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London Mathematical Society Lecture Note Series.
63
Continuous Semigroups in Banach Algebras
ALLAN M. SINCLAIR Reader in Mathematics University of Edinburgh
CAMBRIDGE UNIVERSITY PRESS CAMBRIDGE
LONDON
NEW YORK
MELBOURNE
SYDNEY
NEW ROCHELLE
CAMBRIDGE UNIVERSITY PRESS Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, Sao Paulo
Cambridge University Press The Edinburgh Building, Cambridge C132 8RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521285988
© Cambridge University Press 1982
This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 1982 Re-issued in this digitally printed version 2007
A catalogue record for this publication is available from the British Library Library of Congress Catalogue Card Number: 81-2162 7 ISBN 978-0-521-28598-8 paperback
CONTENTS
1.
Introduction and preliminaries
2.
Analytic semigroups in particular Banach algebras
12
3.
Existence of analytic semigroups - an extension of Cohen's factorization method
35
4.
Proof of the existence of analytic semigroups
50
5.
Restrictions on the growth of
70
6.
Nilpotent semigroups and proper closed ideals
1
jatll
Appendix 1. The Ahlfors-Heins theorem
91
111 I
Appendix 2. Allan's theorem - closed ideals in L ( R+,w)
131
Appendix 3. Quasicentral bounded approximate identities
134
References
138
Index
143
1
1
1.1
INTRODUCTION AND PRELIMINARIES
INTRODUCTION
The theory of analytic (one parameter) semigroups
t F* at
from the open right half plane H into a Banach algebra is the main topic discussed in these notes. Several concrete elementary classical examples of such semigroups are defined, a general method of constructing such semigroups in a Banach algebra with a bounded approximate identity is given, and then relationships between the semigroup and the algebra are investigated. These notes form small sections in the theory of (one parameter) continuous semigroups and in the general theory of Banach algebras. They emphasize an approach that is standard to neither of these subjects. A study of Hille and Phillips [1974] reveals that the theory of Banach algebras has been used as a tool in the study of certain problems in continuous semigroups, but that semigroup theory has until recently (1979) not impinged on the theory of Banach algebras. These lecture notes are about this recent progress.
Throughout these notes we use 'semigroup' for
one parameter
semigroup' when discussing a homomorphism from an additive subsemigroup of
Q into a Banach algebra, and we write our semigroups the power law
at+s =
at. as
t F-* at
to emphasize
and function property of the semigroup. In
the standard works on semigroups much attention is given to strongly continuous semigroups and their generators (see Hille and Phillips [1974], Dunford and Schwartz [1958], and Reed and Simon [1972]). In these works the generator itself is important, plays a fundamental role, and is often an object of considerable mathematical interest (for example, it may be the Laplacian). As the theory is developed here the generator is useful only in Chapter 6, and even there it is the resolvent
(1 - R)-1, not the
generator R, that occurs in our Banach algebra results. It is possible in the Banach algebra situation to develop lemmas corresponding to the HilleYoshida Theorem totally avoiding unbounded closed operators and working
2
with what is essentially the inverse of the generator. This seemed artificial and we do not do it here. In the standard works on semigroups (ibid.) most of the emphasis is on semigroups that are not quasinilpotent, and there is little or no space devoted to quasinilpotent semigroups (see Hille and Phillips [1974], p.481). However Chapters 5 and 6 of these notes concern radical Banach algebras, perhaps indirectly. In these radical algebras we are studying quasinilpotent semigroups. The general theory of Banach algebras has mostly been developed for (Jacobson) semisimple algebras, and the most studied families of Banach algebras are semisimple: C*-algebras, group algebras, and uniform algebras. A brief glance through the standard references (Rickart [1960] and Bonsall and Duncan [1973]) illustrates this. Radical algebras and quasinilpotent elements play a very important role in Chapters 5 and 6 of these notes. However we do not attempt a study of radical Banach algebras or even discuss the role of non-continuous semigroups in the classification of radical Banach algebras. Various weaker assumptions on the domain of a semigroup
t f at,
for example, to the rational numbers, are related to
the structure of certain radical Banach algebras (see Esterle [1980b]). Strongly continuous (one parameter) groups of automorphisms on a C*algebra are fundamental in C*-algebra theory (see Pedersen [1979]). Except for this there had been few applications of semigroup theory to Banach algebras until 1979.
The standard references on the theory of semigroups (Hille and Phillips [1974], Dunford and Schwartz [1958], and Reed and Simon [1972]) contain much of Chapter 2 and the Hille-Yoshida Theorem of Chapter 6. The approach here is also basically different from that in Butzer and Berhens [1967], and Berge and Forst [1975]. The modification of the Cohen factorization theorem discussed in Chapters 3 and 4 is covered in considerable detail in Doran's and Wichman's lecture notes [1979] on bounded approximate identities and Cohen factorization. Even here our account differs from the original version, which is what they give. These notes are elementary and the results are proved in detail. As background for the main results we assume standard elementary functional analysis, the complex analysis in Real and Complex Analysis by Rudin [1966], and the Banach algebra theory in Complete Normed Algebras by Bonsall and Duncan [1973]. We shall use the Titchmarsh convolution theorem (see Mikusinski [1959], Chapter 2) a couple of times. In a few corollaries and applications considerably more is assumed (for example, there are
3
results applying to
L1(G)).
standard functions in
Ll(]R)
Calculations are given in detail even when are being considered. The main tools in our
proofs are techniques from Banach algebra theory and semigroup theory, the Bochner integral, and some classical results of complex analysis. Although the Hille-Yoshida and Ahlfors-Heins Theorems are standard results readily available in books, they are not in the assumed background and so they are proved in suitable forms in these notes (Theorem 6.7 and Appendix A1.1). In the introduction the Bochner integral is briefly discussed. The notes are not polished. Each chapter beyond the first ends with notes and remarks where brief reference will be made to the literature, related results, and open problems. The bibliography is not comprehensive. These notes are an expanded and revised version of lectures that I gave at the University of Edinburgh in January, February, and March 1980. The lectures and notes were both influenced by a course that J. Esterle gave in the University of California, Los Angeles, in April, May, and June 1979. Some parts of my lectures appear as they were given, others have been extensively revised, and occasionally a single verbal remark in a lecture has become a whole section here. The concrete semigroups in L1(IR) and Ll(]R °) were covered as here (Chapter 2) as was the Wiener
Tauberian Theorem, (Theorem 5.6), Theorem 5.3, and the whole of Chapter 6. Chapters 3 and 4 were a single unproved result in lectures, but several of the audience had suffered talks from me on these subjects in a seminar. I am grateful to many mathematicians for preprints and odd half forgotten conversations, which have influenced the development, and to the audience who survived my lectures. I am grateful to P.C. Curtis, Jr. and F.F. Bonsall for encouragement, to T.A. Gillespie for useful criticism of an early draft, to S. Grabiner for many discussions about Banach algebras, and to A.M. Davie for suggesting several improvements to results and proofs. H.G. Dales read the complete notes, and his detailed and careful criticism has enabled me to correct several errors and improve the notes. I am indebted to him for this and other suggestions. During 1978-9 J. Esterle and I had many discussions about radical Banach algebras and semigroups, and his U.C.L.A. lectures and seminars influenced my ideas. He has kindly given permission for me to include his results on nilpotent semigroups in Chapter 6 before he has published them. I am very grateful and deeply indebted to J. Esterle. Without his results in Chapters 5 and 6 these notes would not exist.
4
DEFINITIONS AND NOTATION
1.2
We shall now give some definitions, fix various notations, and prove a couple of useful little lemmas. Throughout these notes we shall consider complex Banach spaces and Banach algebras, and linear operators
will be taken to be complex linear. The Banach algebras will not be assumed to have an identity, and these notes deal mainly with algebras without identity. If A is a Banach algebra, then
A $ C 1
is the Banach
algebra, obtained from A by formally adjoining an identity; note that the
for all
norm is Ila+AII =flat! + IAI Banach algebra with identity identity
A# = A 0 C 1.
A# = A,
and As C. If A is a
aEA
and if A is an algebra without
The algebra A# is the algebra in which the x E A is de-
spectra of elements of A are calculated. The spectrum of noted by
a(x)
and the spectral radius by
v(x).
If f is a function from a set X into a set Y, we shall often write
x F> f(x)
:
If X is a Banach space,
X - Y.
denotes the
BL(X)
Banach algebra of bounded linear operators on X. For a commutative Banach algebra A the multiplier algebra such that
T E BL(A)
is defined to be the set of
Mul(A)
T(ax) = a T(x)
for all
Clearly
x,a E A.
Mul(A)
is a unital Banach algebra, and there is a natural norm reducing homo-
morphism Q
a I* L
: A + Mul(A),
a
where
L
a
x = ax
be a locally compact Hausdorff space and let
algebra of continuous complex valued functions on infinity. Then
is isomorphic to
Co(0)#
the one point compactification of C(gQ),
where
$52
St,
for all
be the Banach
Co(S2)
0
vanishing at
C(Qu{-}), where
and
Let
x E A.
52u{°}
is
is isomorphic to
Mul(C0(S2))
is the Stone-Cech compactification of Q.
Most of the Banach algebras we study have bounded approximate
identities. A Banach algebra A has a bounded approximate identity bounded by set
F c A
d
if
If II
and each
for all
<_ d
c > 0,
Ilea - all + Ilae - all
< c
f E A,
there is an
tive,...) bounded approximate identity. If A.
e e A
such that
for all a E F. If the set A can be chosen to
be countable (commutative,...), we say that
shall suppress
A
and if, for each finite sub-
A has a countable (commutaA = {a E A
:
we
halt<- d)
In Chapter 3 the countability of the bounded approxim-
ate identity is important. Here we note a couple of folklore facts which indicate that this hypothesis is not too restrictive for our purposes.
If a Banach algebra A imate identity, then
is separable and has a bounded approx-
A has a countable bounded approximate identity.
This can be seen by choosing a countable dense subset
{yn}
of
A,
and
5
then choosing a sequence
that
from the bounded approximate identity such
(en)
for 1 < j
enY j - yj II + IlYjen - yj 11 < n-1
II
and all n. The set
<_ n
is a countable bounded approximate identity in
{en:n a I1}
A.
If A is a Banach algebra with a bounded approximate identity and is a separable subspace of
if Y
algebra Let
of A
B
that contains
A,
then there is a separable Banach sub-
Y
and has a bounded approximate identity. Y
be a countable dense subset of
{yn}
II
ny j - yj II + IlYjen - yjll < n
for
1
<-
<_ n - 1
j
generated by
and all
n.
1
and choose a sequence
Ilenej - ejll + Ilejen - ejll < n-1
and
The Banach subalgebra
B
of A
has the required properties.
Y u{e :n E IN) n
If a commutative Banach algebra
A has a bounded approximate
identity bounded by 1, then the natural homomorphism
from A
a I* La : A -> Mul(A)
into the multiplier algebra is an isometric embedding from
a closed ideal in
(en)
A such that
from the bounded approximate identity of
A onto
Mul(A).
The complex numbers, real numbers, integers, and positive integers are denoted by fi, ]R, 7, and t, respectively. The open right half
plane
{zed: Re z > o}
is denoted by H, and the closed right half plane by
H . The reader will be reminded of this notation periodically.
A function space
X
f
from an open subset
is said to be analytic if for each
lim h-1(f(z+h) - f(z)) exists in h-o
X.
U
of
z e U
C
into a Banach
the limit
This limit is denoted by
(Df)(z).
The Hahn-Banach Theorem may be combined with results of complex analysis to yield results about analytic functions into a Banach space. The following result illustrates this technique and shows why it is not necessary to consider separately semigroups which are weakly or strongly analytic.
1.3
LEMMA Let
f
be a function from an open subset
plane into a Banach space
X.
Then conditions (a),
equivalent. (a)
f
(b)
Ff
(c)
f
is analytic.
is analytic for all
F E X
is continuous, and
f(z) = 1 fy fE) 2ni
(&-z)-1
d
U
of the complex
(b), and (c) on
f
are
6
for each z E U and each closed path such that the winding number of respect to
in
y
U
with
z
is 1 and of each point in
y
C \ U is O. for a Banach space
X = BL(Y)
If
(d)
then
Y,
the above conditions are equivalent to z J* F(f(z)y)
:
being analytic for all
C
U
y E Y and F E Y Proof
Clearly (a) implies (b) and (d). z, z
If (b) holds and if F((z
n
- z)-1(f(z
n
n
c U with
converges in
- f(z)))
)
T
boundedness theorem implies that the sequence is bounded. Thus
converges in
is continuous on
f
U.
z
z,
n
for each
then *
F E X
The uniform
.
(II(zn - z)-1(f(zn) - f(z))II)
The integral in (c) now
(see 1.6), and using the classical Cauchy integral formula
X
for a complex valued analytic function we obtain
F(f(z)) = 1
y (Ff)(E)( - z)-1 dE
2-71-
=
j
F(127
dal f(S)(E - z)1 d)
fy
*
for all
F E X
.
An application of the Hahn-Banach Theorem gives the
equality of (c).
Now suppose that (c) holds. Let circle with centre
contained in Ihi
< r/2,
U.
z
and radius
z E U
so that
r
Let M = sup {11f(E)II
y
E y}.
and let
f
If h E T with
z)-2 dE
27ii fy
= h2
2TIi
so that
ly f(E) 11
be a small
and its interior are
then
f(z + h) - f(z) - h
y
(E - z)-2 (E - z - h)-1 dE
7
fO(- z)2 dEII
If(z + h) - f(z) - h
2ri Ihi2
2irr
.
This shows that
. 1
. 1
2ir
J
r
has derivative
f
. M.
r/2
2
1
(E - z)-2 d at z,
jY
2,ri as we would expect from classical complex analysis. Hence (a) holds. Suppose that (d) holds. Using the equivalence of (a) and (b) we find that
z I* f(z)y
:
is analytic for each y E Y.
U -> Y
of the uniform boundedness theorem as above shows that on
An application is continuous
f
The integral in (c) now exists, and the equality in (c) is
U.
obtained by an application of the Hahn-Banach Theorem. The proof is complete.
A (one-parameter) semigroup in a Banach algebra A from an additive subsemigroup of
t N at
function
right half line
(O,-)
into A
T
containing the open
at+r = at.ar
satisfying
is a
for all
t,r
in the domain of definition. We shall be concerned mainly with semigroups defined on the open right half plane
and on the open right half line
H
The semigroup is said to be analytic or continuous (in some topo-
(O,").
logy on
A)
if the function is analytic or continuous. Lemma 1.3 shows
why we restrict attention to the norm topology when considering analytic semigroups defined on semigroup if
1.4
IlatII<_
A semigroup
H. 1
for all
LEMMA Let
t I* at
= (A a1)
Proof.
Let
be an analytic semigroup from the open
H -> A
:
for all
F E A
with
Re t > 1
for all (at A)
t E H
because and
c(a1 A)-,
A.
Then
(at A)
1
A) = {O}.
x E A.
Then
t I+ F(at x)
:
H - C
This function is zero for all
Flat X) = F(a1 at-1 x) = O.
X E A.
and
= (a1 A)
t E H.
F(a
analytic function for each with
is said to be a contraction
t.
right half plane into a Banach algebra (A at)
t I* at
Hence
is an
t E T
Flat X)
= 0
By the Hahn-Banach Theorem it follows that
and a similar argument yields the reverse inclusion.
Semigroups
t I; a t
:
(O,") - A
that are not analytic may have
8
strictly decreasing, and Chapter 6 is devoted to the study of such
(at A)
semigroups. However even there our aim is to study continuous rather than strongly continuous semigroups.
1.5
AN OUTLINE OF THE NOTES This section is a synopsis of the notes. In Chapter 2 examples
of analytic semigroups from the open right half plane into several concrete Banach algebras are discussed. These semigroups will suggest abstract properties to be investigated in Chapter 3 and will provide examples for the results and proofs of Chapter 5. The structure of C -algebras is very *
rich, and this makes it easy to construct analytic semigroups in C algebras by using the commutative Gelfand-Naimark Theorem to define for a positive element in the algebra and
t
in
H.
at
However these
*
semigroups in C -algebras are not useful tools to study the algebra. *
Semigroups in various convolution algebras are more interesting than in C algebras and throw more light on the structure of the algebra. The fractional integral semigroup t the backward heat semigroup
Ct()
t(2Tr/)- 1 w-3/2
=
semigroups into
L1(R+)
It
where
t f Ct,
where
It(w)
wt-1
= P(t)
and
e-w,
exp(-t/4w), 2 are given in detail as examples of (see Theorem 2.6, and Lemma 2.9).
The Banach algebra
L1(R+)
is very important in the study of
semigroups in Banach algebras for the following reason. If t N at
:
algebra L1(R+)
(O,-) - A
is a continuous contraction semigroup into a Banach
then there is a natural norm reducing homomorphism
A,
into
A
defined by
0(f) = f-0 f(t)at dt.
Bochner integral (1.6), the norm reducing property from
Ilat1I5 1
for all
t > 0,
The integral is a <-
(IIOII
L1(R+)
0
follows
may be used to
into analytic semigroups in
image semigroups are called subordinate to the semigroup Gaussian semigroup
Gt(w) = (4Trt)-n/2 exp (-Iwl2/4t)
and Poisson semigroup
Pt(w) = r((n + 1)/2). t (n + 1)/2 (t2 + 7T
1)
and the homomorphism property follows by
a change in the order of integration. The homomorphism map analytic semigroups in
from
0
2) (n + 1)/2 IWI
t F* at.
A.
The The
9
in
Ll Otn)
are discussed, and the Poisson semigroup is studied via its
subordination to the Gaussian semigroup. Certain growth properties of IIGtII1
are substantially better than those of
properties of
JIG tIll
IIIt111 These strong growth
are crucial in the proof of the wiener Tauberian
Theorem (Theorem 5.6).
Chapter 3 contains a theorem that gives the existence of analytic semigroups in a Banach algebra with a countable bounded approximate identity. The result is proved by modifying the proof of Cohen's factorization theorem so that the proof resembles the way in which a strongly continuous semigroup is generated in the Hille-Yoshida Theorem (see 6.7). The semigroup constructed in Chapter 3 has growth and structure more like the fractional integral semigroup than the Gaussian semigroup. The general semigroup result is applied to the group algebra
of a metrizable
L1(G)
locally compact group, and to obtain commutative bounded approximate identities in Banach algebras with countable bounded approximate identities. The proof of the main result in Chapter 3 and the lemmas required in the proof fill Chapter 4. Two of the properties (6 and 15) of
Theorem 3.1 I have not been able to prove by the exponential methods of Chapter 4. These two properties require the factorization results
developed in Sinclair [1979a], although I believe they may be obtained from exponential calculations. I have attempted to prove the most general factorization result for semigroups that I know. In other chapters generality has often been sacrificed to obtain an elementary account. The properties of the semigroups near the boundaries of their domains of definition are interesting, and are closely related to the fine structure of the semigroup and the algebra. In Chapter 5 we investigate
A by the assump-
the restrictions imposed on a commutative Banach algebra tion that it contains an analytic semigroup (a1 A)
= A
such that the growth of
Ilat11
H + A
t I; at :
with
is suitably restricted. The
restrictions we consider are: (i) to the growth of
II atll
along rays in
H
emanating from 0 (5.2); (ii) to the growth of
t II a
II
along a vertical line (5.5);
(iii) the boundedness in the semidisc
{z E H:lzl<_l}
In each of these cases we prove a result due to Esterle
:
(5.12).
these are,
respectively, a result about radical Banach algebras, a Tauberian theorem, and a result on the non-separability of the multiplier algebra. The
philosophy underlying the proofs in the first two cases is to define a
10
suitable classical analytic function
(by using the analyticity of the
F
semigroup and a continuous linear functional on the algebra), and to apply the Ahlfors-Heins Theorem to
F.
