that
Y
zI E ZI
and
Yi E Y,
yl = z2 + w
(I - P2)S(I - PI)y
(I - P2) (z2 + w + Fy1)
=
T
is an inverse for
F(X)n B.
modulo
Thus
where
Since
Then
z2 C Y.
=
P11 P2 E F(X)1 B,
and as
(I - P2)S(I - P1) C B
y = z1 + Ty1
and w E Y r)F(Y).
z2 C Z2
where
ker(P2) DZ2.
and
y £ Y,
If
S(I - P1)y = STy1 = y1 + FyI.
therefore
y1 E Y,
and
Y = Z2 ® (Y 1 F(Y)),
P2(X) = F(Y)
such that
Again
ker(P1) > T(Y).
and
(I - P2)S(I - P1) E B.
We verify that
Thus
P1(X) = Z1
with
PI E B(x)
a projection
(I - P2)S(I - PI)
The converse is
T £ D(B).
obvious.
F E F(X)
But,(0.2.8), there exists
giving
ix(T) = iX(T + F) = 0,
(the case
i.(T)(Pq) > 0
F £ F(X)A B
T + K
hence
TIY c $°(Y);
then, by (i),
Suppose
by hypothesis.
is similar).
is left invertible in
T + F E Inv(B(X)).
T + F E Inv(B).
such that
Thus
13(X).
Tie inverse of
But since
T + F
T c O(B). < 0
t(T)(Pq)
there exists
t(T) < 0,
Since
T + F has a left inverse
such that
TIY E $°(Y),
T £ (D°(X).
t(T)(Pr) = iY(TIY) = 0,
Further
B
and
T £ $°(X)
Conversely, let
so that
0 = t(T)(Pr) = iY(TIY),
then
T E (D°(B),
If
(ii)
S C B (F.3.11).
Thus
iX(T + F) = ix(T) = 0,
must be
S
T + F £ Inv(B)
thus
.
T E (D°(B)
The next result is a Corollary of F.3.11 and A.3.3. A.3.4
THEOREM.
Let
V e Inv(B(X)),
and
T E B(X)
and T(Y)CY.
is invariant under
Y
and
TIY £
A.4
Triangular operators on sequence spaces
(1 < p < c)
where
and
where
F <=> T £ V(X)
X
will denote one of the sequence spaces
c
O
or
Q
0 will be the usual Schauder basis for X. If {en}1 Then <x, a> = 00)xnan put <x, a> = c(x) and an = B(en}.
and
u £ X,
x = E a e 1
n n
.
If
T C B(X)
defined by
tij =
t, = t. . i ii
T £ B(x)
92
V, V-1
°(Y),
In this section
x E X,
Then, T = V + F
the corresponding matrix
(1 < i, j < '),
is upper-triangular if
[ti,]
isl
and for convenience we write t
i]
= 0
for
i >
In this section A = L(X), algebra of
A
is a sequence in
It is easy to check that
The first lemma is elementary.
Suppose that
LEMMA.
C
T E B
such that
and that
t
o
and
x,
=
n
(3 > 1),
A.4.3
=
for some
E
1=1
T E
then
4) °(X)
(i > 1),
then
T(X)
-
(n > 1) . and
a = 0,
T(X) = T(X) = X
T E B R O°(X) => t = 0 i
LEMMA.
-
1
then
hence, by F.2.8,
n(T) = 0,
so
{A1}1 (1 > 1).
A. = 0
is invertibZe.
T
a c X',
a1tin
then
t # 0
and that
(1 > 1), by A.4.1, hence
a1 = 0
If, in addition, o = ix(T),
CX
T E B
if T - (D°(X),
a(T(X)) = 0
If
Proof.
Thus
Suppose that
LEMMA.
is dense in
If
(1 > 1).
0
-
E A t = O 1 13
i=1 A.4.2
denotes the closed sub-
B
of upper-triangular operators.
Inv(B) = Inv(A) h B.
A.4.1
and
KA = K(X),
is dense in
T(X)
so
d(T) = O.
X.
But
is invertible .
T
for at most a finite number of indices
i.
Proof.
Suppose that the set
such that
and
S E B(X)
W = {i
11SII < E
t
1
is infinite.
= o}
T + S E
S.. = 0 thus
(1
and
W)
= E1-1
s
T1 = T + S E BZ °(X),
of the matrix
T1
(1 E W).
Take
°(X).
operator corresponding to the diagonal matrix
Is1j]
Then
Choose
where S E B
to be the
S
sib = 0 and
c > O
(1 # 3),
IISII < E,
and, by construction, all the diagonal entries
are non-zero.
By A.4.2,
Tl
is invertible, but its
diagonal entries are not bounded away from zero which gives a contradiction* If
T E B
those of A.4.4
-
Proof.
Define
let
denote the diagonal operator whose diagonal entries are
TA
T.
THEOREM.
(D (B) =
Suppose that S
°(B) = B A 1D°(X).
T E B
Tl e B n°(X), T
By A.4.3,
W = {i
:
t
1 to be the diagonal operator with diagonal entries
si = 1 (i c 4) and s1 = O A.4.2,
°(X).
(1 V W) .
is invertible in
13(X),
{s1}i
If T1 = T + S , since
also the diagonal entries of
T1
and hence in
is finite.
= O}
where
S E B,. K(X),
are all non-zero so, by B.
Thus
T = TI - S e
°(B)
93
so
(X)
Now suppose that such that
This implies that
Phen there exists
T E (D(B).
TS = I + L.
ST = I + K, T
1Q(TQ) = 0
and so
T
A
where
then R
R = T - M,
1Q
(where
= V + M
A
= T
A
X
K, L E KB
TASA = I + I,
A
and
- M = V E Inv(B)
M E KQ, since
By A.4.2,
by F.3.11.
thus
1(T) = 0.
R
So
Q C B.
A)
If we put and
E c(B)
A
R(X) = R(X) = X.
The same argument applied to
1(T),
1(S)
KQ = K(X), A.
denotes the index function in the algebra
V E inv(A),
i(T) = 1(R + M) = 1(R) > O.
But
and
relative to the ideal
all its diagonal entries are non-zero.
i(S) > 0.
S E B
SATA = I + KQ,
is a Fredholm element of the commutative Banach algebra
A of all diagonal operators on Hence
Hence
Hence
gives
S
Thus we have
B() 'D°(X)C0°(B) = (D (B) , A.4.5
If T iS an upper triangular then W(f(T)) = f(W(T)).
COROLLARY.
f e Hol(0(T)) Proof.
T E B - f(T) E B,
A.5
Algebras of quasitriangular operators H
is a Hilbert space let ordered by
F(H)
Also
P < Q
if
P
denote the set of hermitean projections in (<_> P(H)C Q(H))
PQ = QP = P
for
lim inf JIPTP - TP11 P
=
0,
and the set of quasitriangular operators in
is denoted by
B(H)
operators were first studied by Halmos (39) who showed that Note that
Proof. an
QA
LEMMA.
is not an algebra.
such that
and
QQ
These
K(H)C k
KA = K(H).
T E QAn D(H) => ill(T) > O.
such that
P(T + F) = O.
+ K(H)C QQ.
A = B(H)
Let
Suppose, on the contrary, that
F E F(H)
GA
P, Q E P.
is quasitriangular if
T E B(=H)
A.5.1
and
and the result follows from the spectral mapping theorem
wB
If
x
0
for
(R.5.2)
by A.4.4.