Chapter 6 begins with a standard account of the Hille-Yoshida
Theorem relating a strongly continuous contraction semigroup
Banach space with the closed operator R
on a
at
that is its infinitesimal genera-
tor. The relationships between the nilpotency of the semigroup and the growth of
(1 - R)-n11
II
n
as
tends to infinity is investigated. This
result is applied to a hyperinvariant subspace theorem for suitable
operators on a Banach space, and to prove the existence of a proper closed ideal in a commutative radical Banach algebra containing a non-zero element
such that
u
unII 1/n {nII
:
1j u (A - u)-l II
<-
1
and
A > 0
for all
is bounded. This process is seen to give abstractly
n E IN}
the obvious ideals in the Volterra convolution algebra
L* [0,1].
In the appendix we give a proof of a special case of the
Ahlfors-Heins Theorem, and prove a theorem of G.R. Allan [1979] on certain closed ideals of
for w a radical weight. It is also shown that
L1 (R+,w)
an Arens regular Banach algebra with a bounded approximate identity has a bounded approximate identity that is well behaved with respect to derivations (and is quasicentral).
INTEGRALS
1.6
We shall frequently integrate continuous functions on
(0,°°)
with values in a Banach space, and in this section we briefly define the elementary integrals used. There are extensive discussions of the integra-
tion of Banach space valued functions in Dunford and Schwartz [1958] and in Hille and Phillips [1974]. Let
space
f
with
X
be a continuous function from finite.
IIf(w)II dw
For
into a Banach
(o,-)
o < a < $ < °°
we define
J-
f(w)dw
as the limit of Riemann sums using partitions
a = a° < al < ...
< an = S with max
{a . 7
to zero. The limit may be shown to exist in
- a j-1 X
1 <_
:
j
<_ n}
tending
in the same way that the
classical Riemann integral is shown to exist by using the uniform continuity
of the integrand. Further the integral JBf(w)dw a s f, and to satisfy I J f (w) dw I <_ f'11 f (w) I I dw
is seen to be linear in
I
a
fa
and
f(w)dw) =
F(
a
(Ff)(w)dw
I
for all
F E X
.
Using the
J
a
S
a
observation
that
r
+
IIf(w)Ildw 1
a
tends to zero as
a
tends to
11
tends to infinity, we define
zero and
f
fo f
to be the limit of
(w)dw
fo-
laf(w)dw
.
t f b(t) II b (t) II
<M
from the space
Jg(t) b(t) dt
(Co(0,'), II.II1)
0
and
-
of continuous complex valued functions with the II.II1-norm into C (O,-) o
may be defined for all
by continuous functions. Further
F( lo g(t) b(t) dt) = Io g(t) *
(Fb)
11
in L1QR+)
g e L1OR+)
X
the integral
by approximating
I og (t) b (t) dt I I
M
<_
(t) dt for all g e
,
IIgII1
g
and
and all
L1OR+)
where the last integral is a Lebesgue integral. If :
t
is a strongly continuous function from
-+ BL(X)
(0,00)
into the Banach space of bounded linear operators on
X
with
ra.
finite, then we define
T(t)dt) x
T(t)dt
T(t) x dt
T(t) dt
11
5
f all
as an operator in
for all
fo
f'0
is
JJJ
t I* T(t)
II
Let
--> X
continuous. Using the density of
o
.
be a continuous function into X with for all t > O. The linear operator g F' log (t) b (t) dt
(O,-)
on (O,-) vanishing at
F e X
fB
satisfies similar properties to
The integral
T(t) II dt.
x e X.
BL(X)
Note that
by
(O,°°)
f'011 T(t)II
dt
12
2
2.1
ANALYTIC SEMIGROUPS IN PARTICULAR BANACH ALGEBRAS
INTRODUCTION In this chapter we introduce various well known semigroups from
the open right half plane
H
into particular Banach algebras. We discuss
the power semigroups in a separable C -algebra, the fractional integral and backwards heat semigroups in groups in
L1(Rn).
L1
+), and the Gaussian and Poisson semi-
While doing this we shall develop notation that is used
in subsequent chapters. The discussion is very detailed throughout the chapter, and is designed to introduce and motivate following chapters dealing with more abstract results for analytic semigroups. For example we are concerned with the asymptotic behaviour of
IIal + iyII
as
tends to
lyl
infinity, but not with the infinitesimal generators of our semigroups even though they are important. We shall discuss generators in a different context in Chapter 6. *
2.2
C -ALGEBRAS The functional calculus for a positive hermitian element in a
*
C -algebra that is derived from the commutative Gelfand-Naimark Theorem *
enables us to construct very well behaved semigroups in C -algebras. We *
shall briefly discuss the case of a commutative C -algebra before we state *
and prove our main result on semigroups in a C -algebra. The commutative Gelfand-Naimark Theorem (see, for example, Bonsall and Duncan [19737) *
enables us to identify the commutative C -algebra with
which is
C (C),
*
0
the C -algebra of continuous complex valued functions vanishing at infinity on on the locally compact Hausdorff space C. C
(0)
It is easy to check that
has a countable bounded approximate identity if and only if
C
is
0
a-compact (that is,
is a countable union of compact subsets of itself).
C
By using a countable bounded approximate identity in the a-compactness of 1
>_ f(o) > 0
C,
an
f a C (C)
C (C), 0
or by using
may be constructed so that
for all 0 e C. The analytic semigroup t F' f t
:
H -> C
0
(S2)
13
t
is given by defining
2.3
f
W = f(¢)t
for all
0 e
S2
and
t E H.
THEOREM *
A C -algebra A has a countable bounded approximate identity if and only if there is an analytic semigroup (atA)
= A = (Aat)
t > o,
and
Proof. 1/n
If
{a
and
Ilatll
I I atx - x l l + I I xat - x I
t }* at
for all
1
<_
at 2 0
+ o as t -> o in
I
such that
H -r A
:
t e H,
for all
for all
H
x E A.
contains a semigroup with the required properties, then
A
is a countable bounded approximate identity in
n E IN}
{g
Conversely suppose that
n
*
To show that {e n n E IN} en = gngn is a bounded approximate identity in A, it is sufficient to show that. identity in
A.
For each
A.
is a countable bounded approximate
n e N}
:
n
let
:
.
Ilx(e n - 1)II
o
II (en - 1)xlI
=
as* n - for II x (en -l)11 .
all
x e A
because
Now
Ilx(e n - 1) II
II x(gn - 1) II
5
5 11x(gn - 1) 11
+
II xgn(gn - 1) II
+ 11 x(gn
1)
(IkJnlI
+ 1) + 11 x(gn
II x(gn - 1) II = II(gn - 1)x II for all n E IN and x E A. Hence llx(en - 1)11 tends to 0 as N tends to infinity. Let a = 4j=1 e2 2-I II ej II -2. Then 0 5 a and II a II 5 1. and
*
We apply the Gelfand-Naimark Theorem for a commutative C -algebra to the *
C -algebra generated by 0
from the C-algebra
algebra generated by zt(w)
at
= wt
for all
a.
This gives a norm reducing *-homomorphism
{f E C[O,1] a
with
w E [0,1]
and all
and properties of semigroups in
is an analytic semigroup such that
for all t > 0, Ilenat - enll
and
f(O) = O}
0(z) = a.
C, Ilat1I
onto the commutative C
We take
at = 0(zt),
From the definition of
t E H.
we observe that
where
tends to zero as
t
A
H
II ata - all- 0 as t + 0, t E H. If we show that t
tends to zero,
t e H,
have completed the proof for the following reason. If t e H
t f+ at :
t E H, at t 0
for all
5 1
where
is the complex conjugate of
Ilaten - en 11 = II en (at) * - en 11
= 11
enat - en II
t
and
.
Also
then we shall
t E H,
{en
:
then
n E IN}
is a
14
bounded approximate identity for and
{Ae
n
tends to zero as
Ilen at - e nil
:
n E IN}
To prove that
A.
tends to zero,
t
{enA
Thus the closures of
A.
are both equal to
n E IN}
:
we require the
t E H,
following standard little lemma on C -algebras.
LEMMA
2.4
x, y, b
Let
If
Proof.
If
Ilxb II
sup{f(c)
:
*
f 2 0, IIfll s 1}, we have >
II
(xb)
then
A,
0 <_ y.
* f(b zb)
z
* * * f(b z zb) = f((zb) zb)
since
for all positive linear functionals
f(b*y2b) ? f(b*x2b)
yb I I
and
0 <- x
II yb II
<_
Since the norm of a positive element
A.
(yb)
A with
be in a C -algebra
is a positive linear functional on
f
Hence
0.
on
then
is a positive linear functional on A
A -> C
II
y2,
0 <- x2
*
f
*
in a C -algebra A is
IIb*y2biI ? Il b*x2bit
Il ybll2 =
so
This proves the lemma.
.
xb II = I1 xb I
c
*
We apply the lemma in the C -algebra obtained by adjoining an
identity to Then
A.
I l en (1 - at) 11
a1/2. and y =
b = 1 - at, x = en.2-n'2Ilenll-l,
Let
<_ 2n/211 en l . l
11 all - at) I I
for all t E H, and the
proof is complete. *
The proof of Theorem 2.3 shows that a C -algebra with a countable bounded approximate identity has a commutative bounded approximate identity.
THE CONVOLUTION ALGEBRA
2.5
L1OR+)
In Corollary 3.5 we shall see that there is a norm reducing homomorphism
8
into a Banach algebra
from L1OR+)
A with a countable
bounded approximate identity bounded by 1. Under this homomorphism (analytic) semigroups in A.
L1 OR+)
are mapped into (analytic)semigroups in the algebra
This is an old idea first systematically exploited by Bochner [19551
and called the subordination of one semigroup to another. We shall subsequently use the homomorphism
8
from
The growth estimates on semigroups in groups in
L1CR+)
L1OR+)
into
A
several times.
will hold for some semi-
A.
Recall that of) integrable functions
L1OR+) f
With the convolution product
is the Banach space of (equivalence classes
on 3R+ = [o,-)
with norm
If(w)Idw.
llfIll =
fiz+
15
It
(f*g) (t) = 1
f (t - w) g (w) dw
0 defined almost everywhere,
is a commutative Banach algebra without
L1 O2+)
identity. The Banach algebra
is semisimple and its carrier space
L1(R+)
may be identified with the closed right half plane
(D
A }r mX
I + )R
OX (f) = for all
Laplace transform n f(OX) _
f (w) e-aw dw
where
L,
:
{e
because
+ f(w)e-Xw dw,
-
is continuous on
Lf
H
has a countable bounded approximate
L1(R+)
for example,
is just the
L1OR+) -' Co((D)
R
n
:
where
n E IN},
for O <_ w <_ 1/n
n
e n (w) =
:
= J
(Lf)(A)
The algebra
H.
identity bounded by 1
f F'
The Laplace transform
X(f) = (Lf)(X).
and analytic in
A
The Gelfand map
f e L1 Ot+).
by the mapping
H
X is defined by
where
(D
I
The Titchmarsh convolution theorem (or properties of analytic functions via the Laplace transform) implies that with
f, g e L1OR+)
if
algebra Mul(L1 O2+))
f*g = 0
of
L1(R+)
is an integral domain; that is,
L1OR +)
f = 0
then
or
is naturally isometrically isomorphic
with the convolution measure algebra
on R+.
MOR+)
Banach space of bounded regular Bonel measures h
u
hI
f
XE (w+u) dp(w) dv(u),
function of the Borel set Mul(L1 O2+))
Tpf = p*f
p
Here
is the
MQR+)
on R
with norm
and product p * v defined by
= I u I (R+)(r
(p*v)(E) =
The multiplier
g = O.
is defined by
E.
where
XE
is the characteristic
The isometric isomorphism from
p F' T11
:
MaRt) - Mul(L1CR+)),
M(R+)
onto
where
(see Johnson [1964]).
Recall that the Gamma function
r(t) =
r
is defined by
Jwt_l e-w dw 0
for all
t e H,
F(t) =
and that asymptotically exp(-t + O(Ith-1))
as
ItI
-'
for all
Erdelyi [1953, p.21] or Olver [1974, p.294]). Further H, r(t+l) = tr(t)
for all
t E H, P(n+l) = n!
for all
F
t E H
(see
is analytic in n E IN,
and
16
r(1/2) = Tr
1/2
2.6
THEOREM if
all
w e (O,-)
It a L1 pt+)
and all
is defined by then
t e H,
= wt-1 e -w r(t)-1
It(w)
t N It
H # Ll Ct+)
for
is an analytic
semigroup, called the fractional integral semigroup, with the following properties. (i)
(ii)
(It *
L 1
r(x+iy)
1 (iii)
(LIt)(z) = (z + 1)-t
for all
a(It) = {O} u {(Z+1)-t (iv)
If
x+iy a H,
where y,
z e H-1
:
K(x) (l + y2x-2)
and all
for all
2
1
4. expr
+ O(IYI-1) l
Z
)
is independent of
0
and
t e H,
t e H.
is a constant depending on
K(x)
and
z e H
y e Il2.
then
-x IIIx+iylll=
t e H.
for all x > 0 and all
= r (x)
II Ix+iyII
for all
= LI R+)
) )
x
but not on
x.
Before proving Theorem 2.6 we prove two little lemmas, which will be useful in proving the analyticity of the semigroup and in checking condition (i). We shall use these lemmas several times in this chapter.
2.7
LEMMA Let
let
1 <_ p 5 w,
(W, J,u)
and let
function such that
be a measure space with (t,w)
w I; F(t,w)
I} F(t,w)
is in
a positive measure,
u
be a measurable
: H x W -- T.
for each
Lp(W)
t e H
and
is continuous. If t I} F(t,w) : H -> T is analytic t I+ for each w e W, then t F' H I LP(W) is analytic. H +]R
Proof of Lemma. Let circle with centre
t = x+iy a H,
0
and radius
let r.
and let
0 < r < x/2,
C
be the
By Cauchy's integral formula,
F(t+h, w) - F(t,w) - h aF(t,w) = h2
2t
F(t+z, w) dz
2iri
f
Cz
(
so that
IF(t+h, w) - F(t,w) - h DF(t,w)I 3t
<_
IhI2
f2
2
0
irr
IF (t+reie, w)Ide
17
w e W
for all
with
h E T
and
Since w }> aF (t,w)
< r/2.
Ihi
: W
at is a pointwise limit of a sequence of measurable functions, it is measurable. Raising the last inequality to the power
p
W
and integrating over
we
obtain
IIF(t+h, ) -
h aF
2rt
2 IhI
I F (t+re
rr2
i8P/ ,
and using Holder's Inequality on the
IhI2 if J
n
0U
du (w)
w) I
de)
d0
integral
l/P
du (w) l/P
d9.(25)(1-P)P
JF(t+re10, w)IP
rr2
Ihi2(2,O 1-1/p ((2n F(t+rei8, ) IL 2 IJo
<-
by Fubini's Theorem. We have proved the case p = =
p = 1
and
Hence
aF(t,
it/P
and the cases
1 < p <
are similar but do not require Holder's inequality.
E LP(W)
)
d8
and
t f* F(t,
)
H -} LP(W)
is analytic. This
at completes the proof.
A standard argument involving bounded approximate identities in convolution algebras is used in the proof of Lemma 2.8.
2.8
LEMMA Let
and let
A be one of the convolution algebras
t N at
H - A
be an analytic semigroup in for all
everywhere and Hat Ill = 1
t e H 6
if and only if
t > O.
at(w) dw } 0
Then as
Ll(R+)
or
L1 OR n)
A with at ? o
(at * A)
= A
t -> 0, t > 0,
almost
for all for all
IwI?5
> O.
Proof. We shall only use and prove the if implication of this lemma. By Lemma 1.4 to prove that show that t IIa
Ill = 1
for all
t > 0
= A
(at*A)
as
Ilat*g - gII1 ; 0
for all
t -> 0, t > 0,
it is sufficient to
t e H,
for each
g e A.
Since
and since the set of continuous functions with
compact supports is dense in compact support. The function
A, g
we may assume that
g
is continuous with
is then uniformly continuous so for e > o
18
there is a lu - wl
1 > 6 > 0
< 6.
sphere in
or
) n
such that
lg(u) - g(w)l < c
for all
be the sum of the support of
C
Let
g
u,w
with
and the closed unit
(in the latter case the sphere is [0,11). If
II2+
t > 0,
then
flat*g - gII =
1
(g(w - u) - g(w)) at (u) du l dw IIJ
- g(w)l at(u) du dw
flul<6lg(w-u)
< 1
+
f flul?6 lg(w-u) - g(w)) at (u) du dw flul<6 c at(u) du dw + fl
<_ 1JC
<_
Choosing
II--
at(u) du.
fC dw + 2IHlll
c
2 11 g111 at (u) du
ulz6
small enough and then letting
c
t -> 0, t > 0,
completes the
proof.
Proof of Theorem 2.6. If
t = x + iy e H
is a continuous function on
Thus
It
(E
:
DIt (w)
for each
= wt-1 e -w
By Lemma 2.7,
s,t,w > 0,
t f* It
:
H } Ll CR+)
then
(w-u)t-l e-(w-u) us-1 e-u
0 - wt+s-l e-w
r(t) r(s)
H + ]R
t I
is continuous.
r (t)
It * Is (w) = fw
:
log w - r'(t)
r (t)
w > 0.
function. If
and t F ' II It Ill
is analytic and
H-C
at
real, then
r(x+iy)
for all t e H,
L1(R+)
Further t N It(w)
y
dw = r (x)
r(x+iy)
0
and
x
(0,'), and
wx-1 e-w
III till = f
with
du
r(t).r(s) 1(1 -
fo
v)t-1 vs-1
dv
is an analytic
19
and the last integral is the Beta function
r(t) r(s) r(t+s)
6(t,s) =
(see Erdelyi [1953]), where the equality is by the standard formula linking It * Is = It+s
the Beta and Gamma functions. Thus so for all
because of the analyticity of
s,t e H
0 < t < r = Se-1 < 1
If
for all
by considering the derivative of
and
6
and
t [' It
then
<_ w,
s,t > 0,
wt <-
(w/t) t <-
(w/r) r
Thus
t F' log (w/t).
f It (w) dw
*) lwl?6
wt-l e-w
dw
r( I
r -r
r(t) <
r -r
wr-1 e-w dw
J S
r(r) r(t) - l
which tends to zero as and since
decreases to zero for all
t
for all
llItill = 1
t > 0,
Since the Gelfand map on
6 > O. Since
It(w) > 0
property (i) follows from Lemma 2.8. is essentially the Laplace
L1CR+)
transform,
a(at) = {O}
for all
= (z +
t E H. 1)-t
z N LIt(z)
It^(o = {o}
u
{Lit(z)
:
z E H-}
To prove (iii) it is sufficient to show that z E H
for all
and
a
z }*
and
For fixed
t E H.
are analytic in
(z + l)-t
H
t E H,
(LIt)(z)
the functions
and continuous in
and from properties of analytic functions we need only show that (z + 1)-t
z ? 0
t E H
for all
fixed and varying
LIt(z)
and t,
Repeating this argument holding
z >_ O.
we are done because
wt-l e-w e-wz
(
_
H
dw
r (t) ut-1 e-u 0
From the asymptotic formula for obtain
du (z + 1)-t
r (t)
= (z + l)-t
for all
t > O.
r(x + iy)
,
LIt(z) _
given before Theorem 2.6, we
20
X+iy111
= r (x) . I r (x+iy) I-1
II
I(x+iy)-x-iyl.
(2,r)-1/2. r(x). Ix+iyll/2. exp(x + 0 (lx+iy1-1))
1 x = (x2 + y2) 2 + 4
Since
1/2.
y Arg (x+iy) = IyIir/2 - 0(1)
lyl
-> -
r (x) . exp (y Arg (x+iy) + x + 0(Ix+iy
as
we have
1
2 + 4. exp(7rlyl/2 + 0(1))
= K(x) (1 + y2/x2)
for each
- -,
I y I
_ x
IIIX+iYlll
as
(2n)
This proves Theorem 2.6.
x > 0.
The following semigroup will be used in the proof of various properties of the Poisson semigroup in Theorem 2.17.
2.9
LEMMA
If
Ct E L1OR +)
Ct(w) =
is defined by
w 3/2 exp(-t2/4w)
t 2n 1/2
for all
w > 0
and all
t F' Ct : Q + L1CR+)
(i)
(ii)
(iii)
t E Q = {z E T : z X 0,
IArg zl
< it/4},
then
is an analytic semigroup with the following properties
(Ct * L1OR+))- = IlCtl _ (x2 + y2
(LCt)(z)
=
for all t e Q.
L1OR+) \x)1/2
e-tzl/2
for t = x+iy e Q. H-
for
t e Q
and
z e
.
Before proving this Lemma we evaluate an integral that is closely
related to the Laplace transform of C.
2.10
LEMMA If
a > 0,
then
21
a
e
I
= 1
u 1/2 exp(-u - a2 /4u) du
,1/210 = a 2
Let
Proof.
u-3/2 exp (-u - a2/4u) du.
I
'1/2 ) o J_1/2
F(a) = 1
/2
1/2
exp(-u - a2/4u)du
a > 0
for all
0
By the dominated convergence theorem, or uniform convergence,
dF (a) = -a
for all
u 3/2 exp(-u - a2/4u) du
I
271/2 1 0
da
a > O.
w = a2/4u
Let
in this integral and on simplification we
obtain
W-1/2 exp(-a2/4w - w) dw.
dF (a) _ -1 J
T a-
0
71
Thus
F
satisfies the differential equation
F(a) = C e-a
for all
and so
The dominated convergence theorem may be
a > O.
used to check the continuity of F(O) = 1
dF + F = 0, da
at
F
0,
and since
the proof is complete.
r(1/2) = 1
r1/2 t = x + iy e Q with
Proof of Lemma 2.9. If
is a continuous function on x2 - y2 = u2 4w
(O,=),
x
and
y
I W-3/2 exp(-(x2 - y2)/4w) dw
271 1/2 1 0 =
2
Iti
(x2
_ 1x 2 x
2
- y2)1/2 + Y2
-
y2
)1/2
n
Ct
and after making the substitution
we obtain
lictill = ti
real, then
/fe_u2 0
22
Thus
and
Ct E L1 QR+)
Further
t 1 Ct(w)
for all
w > 0.
:
t N 11C4 1
t N Ct
Thus
:
Q -]R is a continuous function.
is analytic and
H + T
Q - L1OR+)
aCt(w) = (1 - t/2w) at
Ct(w)
is an analytic function by
Lemma 2.7. If
then the substitution
t,z > 0,
gives
u = wz
w 3/2 exp (-wz - t2/4w) dw
(LCt)(z) = t 1
27r1/2
0
= tz1/2 Iu 3/2 exp (-u -zt2/4u) du
o
2Tr
exp(-z1/2
t)
=
by Lemma 2.10. The analyticity of the functions implies that they are equal for
t E Q
and
(LCt)(z)
z e H
and
exp(-zl/2t)
The semigroup
.
property follows from the one-to-one property of the Laplace transform and exp(-(t+s)z1/2) = exp(-tz1/2). exp(-sz1/2)
for all
z c H_.
t,6 > 0,
If
then
I
w
w 3/2
Ct(w) dw <_ t
2,1/2
dw
w>_6
t 2(TrS)1/2
which tends to zero as
t
decreases to zero for fixed
6 > 0.
Thus
property (i) holds by Lemma 2.8. We have already checked (ii) and (iii).
2.11
REMARKS, AND OTHER SEMIGROUPS IN L1(R+) The order of growth of
lyl
is the same as
Ilax+iyll
IIIx+iyIil
for fixed
x > 0
as
in Property 6 of Theorem 3.1. it is
possible that this is essentially the best order of growth along a vertical line for an analytic semigroup in a Banach algebra in general. Special algebras like C -algebras (Theorem 2.3) or
L 11R)
have semigroups with
considerably better orders of growth. Note that each non-zero analytic semi-
group t f' at: H - L1OR+)
has
(1 + y2 )_ 1 log+ l I al+iy I ll dy
23
divergent by Theorem 5.6 because there are continuous monomorphisms from L1OR+) into radical Banach algebras (see 2.12 and 3.6). We shall mention some other semigroups into details. Given a function
t F* ft
from some additive subgroup of
to prove the semigroup property
L1 OR+)
to show that
(Lft )(z) = exp tG(z)
open right half plane
without
L1OR+)
into
H
it is sufficient
fs+t = fs * ft
for some analytic function
on the
G
A study of the Laplace transforms in the Bateman
H.
Project table of integrals (Erd4lyi [1954]) shows that the following tables give semigroups from
2.12
into
H
L1(R+):
#20, p.238 with
t = A= v
H
e
#13, p.239 with
t = -v e H
#15, p.239 with
t = v e H
#21, p.240 with
t = v e H.
THE RADICAL CONVOLUTION ALGEBRAS L18R+,w) Let
satisfying
be a continuous function from
w
for all
w(O) = 1, w(s+t) <_ w(s).w(t)
w(t)l/t -r 0
as
e-t log t
w(t) = e-t2, w(t) =
large
are radical weight functions. Let
w(t) =
and
L1QR+,w)
of equivalence classes of locally integrable functions If(w)I
Ilfll =
into and
(O,-)
and
t e IR+,
Such a function is called a radical weight. For
t -> -.
example, t
II2
s
w(w) dw
e-t log log t
for
be the Banach space f
on
such that
]R
is finite. With the convolution product
f * g(t)
J
ft
]R+
f(t - w) g(w) dw
for almost all
t e ]R
,
this Banach space becomes a
commutative radical Banach algebra. The Titchmarsh convolution theorem may
be used to show that algebra
L1OR+,w)
1 (because
L1 Ot+,w)
is an integral domain. Further the Banach
has a countable bounded approximate identity bounded by
w(O) = 1),
and the identity map from
is a continuous monomorphism from
L1 R ,w).
L1QR+)
L1OR+,w)
L1OR+)
into
L1(R+,w)
onto a dense subalgebra of
This monomorphism has norm 1 if and only if
t > 0. The analytic semigroups in groups in
L1(R+)
w(t)
<-
1
for all
are mapped into analytic semi-
by the identity map; however the spectra and some of
the norm properties change in the process.
2.13
THE VOLTERRA ALGEBRA L#1,[O,1]
The Banach space
L1[O,1]
becomes a commutative radical Banach
24
((t
algebra with the product
f(t-w) g(w) dw
(f*g)(t) = 1
t E [0,17.
for almost all
0
We denote this Banach algebra by
and call it the
L'* [0,1],
Volterra algebra. This Banach algebra has a countable bounded approximate identity bounded by 1, and the restriction map
f E* fI[0,1]
Ll(R+) -> Ll*[0,l] is a norm reducing algebra epimorphism. In fact the
restriction map is a continuous algebra epimorphism from each radical algebra
Ll(R+,w)
of 2.12 onto
L1* [0,l].
The analytic semigroups in
are carried into analytic semigroups in
L1 OR+)
by the restric-
Ll*[0,1]
tion map. Note that the Volterra algebra has a dense ideal of nilpotent elements
:
the set
If E L1* [0,17
f = 0
:
for some
on [O,e]
a.e.
e >o}.
2.14
THE CONVOLUTION ALGEBRA L1(Rn)
We shall now discuss two very important semigroups, the Gaussian (or Weierstrass) and Poisson (or Cauchy) semigroups, in the convolution
Banach algebra
where
Ll Otn),
n
is a positive integer. These two semi-
groups were the motivation for several of the properties of Theorem 3.1, and for the Tauberian Theorem 5.6. They play an important role in
probability theory and in several branches of analysis but we shall restrict our attention here to properties related to the Banach algebras shall not discuss the corresponding semigroups in
L1(Rn). We G
for
L1(G)
a Lie
group; for an excellent account of this topic see Chapter 2 of Stein [1970].
We shall recall some well known facts about
and at the
L1 ORn)
same time introduce notation. The Banach algebra
LlORn)
(equivalence classes of) integrable functions
on Mn with norm
Ilflll = Ln If(w)jdw,
R
and convolution product
n w e R
for almost every
.
The algebra
f
consists of
f * g(w) =
(7Rn
f(w-u) g(u) du
is a commutative Banach
Ll stn)
algebra without identity and with a countable bounded approximate identity bounded by 1. There is a natural involution f *(w) = f(-w)
for all
f e L Otn),
Banach *-algebra. The algebra
L1(Rn)
on
*
Ll stn)
and with this involution
(f) = J
f(w) exp(-2wri<X,w>) dw
is the usual inner product of
A
and
identification the Gelfand transform from Fourier transform
is a
L stn)
is (Jacobson) semisimple, and its
carrier space may be identified with Mn by the mapping defined by
defined by
w
L1(Rn)
a f' 0X
:
n ,
for all
f e L1(Rn)
in
With this
]R
into
.
C (0) 0
where
becomes the
25
fA(mx) = oa(f) = J
f(w) exp(-27ri<1,w>) dw = fA(A).
n
R The inverse Fourier transform is
for all
fv(A) = fn(-A) of
Mul(L1(Rn))
The multiplier algebra
A E Rn.
f E L1(Rn)
isometrically isomorphic with the convolution measure algebra is defined in a similar way to
The Laplacian
M(R+).
and
is naturally
L1 stn)
M ORn),
which
is the differen-
4
n
tial operator
defined on suitably differentiable functions on R n
2
32
1 9x. 3
2.15
THEOREM
w E Rn
2
is defined by Gt(W ) = (47rt)-n/2
Gt(w)
If
and all
then
t E H,
t [* Gt
:
for all
is an analytic semi-
L1(Rn)
H
a-Iwt /4t
group, the Gaussian (or Weierstrass) semigroup, with the following properties.
(i) (ii)
(iii)
(Gt * L1 ORn)) IIGx+iylll = (1 +
(v)
y2/x2)n/4
_ {O} u {exp(-ta)
x + iy E H.
for all for all
GtA(A) = exp(-4,r2lal2t) a(Gt)
(iv)
for all t c H.
L1 OR n)
:
a >_ O}
A E Rn, and for all
t E H.
Gt ? 0
as a function and as an element of the *-algebra
L1(Rn)
for all
t > O.
for all
(at - A)(Gt * f) = 0
f E L1
cn)
Proof. For notational convenience we shall prove this theorem for
n = 1
only. Minor changes give the general case, and when we discuss the Poisson semigroup in Theorem 2.17 we shall consider the case
n
a positive integer.
In this proof we shall omit the range of integration if it is R. substitution turns each a > O.
exp(-w2)dw =
,T1/2
into
f
(exp(-aw2)dw
A trivial
= (7r/a)1/2
for
J
If
t = x + iy E H,
IIG
t II1 = (4TrItI) -1/2
then
Gt
is a continuous function on R,
and
(47T
Je _w2x/4hth2 dw
ItI)-1/2 (47rIt12/x) 1/2
_ (1 + y2/x2)1/4.
Thus
Gt
e L1 (R)
for all t E H,
and
t J*IIG tlll
:
H 13R
is continuous.
26
Further
Gt(w)
t f
:
is analytic and
H -) C
t
for each
t N Gt
By Lemma 2.7,
w E ]R.
_ 1
4t2
2t
is an analytic
H - L1 O2)
:
w2
8Gt(w) = Gt(w)
function.
t,r > 0
If
then we complete the square
and w,u a ]R,
\2 {(w-u) 2t-1 + u2r 1} = t+r (u - rw + w2 t+r tr t+r) 1
and using this we obtain
Gt * Gr (w) exp{-(w-u)24-1 t-1
= (4Tr)-1 (tr)-1/2 J (47r)-1
(tr)-1/2 exp(-w2/4(t+r))
-1 -1 -1 2 -(t+r)t r4v} dv
J exp{
where
- u24-1r 1} du
v = u - rw t+r
_ (4r)-1(tr)-1/2 exp(-w2/4(t+r))Tr1/2(tr/(t+r))1/2 = Gt+r(w).
Since
t N Gt
extends from
:
is an analytic function, the semigroup property
H + L1(R) to
(0,°0)
Alternatively the semigroup property may be
H.
checked using the Fourier transform of
dw
Gt(M )
I
(4Trt)1/2
which tends to zero as
(Gt * L1(R))
Let and
decay y of
w
jw2 dw
tends to zero for each
t
= L1(R)
for all
F(t,z) = (4irt)-1/2
L1(R).
and so
1
J
z e S
and the semisimplicity of
t,6 > 0, then exp(w2/4t) ? w2/4t
If
follows that
Gt
= 4t2//2 6,
6 > 0.
By Lemma 2.8 it
t c H. -
w24-1t-1) dw
for all
Note that the integral converges because of the rapid
t > 0.
1
exp(-w2 4- t-
entire function for all
1 )
F(t,z) = (4Trt)-1/2
=
near infinity, and that
t e H.
If
z
is in
Bt,
Jexp(_(w+4tz)241tl)
exp(4772z2t)
.
z J* F(t,z)
is an
then
dw exp(47r2z2t)
27
Thus
for all
z e T,
and hence
A E R and
t > O.
Using analyticity again this
F(t,z) = exp(4Tr2z2t)
for all
= exp(-47r2a2t)
formula holds for all
t e H.
From the definition of function for all for all
Gtt(A) = F(t,iA )
Also
t > O.
Gt >_ 0 Gt = (Gt/2 )
so that
= Gt
(Gt)
as a * Gt 2
it is clear that
Gt
>_ 0
Part (v) follows from the definition of the convolution
t > O.
and the formula
3Gt (w
t
- u) = Gt (w - u) )
(w - u) 2 - 1 4t2 2t
= 92G (w - u) .
This completes the proof.
Before we discuss the Poisson semigroup we give a standard little lemma for evaluating spherically symmetric integrals over Rn. we had proved Theorem 2.15 on the Gaussian semigroup for
If
this
n > 1,
lemma would have been useful.
2.16
LEMMA
The area of the surface of the closed unit sphere in Rn w
n
= 2Trn/2 r(n/2)-1.
L1Ct+),
If the function
then w f f(jwj) : Rn -> T
r ' f(r)rn-1
is in
L1(IR n)
is
is in
(O,-) -> T
:
and
fk
I IwI
Owl) dw = w
n 11)o
f(r)rn-1 dr
for 0 < k Proof. Let
V(r)
denote the volume of the sphere of radius
The surface area of the sphere of radius
in Rn
r
is
rn-1
in Rn
r
wn.
From
the definition of the derivative and the volume of a thin shell being approximately the area of the shell times its thickness, we have dV = 7r-
rn-1
w .
Since
w f* f(IwI)
: Rn i c
is constant on spheres with k
centre the origin,
f(IwI) dw = w IwI_k
f(r) rn-1 dr n fo
28
from the definition of the integral. Alternatively we can write each uniquely in the form
w E Rn\{O} {u E Rn :
on
where
dw = rn-1 dr da(s),
so
lul = 1),
w = rs,
0 < r < -
where
da
and
s E S =
is the area measure
The equality of the integrals then follows from this.
S.
We calculate
wn
by applying the integral equality with
e_nr 2
f(r) =
for
e Trlwl2
r
Letting
r 2 O.
dw _
Rn
nr2 = u
wn
r-e-u un/2-1
du
o
2nn/2
=
in the r-integral we obtain
wn r (n/2) 2nn/2
If
then
w = (wit ...,wn) E Rn,
2
dw f3R n
ej3
n =
-nw2 dw
IT
j=1 R n/2
e-n(wl+wz)
d(wl,w2)
R2
w2 r(2/2)
In/2
2n by our formula above with
n = 2.
ference of the unit circle, ference
1
Observe that n
because
w
n r(n/2)
odd, and is an integer if
2.17
n e-Tlwl
w2
is the length of the circum-
dw = 1.
This completes the proof
R
of the Lemma.
power of
Since
n
is always a rational multiple of an integer is a rational multiple of is even.
THEOREM if
Pt(w)
is defined by
nl/2
if
n
is
29
Pt (w) = r ((n+l) /2)
t
.
(t2+1-12)(n+l)/2
TT(n+l)/2
for all
and all
w e Iltn
t N Pt
then
t e H,
:
is an analytic
H - L1 ORn)
semigroup, called the Poisson (or Cauchy) semigroup, with the following properties. (Pt * L1(Rn))- = L1 fin)
(i)
for all
t E H.
(ii) II Ptlll = 1 for all t > 0, 1-n
{Iyl
1+iy
IIP
2
2,
n
and
bounded for
Pt > 0
(v)
Pt =
J0
for all
t E Q = {z e H
then the function r N It2+r21-(n+l)/2 rn-l = O(r-2) .
By Lemma 2.16 it follows that Pte LloRn)
1
y
1} is
I
r > O}
:
as a function and as an element of the *-algebra
for all t > O.