(T E B),
W(T) = wB (T) = G (T + KB)
operator on
Let
T + F
iH(T) < O.
has a left inverse
Put R = T + F,
Q E P with
Q > P;
By F.3.11, there exists S,
and a P E P,
then R E QQ, then
PQRQ = 0,
since
and since
0 # P
QB(H)Q
is finite dimensional (C*.1.2), QRQQo = O.
such that that
P(QRQ) = O.
there exists a
(To verify this observe that QRQ
So
So there is a non-zero
we can choose
ISI
I
!
Q0
Now let
with
T E B
QA
B
B
=
and let
(T, T1 a B(H)
=
I
such that
[
QRQQo = 0,
and
Now
= 1.
Q
contradicting
Q > P),
-
n
and
be any unital inverse-closed
B
{O},
where
which contains
T e B => T-1 e B)
Note that in such an algebra 1B(T)(P) = 0
for
B, the index function
P E IT(B) except perhaps
First let us see that
1B(T)({O}) = iH(T).
Let I
be separable and fix an increasing sequence
H
Define
strongly.
{T c B(H)
:
1
IP
n
B
T - TP n 1 1- >0
Routine computation shows that x, y a H,
QB(H)Q,
exist.
EXAMPLE. P
Q e P
and
is not right invertible
such that
I S (RQ - QRQ) Qo I
(for
has the property that
such algebras
such that
QRQ
Q0 E: QB(H)Q
1>I
KA = K(H)
KB = KA r\ B.
at the zero ideal
A.5.2
I Qo 1
lIS1l-1
A = B(H),
C*-subalgebra of K(H),
o # P c QB(H)Q,
R e Q 0
the fact that
of
I
I
IIRQ - QRQII >
Thus
0 -
to be a projection which is therefore < Q).
QRQ I
F RQ
I
< Q
a P,
0
is not left invertible in the algebra
and, since this algebra is finite dimensional, therein.
Q
B
P
n
C P
by
(n ± -) }.
is a closed *-subalgebra of
Let
B(H)
then
flPn(x a y) -
(x 2 y)PnII = IIx Q (Pny) - (Pnx) a YlI,
< Ilx a (Pny) -x51ylI + IlxSay - (Pnx) ayfI.
Hence
x51 y e B,
C*-subalgebra of
Ixl
I
I'Pny -Y '
and it follows that A,
B
'
K(H)G B.
is inverse closed
+
I 'x - Pnxl l
Since
B
. I lY1 l
-r 0
is a unital
(BA.4.2).
95
(i)
(D(B) _ (D°(B) _ (D°(H) A B. T £ B => WB (T) = W (T) = WB(T) = W(T).
A.5.3
THEOREM.
(ii) Proof.
If
(i)
TS - I £ KBCK(H) . where
so
V £ Inv(B(H))
and
implies that V £ Inv(B),
iH(T) = 0.
Thus
K £ K(H)
(0.2.8).
iH(T) , i1I(S) > 0 T = V + K,
Hence
T e (D °(H).
But the hypothesis on
1B(T) = 0,
therefore
ST - I,
B
and we have shown that
is guasidiagonal if
T E B(H)
lim inf IITP - PTII P
A.5.4 COROLLARY. f(W(T)) = W(f(T)). If
such that
This proves (i), and (ii) is an easy consequence
4)(B)C B 0 (D° (H) C (D°(B) .
Proof.
S £ B
and, by A.5.1,
S, T E c(H)
Hence
iH(S) = - iH(T)
but
there exists
T E c(B),
=
0
If T is quasidiagonal and if
then
f £ Hol (o' (T))
is quasidiagonal it is quasitriangular, hence there exists a
T
C*-subalgebra
containing
B
K(H)
and
Then the result follows from
T.
A.5.3 and R.5.2 0 Note that
T
normal,
K
compact => T + K
is quasidiagonal, so this
result applies to a large class of operators in A.6
B(H).
Measures on compact groups
The background for this section may be found in (45). group and on
G.
M(G) )(G)
denotes the set of equivalence classes of irreducible strongly
metric polynomials on 6 E E(G)
let
X Q (x) = tr((T(x)) L1(G)
dimensional minimal ideal in
M(G).
that
e6 = d6X0
to Haar measure on
96
LEMMA.
and
G.
and
T(G)
the set of all trigono-
M(G)
be the corresponding character; then
M
6
= XQ * L1(G)
is a finite
There exists a constant
is the identity of M 6.
the set of measures in
A.6.1
G
G.
is a central function in
X.
be a compact
G
the convolution algebra of complex regular Borel measures
continuous unitary representations of
For
Let
d
As usual we identify
a
> 0 L1(G)
such
with
which are absolutely continuous with respect
Note that
T(G) = soc(M(G)).
i(G) = L1(G).
Since
Proof.
MG
T(G) = span{MG If
let
T
e*)(,,,
Let
relative to
that
If
Let
G E E(G)
such that
So
(e*)(G}
is a closed ideal of
which is a Riesz
M(G)
u E M(G)
TP E B(L1(G))
define
be the identity measure on
by
M(G)
T x = u*x P
M(G).
11 E '1(G) => G(T) = a(p) .
LEMMA.
Tu E Inv(6(L1(G))),
Suppose
Proof.
then there exists
denote the set of Fredholm elements in
(D(M(G))
L1(G).
(x E L1(G)).
A.6.2
be a fixed Fourier-Stieltjes trans-
(G E )(G))
Hence
LI(G) = soc(M(G))
algebra.
G, hence
(a) # 0 ((45) 28.39), thus e*) # 0 It follows that soc(M(G)) CT(G) .
((45) 28.36).
e (G} # 0 Thus
p(G)
e E Min(M(G)),
Let
p.
for each
It E soc(M(G))
6 E E(G)}C soc(M(G)).
:
U E M(G),
form of
and
is finite dimensional,
T13 S = TSo = STV.
then there exists
If x, y E L 1 (G), then
S E 6(L1(G))
such
T11((Sx) *y) = 11* (Sx) *y
Thus = (TuSx)*y = x*y = T (S(x*y)). (Sx)*y = S(x*y) (x,Y E L1(G))_I By Wendel's Theorem ((4S) 35.5), S = TV, for some v E M(G), thus V = U in M(G), THEOREM.
A.6.3
V*L1(G)
has finite co-dimension in
L1(G)
is a
<=> T p
Riesz-Schauder operator. Proof.
U*L1(G) = T (L(G))
hence, by (25) 3.2.5, since
u
co-dimension it is closed in of distinct elements of (1 < k < m).
(Alyl + ... + Amym)*ea
where
is a set
{al,..., am}
and that there exist
A1y1 + ... + Xmym = 0
If
=
a .YJ
Suppose that
L1(G).
E(G),
has finite
u*L1(G)
yk ¢ 11*L1(G)
Yk E MG '
"k E G
(1
then
0
=
J
Thus
a E E(G),
that
If
0
= span{M
e = 0).
Also
Then
G E E(G)\E Hence
11*y = x. X
Tu(X) = X,
:
6
0 E E(G)\E},
x c M,
there exists Thus
and e =
X=X
y E L1(G)
P*MG = MG
Ee
(if
such
(G E E(G)\E).
E_
put
GEE G is the projection of L1 (G) onto MG and ker(Te) = X. GEE Now since U*MG = MG (G E E(G)\E), then XoC p*L1(G),
X C11*LI(G).
Let
and
p*(ea*y) = eG*x = X.
E CE(G),
°
Te
TeTU = TpTe.
and hence
It follows that, for
with the possible exception of a finite subset
MG C p*L1(G).
Put
is a linearly independent set.
{y1,..., vm}
Therefore
Y = {x s L1(G)
:
X0 = (do - e)*p*LI(G) = P*X. 11*x = 01.