IIPtII
I
a(Pt) _ {O}u {exp(-rt)
and
Ct(u) Gu du
continuous and
y E II2,
and
L1(R n)
t e H,
:
n = 1.
for all to H (iv)
is bounded for each
? 1}
IYI
{(loglyl)-l lip l+iy111
PtA(X) = exp(-2:tIalt)
(iii)
Proof. If
: y E IR,
IIl
(t2+r2)-(n+l)/2
as
Fo
. IR
+
, X
is
tends to infinity.
and that
It, rn-1
= 2r((n+l)/2) 1/2r(n/2)
r
< w/41.
IArg zI
dr.
(1)
It2+r 21 (n+l)/2
TT
From (1) we see that
each w e]Rn (t,w)
1
t NIIPtIIl
the function
Pt(w)
H x 1Rn ; d
:
:
is a continuous function. Also for
H --]R
t E Pt (w)
:
H -> T
is analytic, and
is continuous. By Lemma 2.7
t E
Pt
:
Ll Un)
H
is an analytic function.
We shall now check property (v) from which the semigroup property and the Fourier transform of the substitution
=
(t2+lwl2)/4u
Ct(u) Gu(M )
I
Pt
will follow easily. If
reduces
du
0 = 2-1
-1/2 F0 t u-3/2 e t2/4u
(47Tu)-n/2 e-1w12/4u
du
t > 0,
30
to
Tr-(n+l)/2(t2+1.12)-(n+l)/2
(n-l)/2
I
t
Pt(w)
0
by definition of the Gamma function and Poisson semigroup. Since the integral
Ct E LI(R+),
exists as a Bochner integral
fo Ct(u) Gu du 0
in
and is equal to
LlORn)
and the continuity of the operator
t F* Ct : Q - L1OR+) f (u) Gu du :
8: f * t F*
Pt by the above. The analyticity of
J- Ct(u) Gu du
imply that the function
L1 OR+) - L1 ORn)
is analytic. Property (v) follows from
Q - LlURn)
this and the analyticity of t [-* Pt. The operator
8
from
Pt = 8(Ct)
hence on
for all
t E Q,
we see that
t F* Pt
is a semigroup on
Q
and
H.
The continuity of the Fourier transform from C ORn) 0
into
L1OR
may be seen to be a homomorphism using Fubini's Theorem. Since
L1ORn)
L1ORn)
into
(or Fubini's Theorem) shows that
PtA(a)
=
fCt(u)
GuA(A) du
iCt(u) exp(-47r21AI2u) du 0 =
(LCt) (4Tr21A12)
= exp(-2TTIXIt)
for t E Q and
aEIRn
by Theorem 2.15 and Lemma 2.9. Using analyticity all
A E ]R
and IIPtiil
t E H.
< J:Ct.1Guttl du <_
for all t E Q since t > 0
because
we see that
G. t
I
I Gu I Il = 1
2: PtA(O) IlPtlll
Pt ? 0
for
Ptn(a) = exp(-2nIXIt)
for
By Fubini's Theorem we have
t > 0
1k11,
for u > 0.
= 1.
Since
Hence
Ct(u) ? 0
11P t I
Il = 1
for all
for all t, u > 0,
follows from the corresponding property of
31
To prove property (i) it is sufficient to show that for each d > 0,
Pt(w)dw -+ 0
as
(Lemma 2.8). Discarding the
t -+ 0, t > 0
IwI?d
constants in
and using Lemma 2.16 we see that it is sufficient to
Pt
prove that
I-t(t2+r2)-(n+l)/2 rn-1
dr - 0
d
as
for each
t } 0, t > 0,
d
2)-(n+l)/2
t(t2+r
For
> 0.
we have
0 < t < d,
ft2-(n+l)/2.r-(n+l)rn-ldr
rn-ldr <
d
d
t-+0.
-+0 as
l+iy
We shall now obtain an upper bound for
IIP
II1
I1pl+1YII1
this will complete the proof of (ii). From (1)
for
is a constant
times
I
J l+iy I rn-1
I
o I(1+iy)2 +
dr r2I(n+l)/2
and we estimate this integral. Since
2
2
IyI
2
I1 = 2y
we have
if
Using these inequalities and replacing I < 2(n+1)1/2 (I1 + I2)
> 1
r
11 + iyI
(y
by
1)
2y,
where
((y2-1)1/2
rn-1
0
dr (y2-1-r2+2Y)(n+l) /2
and n-1 12
dr .
2y
(y2-1)1/2 (r2-y2+1+2y)(n+l)/2
The substitution
r = (y2+2y-1)1/2cos
E
reduces
1 1
to
y ? 4 -
we have
32
(y2+2y-1) 1/21/2 sin -nE dE,
2y
v
where
sinv
tion
In the second integral we use the substitu-
= (2y/(y2-1))1/2.
r = (y2-2y-1)1/2 sect
2y(y2-2y-l)-1/2
y > 3
for
(7r/2
to reduce
1
to 2
sin nC d;.
Jv From the graph of the sine function we obtain the inequality 2E <- sin E
if
J/2
0 5 E <- n/2. Using this inequality
for
if
(Tr/2) log (v-1)
(or
n >- 2
from
x/2. We now use
d<
sin
and
to deduce that
y ? 4,
for
n+l (n-l)v
neglecting the term arising
n = 1)
(2y/(y2-1)1/2 = sinv ? 2fr-1v
y2+2y-1 >- y2-2y-1 2, y2/4
(/2)n
1 < 2(n+l)/2.2.2y(y2/4)-1/2(n/2)n(n-1)-10,/2)-n+l((y2_1)/2y)(n-1)/2 8ir(n-l)-l y(n-1)/2 <-
for all
and
y ? 4
2.18
n ? 2
(and
IIPl-lylll
IIP1+iylll
(ii) since
=
REMARKS IIP1+iy1ll
Compare the rates of growth JIGl+i1l1
as
IyI
when
<- 4nlog y
I
= 0(Iyln/2)
for
n-1 = O(IyI -2)
the Poisson growth is
n = 1
Intuitively the subordination of
to
P t
giving a lower rate of growth for
given by
(4yf) (x) = f(x)
and we shall regard embedding. Since
L1 O2+)
loglyl).
Gt has smoothed out
Gt
so
Pt.
from
There is a natural isometric embedding
L1 R)
and
for the Poisson and Gaussian semigroups
n ? 2
tends to infinity (for
This proves
n = 1).
for all y > O.
if x ? 0 and
as a closed subalgebra of
(L1(R+) * L1 O2))
= L1(R)
L1aR+)
(x) = 0 Ll(R)
into
if x < 0, via this
(by Cohen's factorization
theorem - see the introduction), we see that the fractional integral semigroup
t -* It
:
has many of the properties of the Gaussian and
H - L1 OR)
Poisson semigroups. However L 1 (1R+)
n L1cR)+)
= {O},
(It)
x It
for
t > 0,
and the order of growth of
because
II11+iyII,
namely
33
O(y-1/2
exp(IIlyl/2)),
as
jyl
tends to infinity is substantially worse
than the polynomial growth of the Gaussian or Poisson semigroups. Is there a natural isometric algebraic embedding of n > 1
LIOR+)
into
L1 OR )
for
not of the type discussed in Corollary 3.5?
2.19
NOTES AND REMARKS Many of the semigroups discussed in this chapter are considered
in depth by Hille and Phillips [1974]. However we do not discuss generators and we emphasize different aspects of the theory than Hille and Phillips [1974], where there are further references and remarks.
C -algebras. Aarnes and Kadison [1969] prove that a separable C -algebra has a commutative bounded approximate identity. There is a discussion of *
commutative approximate identities in C -algebras in Pedersen [1979] (see Corollary 3.12.15) that essentially gives Theorem 2.2. See also Doran and Wichman [1979]. See also Lemma 4.3 of Haagerup [1981].
The convolution algebra
L1 OR+). The fractional integral semigroup
t J+ It
has played a fundamental role in harmonic analysis - see Hardy and Littlewood [1932] and stein [1970]. This semigroup is used in Stein's treatment of the general maximal function (see p.77 and p.117 in Stein [1970]). When acting on
LpOR+)
by convolution this semigroup is also called the
Riemann-Liouville operator. The semigroup
Ct(w) = t w
3/2
(27r1/2)-1 e-t2/4w
occurs in the backwards heat equation and there is a discussion of it in Widder [1975]. This semigroup is also used in the subordination of the Poisson semigroup to the heat (Gaussian) semigroup in the group algebra of a Lie group (see Stein [1970], p.46). Esterle [1980e] uses it in the construction of infinitely differentiable semigroups in a radical Banach algebra that tend to zero very fast as
The convolution radical algebras
t
L1 OR +,m)
tends to infinity (see 5.4).
and
L1*[O,1]. These Banach
algebras have been extensively studied since the early developments in Banach algebra theory (see Gelfand, Raikov, and Shilov [1964], and Hille and Phillips [1974]). Recently there has been renewed interest in these algebras' closed ideals (Allan [1979]), derivations (Ghahramani [1980]), and rates of growth (Allan [1979], Bade and Dales [1980], and Esterle [1980e]).
34
The convolution algebras L1QRn).
The Gaussian and Poisson semigroups play
an important role in harmonic analysis and function theory on Mn
as may
be seen by examining the books Stein and Weiss [1971] and Stein [1970].
The Gaussian (or normal) semigroup plays an important role in probability theory and the Poisson semigroup in the theory of harmonic functions. However these are outside the scope of these notes and remarks. Computationally the Gaussian semigroup is easier to handle than the Poisson semi-
group. For an excellent account of semigroups corresponding to the Gaussian and Poisson semigroups in the group algebra of a Lie group see Stein [1970] (also Hulanicki [1974]).
Lemma 2.7. H.G. Dales has proved the following Lemma, which is simpler to prove and stronger than Lemma 2.7.
Lemma. Let let
1
:- p < -.
t F' F(t,w)
K
(W,1,1,)
:
Let w N F(t,w)
<_ pk(w)
be in
be analytic for each
H -> C
contained in
IF(t,w)I
be a measure space with
there is a function
H
for all
t E K
a positive measure and
p
for each
LP(W)
w E W.
pk E LF(W)
and w E W,
t E H,
and let
If for each compact set
then
so that t [*
H -} LL(W)
is analytic.
I have used Lemma 2.7 instead of this Lemma because the hypotheses of Lemma 2.7 seem more natural and are easier to check. This Lemma is proved by using Lemma 1.3, the Cauchy estimates, and the dominated convergence theorem.
35
3
EXISTENCE OF ANALYTIC SEMIGROUPS - AN EXTENSION OF COHEN'S FACTORIZATION METHOD
Throughout this chapter
A
will denote a Banach algebra with
a countable bounded approximate identity, which will usually be bounded by 1. The main theorem of this chapter ensures that such an algebra contains a semigroup analytic in the open right half plane. This theorem is proved by an extension of Cohen's factorization theorem for a Banach algebra with a bounded approximate identity. In this chapter we shall discuss the properties of the semigroups that may be obtained by these methods, and shall investigate some applications of the existence of the semigroups. The proof of the existence of the semigroups is given in detail in Chapter 4, and is discussed there. The basic properties of the semigroup are stated in Theorem 3.1, and additional properties relating to derivations, multipliers and automorphisms are dealt with in Theorem 3.15. Many of the properties of the semigroups given below are generalizations of those of the fractional integral semigroup in Gaussian and Poisson semigroups in constructed in
A
L1ORn).
L1(R+),
or the
We should like the semigroups
to be as nice as possible: to have good growth, norm
and spectral behaviour. In Chapter 5 it is shown that certain polynomial growth properties of the Gaussian and Poisson semigroups cannot hold in radical Banach algebras.
The semigroup properties of
at
are emphasised rather than the
factorization properties, and all the semigroup properties that I know are included in Theorems 3.1 and 3.15. However in any given situation one is usually interested in only two or three properties of the semigroup, or of the module factor of
x.
Most parts of Theorem 3.1 have been used in an
application or a stronger version has been used as the hypothesis of a result restricting the structure of the algebra (see Chapter 5).
3.1
THEOREM Let
A
be a Banach algebra with a countable bounded approximate
36
identity bounded by 1, and without an identity. Let
X
(and a right) Banach A-module, and let
be in the closed linear
span of
group
X = {a.w
A.
t E' at :
y)
(and
t F* xt
(and
t N yt
for all
t E
T - X
:
T -
:
be a left
Y)
Then an analytic semi-
(and of Y.A).
from the open right half plane
H -* A
entire functions
a e A, w e X}
:
(and
x
A
into
H
and
may be chosen to
Y)
satisfy the following properties. 1.
x = at.xt
y = yt.at)
(and
H.
Properties of the semigroup
= A = (A.at)
2.
(at.A)
3.
For each
i(
:
t E H.
the function
(O,Tr/2)
E
U(f) = {zET
sector
for all
0 <
IzI
1,
<_
t J*
is bounded in the
IIatII
IArg zI<*},
where Arg is the
principal argument.
4.
For each t -> 0 in
5.
I I at I
6.
if
I
C
= 1 + 3(1 - 4-a)-1
a
ax+iy
x > a
for all
for all
9.
If
A
Jf(t)atdt
f F'
y e IR.
where
t E H, :
0 < x < 1,
for
= {O}U U(7rx/2)
exp(lTIIm tl/2) :
)_(x_a)/2 eTr I y I /2 2
(x-a)
and all
a(ax) E U(Trx/2)
If 6
bat + b as
then
for a > 0,
y2
<_ Ca . 2x1 1+
I I
1\
8.
and
UM.
I I
7.
a b -> b
for all t> 0.
1
<-
t
and each b e A,
E (O,Tr/2)
v(at)
is the spectral radius.
v
then
L1 C2+) - A,
and
is one-to-one.
6
has a closed convex bounded approximate identity
by 1 such that integers
n,
b e A then
implies that
at e A
for all
bn E A
bounded
A
for all positive
t > 0.
Properties of the factors in the module 10. x0 = x
and
x-t = at .x
yo = y
(and,
and
y-t = y.at)
for all
t E H.
11. at.xz+t = xz 12. xt e
(A.x)
13. If 6 > 0
t --,
(and (and
and if
yz+t.at = y yt E (y.A) t 1+ at
:
)
)
for all
t E H
and
z E T.
z
for all
t e
(O,-) - C1+6,-)
then Ilxtll5 (aIt,)ItIIIx1I(and
t e T with ItI ? 1. 14. If C > 0 and 6 > 0, then IIx - xtll <_
(t.
such that Ilytll<_
6
at }
as
(altl) ItItIlyll) for all
(and
Ily - ytll
<_ 6)
for
37
all
t e T
with
<_ C.
Itl
A functional calculus property 15. If
0 < a < S < 1,
H + A
then there are analytic semigroups
t f' (aa - as)t:
(aa - aa)1 = as - aa.
such that
A subsidiary hypothesis in one of the properties, for example, of the form "if..., then..." is assumed to hold before the semigroup is chosen.
3.2
REMARKS
If the bound on the approximate identity in the hypotheses of Theorem 3.1 is
then properties 1 to 4 of Theorem 3.1 still hold
d (2!1),
though Property 5 fails (see Dixon [1978] for this). This slightly stronger version of 3.1 with
d >_ 1
is proved in 4.7. However the stronger form
enables us to pass to an equivalent norm and choose a new approximate identity for which
3.3
d = 1.
COROLLARY
A Banach algebra with a countable bounded approximate identity has a commutative bounded approximate identity bounded by 1 in an equivalent Banach algebra norm.
In the proof of Corollary 3.3 we require the following standard little renorming lemma.
3.4
LEMMA Let
be a bounded multiplicatively closed subset (that is,
S
an algebraic semigroup) in a Banach algebra B. Then there exists an equivalent Banach algebra norm
on B such that
Ibl
<_
for all
1
c e S} for all
:
Straightforward calculations show that
q
is an algebra norm on
satisfying
x e B,
bound for x E B. l'I
Let
<_ q(x) Ilxll
11,11C11
:
for all
<_ Ixll
c e S}.
Ibl = sup {q(bx)
Also :
is the required norm on B.
q(cx)
<_ q(x)
x e B, q(x)
<_ 1}
where
M
b e S.
x e B.
Proof of Lemma. Let q(x) = sup {Ilxll, Ilcxll
B
is an upper
for all
c e S
for all
b e B.
and all
Then
38
Proof of Corollary 3.3.
Theorem 3.1 for
A.
{IlatII exp(-Mt)
t > O}
Property 3 that
t F' at
H -* A
:
be a semigroup provided by M
such that
is bounded. This is because it follows from
{Ilatll
is bounded - we could choose
0 < t < 1}
:
The multiplicatively closed set
M = logllalll.
A
Let
There is a real number
{exp(-Mt) at
gives an equivalent Banach algebra norm
lexp (-Mt).atl
{exp (-Mt).a
t > 0) in
:
such that
(Lemma 3.4). Property 4 implies that
t > 0
is the required commutative bounded approximate
t > O}
:
identity for
for all
1
<-
t
A
on
A.
Examples due to Dixon [1978] show that the hypothesis of countability cannot be omitted from Corollary 3.3. From now on we shall assume that our approximate identities are bounded by 1. Property 3 is essentially the best possible as there are Banach algebras
with
B
bounded approximate identities in which we cannot have an analytic semigroup
{Ilatll
t F* at
:
:
such that
H -> B
t e H, Itl
(atB)
= B
for all
t e H
and
is bounded (see 5.14).
5 1}
The convolution algebra
Ll(R+) plays some of the roles in the
class of Banach algebras with countable bounded approximate identities
bounded by 1 that the Banach algebra
does in the class of unital Banach
algebras. This is the intuitive idea behind the next three corollaries.
Corollary 3.5 corresponds for Banach algebras with a countable bounded approximate identity bounded by 1 to the following trivial observation for unital algebras. A Banach algebra
B
homomorphism
such that
from
0
into
T
B
is unital if and only if there is a 11811 = 1
and
e(C).B = B = B.e(T).
COROLLARY
3.5
A be a Banach algebra. Then A
Let
approximate identity bounded by 0
from
into
L1OR+)
Proof. If
bounded by
A
has a countable bounded
if and only if there is a homomorphism11
1
such that
11811 = 1
and
0(L1(R+)).A = A = A.0(L7R+)).
exists, then a countable bounded approximate identity
0
in
1
is easily transferred to a countable bounded
L1(R+)
approximate identity
('(fn))
in
A.
Conversely suppose that
A
has a countable bounded approximate
identity bounded by 1. Using Theorem 3.1 we choose a semigroup and we
define
0
:
f(t)at dt
f [* J
using
llatll
<-
1
for all
(fn
:
L1(R+) -> A.
t F' at :
Direct calculations
0
t > 0
and the convolution product of
L1(R+)
H-*A,
39
show that
is a norm reducing homomorphism from
0
we regard A
as a Banach for all
a.f = a0(f)
Banach algebra and
f e L1(R+).
A.L1(R+).
A0(L1(R+)) are closed linear subspaces of
and
We apply Theorem 3.1 to the
and its left and right Banach
and similarly for
L1(R+).A,
Now
A.
f.a = 0(f)a
is
L1(R+).A
Therefore
A.
A
L1(R+)-modules
Property 1 implies that the closed linear span of
A.
equal to and
and
a e A
L1(R+)
into
L18R+)
L1(R+)-module by defining
0(L1OR+))A
We have essentially
proved a standard corollary of Cohen's factorization theorem in doing this (see Hewitt and Ross [1970], p.268). To show that these two closed linear subspaces of then
b e
A
are equal to
A
Let
f
(e(L1OR+))A)
of [0,1/n].
.
it is sufficient to prove that, if be
n
times the characteristic function
n
For each positive integer
b e A,
and each
n
b e A,
Ilb- 8(fn)bI1
llJofn(t) (b - atb) dtll n
Jn jib - atbil dt 0
<- sup {I l b - atb I I and this tends to zero as
n
0
:
<_ 1/n} ,
tends to infinity (Property 4). This proves
Corollary 3.5.
If the algebra
be one-to-one because
morphisms from
L1(R+)
For example let
J =
quotient map from
A
had an identity, then the map
onto radical Banach algebras that are not one-to-one. f = O
L1OR+)
onto
A
and let
a.e. on [O,l]},
be the
i
which is isomorphic to
L1(R+)/J,
(see 2.13 for the definition of the Volterra algebra if
would not
8
does not have an identity. There are homo-
L10R+)
is a radical Banach algebra, and if
0
L
1
L1*[O,l]
However
[0,1]).
is constructed as an integral
with a continuous semigroup as in the proof of Corollary 3.5, then
0
is
one-to-one.
3.6
THEOREM Let
A
be a radical Banach algebra, and let
be a continuous semigroup such that
for all t > 0. If
0 :
llatll
f F' J 'of (t) atdt 0
:
<_
1
and
(atA)
L1 QR+) + A,
t F' a
t :
(O,a) - A
= A = (Aat)
then
0
is a
40
monomorphism from
A
into
L10R+)
and
satisfying loll = 1
8(L1(R+)).A = A = A.6(L1(R+)).
From the proof of Corollary 3.5 and the observation that {al/n
:
is a bounded approximate identity in
N E 1N}
sufficient to show that
we see that it is
A,
is one-to-one. This will follow easily from the
0
following theorem of Allan [1979], which is proved in appendix A2.
THEOREM
3.7
Let
a
w
be a radical weight on ]R+,
L10R+,w). If there is a non-zero
ideal in
C ? 0
such that
I =
OR+,w)
llanlll/n
for
0 < t <- 1,
we obtain
weight. We now extend 8(f) = Since
f
f(t) at dt
6
- O
w(t) 1/t ; 0
for all
g = 0
ker 6 =
aC = lim f ng(t)atdt = 0. n o the proof.
n , -. as
into
[C,C+l/n] for
there is a
by ker6nL1 OR
C ? 0
n
a positive integer, then
-> A
0
from
t
at
L10R+).
PROBLEM
Let A be a commutative radical Banach algebra, let (O,°)
x W.
If gn is the characteristic
In the proof of Theorem 3.6 we used the homomorphism
3.8
)
such that
contradicts the hypotheses and completes
but only proved it is one-to-one on
L1GR+,w)
is a llatll <- 1
is a radical
w
A
Suppose that
L10R+,w),
A
Since
Thus
t
a.e. on [O,C]}.
This
then there is
Using this and
L10R+,w)
f E L1(R+,w).
function-of the closed interval
t > 0.
for all as
to a map from
is a closed ideal in
ker 0
be a closed
I
g = 0 a.e. on [O,C]}.
:
Proof of Theorem 3.6. Let w(t) = llatll
radical Banach algebra,
and let
f e L10R+) n I,
be a continuous (bounded) semigroup such that
(atA)
= A
for
all t >if(t)atdt 0, and let w(t) = llatll for all t > 0. If 6 : f f* : L1,w) ± A, is 0 one-to-one? Property 8 ensures that by suitably choosing the semigroup t I+ at
H -> A
the conclusion of Theorem 3.6 holds without the hypothesis
that the algebra is radical.
Property 6 implies that the function
t 1+ a t+1
:
H + A
is of
41
exponential type in the closed half plane e7TIIm tI/2
21/2C1/2 e"Itl/2
for all
t e H
x+iy
Note that the order of growth of
!5
(we are taking
a = 1/2).
x > a > 0
for fixed
II
lla
21/2C1/2
IIa1+tII
because
H
as
tends to infinity is that of the fractional integral semigroup in
IYI
Ll m ).
In Chapter 5 we shall observe that a substantial improvement on the size of is not possible in general.
IatII
Property 7 is essentially the best possible. There are commutative Banach algebras with a countable bounded approximate identity such that the spectrum of each element in the algebra is the closure of the interior of the spectrum; for example, the maximal ideal of functions vanishing at 1 in the disc algebra has this property. We shall apply property 9 to certain group algebras and algebras *
of compact operators on suitable separable nuclear C -algebras.
COROLLARY
3.9
Let
be a locally compact group. Then
G
and only if there is an analytic semigroup such that
IatII = 1,
algebra llf*at
L1(G)
as
- fIll - 0
t > 0,
t -; 0
Proof. Suppose that
G
such that
t N at : H -> L1(G)
as a function and as an element of the *-
at - 0
for all
is metrizable if
G
and such that
Ilat * f - fll
non-tangentially in
1
+
for each
H
f E L1(G).
is a metrizable locally compact group. The
metrizability of the topology of
G
implies that there is a countable
open base for the topology at the identity of
G.
This countable base of
the topology at the identity gives a countable bounded approximate identity A
1
in
L1(G).
If
Un
is one of the countable open base sets, which may
be assumed to have compact closure, then
fn = u(Un)
is a typical
XU
n
element of p
Al,
where
XE
denotes the characteristic function of
and
E
is left Haar measure (see the proof of Theorem 20.27 of Hewitt and Ross
[1963] p.303 for further details). The elements of
are bounded by 1 in
A 1
I-III - norm and are positive as functions. The involution on
L1(G)
is
isometric and preserves the positivity of functions, and convolution
preserves the positivity of functions. Thus the set is a countable bounded approximate identity in ing of positive functions. Let g ? 0 and
almost everywhere on
llglll
<_
1.
Since
G,
A 2 A2,
L1(G)
denote the set of
A
g
A2 = {g *g
:
g c A1}
bounded by 1 consistg E LI(G)
such that
is self-adjoint in the *-algebra
the set
A
1
L (G),
is a bounded approximate identity
42
in
Also
L1(G).
where
gn e A
is closed convex and
A
is the n-th convolution power of
gn
g e A
if
We apply Theorem 3.1 to the Banach algebra bounded approximate identity bt
t
:
with
t-
as
--> O
0 < * < w/2.
bt e A
at = bt (f
Let
ff
function
f [' JGfdp
for all
and
t > 0,
bt du)-1
G
for all
is a character,
L (G) ; T
:
and the
L1(G)
Ilbt * f - fill +
in any sector {z e C : z x O,
O
analytic semigroup. The order properties of at at = (at 2) * at/2 the definition of A and
I Arg z j
compact neighbourhood of the identity in where
is
shifted by
al
x
in
follow from
6
[1963] p.285). Now
U
a homeomorphism from 6
x
* a1 = 6
so that
y
6
x
topology of
- 6
y
x
is compact and U
then
* a1,
= 0
and
onto (6
x
y
is Hausdorff so
j
will be
i
is one-to-one. If
((6x
-
* L1(G) c
)
-6
y
)
* a1 .' L1(G))
= {o}
is a homeomorphism and the
Thus
x = y.
(by Hewitt and Ross
L1(G)
L1 (G)
if
i(U)
- 6
* a1
6x * a1
in the notation of Hewitt and Ross
* al = al -1
x
x
is a continuous function from
into
G
x k 6
4
be a
U
Note that
x e G.
Then
L1(G).
U with the relative topology of [1963] p.285 since
and let
G,
is the point mass at
6x
is an
H - L1 (G)
t > 0
for
<}
Since the
t e H.
t [* at :
Conversely suppose that the semigroup exists. Let
U -> L1(G),
n E IN,
in it. This gives an analytic semigroup
A
such that
H -> L1(G)
1 1 f * bt - f I Il
and
g.
is metrizable.
G
We shall use Theorem 3.1 to show the existence of semigroups of *
completely positive compact operators on suitable nuclear C -algebras. It will be clear from the proof that analogous results hold for suitable Banach spaces satisfying the metric approximation property (for example, if the Banach space and its dual are separable and satisfy the metric approximation property). However in the Banach space case the order properties are lost. We start by recalling the definitions of a completely positive operator *
and of a nuclear C -algebra. If
X
is a Banach space, let
of compact linear operators on
denote the Banach algebra
CL(X)
and let
X,
FL(X)
denote the algebra of *
continuous finite rank linear operators on algebra
B
is said to be positive if
X.
Tb ? 0
An operator for all
T
b ? O.
on a C Let
Mn(B)
*
denote the C -algebra of B,
and let
onto
M (Q). n
In
n x n
matrices with entries from the C -algebra
denote the identity operator from the C -algebra
We shall think of
M
n
(B)
as
M (f) 0 B. n
An operator
Mn(C)
T
on
43
a C -algebra
is said to be completely positive if
B,
positive operator on
for all
M (B) * n
T 0 I
is a
n
One of the equivalent
n E IN.
formulations that a C -algebra is nuclear is that
has a left bounded
CL(B)
approximate identity bounded by 1 consisting of completely positive continuous finite rank operators (see Lance [1973] and Choi and Effros [1978]). *
On a commutative C -algebra each positive operator is completely positive (Stinespring [1955]).
3.10
COROLLARY *
Let
be a separable nuclear C -algebra, and suppose that CL(B)
B
has a bounded approximate identity of completely positive (continuous) finite rank operators bounded by 1. Then for each separable subspace
Y
there is an analytic semigroup
I1atll
at
t
is completely positive for each
at
H -> CL(B)
and that
t > 0,
IIR.at - RIl -> 0 as t -> 0 non-tangentially in
all
such that
CL(B) <-
and
1
Iat.T - Tll +
for all
H
of
and
T E CL(B)
R E Y.
Proof. We require a compact linear operator
theses. Here is one way. Let in the unit ball of the sequence
(x
n
B.
in
)
on
o
B
such that
There are several ways to obtain this from the hypo-
= CL(B).
(T0 CL(B))
T
{x
n
E B
By regarding B
be a countable dense subset
n E IN}
B
as a left Banach
CL(B)-module,
may be factored by the Varopoulos-Johnson version
of Cohen's factorization theorem (see Hewitt and Ross [1970], p.268) in the form
= B. If T y = x for y E B and T E CL(B). Hence (T B) * o n n 0 0 n and x E B, then T .(fDx) = fOT x. From this it follows that 0 _ 0 (T0 CL(B)) Because FL(B) it follows that = CL(B), z, FL(B). (T .CL(B)) o = CL(B).
f E B
By induction on identity
E
we choose a countable bounded approximate
n
of completely positive finite rank operators bounded by 1
n
acting as an approximate identity for 11E E
n
m
To, Y,
choice we use the separability of
algebra of sequence
CL(B) (E
n
)
generated by
n
:
Y.
Let
n E :]N}, T
o
,
A and
be the closed subY.
Then the
is a countable bounded approximate identity for
apply Theorem 3.1 to A
{E
In making this.
to ensure we can choose a sequence
Y
to act as a right approximate identity for
in
and themselves (that is,
- EM 11 111Em En - Emll 3 0 as n - - for each m).
A
with
A
A.
We
the set of completely positive operators
with norm less than or equal to 1, and with
X = CL(B)
and
Y = Y.
Since the product of two completely positive operators is completely positive,
A
satisfies the hypotheses of Property 9. This completes the proof.
44
PROBLEM
3.11
have the property that
B
Which separable nuclear C -algebras
has a bounded approximate identity of completely positive finite rank
CL(B)
operators bounded by 1?
The following theorem due to Johnson [1972], Lemma 6.2, shows *
that certain commutative C -algebras satisfy the hypotheses of Corollary *
3.10, because a positive operator on a commutative C -algebra is completely positive.
3.12
THEOREM Let
be a compact Hausdorff space. If there is a positive
2
regular Borel measure
on
A
H
with the support of
A
equal to
then
Q,
has a bounded approximate identity of finite rank positive
CL(C(H))
operators bounded by 1.
Proof. Let
denote the set of partitions of unity
G
such that
C(f)
0 <- g,
each
1,
in
G = {gl,...,gn}
is non-zero, and
g.
n G E G
For each
g. = 1.
and each regular Borel measure
on
u
with
Q
1
u ? a, we let
G,u _
EG,L f =
for all dual
n C -1 gj 1
f E C(Q). Note that
Q, s 0
positive with support using the positivity of
:
is
Direct calculations
j.
is a positive finite rank
EG,U We shall now show that
G E G, U E M(D), u
A}
is the required bounded approximate
CL(C(Q)).
Since each
{EG,u
for all
show that
u
1.
identity for
dense in
U
and its
IIEG,u1I
operator, and that :
is the pairing between C(Q)
the space of regular Borel measures, and that since
M(Q),
{EG u
be defined by
E CL(C(Q))
E
CL(C(c)),
EG,u
satisfies
IIEG,uII <_
1
and since
FL(C(f))
is
it is sufficient to show that
G E G, P E M(c2), u ? X}
is a bounded approximate identity for the
continuous finite rank operators. Each continuous finite rank operator
T
n
on
C(W)
may be written T=
fj 0 uj, 1
where
f. E C(Q)
and
p. E M(W).
45
and TEG,u , we see that it is EG,VT and sufficient to prove that for each e > 0, each f1,...,fn e C(SZ),
Calculating the two products
each II EG, E
there is a
p1,...,un E M(S2)
fj - fj II < e
U
:
1 <_ j
Then
u
= 0
.
for
.u
7
7
Now
L1(ii)
1
<_
j
<_ n.
Since
<_ n.
for all
j-
IIEG,p
j.
e/2, e/2
.
n.
<_
Here
110
and
u > IV iI
ml
is dense in
< e/4 i - hj 111
is a closed invariant subspace in
M(S2)
<_ e/2 +
I
e/2 +
II
1
for
E
G,u
ujll
hj - h .
I EG* u
EG,
I I1
hj - h.II -.u(S2)
-1
for all
G so that IIEG,u h - hil . < v = min{ h E If ,...,f , h ,...,h } = K. For each
n
1
J.
n
defines a probability integral on
is in the closed convex hull of
{h(w)
C(S2)
I- 1
- h(x)l
5 sup {Ih(y) - h(x)I
For each
x E Q.
IE
:
h E C(0)
y E supp g
and
X E Q,
we have
h(x) - h(x)I g,u
m < c
I
- 1
- h(x)I g.(x)
1
G = {g1,...,gm}
is a partition of unity in
C(0).
j,
and thus
: w E supp g.}
that
since
for
We must thus choose p(S2)-1}
h 1*
for all
n E L1(u) L1(u) in
j _ jlll
IIEG,u*
=
j
Further
G E G.
for all
u ? A
C(G)
there are h1,... ,hn E C(S2) such that
II.111,
1 <_
so by the Radon-Nikodym Theorem there are
j
such that
such that
A
E.
V = IV1I +...+ IV nI + A.
Let
u >_
uj - uj II < e for
u
II EG
is the adjoint of
M(c2) + M(S2)
for each
and
and a
G E G
Hence
so
46
IE
h(x) - h(x) I
G, V
m <
: w,y E supp g.}
sup{ Ih(w) - h(y)I
: w,y c supp g.}
<- max {sup{lh(w) - h(y)I
K
Since
gj(x)
1 s j s m}.
:
G
is a finite set, we can find a partition of unity
such that
sup{lh(w) - h(y)l
:
for
w,y E supp g.} < v
in
m
1
C(S2)
and
This completes the proof.
h E K.
Property 13 shows that
t F' xt
:
T -
may be chosen to be an
X
exp(ltll+6)
entire function bounded above in norm by a constant times In general the function
6 > 0.
following argument shows. Let
t N xt A
I xt I <- K1 exp (K2 I t I) I
I
for all t E H.
for all t E H so that
exp(KZItI)
A
contradicts
IIatI 1/t
lim
and
K1
We now have
with
an analytic function
t N xt : H - X
of exponential type. Then there are positive constants
that
x = at.xt
can be factored
x E X
an analytic semigroup and
H ± A
:
is not of exponential type as the
be a commutative radical Banach algebra,
and suppose that a non-zero element t N at
for
I
lxI
I
<_
? exp(-KZ) > O.
K IIa
such I
I K1
This
t-
being a radical Banach algebra.
Here is another result obtained by a variation of this argument.
COROLLARY
3.13
Let
A
be a Banach algebra with a countable bounded
approximate identity (bounded by 1). If continuous function such that t f at
semigroup
tI
H i A
such that
t N yt
:
is a
[O,-) - (O,1)
t +
as
then there is an analytic
Ilatlll/Itl
for all
yltl
t E H
with
1.
>-
Proof. Let
at = yt
function from at ? 1 + 6
with
:
yt -> 0
I
IxI
l
[O,-)
for some
= 1,
for all t E C
1
for all into
t E [O,°°).
(1,-)
6 > 0
and factorize
with
and all
x
Then
at ± .
t > 0.
Let
t N at
x
and hence
be an element of
as in Theorem 3.1. Then
ti ? 1 so that IIathI ? Ilxtll -1 ? (yltl) Itl for all t E H
is a continuous
t - -,
as
I Ixt I
I
< (.It,)
A Itl IxI
with
with
Iti ? 1.
This corollary says that in a radical Banach algebra with a bounded approximate identity there are analytic semigroups with
I1-till/It,
l
I
47
tending to zero arbitrarily slowly as Ilatlll/t
In 5.3 we shall see that
tends to infinity with
Itl
t E H.
cannot tend to zero arbitrarily fast as
tends to infinity.
t
We know from the analyticity of to
x
as
t [' xt
:
that
T -> X
tends to zero. Property 14 says that for a preassigned
t
and bounded region of
we can ensure that
tends
x t
S
lix - xtll < d.
The only motivation that I have for extracting property 15 is that it gives the Banach algebra generalization of a type of bounded approximate identity that has played a crucial role in two deep results in C -algebras (see Arveson [1977] and Elliott [1977]). We shall briefly define the type of approximate identity obtained from property 15 but we shall not consider its use in C -algebras. Let
(X
n
be a strictly decreasing
)
sequence of positive real numbers converging to zero with E 0 = aX1/2
and
E
n
n
(aXn+1 - aAn)1/2
=
for all
is a bounded approximate identity for
E,2
n E IN.
al < 1,
and let
Then the sequence
It is this form of the
A.
3
approximate identity together with the special order properties of C algebras inherited by the sequence
(En)
that are used in Arveson [1977]
and Elliott [1977]. Let
B
be a Banach algebra containing the Banach algebra
as a closed ideal. Then quasicentral for subset
and
K
of
A
if for each finite subset
B
and each
B,
e > 0,
F
there is an
Ilea - all + llae - all < e for all a E F,
b e K.