Since
MG
Hence
is always finite 97
dimensional and
u*Ma = Ma
(a 0 E),
it follows that
Then Y*MaCY AMa = (0) if 6 V E so YClan(X0) . 2 Therefore (Y() X) = (O) , hence Yn X = (O) , since
Y
Ma = (0)
YClan(X) .
Thus 1
is semisimple.
L (G)
We have proved that ker(Tu) X = (0) and that T(X) = X. K = Te (TV - I) E
F(L1
and Tu = S + K with
(G)),
If
COROLLARY.
(ii) p E
(iii)
U = (60 - fl *,j
Proof.
rl
L1(G);
V
E Inv(M(G)),
K ET(G) and and
Where V E Inv(M(G))
_
then Tu E $(L1(G)),
Obviously (v) _> (iii). there exists
A.6.5
Conversely, if such that
K E
- l)*(p + K) COROLLARY.
Proof.
E T(G).
p E (o(M(G))
A.6.4 implies that if
AS
( (25) 1.4.5) 0
0
then, by F.3.11,
p + K E Inv(M(G)),
is a factorisation of
p E M(G) _> w().1)
If
(ii) => (i).
hence
and l, 2 are idempotents in soc(M(G)) = T(G) 11 = (S
p*K = K*p;
2
(i) _> (iv) by A.6.3, and (iv) => (iii) => (ii) clearly.
U E O(M(G)),
= W(}.1)
}1
u*¢l = 0 = ¢2*U,
Now
(A.6.1).
as in (v) ,
= 6(U) .
- u E (D(M(G)), then
A
is a Riesz point
Notes
A.7
Spectral mapping properties.
I
T
(D (M(G)) ;
V=V +K Where
(v)
11
Thus
}1 E $o (M(G)) ;
(iv)
of
Now
p E M(G) the fol7oWing are equivalent:
U*L1(G) has finite co-dimension in
(i)
SK = KS.
0
is a Riesz-Schauder operator A.6.4
Thus
S = Te + (I - Te)Tu E Inv(B(L1(G))).
hence
TuIX E Inv(B(X)),
(0 ¢ E).
Gramsch and Lay (38) prove spectral mapping
theorems for essential spectra in a general context.
open semi-group in a unital Banach algebra A for
x E A
Assume that
which contains
Inv(A)
S
is an
and
define
aS(x) ={AEC: A-x¢S}. Then Gramsch and Lay say that the spectral mapping theorem holds in this context if for each 98
f E Hol((5 (x)) S
and
x E A
0s (f(x))
=
f(as (x)).
They show that the spectral mapping theorem holds for a number of essential spectra of interest in operator theory including those of Bowder (16)
(S(x)
in our notation), Kato (52) and Schechter (79) and an example is given to show that it fails for the Weyl spectrum II
Lifting theorems.
W(x).
The decomposition theorems of West and Stampfli,
proved in §C*.2, extend the corresponding lifting theorems from the Calkin This observation applies to the results
algebra to a C*-algebra setting.
of Olsen and Pedersen mentioned in §C*.5.
For a Banach space
°(X) = Inv(B(X)) + K(X)
extension of the lifting theorem
X
the
due to Pearlman
Lay (56) pointed out that if
and Smyth has been discussed in §F.3.
has a commuting decomposition into the sum of an invertible and
T = c)°(X)
a finite rank operator then zero is a Riesz point of
T
(the Riesz-Schauder
case). III
Compact perturbations.
{A S C
:
If
T E B(X), Schechter (79) shows that
is the largest subset of
A - T ¢ D°(X)}
under all compact perturbations of
{SC
A-T
° (X) }
=
n
6(T)
which is invariant
equivalently,
T;
9 (T + K) .
KEK(X)
The generalisation of this result to Banach algebras is given in R.2.2. Lay (56) proved that
(T)
= ("{0(T + K)
:
K S K(X)
and
KT = TK}.
The verification of this result in Banach algebras is contained in the proof of R.5.2.
99
BA Banach algebras
This chapter lists the information required from algebra theory, and in particular deals with Banach algebras over the complex field.
Where the
results are known they will be referenced in one of the standard texts (14), (75),
The algebras will always be
Otherwise proofs are given.
(48).
unital and complex, the non-unital case may be dealt with by adjoining an identity.
§1 deals with basic spectral theory and §2 with the space of primitive ideals in the hull-kernel topology, in the commutative case with the space §3 gathers information on minimal ideals and the socle,
of maximal ideals.
while basic results on C*-algebras are listed in §4. BA.1
Spectral theory
Let A be a unital Banach algebra and let invertible elements in
A
A.
x E A,
If
Inv(A)
denote the set of
the resolvent set
p(x)
of
x
is the set
p(x)
=
_
pA(x)
while the spectrum
{A E C
:
A - x E Inv(A) },
0(x) = OA(x) = C"p(x).
The subscripts may be omitted if
the algebra in question is unambiguous. BA.l.l
Inv(A) is an open subset of A
and if x E A,
QA(x)
is a non-
empty compact subset of e. {
(14) 2.12 and 5.8). Ile use
BA.1.2
to denote the topological boundary of a set.
3
Let
B
6A(x), (14) 5.12).
0B(x) {
in
For
x C A
be a closed subalgebra of A containing
while
36B(x)e3aA(x).
the spectral radius is defined by
1, x.
Then
r(x)
supfIxI
=
A E Q(x)'I.
:
limllxnlll/n_
r(x) =
BA.1.3
n-o It follows that the spectral radius is independent of the
((14) 2.8).
containing algebra.
BA.1.4 If B
is a maximal commutative subaZgebra of A
containing
it
x
will be -rcnitai and aB(x) = QA(x) . ((14) 15.4). is a non-empty compact subset of
a
If
C,
denote by
the class
Hol(a)
of complex valued functions which are defined and holomorphic on some neighbourhood of
may be regarded as an algebra if we restrict a
Hol(Q)
a.
combination of functions to the intersection of their domains. and
then an element
f E Hol((Y(x)),
f(x) E A
If
x E A
is defined by means of an
A-valued Cauchy integral ((14) §7).
BA.1.5 If Hol(6(x))
x E A the map
into
polynomials in
f -
is an algebra homomorphism of
f(x)
A, mapping complex polynomials into the corresponding x.
((14) 7.4).
BA.1.6
If x e A
(Spectral mapping theorem).
and
then
f E Hol((Y(x)),
G(f(x)) = f(Q(x)).
((14) 7.4). Note that if
x E A
satisfies
III - xII < 1,
then we can define
log x
by means of an A-valued Cauchy integral. BA.2
The structure space
From BA.2.1 to BA.2.5 we shall be dealing with a purely algebraic situation. Banach algebras are introduced after 2.5.
The term ideal without either
adjective right, or left, means a two-sided ideal.
Let A be a unital algebra and on
X
operators on
X.
A representation
ally) irreducible if under
X
is an algebra homomorphism of
c,(x)
for each
# 0,
x E A
A
a linear space, a representation of
A
into the algebra :
A - L(X)
of linear
L(X)
is (strictly or algebraic-
and if the only subspaces which are invariant are
(0)
and
X.
An ideal
P
of
A
is
101
primitive if there exists a maximal modular left ideal
of A
L
such that
P = {x E A : xACL}. BA.2.1
is a primitive ideal of A <--> P
P
(i)
is the kernel of an
irreducible representation of A; is the kernel of the (irreducible) left regular representation on
P
(ii)
A/L;
the quotient space
is a primitive ideal of A and
(iii) If P
with L1L2C P then either L1GP or Note that
((14) 24.12).