A
is said to have a bounded approximate identity
and
of
each finite
A,
e E A
with
llell s 1,
llbe - ebll < e for all
This definition can easily be translated into one about nets. In
appendix A3 we show that an Arens regular Banach algebra
A
with a bounded
approximate identity has a quasicentral bounded approximate identity for all enveloping algebras of
3.14
However the following problem seems to be open.
PROBLEM Let
A
be a Banach algebra with a bounded approximate identity
bounded by 1, and let A
A.
Mul(A)
be the multiplier algebra of
A.
have a bounded approximate identity that is quasicentral for
When does Mul(A)?
If the bounded approximate identity in the Banach algebra
A
has nice properties with respect to a suitable set of derivations, multipliers or automorphisms, then these properties may be inherited by
at
as
48
t -. 0 by suitably choosing
at.
This is the intuitive idea behind
Theorem 3.15 which continues the properties of Theorem 3.1.
THEOREM
3.15
A
Let
be a Banach algebra with a countable bounded approximate
identity bounded by 1 and without identity. Then an analytic semigroup t }* at :
may be chosen so that properties 1 to 5 and 7 to 14 of
H -> A
Theorem 3.1 hold, and that one of the following properties hold. 16.
Let
Z
be a separable Banach space of continuous derivations on
If there is a bounded approximate identity
II D (gn) II - O
t > O, 17.
18.
If
B
for all
as n -> -
for all
is a Banach algebra containing A
quasicentral for
then
b E B.
Let
G
G
then
D E Z,
satisfying
as t -; 0,
II D (at)II -> O
D E Z.
is separable, and if
all
(gn) c A
A.
B,
A
as a closed ideal,
if
has a bounded approximate identity in Ilbat - atbll- 0
as
t , 0,
be a group of continuous automorphisms of
A,
B/A
A
t > 0,
for
and suppose that
contains a countable dense subset (in the uniform norm topology).
If there is a bounded approximate identity
II S (gn) - gn 11 , 0 as
t -> 0, t > 0,
3.16
as
n , - for all
for all
$ E G,
(g
n
)
c A
in
A
satisfying
then 116(a t) - at II - 0
R E G.
NOTES AND REMARKS Most of the properties in Theorems 3.1 and 3.15 are in Sinclair
[1978], [1979a] but in several cases the results are only implicitly there. For example, property 3 is proved in Sinclair [1978] for the interval (0,1] in place of the sector
U(*),
and I first saw the sector result in Esterle's
U.C.L.A. lecture course. We shall prove the properties using the exponential methods of Sinclair [1978] except that we shall deduce 6 and 15 from the functional calculus results of Sinclair 11979a]. Property 15 is essentially a functional calculus property but 6 is not. I do not know how to prove 6 using exponential methods, and the functional calculus factorization is not proved in these notes. The proofs of Theorems 3.1 and 3.15 are discussed in detail in Chapter 4, and are variations and extensions of Cohen's factorization theorem. There are good accounts of various forms of Cohen's factorization theorem in Hewitt and Ross [1970], Bonsall and Duncan [1973], and Doran and Wichman [1979]. The latter notes contain a detailed account of
49
bounded approximate identities and factorization of elements in Banach
modules including the results of Sinclair [1978], [1979a] with little modification. Neither of the books nor lecture notes touch the
nl-
factorizations of Esterle [1978], [1980b], and Sinclair [1979b]. Corollary 3.3 was first proved for a commutative Banach algebra by Dixon [1973]. The form here is in Dixon [1978] and Sinclair [1979a].
Aarnes and Kadison [1969] showed that a separable C -algebra has a commutative bounded approximate identity, and Hulanicki and Pytlik [1972] (see also Pytlik [1975]) proved that
L1(G)
has a commutative bounded
approximate identity. Property 8 seems to be new.
Properties 7 and 9 were first proved using the functional calculus methods (Sinclair [1979]) but here are proved using exponential methods. Corollary 3.9 is due to Hunt [1956] (see Stein [1970] Chapter III) for
G
a connected Lie group. There have been some investigations of semi-
groups of completely positive operators on C -algebras - see Evans and Lewis [1977], for example. Theorem 3.12 is proved for
0
the circle T
in
Johnson [1970], but his proof gives what we state here. See also Herbert and Lacey [1968].
Corollary 3.13 is the semigroup version of a result in Allan and Sinclair [1976] that in a radical Banach algebra with a one sided bounded approximate identity there are elements arbitrarily slowly. Rates of growth of
a
with
Ilanlll/n
Ilanlll/n
tending to zero
and semigroups in radical
Banach algebras are discussed in Bade and Dales [1980], Esterle [1980], and Gronbaek [1980]. *
Quasicentral bounded approximate identities in C -algebras are used in Arveson [1976], Akermann and Pedersen [1978], and Elliott [1977]. See also A3.
So
4
PROOF OF THE EXISTENCE OF ANALYTIC SEMIGROUPS
In this chapter we shall prove Theorems 3.1 and 3.15, and the
various lemmas required in the proofs. In 4.1 we sketch the ideas behind the proofs, and after proving all the lemmas we prove 3.1 in 4.7 and 3.15 in 4.8. Throughout this chapter
will denote a Banach algebra with a
A
d(?l), X
countable bounded approximate identity bounded by
will denote
a left Banach A-module satisfying Ila.xll <- Ilall.llxll for all a E A and x E X, {a.x
:
and
will denote the closed linear span of the set
1A.X]
Taking
a E A, X E X}.
We assume that
4.1
simplifies the calculations slightly.
d = 1
does not have an identity.
A
SKETCH OF THE PROOF The proof is a variation of Cohen's factorization. theorem
(Cohen 11959]) with the analytic semigroup obtained as a limit of exponential semigroups in the unital Banach algebra
The variation is
A#.
influenced by the proof of the Hille-Yoshida Theorem. If the algebra
A
had
an identity, then the factorization results would be trivial as we could take
for all
at = 1
Though our algebra does not have an identity,
t E H.
we shall use the case when there is an identity and an approximation to prove 3.1. We work in the algebra identity to
A,
and we regard
modules by defining
l.w = w
A# = A ® Cl and
X
and
as left and right Banach
Y
u.1 = u
for all
In the proof we choose inductively a sequence approximate identity
b
t
n
A
= exp(t
converges to an element
of A n 1
)
n
and
A#-
u E Y.
from the bounded
(e. - 1)) E A# J
at E A
xt
(e
w e X
such that
for all
on the inductive choice of the sequence converges to an element
obtained by adjoining an
in
X
t E H. (en),
for all
With suitable restrictions the sequence
t E C.
(b
_t. x)
n
The analyticity of the
51
functions
H - A
t 1+ at :
that show that
and
at
and xt
t F' xt :
usually does not converge and
C - X
follow from the calculations (bnt)
In a Banach algebra the sequence
exist.
should diverge rather badly as
(b -t.x)
n
n
tends to infinity, but this does not occur with the correct choice of the sequence
because
en
acts on
(en - 1)
approximate zero divisor. That than in
A#
for all
t c H.
is because
(b t)
n
and
x
like an
ell..., en-1
converges to an element in
b t E exp(-nt) ® A
and
n
exp(-nt)
The crucial semigroup property of
-> 0
rather
A
n - -
as
is obtained
t ' at
from the corresponding semigroup property of the exponential groups t 1 b t in the unital Banach algebra A To obtain further properties of n at or xt one simply imposes further restrictions on the choice of the .
sequence through
and has calculations linking
(en),
b t
b
and
n
en
with
at
and
xt
Convexity and convex combinations play a vital
-t. x.
n
role in the proof, and the semigroup
at
for
is essentially just
t > 0,
a weighted average of a suitable approximate identity. The averaging smooths the given approximate identity into a nicer one. This idea is clearly illustrated in the proofs of Lemma 4.5 and property 9 of Theorem 3.1. The
b t
n
corresponds to
exp t(A2(A - R)-1
in the proof of the
- A) n_
Hille-Yoshida Theorem (6.7), and heuristically
1(e1 +...+ en - n)
approaching the infinitesimal generator of the semigroup
is
t F at.
The first lemma is the standard opening to the stronger forms of Cohen's factorization theorem.
LEMMA
4.2
w
If
then there are
e E A with
is in the closed linear span of
al,...,am E A
hel
l
<- d
and
I
and
such that
l eaj - aj I I< 6 for all
al,...,am E A
Proof. There are
d > 0
A.X
and
and if
Ilew - wII < E for each
j.
wl,...,wm E X
so that
m
II w -
Thus
I
i ajwj II
I ew - w I
< E/3d.
l
m
m Iledj-
for all
e E A.
ajll IIwjII+(IIeII+1)II w - X a .wjll
We may now choose
6
c > 0,
to give the result.
52
When the algebra is non-commutative Lemma 4.3 is required in the proof of 4.4 : however 4.3 is not required if the algebra is commutative. Lemma 4.3 is used to get around the failure of the formula exp a.exp b a
and b
in
A
if
A
exp(a+b) =
is non-commutative. This formula does hold if
commute.
LEMMA
4.3
If f e A and if n =IIfII + d + 1, then (a)
II (f + e - 1)k - fk - (e - 1) kII
(b)
nk {II (e - 1) f II + Iif(e - 1) II}, II (f + e - 1) kw - f.w ll <
n
for all
k
and
{ II(e - 1) fli I _ II + II f (e - 1)II IIwII + II(e - 1) wII }
e e A with
all
IIeII <_ d,
and all
k e IN,
Proof. We multiply out the power (f + (e - 1))k, (e - 1)
do not commute, and after taking
same side as
(f + (e - 1)) k
fk
w e X.
remembering that
and
(e - 1) k
f
and
over to the
use the norm inequalities to obtain
II (f + (e - 1)) k - fk - (e - 1)k II k-1
Ik-1-1
IId
1
((k) j- 1){II(e - 1)fII+IIf(e - 1)II }
1113-1
IIe -
1
k--1
-1-J
IIfII
L
/
I\`
k--1 1
<
j) -
e-1113-1
II
1
I
-1-3
1)
/
(d+1)3-1 (j) 1
1
(11f11 + d + 1)k
This proves (a). From (a),
II (f + e - 1) kw <_
fkwll
II (e - 1)kwl1 + nk{II (e -
IIf(e - 1)II IIwII}
53
< nk {lle - 1).wii + II (e -
1)
The following is the main technical lemma used in the proof of Theorem 3.1.
4.4
LEMMA
f + ul E A# with f
Let (a)
A,
and let n = Il fII + d + 1.
Then Ilexp t(f + e - 1) - exp tf - exp t(e - 1) + 111
(exp (nltl) - 1) (ll(e - 1)fil + Ilf(e - 1)II)
<-
for all e e A with
W E X
If
depending on
(b)
and all
Ilell < d
and
Ilfll,Iul,
M > 0,
M,
and
t E T.
then there are constants d
and
such that
llexp t(f + ul + e - 1) - exp t(f + ul)II
s exp Re(tp)
.
{expltl (d+l) - 1 + C(II(e - 1)fii +llf(e - 1)11)
and (c)
llexp t(f + ul + e - 1).w - exp t(f + ul).wll
E{ II(e - 1) for all
with
t e C
Itl
II f(e - 1)11.11wII + II (e - 1) .w II} <_ M
and all
e e A with
llell
<_
d.
Proof. Expanding the three exponentials in power series in the algebra A we obtain
llexpt(f+e - 1) - exp tf - exp t(e - 1) +III L
k=l
k=l
J_ L II (f + e - 1)k - (e - 1) k - fkll k!
l nk{II (e - 1) f l l + II f (e - 1) II
}
k!
by Lemma 4.3. This proves (a) Using (a) we have
Ilexp t(f + u1 + e - 1) - exp t(f + ul) ll = exp
exp t(f + e - 1)
- exp tfll
<_ exp Re (tu){Ilexp t(e - 1) - liI + (exp(ItIv) Il f (e - 1)11)}
(e - 1)fhI +
54
<_ exp
with
Re(tM) {expltl (d + 1) - 1 + C(II(e - l)fII + IIf(e - 1)II)
= exp(Mn) - 1.
We have used the inequality
Ilexpt(e-1) -111
exp (Itllle-l1I) -1 exp (ItI(d + 1)) - 1.
This proves (b).
There is a proof of (c) similar to that of (b) using (a). However we proceed by expanding the exponentials in series as in (a) to obtain
Il exp t(f + ul + e - 1).w - exp t(f + ul).wll <- exp Re(tu).
ItI
k=1
< (exp MIpI).
II(f + e - 1)k.w - fkwll
k!
Y
k=l
Mknk {II (e k!
- l)fII.IIwIl + Ilf(e - 1)
II(e - 1).wII } for all
t e C
with
= exp M(n + IpI).
t < M
and all
e e A with
Ileli < d.
We choose
This proves Lemma 4.4.
We now sketch a proof of the commutative case of Lemma 4.4(a) avoiding the use of 4.3. If
f
and
e
commute, then
Ilexp t(f + e - 1) - exp tf - exp t(e - 1) + 111
= II(exp t(e - 1) - 1) (exp tf - 1) II k=l <-
for all
1tl
IIe-IIIk-l
II (e - 1) (exptf-1)II
k!
(exp (Itl(d+ 1)) - 1).(exp ItIIIfIl-
t e T.
1)fII
This is an estimate like 4.4(a), and may be used in place
of 4.4(a) to deduce (b) and (c). The following lemma is used in the proof of Property 7 of Theorem 3.1, and provides control of the spectrum of a bounded approximate identity.
55
4.5
LEMMA
A be a commutative Banach algebra with a bounded approxi-
Let
mate identity
If
A.
there is
a > 0,
A
o
such that
where
Jim a(f)I 5 a,
Proof. Let
is convex, then given
A
f e A
is the spectrum of
0(f)
= {g e A :IJgaJ . -
aJ.II_a for
convex bounded approximate identity for a positive integer
al,..
A
1
am e A
,
for
- aJ,Il <_ a
IlfaJ .
and
j
<_ m
Then
A o
<-
1
and
f.
<_
<_ m}.
j
bounded by
is a
(say). We choose
d
with
n
n > 2da-1,
(1)
IIeke j - e7 .II
2
/4 for 1
<_ a
Then
f = n 1(e1+...+en).
Let
such that
ell...len e A0
and by induction
j < k <_ n.
<_
f e A
and
Ilfa
(2)
- a
.
J
since
A
0
is convex. We check that
by showing that if If
0 e
and
(D
e
for
If(e .)I
for
1
<_
j
then
A,
j 5 n
IImo(f)I <_ a.
<_ n.
11
Ileke. - ejll >-
which contradicts inequality (2). Now <_ a/2
<_
j < k.
1
If both of these inequalities fail, then > a 2/4,
1
or
<_ a/2
<_ a/2
Ik (e.)I
for
then either
1 <_ k < n,
If(ek) - 11
<_ a
has the required spectral property
f
the carrier space of
(D,
,ll
J
I¢(f)I
a/2 < a
So suppose that there exists
j
provided such that
J
I4(ek)I
<_ a/2
for
1
<_ k <
for
j
and that
j
(k(ej)I
> a/2.
(3)
Then
-11 and so
Jim
( e k )
<_ a/2
for
< a/2
k a j
< k <_ n,
and
ek (k x j )
1 < k < n.
Let
I
w = (n - 1)1(e1+...+ eJ,'1 + ej+1+...+en). of the
(4)
and I
I w l
< d.
Thus
So
w
is a convex combination
f = ( 1 - n 1)w + n-le,
i
3
and
56
IIm 4(f)I IIm 0(w)I + 14(w) - ¢(f)I
<_
<_ a/2 + n 1 If(e.) - $(w)I
<_ a/2 + n 1(d + d) <_ a.
This completes the proof of Lemma 4.5.
REMARKS ON THE PROOF OF THEOREM 3.1
4.6
In the proof at certain stages simpler choices are indicated for the sequences constructed that provide a proof of all but a couple of the properties. At a first reading of the proof it is easiest to take the simplest choice at each stage. We shall use Lemmas 4.2 and 4.4 for right
Banach A-modules as well as for left ones. The right versions may be deduced from the left ones by reversing the products and turning right modules over the algebra into left ones over the reversed algebra.
PROOF OF THEOREM 3.1
4.7
To prove properties 13 and 14 we require a suitable increasing sequence
(y
n
of positive real numbers, which will probably be tending to
)
infinity very fast. If one is not interested in properties 13 and 14, then the choice
y
at ; -
(y
such that
)
gives all the other properties. Using the hypothesis
= n
n
that
as
we choose an increasing sequence of real numbers
t
(1) Yo = O, yn Z max (C,n), and 1 + exp 2n <_ at for all t 2 yn.
Here
is the constant of Property 14. We suppose that
C
(1)
(and
IIxii 5 1
IIyMI 5 l) . Let for
A.
{g
n
:
n E 3N}
be a countable bounded approximate identity
We shall be choosing a sequence
identity for
A,
(e
n
)
from a bounded approximate
but probably not from the sequence
we choose a sequence such that
(2) IIen1I <_ 1 (3) en e A,
(en)
(gn).
By induction
from a bounded approximate identity
A
for A
57
(4)
IIm a(en)I
1
<_ 2
provided A
ll(en - 1) (e1+...+en-1)
II
is commutative, and
(el+...+en-1) (en 1)
II
(en - 1) (e1+...+en-1)11 IIuII + II(e1+...+en-1) (en - 1)II IIuII
+ Ilu. (en -1) II M
are so small that if
bot = 1
for all t o T and all
and
bmt = exp t
X
(e. - 1)
(j=1
m E F,
J
then
11b
(5)
t11 5 exp(-(n-1)Re t). {exp(2Itl) -1 + 2-n-1} n-lt - bn
and
(6) Ilb1 n-'w Ilu
and
bnt.wll 5 2-n
bn-lt- u.btllc 2-n d n
for all
t E C
with
and all
u E {y,gl " ...gn}.
Iti
<_ yn,
Note that in (2) we have taken
and all
lie n
11
:5
w E {x,gl,...,gn}
1 <_ d but that the
choice of the sequence and the first few calculations just require lie
n
11
:5
In (4) and (6) one of the norms is evaluated in a Banach module
d.
and the others are calculated in
We shall use
A.
en e A
in the proof of
Property 9 (of Theorem 3.1), and we shall use inequality (4) in the proof of Property 7 and hence of Property 8. We shall now choose
by induction so that (2) to (6)
el,e2....
are satisfied. By Lemma 4.4(c) applied with
constant Iti
e1 e A
C
and all
5 yl
e e A
with
f = 0
and
y = 0,
there is a
5 II(e - l) .wlI for all t e C with
such that Ilw -
d.
lie II
Then, by Lemma 4.5, we choose
so that inequalities (4) and (6) are satisfied. We require
convex to apply 4.5. Recall that
IIAII 5 1
so (2) holds. If we are not using choice of
e1
is possible because
A
to be
by the hypothesis of Property 9
we must ensure that
A
A
(Ie11I <_
1. The
is a convex bounded approximate
58
identity for
A
bounded by
and the inequalities for w = x
1,
are handled using Lemma 4.2. Suppose that satisfy (2) to (6). We choose +e
and
n
by applying Lemma 4.4 with
en+l
u = y
and
have been chosen to f = e1+...
and using Lemma 4.5 to take care of inequality (4). The
u = -n,
choice of
el,...,en
such that (2) to (6) hold is possible because
en+l E A
is a
A
convex bounded approximate identity bounded by 1. Here we are again using Lemma 4.2 to ensure that
and
II(en+l - 1).xII
can be chosen
1)II
to be very small. This completes the inductive choice of the sequence We now define semigroup. For each Cauchy for
t f' at
in the set
t
inequality (5) implies that
v > 0,
{z E Q
Re z ? v-1,
:
(en).
and check that it is an analytic
H } A
:
(b t)
n
< v}.
IzI
is uniformly
This is because
we have
Ilbnt - bn-tII <_ exp 2v.exp(-(n - 1) Re t) <_ exp 2v.exp(-(n - 1)v-1)
yn ? v
for
and
with
t c T
and
Re t >- v-1
Hence
v.
:-
Iti
lim b
exists in
A#
t c H =
for all
{z E T
u
:
Re z ? v-1,
v}.
IzI
t
n
n
Writing
v>O
out the power series for lim b n
t
E A
for all
we see that
b t n
n
We denote this limit by
t E H.
A
b t E exp(-nt) t
a
.
and thus
Each function
n
t
t
b n
t
at
t
b t
to
at
H - A
:
n
t N at
H -
:
A
H - A#
is a semigroup homomorphism from
H } A
is analytic, and the sequence
v > O.
n
converges uniformly
<_ v}.
Therefore
t f yt
:
xt = lim
bn t.x
and we note that there is
xt : C ± X,
t
C -> Y.
is uniformly Cauchy for Let
so
(A#,-)
is analytic.
a similar definition of -t. x)
(b t)
Re z ? v-1, IzI
{z E C
We shall now define
(bn
into
is a semigroup homomorphism. Also each function
on the compact set :
(H,+)
t
in
for each
By inequality (6) the sequence {z E C t c C.
IzI
:
Then
<_ v)
for each
t f- xt
:
T - X
is
n
analytic. We can also define
t F' G
m,t
where
Gm t = lim bn t.gm,
n
Km't = lim gm.b
n
-t
and
and
t F* K
m,t
from
C
into
A.
59
for all t E C
and m E IN.
With the construction of
at, xt
yt
and
complete, we check
the various properties of Theorem 3.1.
Property 1. This property follows from the equality and
t E H
and from the definitions of
n E]N,
Property 2. Because
{gn
gn
it is sufficient to show that since for
A = (ugnA)
Property 3. If
t E H
Ilat
-
E
gn e atA
so
with
Iti 5 Ym+l'
11 bk+l
L
- bk
for all
X
and bot = 1
? 1
llatll
t E H
<_
2-k-2
exp(-k Re t) + k=m
for all
t E T,
we have
2 + (exp(2It1) - 1)(1 - exp(-Re t))-1
with
Itl
s 1.
t E UM = {z E then
t E H.
(exp(21tl) - 1)(1 - exp - Re t)-1 + 1.
<-
(C
if
:
0<
IZI
1, IArg zI 5 U},
Re t ? Iti cos P so that llatll <_2 + (exp 2ltl - 1) (1 - exp(-Itl cos ))-1 2+
2
cos J
Hence
{I I at I
I
:
t E U()) is bounded.
as
and
I t I -> O.
t E H
(Property 1)
then by inequality (5)
k=m
for all
n E IN
x = at.xt
ll
5 (exp(2ItI) - 1)
y
xt.
- bmtll
k=m
Since
and
for all
(at.A)
The equality equivalent to
.
gn = at.Gn t
is
gn
at
for
is a bounded approximate identity for
n EIN}
:
x = b t. (b -t x) n n
A,
60
b E A
Property 4. If
with llb - a1.cll <-
and
then by Property 2 there is a c E A
c > 0,
2-1 (1 + sup {IIatII
e
t E U(tp)})-l. We are using
:
Property 3 to ensure that the supremum is finite. Thus
Ilat.b - bil (IIatII + l)Ilb - alcII +
Ilat+l
<_
-
E/2 + E/2 for
with
t E U(i)
Property 5. If
sufficiently small.
Itl
t > 0,
then n
I1bnt11 < exp(-nt). exp(t III-; II) 1
<- exp(-nt).exp(nt) = 1
for all
n E IN. Thus II atll <_ 1
for all t > 0.
Property 6. I do not know how to prove this property by the exponential methods that we are using here. We shall see how it can be deduced from the factorization theorem using the analytic functional calculus in Sinclair 11979a]. However we must then check that these functional calculus methods
will give all the other properties of Theorem 3.1. The functional calculus methods require a commutative Banach algebra with a bounded approximate identity, and we obtain this commutative algebra by an application of Theorem 3.1 to
A.
By Theorem 3.1 with all the properties except 6 and 15
we obtain an analytic semigroup
t F at : H -> A
Property 6 to prove 7 to 14). We let generated by the set Property 4,
x E B.X,
{at
:
t > 0).
Y E Y.B,
and
B
(of course we must not use
be the Banach subalgebra of
Then B n A
(B.A)
= A = (A.B)
A
by
is a bounded approximate
identity for B by Properties 3, 4, and 9. We shall apply Theorem 1 of Sinclair [1979a] to B and the left and right Banach B-modules X
and
Y.
An examination of the statement and proof of Theorem 1 of Sinclair [1979a]
will show that many of our properties are already included in that theorem either explicitly or implicitly in the proof. We are taking for all
t E H,
where
a
at = 2t O(wt)
and w are as in Theorem 1 of Sinclair [1979a].
From the working after inequality (8) of (iv) of that Theorem, for
0 < a < x and y E]R, we have
61
(wx+iy)
2x
II
110
< Ilwx+iylla .
and
E > 0
Here
K > 1
2x.{£/2 + 201.(2K - 1)(1 - K a)- 1},
are constants chosen at the beginning, and
Ilwx+iylla
sup{Izx+iyl.lzl-a
=
We take
Iz - 1/21 < 1/2).
z E T,
and using the maximum modulus theorem write the supremum in
K = 2
terms of
:
on the boundary
z
{z E C of the open disc
z = r elV, r = cos iy, - 7/2 < V < a/2}
:
{z c
Iz - 1/21
:
llax+iyll
<_ Ca .2x
C
a
.
< 1/2}.
sup {(cos
x
y \12
Ca.2x. rl +r y l2
Then
f)x-a.
7r/2
-(x-a)/2 -(x-a)/2.
7T/2}
<_
exp (y arctan
y
exp (lrIyl/2).
\x-a/ The most straightforward inequality would be
Ilax+iyll <_
Ca.2x exp(Trlyl/2)
and even this I cannot obtain by exponential methods.
Property 7. This property requires an initial application of Theorem 3.1 except for properties 7 and 8 so we can pass from subalgebra
B
of
A.
A
to a commutative
The reason for this is that our spectral calculations
require an application of Lemma 4.5, which has the hypothesis that the algebra is commutative. By Theorem 3.1 with properties 1 to 5 and 9 to 14 we choose an analytic semigroup
t F* at
:
H 3 A.
Note that we must be care-
ful not to use 7 or 8 in the proof of the other properties. Let Banach subalgebra of y E
(Y.B)
in
B
as
(A.B)
,
A
A
generated by
= A = (B.A)
does in
A.
,
and
(at
:
A n B
t > 0).
be a character on B#.
If
t > 0,
X E
B be the (B.X)
,
satisfies the same hypotheses
We now choose the sequence
satisfy (2) to (6), and proceed as before working in Let
Then
then
(en)
B#
in
B
instead of
to A#.
62
IArg ¢(bnt)I n
IArg exp (t I
-1))I
n
Im
= t n
1
1
Thus
IArg (at)I 5 7Tt/2
for all
t > 0
Since the boundary of the spectrum of the spectrum of
at
in
A, o(at) c_ U(7rt/2)
in
same idea we obtain
and all characters
at
If(at)I
_< exp (7r Imt/2)
on
B
.
contains the boundary of
B
for all
t > 0.
Using the
for all
t E H.
This proves
property 7.
Property 8. As in the proofs of Properties 6 and 7 we pass to the commutative Banach subalgebra
generated by
B
{at
If
t > O}.
:
algebra, then we can choose any semigroup
t f' at
is a radical Banach
B
satisfying all
H ; B
:
of the properties of Theorem 3.1 except for 8, and it will then satisfy Property 8 by Theorem 3.6. Thus we may suppose that the carrier space of 1
is not empty. We consider separately the two cases
B
'
compact and
(D
locally compact.
Assume that
(e) = 1
'
is compact. Then there is an idempotent
for all
e
(P
and Duncan [19737, Theorem 21.5). Now is a continuous semigroup bounded by algebra
(1 - e)B.
were zero, then since
= (1 - e)B
(1 - e)
because
,
:
(O,-)
t > 0.
(1 - e)B
-
e
If
with
f E L1 O2+)
(1 - e)at dt = 0
(1 - e)B
so A would have an identity, ((1 - e)at 8(f) = 0,
f = 0 by Theorem 3.6
so
is a non-zero radical Banach algebra.
Now assume that
(D
:
contrary to hypothesis. Further
(D
is not compact. Let
countable bounded approximate identity in
Wn = { e
such
into the radical Banach
is non-zero because if
(1 - e)B
for all
8(f) = 1 f(t)
(1 - e)B
(1 - e)at
t
(1 + IIell)
would have an identity
= A = (B.A)
(A.B)
(1 - e)B)
then
B
Note that
e E B
(see Rickart [19607, Theorem 3.6.3 or Bonsall
1/2}
for each
B, n E IN.
{h
n
:
n E IN}
be a
and let
Then each
Wn
is an open
63
subset of
with compact closure, and
'
OWn = (D because n
an approximate identity in B. Therefore sequence
in
(fi) n
such that
(D
n E]N}
:
is
a-compact. We now choose a
is
0
{hn
for all
n
0n (b) - 0 as n -
b c B.
adjusting for the fact that
For example, we could take n E Wn+l \ u W, 1
n
may often be empty but is infinitely often non-empty; if
Wn+1 \ U W. 1
b E B n
and
for some
u W.
in
1
than
c > 0,
{¢ E 0
and hence
n,
is compact so is contained
E}
:
for all
e
We now
k > n.
3
choose a continuous function
h( ) ? 1/n sequence
on
h
vanishing at infinity and satisfying
n E IN - we are assuming that the points in the
for all
are distinct. We regard the Banach space
(¢n)
continuous functions vanishing at infinity on a.f = Af
defining
Gelfand map from identity,
a E B
for all into
B
C
0
((D).
is dense in
B.C0((P)
and
of
as a Banach B-module by
f E C0(f),
Since
CO(D)
A
where
is the
has a bounded approximate
B
We apply Theorem 3.1 with all the
Co((P).
properties except 8 to the Banach algebra B, the left Banach B-module C ON ® X, 0 in
the right Banach B-module and
C (f) ® X 0
h ® x
and the elements
Y,
Then there is an analytic semigroup
Y.
and
t f bt
y
H + B
satisfying Theorem 3.1 except for 8, and an entire function t }+ ht
T - Co((D) :
h = bt.ht
such that
The semigroup
r > 0.
the map is a
t [* ¢n (b
5 1
t
tends to zero as
infinity as
n
n.
Let
of
=
8)1 >
f E L1CR+)
Then
for all
for all
n.
Since
t E H. A
b 1
n
n E]N,
Since
E Co("), n(b1)
tends to infinity. Thus
I h (an) l I I h1 l I-1 > so that
n(b1)
Hence the series
f.
is the required semigroup. For each
Re a
n
tends to
tends to infinity. Also
l exp (h(en)
for all
a(bt) c U(rT/2)
is a non-zero analytic semigroup, and so there
t > 0, Re Sn ? 0
for
Further
n(b t) = exp(-t6 n)
= exp(- 8n)
since
t E H.
bt
t
H } C
such that
Sn E T
Ilbtll
for all
with
1/Re
an
0(f) = 0,
-1 n-1 I I h1I I
Re an 5 log n + log Ilhlll
summed over and let
Lf
n
with
Re 8
n
x 0
for all diverges.
denote the Laplace transform
64
f(t) exp(- Snt) dt
(Lf)(Rn)
f(t) bt dt) ='n (I0 J)o
_ n 6(f) =0 for all
n.
By Corollary A 1.4 it follows that
Lf (and so f) is zero.
This proves property 8.
Property 9.
The sequence
convexity of
A
(en)
implies that
was chosen in (3) to be in = n
fn
e
1
1
under powers ensures that
fn3 e A
e A,
The
A.
and the closure of
A
3
for all
j, n E IN.
Thus n bnt = exp(t
(e. - 1)) 1
exp(-tn) exp(tnfn) exp(-tn)
(tn)' fn
I
j=0
E exp(-tn) {l + (exp (tq) - 1) Al for all
t > 0
and all
n E IN.
Taking the limit as
we have
n
at e A for all t > 0. Property 10.
and
This property follows directly from the definitions of
xt.
Property 11. If n,
t e H
and
and taking limits as
n
z E T,
bn-z.x = bnt.(bn z-t x)
then
x e (A.x)
.
n E IN.
Thus b tx e exp nt. (x + A.x) E- (A.x) n Therefore xt a (A.x) for all t e T.
Property 13. If
t e T
with
for all
tends to infinity gives the result.
Property 12. Lemma 4.2 and the hypothesis concerning
all
at
Ym <
itl
<
Ym+l'
then
x
ensure that
for all
t e M
and
65
IIxtII <
- bkt.xll
Ilbk-i.x
1
k=m+l 2-k
exp (2mltl) +
<-
L
k=m+l
by the definition of
the normalization
bm-t,
IIxII
and inequality
1,
<-
(6). Thus
IIxtII S exp(2mltl) + 1 by inequality (1). Thus
IIxtII 5
<_
(exp 2m + 1)
(altI)Itl
ItI < ("It, )Itl
for all
t E C
With
for
C > 0
.
ItI
? 1.
inequality``
1 + r <_ (1 + C)r
Note we have used the
Property 14. If
with
t E T
ItI
<_
C,
then
<- yl
ItI
and
r >_ 1.
and
2-k 6 = 6 1
by inequality (6).
Property 15. The only way I know of proving this property is by the analytic functional calculus methods of Sinclair [1979]. It is really a functional z f' (za - zs)t =
calculus result. Those methods work because the function exp t(log (za - z6)) is analytic on the open disc
and is dominated by
IzI
for some
> O
disc. Thus the function is in the algebra
as C
{z E C
IzI 1 0
:
Iz - 1/21 < 1/2},
with
z
in the
of Theorem 1 of Sinclair
[1979].
This completes the proof of Theorem 3.1 and we turn to the proof of Theorem 3.15.
4.8
PROOF OF THEOREM 3.15 In proving the three properties of this result we have an extra
condition to impose on the choice of the sequence ensuring that
(e
n
)
(en)
- a condition
is nice with respect to the derivation, multiplier, or
automorphisms. We shall deduce Property 17 from 16.
66
Property 16. Let
{D
n
N}
n
:
be a countable dense subset of the unit
ball of the separable Banach space initial choice of the sequence
of derivations on
Z
In making the
A.
at the beginning of 4.7 we require
(en)
the additional inequality
(7)
2-n-l e-1
11D .(e )II <
for 1 5 j 5 n,
n
7
and the stronger version of (5)
(5')
Ilbnt
- bn-1 + exp(-(n - 1)t) - exp t(en - n)II 2-n-1
<- exp(-(n - 1) Re t).
for all
with
t e C
and all
Itl :5n
n e N.
The inductive choice of the sequence
(e
n
proceeds exactly
)
as in 4.7 with the additional inequality (7) built into the choice. We are able to ensure that (7) is satisfied because of the hypotheses on Z.
We obtain (5') from Lemma 4.4(a) with
by choosing
II(e - l)fII
t-1 Ilbnt
and
IIf(l - e)II
f = e1+...+ en-1
and
A
and
e = en
very small. From Lemma 4.4(a).
+ exp -(n - 1)t - exp t(en - n)II
-
= exp (-(n - 1) Re t). Ilexp t(f + en - 1) - exp tf + 1 - exp t(en - 1) II
5 exp(- (n - 1) Re t).(exp (nltl) - 1).
{II(en - l)fII + Iif(en - 1) which gives (5') on choosing
en
so that
II},
II(en - l)fII
and
IIf(en - 1)II
are small enough. This completes the inductive choice of the sequence
The derivations D(l) = 0.
D
in
Z
are lifted to
A# by defining
Using the formula for the action of a derivation on the
power of an element we obtain
I I D (exp t en) II 5
L
Iti'
k
5 Itl IID(en)II exp Itl
IID(e,
II
(en).
kth-
67
for all
t E
and all
(t
D E Z.
From inequality (5') it follows that
IID.(bnt) - D.(bt-1)II
n
<_ exp (-(n - 1) Re t). 2-n-1 + exp(-n Re t). IID.(exp t en)II <_ exp (-(n - 1) t).2-n
for O < t <_
and 15 j 5n.
1
Thus if O < t <-
j < m E N,
and
1
we
have
I
I Dj (at )
<_
I I
II Dj (bmt) II +
IIDj (bnt) - Dj (btn-l )
E
n=m+1
o . (bm t - 1) I I + < 11bm
t
II
exp (- (n - 1)t).2 -n n=m+l
- 111 + 2-m
S exp (2mt) - 1 + 2-m
Let
c > O.
Then there is an
m
and corresponding to this Therefore
0 < t < v.
for
there is
m >
such that
v > 0 for
IID (at)II <_ E
such that
j
2-m < E/2,
exp(2mt) - 1 < E/2
0 < t < v.
Since
7
{Dj
:
j E IN}
{IatII:
is dense in the unit sphere of
0 < t < 1}
Z
and since
is bounded, the proof of Property 16 is complete.
Property 17. We shall deduce this from Property 16. Let countable subset in
such that the set
B
{c
+ a
.
:
{c.
j E N}
be a
E IN}
j
:
is dense in
7
B/A.
Let
Z
derivations
be the closure in
BL(A)
of the linear span of the set of
ad(cwhere ad(cj).a = cja - ac.
for all
existence of a quasicentral bounded approximate identity in implies that there is a bounded approximate identity that
ad(c ,)g
0
as
n -> =
for all
j
a E A. for
A
in
(gn)
A
The B such
From Theorem 3.1 and 3.15,
E IN.
7
Property 16, we obtain an analytic semigroup I ID(at) II
0
as
t } 0, t > 0
for all
of all continuous linear operators t i O.