L1L2
are left ideals of A
L1, L2
L2C P.
n
is made up of elements
I xiyl
yi E L2).
(x1 E L1,
1
is denoted by
A
The set of all primitive ideals of
II(A)
and has a
standard topology known as the hull-kernel topology defined in terms of a If
closure operation. k(P) = (\{P E II(A)
hull of v is closed if
:
semisimpZe if J
BA.2.2
(i)
(ii)
if J if J
(iii)
If
rad(A) = (0).
such that A/rad(A)
the
A,
rad(A) = A.
A
is
We collect some information on the radical.
Jn = (0)
is nilpotent.
is semisimple;
is an ideal of A,
rad(J) = J n rad(A); is a nilpotent left or right ideal of A
x E Inv(A) <=> x + P E Inv(A/P)
(v)
Proof.
For (i) -
see (14)
(iv)
=> is obvious.
show that
then
J C rad(A);
Ax = A.
Then
hypothesis, there exists
u e A
hence
1 C L
inverse
Ax
Thus
A/P
x + P
is invertible for
P E 11(A).
Q = {a C A : aA c L} E 1(A),
ux - 1 E QC L.
such that
has
(P E 1(A)).
y has a left inverse in
We
is a left ideal which is contained in
which is a contradiction.
y E A.
as its inverse in ag above,
x + P
If not then L.
(P E 1(A)).
24.165, 24.20, 24.21.
So suppose
a maximal left ideal
102
is empty,
11(A)
A
is defined to be
A set FCI[(A)
The radical of
((14) p.132).
P E UI (A) } .
is a non-empty subset of
V
P DV}.
:
is
F
z E rad(A) _> 1 + z E Inv(A);
(iv)
(v)
while if
P s F}
h(V) = {P e 11(A)
r = h(k(F))
rad(A) = A {P An ideal
:
the kernel of
P CII(A)
y + P
So
Ax = A
and
and, by But
x has a left
as a left inverse, and therefore
Now the hypothesis holds for A.
ux E Ax C L,
Therefore
y =
x-1 S
y
so,
J be an ideal of
Let
A
A -> A/J be the canonical homo-
and let
morphism.
P 6 h(J)<> (P) 6 11(A/J);
(i)
BA.2.3
(ii) k (h (J)) _ -1(rad (A/J)) ; A/J
(iii)
is semisimpZe <=> J = k(h(J)
((48) p.205).
The next result is frequently used in the text.
BA.2.4 Let such that
A
be a unitaZ algebra with ideal
xy - 1, yx - 1 c J <=> there exists
y c A
then there exists
J,
y c A
such that
xy - 1,
yx - 1 c k (h (J)) . is obvious, so let
_>
Proof.
denote the canonical homomorphism
: A ± A/J.
is invertible modulo
x
y e A
there exists
k(h(J))
(1),
(l) c
(xWy), Q;(Y)Cx) c Inv(A/J) x c A
(BA.2.3),
(BA.2.2),
x' = x + rad(A) 6 A' = A/rad(A).
set
If WGA set 'a' _ {x' BA.2.5
(k(h(J))) = rad(A/J)
c Inv(A/J)
fi(x)
If
xy - 1, yx - 1 c k(h(J)),
such that
:
x c W}.
The structure spaces
(i)
11(A), 11(A')
are homeomorphic under the
map P;P'; (ii)
(iii.) Proof.
has a left (right) inverse in A<=>x' has a Zeft(right) inverse in A:
x
= Inv(A')
Inv(A)'
.
((14) 26.6).
(i)
(ii) > is obvious, so assume that there exist (BA.2.2),
y 6 A, so
x
z c rad(A)
x'
has a left inverse in
such that
has a left inverse in
yx = 1 + z.
But
Then
A'.
1 + z e Inv(A)
A.
(iii) follows at once . Now specialise to the case of a Banach algebra
A.
Let
there exists a maximal modular (and therefore closed) left ideal such that
P = {x C A : xA CL}.
it follows that
P
then
P c 11(A)
is closed in
L
of A A.
103
Further, by BA.2.1,
P
is the kernel of the irreducible left regular A/L
representation on the quotient space
the image of this representation is contained in
B(A/L),
the bounded linear
hence,by Johnson's theorem ((14) 25.7),this represen-
A/L,
operators on
Now
which is a Banach space.
tation is continuous.
Thus, without loss of generality, when dealing with
Banach algebras it will be sufficient to consider the continuous irreducible representations into If
A
is a Banach algebra, then
X.
is a closed ideal of
rad(A)
is a semisimple Banach algebra.
A' = A/rad(A)
primitive ideal of
BA.2.6 If A A/P
An algebra
(x E A).
aA(x) = 6A,(x')
algebras
for Banach spaces
B(X)
and
It follows from BA.2.5 that
is primitive if
zero is a
A.
is a Banach algebra and and
A
A,
A'/P'
P E 11(A)
the primitive Banach
are isometrically isomorphic under the map
x + P -* x' + P'. The map is an isomorphism since
Proof.
rad(A) C P
A
(P E 11(A)).
straightforward computation shows that the mapping is an isometry
BA.2.7 If A
is a semisimple Banach algebra and
closed subalgebra Proof.
B = eAe
P -> P (NB
eAe
e2 = e e A,
then the
is semisimple.
is closed in
A
since
e
is a homeomorphism of 11(A)\h (B)
is idempotent.
onto
II (B)
The map
((14) 26.14), so
rad(B) = rad(A)RB = (0) , The quasinilpotent characterisation of the radical in the next theorem is due to Zeminek (104).
BA.2.8 Let (i)
Z
rad(A)
Q(A)
denotes the set of quasinilpotent elements of A.
be a unitaZ Banach algebra, then contains any right or left ideal all of whose elements are
quasiniZpotent;
(ii) rad(A) = {x E A : x + Q(A)CQ(A) }; (iii)
Proof. (ii),
rad(A) = {x E A : x + Inv(A) C Inv(A)}. (i) follows from (14) 24.18. (iii)
We show that
x + Q(A)CQ(A) => x C rad(A) _> x + Inv(A)C.Inv(A) _> x + Q(A)CQ(A). 104
Let x + Q (A) CQ (A) .
on a linear space
irreducible representation of A such that
E X
Choose
u E A
0 / A E p(u)
and put v = A - u. - x) F.
such that
-1
E
xEP
=
7 (v-1 )Tr(ux
x c rad(A).
Let
u + x E Inv(A).
Thus
71(x)E
(v-1)
Choose
7r(u)7r(x)E _
£;
But
-
=
x E Q(A) _> v-1xv c Q(A),
by hypothesis, which is a contradiction.
(P E li (A)) ,
and
x c Q(A),
0.
71(x)
7r (xv - vx) E,
7r (v 1)
6(v-1xv - x).
x - v 1xv E Q(A)
7r(u)E = 0,
=
7
Thus
and that
x
is an
7
By the Jacobson density theorem ((14) 24.10)
there exists
1xv
then, since
71(x)E 76 0,
are linearly independent.
7r (v
Suppose
0 E Q (A) _> x E Q (A) .
Then
It follows that
x E rad (A) .
hence
(a +
u E Inv(A),
If
hence
x)-1
= u
1(1 + xuhence
x + Inv(A)C Inv(A).
Let x + Inv(A)C,Inv(A). Now Ax + Inv(A)C_ Inv(A) (A E V, and Thus 1 + A(q + x) E Inv(A) q E (2(A) <_> 1 + Xq E Inv(A) (A E V. (A E C), and it follows that q + x E ()(A), therefore x + Q (A)CQ (A) . Minimal ideals and the socle
BA.3
ideal in
be an algebra over
A
Let
J.
J
(0), such that
a nr:nimal right ideal of A
C,
and
(0)
J
A minimal idempotent is a non-zero idempotent
is a division algebra.
A
(If
of minimal idempotents in
is a Banach algebra
is denoted by
A
is a right
are the only right ideals contained
Min(A).
e
such that
eAe = Ce).
eAe
The set
There are similar
statements for left ideals.
BA.3.1 (i)
(ii)
(iii)
If A
is a semisimpZe algebra, then
R is a minimal right ideal of A <_> R = eA L is a minimal left ideal of (1 - e)A,(A(l - f))
where
A <=> L = Af where
e E Min(A); f E Min(A);
is a maximal modular right (left) ideal of A
if, and only if, e, f E Min(A) ((14) 30.6, 30.11).
7nc
(i)
BA.3.2
Let
uJ = (0),
either
If x E A ,
(ii)
ideal of
be a minimal right ideal of A
J or
uJ
and let
u E A.
Then
is a minimal right ideal of A. 0
xe
e s Min(A) and
then
is a minimal right
x eA
A.
((14) 30.7, (75) 2.1.8). If A
has minimal right ideals the smallest right ideal containing them
all is called the right socZe of
A.
ideals, and if the right and left socles of
socZe of A
exists and denote it by
exists, is a non-zero ideal of
has both minimal right and left
A
If
A
are equal, we say that the Clearly the socle, if it
soc(A).
A has no minimal left or right
If
A.
soc(A) = (0).
ideals we put
BA.3.3 Let A be a semisiirrpZe algebra with ideal J. (i) soc(A), soc(J) exist; (ii) Min (J) = J R Min (A) ;
(iii)
soc(J) = J R soc (A) ; if A is a Banach algebra
(iv)
Proof. (ii)
(iii) (iv)
(i)
and
e, f s Min(A)
Then
then
dim(eAf) < 1.
((14) 30.10, 24.20).
straightforward. follows from (ii) and BA.3.1. ((14) 31.6).
BA.3.4 Let A be a semisimple algebra, canonical quotient homomorphism
P E 11(A), and let
A - A/P.
Then
A/P
denote the is semisimple and
q(soc(A) )Csoc( (A)) . Proof.
and the result follows from BA.3.1 i
c(Min(A))
The relationship between minimal idempotents and primitive ideals is important.
BA.3.5 Let A be a semisimpZe algebra. (i)
(ii)
Proof.
If e E Min(A),
If e2 = e E soc(A), (i)
If
e E Min(A)
Clearly e
P
e.
the set then
Pe c 1(A)
{P E 11(A)
A(l - e)
:
If
Q E II(A)
and e ¢ Q,
e ¢ P}
such that
e ¢ Pe.
is finite.
is a maximal modular left ideal
Pe = {x E A : xA C A(1 - e)}
(BA.3.1), therefore
l06
there exists a unique
is a primitive ideal.
then Q,Ae = (0), since
Ae
is a minimal left ideal.
and hence, by BA.2.1, either
( O ) ,
Hence Pe Q, If
(ii)
Now if q E Q,
Qe = (0).
It follows that q E Pe , hence QLPe.
qAe = (0). PeAe =
Thus
giving
Therefore
If
e = e1 + ... + en
then and
P E R(A)
P = Pe
or
On the other hand PeC Q.
But
Q n Ae = (0).
Pe = Q.
e2 = e c soc(A),
(1 < i < n).
Ae C Q,
therefore
qA C Q
then
P,
e
where
ei V P
e1 E Min(A)
for some
i.
and the result follows 1
Information is also required on the set of accumulation points
of
R* (A)
in the hull-kernel topology.
R(A)
BA.3.6
If A
Proof.
Let n
Thus
is a semisimple algebra then
P E R(A)
P ¢ h(soc(A)).
and
where
x = i ay e1
R*(A) C h(soc(A)).
e1 E Min(A)
E A,
a.
Then there exists
x E soc(A)\P.
Hence at least
(1 < i < n).
2.
one
e
say)
(e
does not lie in
{P} uh({e}).
joint union
Now
P.
h({e})
So, by BA.3.5,
is closed in
R(A)
R(A),
is the disso
{P}
is
P 4 R*(A) .
open,therefore
The Gelfand topology on the structure space of a commutative Banach algebra is, in general, stronger than the hull-kernel topology ((14) 23.4).
If
BA.3.7
is a commutative Banach algebra then
A
the GeZfand topology <_> Proof. R(A)
R(A) is discrete in the hull-kernel topology.
Without loss of generality we may assume
A
If
to be semisimple.
is discrete in the hull-kernel topology, then it is clearly discrete Conversely, suppose that
in the Gelfand topology. Gelfand topology. there exists
where p thus
is discrete in
R(A)
By the S.Llov idempotent theorem ((13) 21.5), if
p = p2 E A
such that
p(P) = 1,
is the Gelfand transform of p.
11(A)
is discrete in the
11(A)
is the disjoint union
hull-kernel closed, so
{P}
p(Q) = 0
(Q E R(A), Q 34 P)
Then p e Min(A)
{P}u h({p})
by BA.3.5.
is hull-kernel open, hence
P E R(A),
and p Now
11(A)
P,
h({p})
is
is discrete in
this topology , BA.3.8
If A
is discrete then Proof.
is a semisimpZe commutative Banach algebra such that
From the above proof if
that p ¢ P,
so
R(A)
h(soc(A)) = . P E 11(A),
there exists
p E Min(A)
such
soc (A) ¢'' P 107
A be a unitaZ semisimple Banach algebra such that
Let
BA.3.9
single ton set for each
Let
Proof.
x E A
x e A,
r(x) = 0.
with
y £ A
for suppose there exists
Then we claim that
such that
x a Inv(A),
which is a contradiction.
x e rad(A),
hence
r(y - Al) = 0
so
A 34 0
If A
BA.3.10
Now, if
x = 0.
A = Cl.
r(xy) > 0,
is not zero.
a(xy) = a(yx)
and (14) 5.3,
then
hence
is a
0(x)
r(xy) = 0
then by the hypothesis
yx, xy c Inv(A),
Thus
Thus, by (14) 24.16,
0 # y e A
y = Al
(y s A),
then
hence
r(x) = O =>
0(y) = {A}
where
0
is a seer simple Banach algebra and if p
idempotent which is not minimal, then there exists
is a non-zero
x a pAp
such that
contains two distinct points.
PAp(x) Proof,
is a semisimple Banach algebra with unit
pAp
is a singleton set for each
6pAp(x)
p
(BA.2.7), so if
x e pAp, by BA.3.9,
pAp = Cp
and
p E Min (A) . BA.4
C*-algebras
A Banach algebra A that
is a C*-algebra if it possesses an involution * such
11x*xfl = 11x112
(x E A).
(The terminology B*-algebra is also used).
The Gelfand-Naimark theorem states that every C*-algebra is isometrically *-isomorphic to a closed * subalgebra of ((14) 38.10).
as follows.
B(H)
for some Hilbert space
H
The commutative version of the theorem, due to Gelfand, is
Let A be a commutative C*-algebra then the space
zero characters (multiplicative linear functionals) on
A
in the weak * topology and A
is isometrically-*-isomorphic to
further,
is unital ((14) 17.4, 17.5).
(2 is compact <> A
Q
of non-
is locally compact Co(Q);
BA.4.1. Let A be a C*-algebra, then (i) A is semisimple; (ii)
if i
is a closed ideal of A,
then
i = I*
and A/I
in the
quotient norm is a C*-algebra; (iii)
then
if
is a continuous *-homomorphism of A
(A) is closed in
B.
((75) 4.1.19, 4.9.2, 4.8.5).
108
into a C*-algebra
B
BA.4.2
A
Let
be a unitaZ C*-algebra and let
subalgebra of A
aB(x) = UA(x)
then
B
be a closed unital *-
(x E B).
((75) 4.8.2),
BA.4.3 Let A 7f
(i)
and
be a C*-algebra.
f = f2 E A,
e = e2 = e* E A
there exists
such that
fe = e
of = f. If s
(ii)
is a minimal right ideal of A,
If A contains a right ideal
there exists Proof.
e = e* E Min(A)
R = eA.
such that (iii)
there exists
(i)
R ( - - ? f 1A (fi E Min(A), 1 < i < n)
1
e = e2 = e* E soc(A)
^uch that
R = eA.
Using the Gelfand-Naimark representation this is equivalent to
the elementary assertion that if a projection is contained in a C*-algebra of operators on a Hilbert space then the C*-algebra contains a self-adjoint projection with the same range ((84) 6.1). (ii)
If R
that
R = fA.
is a minimal right ideal there exists By (i)
find
e = e2 = e* C A
Then R= fA = efA C eA = feA C fA.
Thus
such
f2 = f E Min(A)
such that
R = eA, hence
fe = e, of = f. e c Min(A).
Similar argument 0
(iii)
It is elementary to check the uniqueness of the self-adjoint idempotents in BA.4.3.
Since a C*-algebra is semisimple its socle exists.
BA.4.4 Let A be a C*-algebra, then soc(A) _ (soc(A))*;
(i)
(ii)
x E soc(A) < > x*x E soc(A);
(iii)
x E soc(A) <=> x*X E soc(A).
Proof.
(i)
If
x E soc(A),
n then x E R C Ef,A where R is a right ideal of A 1
and each f .EMin(A).By BA.4.3, 1
x = ex, (ii)
hence
e = e 2 = e* E soc(A).
So
x* = x*e E Ae C soc(A).
_> is clear.
there exists
R = eA where
Let
x E A
and suppose that
e = e2 = e* e soc(A)
X*x(1 - e) = 0,
IIx -- xel12
such that
x*x E soc(A).
Then
x*x E Ae (BA.4,3).
Thus
and
=
11x(1 - e)
112
=
II(1 - e)x*x(1 - e)II
=
0,
109
x = xe a soc (A) .
so
Let
(iii)
be a closed ideal of the C*-algebra
I
Then
A.
I = I*
and
is a C*-algebra (BA.4.1), hence
A/I
IIx*x+111
lI(x*+I)(x+I)11
=
=
llx+1112.
x*x E I <=> x E 1 S
so
Finally we need a result on the spectrum of an operator matrix.
denotes the interior of the set Q, and
BA.4.5 If int (0 (U) r 6 (V) )
=
int(Q)
U, V e i3(H).
and
U
T
=
V
0
then 0(T) = 6(U) U 6(V) . This follows immediately from the following lemma.
BA.4.6
(a(U)ua(V))dint(6(U)n6(V))Ca(T)C6(U) jCY(V).
Proof.
Elementary matrix computation shows that
(a (U) U a (V) ) \ (a (U) (10 (V) ) C a (T) Ca (U) ) a (V) .
Now choose
X c a ((j (U) r) a (V))
then
X E 3Q (U),') 96 (V)
A - V is a two-sided topological divisor all bounded linear operators. I1An 11 = l
for each
n,
and
So
each
(l - U)A
110
(A - T)
A E O'(T).
of zero in the Banach algebra of
In the first case there exist A
(
Then
so either A - U or
n
with
; 0.
- U) A n
0
0
0
_*
In the other case, there exist Bn
n, and B (A - V)-l' 0, hence
n
with
0,
IIBn11 = 1
for
a
o
(A-T) 0
B
Thus
0
0
Bn(a - V)
0,
n
again X E 0(T).
0 =
a(o(U)f)ff(V))C6(T) 0
It is easy to see that the result of BA.4.5 fails if we drop the condition that
int(a(U) n 0(V)) =
shift on
Take
H =
Q2 ED
k2
and if
T
is the bilateral
H,
U T
I
=
0
where 0(V)
U
and V
VI axe the forward and backward shifts on
are the unit disk, while
0(T)
k2,
0(U)
and
is the unit circle.
111
Bibliography
1.
Ambrose, W., Structure theorems for a special class of Banach algebras,
2.
Akemann, C.A., Pedersen, G., Ideal perturbations of elements in C algebras, Math. Scand. 41, 1977.
3.
Akemann, C.A., Wright, S., Compact actions on C -algebras, Glasgow Math. J. 21, 1980.
4.
Alexander, J.C., Compact Banach algebras, P.L.M.S. (3) 18, 1968.
5.
Atkinson, F.V., The normal solvability of linear equations in normed spaces, mat. Sbornik 28 (70), 1951.
6.
Aupetit, B., Proprietes Spectrales des Algebres de Banach, Lecture Notes in Mathematics 735 (Springer) Berlin, 1979.
7.
Barnes, B.A., A generalised Fredholm theory for certain maps in the regular representation of an algebra, Can. J. Math.
T.A.A.S. 137, 1969.
*
20, 1968. 8.
Barnes, B.A., The Fredholm elements of a ring, Can. J. Math. 21, 1969.
9.
Barnes, B.A., Examples of modular annihilator algebras, Rocky Mountain J. Math. 1, 1971.
10.
Berglund, M.F., Ideal C*-algebras, Duke Math. J. 40,(1973).
11.
Bonsall, F.F., Compact operators from an algebraic standpoint, Glasgow Math. J. 8, 1967.
12.
Bonsall, F.F., operators that act compactly on an algebra of operators, B.L.M.S. 1, 1969.
13.
Bonsall, F.F., Compact Linear Operators, Yale University Lecture Notes, 1967.
14.
Bonsall, F.F., Duncan, J., Complete Normed Algebras,
(Springer) Berlin,
1973.
(3) 4, 1954.
15.
Bonsall, F.F., Goldie, A.W., Annihilator algebras, P.L.U.S.
16.
Browder, F.E., on the spectral theory of elliptic differential operators I, Math. Ann. 142, 1961.
17.
Buoni, J.J., Harte, R.E., Wickstead, A., upper and lower Fredholm spectrar P.A.M.S. 66 , 1977.
112
18.
Breuer, M., Fredholm theories in von-Neumann algebras I, Math. Ann. 178, 1968.
19.
Breuer, M., Fredholm theories in von-Neumann algebras II, Math. Ann. 180, 1969.
20.
Calkin, J.W., Two-sided ideals and congruences in the ring of bounded operators in Hilbert space, Ann. Math. (2) 42, 1941.
21.
Caradus, S.R., Operators of Riesz type, Pacific J. Math. 18, 1966.
22.
Caradus, S.R., Perturbation theory for generalised Fredholm operators, Pac. J. Math. 52, 1974.
23.
Caradus, S.R., Perturbation theory for generalised Fredholm operators II, P.A.M.S. 62, 1977.
24.
Caradus, S.R., Operator Theory of the Pseudo-Inverse, Queen's Papers in Pure and Appl. Math. 38, Kingston, 1977.
25.
Caradus, C.R., Pfaffenberger, W., Yood, B., Calkin Algebras and Algebras of Operators on Banach Spaces, (Dekker) New York, 1974.
26.
Chui, C.K., Smith, P.W., ward, J.D., <'A note on Riesz operators, P.A.M.S. 60, 1976.
27.
Dixmier, J., C -Algebras, (North-Holland) Amsterdam, 1977.
28.
Douglas, R.G., On majorization, factorization and range-inclusion of operators in Hilbert space, P.A.M.S. 17, 1966.
29.
Dowson, H.R., Spectral Theory of Linear Operators, (Academic Press) London, 1978.
30.
Dunford, N., Schwartz, J.T., Linear Operators, Part I, (Wiley-Interscience New York, 1958.
31.
Erdos, J.A., on certain elements of C -algebras, Illinois J. Math. 15,
*
1971. 32.
Erdos, J.A., Giotopoulos, S., Lambrou, M.S., Rank one elements of Banach algebras, Mathematika, 24, 1977.
33.
Fredholm, I., Sur une classe d'equations fonctionelles, Acta. Math. 27, 1903.
34.
Freundlich, M., Completely continuous elements of a normed ring, Duke Math. J. 16, 1949.
35.
Gillespie, T.A., West, T.T., A characterisation and two examples of Riesz operators, Glasgow Math. J. 9, 1968.
36.
Gohberg, I.C., On linear equations in normed spaces, Dokl. Akad. Nauk. S.S.S.R. 76, 1951. 77I
37.
Gohberg, I.C., On linear equations depending analytically on a parameter, Dokl. Nauk. S.S.S.R. 78, 1951.
38.
Gramsch, B., Lay, D.C., Spectral mapping theorems for essential spectra, Math. Ann. 192, 1971.
39.
Halmos, P.R., Quasitriangular operators, Acta Sci. 1ath.(Szeged)29, 1968.
40.
de la Harpe, P., Initiation a 1'algebre de Calkin. Lecture Notes in Mathematics 725 (Springer) Berlin 1978.
41.
Harte, R.E., Sh ckstead, A., Upper and lower Fredholm spectra II, Math. Zeit. 154, 1977.
42.
Heuser, H., Zur Eigenwerttheorie einer Klasse Rieszscher Operatoren, Archiv. Math. 14, 1963.
43.
Heuser, H., Atkinson-und Fredholm-operatoren, Lecture Notes, Karlsruhe, 1969.
44.
Heuser, H., Funktionalanalysis, (Teubner) Stuttgart, 1975.
45.
Hewitt, E., Ross, K.A., 'bstract Harmonic Analysis Vol. II, (Springer) Berlin, 1970.
46.
Hilbert, D., Grundzuge einer allgemeinen Theorie der linearen Integralgleichungen IV, Nachr. Akad, Wiss. Gottingen Math.-Phys. K1. 1906.
47.
Huruya, T., A spectral characterisation of a class of C*-algebras, Sci. Rep. Niigata Univ. Ser. A 15 (1978).
48.
Jacobson, N., Structure of Rings, (A.M.S.) Providence, 1956.
49.
Kaashoek, M.A., Smyth, M.Z.F., On operators T such that f(T) Riesz or meromorphic, P.R.I.A. 72(A), 1972.
50.
Kaplansky, I., Dual rings, Ann. Math. (2)
51.
Kaplansky, I., Normed algebras, Duke Math. J. 16, 1949.
52.
Kato, T., Perturbation theory for nullity, deficiency, and other quantities of linear operators, J. d'Analyse Math. 6, 1958.
53.
Kleinecke, D.C., Almost-finite, compact and inessential operators,
is a
49, 1948.
P.A.'1.5. 14, 1963. 54.
Kroh, H., Saturierte Algebren, Math. Ann. 211, 1974.
55.
Laffey, T.J., West, T.T., Fredholm commutators, P.R.I.A. 82(A), 1982.
56.
Lay, D.C., Characterisations of the essential spectrum of F.E. Browder, B.A.M.S . 74, 1968.
114
57.
1,ebow, A., Schechter, M., Semigroups of operators and measures of noncompactness, J. Functional Anal. 7, 1971.
58.
Legg, D., r, note on Riesz elements in C*-algebras, and Math. Sci. 1, 1978.
59.
Mizori-Oblak, P., Fredholm elements in Banach algebras, Glasnik Mat. 10, 1975.
60.
Murphy, G.J., West, T.T., Spectral radius formulae, F. Edinburgh Math. Soc. 22, 1979.
61.
Murphy, G.J., West, T.P., Decomposition algebras of Riesz operators, Glasgow Math. J, 21, 1980.
62.
Murphy, G.J., West, T.T., Decomposition of index-zero Fredholm operators P.R.I.P. 31(A), 1981.
63.
Murphy, I.S., Non-compact operators that act compactly on their centralisers, B.L.M.S. 2, 1970.
64.
Hashed, M.Z., Generalised Inverses and Applications, (Academic Press) New York, 1976.
65.
Ogasawara, T., Zfinite dimensionality of certain Banach algebras, J. Sci. Hiroshima Univ. 17(A), 1954.
66.
Ogasawara, T., Yoshinaga, K., Weakly completely continuous Banach *-algebras, J. Sci. Hiroshima Univ. 18(A), 1954.
67.
Olsen, C.L
68.
Olsen, C.L., A concrete representation of index theory in von-Neumann algebras, Lecture Notes in Mathematics 693 (Springer) Berlin, 1977.
69.
Pearlman, L.D., Riesz points of the spectrum of an element in a semi-
Internat. J. Math.
A structure theorem for polynomially compact operators, Imer. J. Math. 93, 1971.
simple Banach algebra, T.r.:I.S. 193, 1974. 70.
redersen, G.K., Spectral formulas in quotient C*-algebras, Math. Zeit. 148, 1976.
71.
Przeworska-Rolewicz, D., Rolewicz, S., Equations in Linear Spaces,
(F.W.N.) Warsaw, 1968. 72.
Puhl, J., The trace of finite and nuclear elements in Banach algebras, Czech. Math. J. 28, 1978.
73.
Rad3avi, H., Laurie, C., On the West decomposition of Riesz operators,
B.L.l.S. 12, 1980. 74.
Riesz, F., Uber lineare Funktionalgleichurgen, Acta. rlath. 41 (1918).
115
75.
Rickart, C.E., General Theory of Banach Algebras, (Van Nostrand) Princeton, 1960.
76.
Ruston, A.F., Operators with a Fredholm theory, J.L.M.S., 29, 1954.
77.
Salinas, N., operators with essentially disconnected spectrum, Acta. Sci. Math. 33, 1972.
78.
Schechter, M., Basic theory of Fredholm operators, Ann. Scucla Nermale Sup. di Pisa, Classe di Sci. 21, 1967.
79.
Schechter, M., On the essential spectrum of an arbitrary operator, J. Math. 'anal. Appl. 13, 1966.
80.
Schechter, M., r'rinciples of Functional Analysis, (Academic Press) New York, 1971.
81.
Smyth, M.R.F., " note on the action of an operator on its centraliser, P.R.I.A. 74(A), 1974.
82.
Smyth, M.R.F., Ideals of algebraic elements of a Banach algebra, Trinity College, Dublin, School of Mathematics Preprints 11, 1974.
83.
Smyth, M.R.F., Credhoim theory in Barrach algebras, Trinity College, Dublin, School of Mathematics Preprints 8, 1975.
84.
Smyth, M.R.F., Riesz theory in Banach algebras, Math. Zeit. 145, 1975.
85.
Smyth, M.R.F., Riesz algebras, P.P.I.A. 76(A), 1975.
86.
Smyth, M.R.F., On problems of Olubummo and Alexander, P.R.I.A. 80(A), 1980.
87.
Smyth, !1.R.P., West, T.T., The spectral radius formula in quotient algebras, Math. Zeit. 145, 1975.
88.
Stampfli, J.G., Compact perturbations, normal eigenvalues and a problem of Salinas, J.L.M.S. (2) 9, 1974.
89.
Tomiuk, B.J., Wong. P.K.t `Jeakly semi-completely continuous A*-algebras, J, Math 16, 1972.
90.
Treese, G.W., Kelly, F.P., Generalised Fredholm operators and the boundary of the maximal group, :.3.'i.S. 67, 1977.
91.
Vala, K., On compact sets of compact operators, Ann. Acad. Sci. Penn. AI, 1964.
92.
Vala, K., Sur les elements compacts d'ure algehre normee, %nr. Acad. Sci. Fenn. AI, 1968.
93.
Veselic, K., On essential spectra in Banach algebras, Glasnrk Mat. 10 (3), 1975.
116
94.
West, T.T., 2iesz operators in Banach spaces, P.L.M.S. (3) 16, 1966.
95.
West, T.T., The decomposition of Riesz operators, P.L.M.S.
96.
Wong, P.K., Modular annihilator A*-algebras, Pac. J. Math. 37, 1971.
97.
Yang, K.-W., The generalised Fredholm operators, T.A.M.S. 216, 1976.
98.
Yang, K.-W., Operators invertible modulo the weakly compact operators, Pacific J. riath. 71, 1977.
99.
Ylinen, K., Compact and finite dimensional elements of nonmed algebras, Ann. Acad. Sci. Tenn. AI, 1968.
(3)
16, 1966.
100. Ylinen, K., A note on the compact elements of C*-algebras, F.A.N.S. 35, 1972. 101. Ylinen, K., Weakly completely continuous elements of C*-algebras, P.A.M.S. 52, 1975.
102. Yood, B., Difference algebras of linear transformations on a Banach space, Pacific J. Math. 4, 1954. 103. Yood, B., Homomorphisms on normed algebras, Pacific J. Math. 8, 1958. 104.
Zemdnek, J., A note on the radical of a Banach algebra, Manuscripta Math. 20, 1977.
SUPPLEMENTARY
BIBLIOGRAPH"
105. Kirchberg, E., Banach algebras whose elements have at most countable spectra, (preprint) Akad. '7iss. DDR, Zentralinstitrt Math. Mech. 1979. 106
Harte, R.E., Fredholm thecry relative tr a Banach algebra homomorphism, Math. Zeit. 179, 1982.
107.
Zemanek, J., The essential spectral radius and the Riesz part of the spectrum, Functions, Series, Operators (Proc. Irternat Cont., Btdapest, 1980) Colloq. Math. Janos Boylai (to appear).
108.
?enanek, _., ,eneralisaticr_s of the spectral radius formula, P.P.I.A.
81(A), 1981. 109.
Kral3evi6, H., Veselic, K., On algebraic and spectrally finite Banach algebras, Glasnik Mat. 11 (31),1976.
110.
Kral3evic, H., Suljagic, S., 7eseli6, K., Index in semisimple Banach algebras, preprint Dept. of Math. Univ. of Zagreb 1980
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-hyponormal 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
decomposition algebra
83
defect of element
F.2.7, F.3.6
defect function
F.3.5
defect of operator
o.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
finite rank element finite rank operator generalised 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
ideal of finite order
F.4.1
index in algebra
F.4.12
index of element
F.2.7, F.3.6
index equivalence
F.4.13
index function
F.3.5
index of operator
0.2.5
index theorem
F.2.9, P.3.8
inessential element
F.3.1
inessential ideal
F.3.12
inessential operator
8
inverse semigroup
49
irreducible representation
101
kernel
102
K-Fredholm element
F.3.12
left completely continuous Banach algebra R.4.4 minimal ideal
F.1.2
minimal idempotent
F.l.l
modular annihilator algebra
68
nullity of element
F.2.7, F.3.6
nullity function
F.3.5
nullity of operator
0.2.5
order of ideal
F.4.1
orthogonal idempotents
F.1.5
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
quasitriangular operator
94
radical
102
rank of element
F.2.3
relatively regular operator
48
representation
101
Riesz algebra
R.3.1
Riesz element
R.1.1
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 Smyth characterisation socle
spectral projection spectral set spectrally finite Banach algebra Stampfli decomposition 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(r)
102
B(x)
1
00
(x)
0.2.1
18
lan (x)
F.1.8
0 (x)
R.2.1
L
62
C
1
CX(E)
15
dim (X)
29
d(T)
0.2.5
def (x)
F.2.7, i .3.6
d (x)
F.3.5
a(c)
100
11(x, s) H
B1
x
L.C.C.
R . 4.4
m( X)
0.2.1
Min(A)
F.1.1
M
96
n (T)
0.2.5
nul(x)
F
.
2
.
7,
v (x)
F.3.5
8
ord(J)
F.4.1
1
P (w,T) , P(X,T)
2
H (A)
77
psoo(A)
F.3.1
Hol(a)
2
P
94
h (V)
102
IT (A)
102
i (T) , ix (T)
0.2.5
fl* (A)
107
ind(x)
F.2.7, F.3.6
Q (X)
1
1Cx)
F.3.5
q(B)
0.3.1
94
2A
94
r(T)
2
P (T)
2
r (x)
101
Q (T)
I (x)
1
I (A)
F.3.1
int (S2)
110
Inv(B(X))
1
P(x)
,
p A (x)
100
Inv (A)
100
R (x)
2
K(x)
1
R(A), RK(A)
R.1.1
K
F.3.1
rad(A)
102
-A
86
F .3
.E
62
R
x
IS, 21 [x , y]
12
53
R.C.C.
64
ran(x)
TA T
17
rank (x)
F.1.8 F.2.3
SAT
20
soc(A)
23
XAX
70
Q (T)
2
ot 41 x
2
0(x), clA(x.)
100
Q(A)
104
E (C)
96
T*
2
T[Y
2
TC
93
T(G)
96
T
97
u
T (T)
15
U
8
W(x)
R.2.1
to (T)
ao(x)
0.2.4 R.2.1
x
1
X*
2
x'
35
B,
7G
2
Z (T)
15
Z(x)
62
Z2(x)
66
$ {X)
1
e (X)
4
(T)
0.2.5
(A)
F.2.5, F.3.2
0K (A)
F . ,, .12
TF (A)
46
ABOUT THIS VOLUME
In this Research Note the authors exploit the relationship between algebra and spectral theory, in order to set into the context of Banach and C*-algebras the classical Riesz and Fredholm theory of bounded linear operators on Banach and Hilbert spaces. Fredholm theory is developed in primitive Banach algebras, and then extended to the general case, while Riesz theory follows as a consequence. Hilbert space results 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 NOTES IN MATHEMATICS
The aim of this series is to disseminate important new material of a specialist nature in economic form. 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 chosen accordingly and will from time to time be recomposed to represent the full diversity of mathematics as covered by Mathematical Reviews. This is a rapid means of publication for current material whose style of exposition is that of a developing subject. Work that is in most respects final and definitive, but not yet refined into a formal monograph, will also be considered for a place in the series. Normally 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. Main Editors
A. Jeffrey, University of Newcastle-upon-Tyne R. G. Douglas, State University of New York at Stony Brook Editorial Board F. F. Bonsall, University of Edinburgh H. Brezis, Universite de Paris G. Fichera, Universite di Roma R. P. Gilbert, University of Delaware K. Kirchgassner, Universitat Stuttgart R. E. Meyer, University of Wisconsin-Madison J. Nitsche. Universitat Freiburg L. E. Payne, Cornell University I. N. Stewart, University of Warwick S. J. Taylor, University of Liverpool ISBN 0 273 08563 8