Because the set
subalgebra of
BL(A)
{IIatII
:
T
t
D E Z.
on
0 < t <_ 1}
A
at
:
such that
H -; A
Finally let
satisfying is bounded,
E
T(at) E
be the set -} 0
as
is a Banach
that does not contain the identity operator. From
68
the construction of E
and
ad(a) e E
the space of derivations
t N at
a e A
for all
Z
is contained in
(by 3.1, Property 4). This proves 17.
Property 18. The proof of this is similar to the proof of property 16 - we give some of the details. We require inequality (5') as in the proof of 16. Let
{8
n
;
n EIN)
be a countable dense subset of
on the approximate identity
A
G.
Using the hypothesis
we may ensure that the sequence
(e
n
)
satisfies
(8)
118j(en) - enll < 2-n for 1
The automorphisms For each
on
A
<-
<- n.
j
are lifted to
n E IN,
a E G,
and
A#
by defining
0 < t,
we have
II6(bnt) - 8(bn-1) - bnt + bn-t l
(l) = 1.
II
II nb t - bn-1 + exp (- (n - 1) t) - exp t (e n - n) +II(8 - 1) exp t(en - n) II. <5
(I 18 II + 1)
The last term is bounded above by
exp (-tn)
I tk II8(en)k - enkll 1 k!
(II8II + 1) k -- exp (-tn) E tk Il8(e) n Iien n -ell 1 k! < exp (-tn) . {exp (t (11 $11 + 1)) - 1} 118(en) - enll Combining these two inequalities,
(8), and (5'), we have
Il8.(bnt) - 8.(bn-t) - bnt + bn-til
<
(Ila1II + 1) exp (-(n - 1)t).
2-n-l +
exp(-tn) {exp(t(II8 .II + 1)) - 1}. 2-n J
for O < t < 1 and. 15 j
<_ n.
If 0 < t < 1
and
j 5 m, then
s A
-
69
116
(at) - atll
II 8j (bmt) - bmtll + <
mIl
II Bj (bnt) - 8j (bn-t) - bnt + bn-1II
{exp (2mt) - 1} +
(II 8j II + 1)
2-m
(118i II + 1 + exp (tj 8j II + 1) - 1)
by working similar to that used in 16. For given large, and then choose
t
small depending on
M.
c > 0 we choose
m
very
The details are similar
to those in 16. This completes the proof of Theorem 3.15.
4.9
NOTES AND REMARKS ON CHAPTER 4.
Doran and Wichman [1979] discuss Cohen's factorization theorem in great detail in their lecture notes on bounded approximate identities and factorization. There is a historical discussion in those notes and in Hewitt and Ross [1970]. Lemma 4.2 is what permits one to show that the subset
A.X of the Banach A-module
X
is actually a closed submodule. This
step was found independently by Hewitt [1964], Curtis and Figa-Talamanca [1966], and Gulick, Liu, and van Rooij [1967]. Lemmas 4.3 and 4.4 are minor modifications of Lemmas in Sinclair [1978]. Lemma 4.5 is a special case of Theorem 2.2 of Dixon [1980] that is just strong enough for our purposes.
Our proof of Theorem 3.1 is a variant of the proof of Theorem 1 of Sinclair [1978] except that properties 6 and 15 are deduced from Theorem 1 of Sinclair [1979]. In the proof of Theorem 3.1 we approximate not only the single element (g ).
n
x
but also a countable bounded approximate identity
This is the technique used in Varopoulos [1964] and Johnson [1966].
Theorem 3.15(a) and (c) are new but essentially like 3.15(b). Theorem 3.15(b) is proved in Sinclair [1979a] by analytic functional calculus methods.
70
RESTRICTIONS ON THE GROWTH OF
5
INTRODUCTION
5.1
t N at
Let
right half plane
H - A
:
be an analytic semigroup from the open
into a Banach algebra
H
A
such that
= A
(atA)
for
In this Chapter we are concerned with relationships between
t e H.
all
IIatII
the structure of the Banach algebra A approaches the boundary of
and the growth of
approach to the boundary of
H
along vertical lines in
and in a semidisc centred at
H,
as
IIatII
t
in some way. We consider three types of
H
:
along a ray in
H
emanating from 0
in
0,
Each
H.
type is discussed in a section of this Chapter. The more complicated boundary of
compared with 1R+
H
means that it is possible to extract more
structure from restrictions on an analytic semigroup defined on continuous semigroup defined on
than a
H
[O,o).
In Chapter 3 we found that in each radical Banach algebra with a bounded approximate identity there are analytic semigroups
A
t [* at
i6 1/r
such that
Ilare
infinity for
II
161
tends to zero arbitrarily slowly as < r/2.
r
tends to
Rather surprisingly for analytic semigroups
Ilarei6
1/r II
cannot tend to zero arbitrarily fast as
r
tends to infinity.
This is the conclusion of Theorem 5.3, and in Theorem 5.5 we shall show that the analytic property is crucial - without analyticity results of this type fail. Ilal+iyhI
There is a link between the behaviour of and the fine local behaviour of near zero in -1
(Re t)
of
H,
at
for
then
t a
t
(Re t) -1 t e 1 + i]R t moves a into {a
as
near zero in
H.
If
Iyl - t
is
so that a large power, y e IR1.
We can also see this
intuitive idea by considering the fractional integral semigroup t N It zero in
:
H ; L1OR+) IR+,
into
L1(IR+).
Much of the mass of
and so in the region of IR+
I1+iy
occurs near
where approximate identities occur
71
in
I1+iy will have complex values as a function on IR+
though
L1OR+)
Iil+iyl
rather than positive values. To see that much of near zero we consider the interval
but
[0,1],
t 1+ ItI: H -+ L1*[0,1]
well. Now
[O,E]
is concentrated
would do just as
is an analytic semigroup because
[O,1]
the restriction map is a natural quotient from algebra
L1*[0,1].
y2)-1 log+Illl+iy
onto the Volterra
I1 dy
( 1 +
fIR
L1OR+)
By Theorem 5.6
[O,1] 1
J
y2)-1 log+
I (1 +
I1+iy (w)I dw} dy
{ fJ
O
1
is infinite. Thus much of the area under w 1*
concentrated near w = 0
for large
lyl.
must be
: 3R+ + ]2+
Il+iy(w)
Of course we could calculate
this directly for the fractional integral semigroup, but the general idea above applies to any analytic semigroup from
into
H
L1OR+).
In Theorem
5.6 we shall see that the finiteness of
JR for
t [+ at
:
(1 + y2)_ 1 log+ al+iY I I dY II
H - A
an analytic semigroup with
implies that
= A
(a1A)
there are no continuous non-zero homomorphisms from
A
into a radical
Banach algebra. Intuitively this is because the semigroup near zero looks so like an identity that it cannot occur in a radical Banach algebra. In Theorem 5.14 we shall show that boundedness of the semigroup in the semidisc
{z e H
:
Izi
<_ 1}
centred at
0
in
H
implies that the
multiplier algebra is non-separable. An example shows that there are separable Banach algebras with bounded approximate identities (and hence analytic semigroups) with separable multiplier algebras. There are other restrictions on the semigroup that we have not investigated: for example, how does
IIatII <_
1
and
(atA)
= A
for all
t e H
of the algebra? Semigroups like this exist in
5.2
affect the structure
C0 OR).
GROWTH ON RAYS - LOWER RATES OF GROWTH IN RADICAL ALGEBRAS In this section we shall discuss the growth of
tends to infinity along a ray emanating from plane
H.
0
IIatII
as
t
in the open right half
Firstly the rate of growth is always at most exponential along a
ray because the boundedness of the set
{IIatII
:
t e ray, 1
<_
Iti
<_ 2}
may
72
be pushed along the ray by the semigroup property to give an exponential bound of the form
What about a lower bound? If there is a
C.MItI.
on the Banach algebra such that
character
then there is a complex number
¢(az) x 0
i8
Now if
t E H.
r > 0
for all
and some
along each ray in
z E H,
for all
i6
)I = lexp(preie)I = exp(rk)
Ilare
then
< 7r/21
161
for some
¢(at) = exp(tp)
such that
p
11 >
In this case the growth is exponential
k E ]R.
However if the Banach algebra is a radical algebra
H.
i6 1/r Ilare
we easily see that
11
-, 0
formula. The rate of growth of But how fast does
as
1l-t11
by using the spectral radius
r ->
along a ray is less than exponential.
tend to zero? Corollary 3.13 shows that in a
Ilatll
A with a (countable) bounded approximate identity
radical Banach algebra
for a prescribed rate of decrease to zero there is an analytic semigroup t f' at
:
A
H
such that
specified rate as
Ilatlll/Itl
tends to zero more slowly than the
tends to infinity in
t
Thus analytic semigroups
H.
may be chosen in radical Banach algebras with countable bounded approximate identities such that
tends to zero very slowly. Can they tend
Ilatlll/Itl
to zero very fast? The answer is no, and we prove this in Theorem 5.3. i6
Though
tends to
11
r 1 log flare
as
r
tends to infinity, the
following theorem shows that this convergence to for any
is slower than
-ra
a > O.
THEOREM
5.3
A
Let
A-module, and let x E X
--
with
be a radical Banach algebra, let
X
be a left Banach
t f' at H + A be an analytic semigroup. If
a1. x x 0,
then
lim r a
0
y > 1
and
for all
r4E
(-7r/2,
Tr/2). The convergence is uniform for
W E C-a,a]
for all
a E (0, 7r/2) .
are
Proof. Since (1)
for all
J E
I
iry
l
11 + O
as
r
we have
i>U lim sup r y log IIare .x11 < 0
r
(-Tr,Tr).
We shall prove that the lim inf is non-negative, and
2 2
except for the uniformity of the limit the result will follow from this.
73
Choose
we choose
such that
F(z) =
f(al+(1+z)S.x)
half plane
then
z E H-,
analyticity of
a E
integer
with n
such that
a,
1/2 < Izl
:
F
b
:
and
<_ a
ensures the
a
and also gets around the
H ,
n <_
IwI
< 2,
IwI ? 1,
IArg zI
w and
For this
<_ n + 1.
< a}
then there is a positive
and IIawII :511a w/nlln < Mn <_ MI"I
.
If
z
we have
n
is in
H-,
M1 +
I1+zls
is of exponential type (that is, there are constants
H -- C
such that
F(z)I
for all
<_ a exp(blzl)
z e H),
and
log+ LF(iy)I dy f3R
1 + y2
+ (1 + y2B/2
<
log+M + 1
1 + y2
f3R
log+ (II
fI
x II)
1+y2
l dy I
is finite. By the Ahlfors-Heins Theorem (Theorem A1.1) there is a real number e
c
such that
(-Tr/2, nr/2).
c cos t
lr m r-1 log
For
4 e
for almost all
we have the inequality
(-ir/2, x/2)
r 1 log IF(reio)I <_
for all
r 1 log I I f I I + r 1 log Ilal+G(r) II + r -l log
r > 0,
then
we obtain
IF(z)I <
and
is an analytic function on
F
in the exponent of
H --> A may not be bounded near zero. Let
IArg wl
w = (1 + z)B,
Thus
Note that if
z.
and let
IArg(w/n) I taking
:
M = sup {IIaZII
W E H
1+
in a neighbourhood of
t E' at
(0, n/2)
(2)
If
F
0,
in a neighbourhood of the closed right
and that
The
H .
z
By the Hahn-Banach Theorem
and we let
is the principal power of
zs
(1 + z)S E H,
a neighbourhood of
problem that
f(a1.x)
for all
where
H-,
y-1 < $ < 1.
such that
S
f e X
where
G(r) _ (1 +
(rel S )
.
xll
a
74
For large
we have
r
is approximately
Srs-1 ei+(S - 1)
infinity. Therefore e
G(r) = rB e1
almost all
r'
so
tends to zero as
G(r)
lim inf r 1 loglla(Yei0)II
and so lim a (-sir/2, STr/2) .
(-Tr/2, Tr/2),
) 6 - 1}
{ ( l + r-1 e-'
1/B log II are
inf r
c cos 4
r
tends to
for almost all
iU
x ll
>- c cos (ry/B)
for
We shall now check the limit required in the conclusion for
= 0, and deduce the general case and the uniformity of the limit from
this. We choose X lim inf r
1/S
E
(0,
61r/2)
such that
iX
logllare
xII
>- c cos (X/S).
see that there are , > 0
From the following diagram we
Ce'X = 1 +
such that
eiSTr/2
1
ifSTr/2 For large
r
we have
llarCe
< 1, II
radical algebra, and hence log II afire lim inf r
From this,
1/S
log
ix
because the Banach algebra is a
.xII <_ log Il ar.x II .
Thus
C1/S c cos (X/$) ? O.
r 1 log Ilar.xll= (r-Y+1/B).r-1/S log Ilar.xll, and -y + 1/R < 0
75
it follows that
This inequality and (1) imply
lim inf r-Y log Ilar.xII ? 0.
that lim inf r
Y
log Ilar.xll = 0.
t e [-a, a], then
If
Ilarv
e-i ? 2 cos a > 0,
v = elf' +
and
.-1
r -Y log
s r-Y log <
for all
r -Y log
r ? 1
are-1
aYe1V
. xII + r-Y log I I
II
Ila
re
11
iW . xII
+ r -Y r.log M
by (2). Hence
i1. r-Y log I I are
xII
> vY (rv) -Y log I I arv .xII - r-Y+l log m
and
r-Y log l l are for all
r ? 1
i) .xl
I
<- r-Y r log M
by (2). Taking the limit as
tends to infinity in these
r
inequalities proves the result. The uniformity of the convergence for e [-a, a] is obtained using
v >_ 2 cos a > 0.
In Theorem 5.4 we shall give a construction that shows that the above result depends in a crucial way on analyticity. We shall construct a continuous semigroup on t
such that
(O,')
llatll
tends to zero very fast as
tends to infinity. Recall that a radical weight
uous function from ]R+
x,y
]R+
and
5.4
w (t)
into
(0,°°)
such that
w on
is a contin-
]R
w(x+y) 5 w(x) w(y)
THEOREM Let
g
be any continuous positive function on
there is a radical weight
w
on
]R1
[2,").
from
Firstly we shall choose the weight g
with
w
Then
and a continuous semigroup
t at (0,-) -> L1OR +,w) such that llatll <_ g(t) for all t ? 2, at *b }b as t - 0, t > 0 for all Proof.
for all
1/t + 0 as t -* -.
w
and
by a constructive process
tending to zero very fast near infinity, and then we
shall show that a natural semigroup in
L1 O3+)
L1OR+,w)
properties. We define the continuous positive function
has the required on
[0,")
by
76
for
1
Ox) =
inf (l,g(x))
We define
4
x ? 2
for
by
[0,00) ->]R+
:
l
0 <_ x <_
linear between 1 and 2
fi(x) = inf {flx1)....,f(xn)
From the definition of $(x) = 1 <-
it follows that
0 <_ x S 1, 0 <_ O(x)
for
4(u) 4(w)
for all
x = x1+...+x, n E IN}.
:
O(x)
0 <_
<_ g(x)
for all
<_
for all
1
We shall now show that
u,w ? 0.
and
x >_ 2, ¢
x ? 0, +(u + w)
is positive and
continuous. Let
and let
x ? 1,
real numbers such that with
m < X.
At most m
than 1, and those
i)
n'
m
Let
denote the largest integer
elements of the set
xj < 1
give
*(xl) .... 14(xn)
since
be a finite set of positive
x = x1+---+x
? (inf {4(z)
are greater
so that
4(x.) = 1
z s x})m > 0
0 <-
:
is a continuous positive function on
4(x) ? (inf {4(z)
with
:
0 <_ z <_ x})m > 0.
y > x
If
Thus
[0,00).
and if
4(Y) < 4(y - x). <- 4(X 1)
.
4(xn) .
i
(xn) .
Taking the infimum over all such finite sets 4(y) s 4(x).
Thus Let
[O,k]
so there is a
4(x).
we obtain
is decreasing.
4
k > 0
and 6 > 0
Now
E > 0.
with
u,w e [0,k] and lu - wl < 6. 4(y)
xn
x =
positive real numbers, then
If
6 < 1
1
4
and
is uniformly continuous on l4(u) - $(w)l <E/2
<_ x < y < k
There are
for
and if y < x + 6, then y =
and
4(y1)..". OYn) < 4(y) + e/2. Because
4(y.) 5 1
for all
j
and
iy(yI + x - y)
< Oyl) + e/2,
we have
77
$ (x) 5 fly l + x - Y). (Y1),....
)
(Y2) .....
lk (Yn)
$(Yn) + E/2
< 4)(y) + E. Hence
is continuous on
4)
Let
since
weight on ]R1
since
monomorphism of norm 1. Let t F 6t
that let
t N at
:
:
= L1CR+)
t
in
CO,-).
is a strongly continuous
into the multiplier algebra of
and
Ilatll 1
<-
1
for all
L11R+).
L1aR+)
t > 0;
such
for example,
t N 6t * at
:
(O,-)
is a semigroup and is easily seen to be continuous, using the
Further
6t * at * f -> f
t F 6t * at : (O,-) -> L1 CR+,w)
norm in
is a continuous
w)
be a continuous semigroup in
6t
strong continuity and boundedness of at.
R+) ; Ll *+
be the fractional integral semigroup. Then
at
L1 Ot+)
of
L1
be the unit point mass at
6t
(O,-) - L1 OR +)
(at * L1 OR+))
is bounded. Further
4)
± M(Qt+) = Mul (L1 C2+))
IR
:
f F f
one parameter semigroup from ]R+ Let
is a
w
x ? 2.
The injective map
The semigroup
Then
x ? O.
are weights, and is a radical weight
and exp(-x2)
4'
for all
g(x)
for all
is a radical weight and
exp(-x2)
w(x) 5
CO,co).
w(x) _ ¢(x) exp(-x2)
5
10
w(x) dx
Iat(x - t) l w(x) dx
w(t) ft at(x - t) I
dx
l
<_ w(t)
5.5
Also
is a continuous semigroup. Calculating the
-I6t * at(x) I
II6t * at II =
t
and the continuity and boundedness
t 3 0, t > O.
we obtain
L1 (R+,w)
= r
as
<_ g(t)
for all
t >_ 2.
GROWTH ON VERTICAL LINES - A TAUBERIAN THEOREM In this section we shall show that restricting the growth of
Ilatll on a vertical line restricts the structure of the algebra. Throughout
this section let
t [* at
:
H + A
be an analytic semigroup from the open
right half plane into a Banach algebra A. To show how the restrictions arise we begin with an elementary result. Suppose that to zero as
IyI
tends to infinity
(y a ]R).
Let
Ilal+iyll
IyI-1 log ¢
tends
be a character on a
maximal commutative subalgebra of A containing the semigroup
at.
Then
78
there is a t F' 0(at)
:
t
for all
$ E T
such that
H - C
is a continuous semigroup. Now
0(a
)
= exp(tB)
t e H,
because
lyl-1 (Re s - y Im $) lyl-1
Re ((1 + iy)s)
=
+ i.y)
= l y l - 1 log l$(a1 + llal
lyl-1 log
<_
tends to zero as for all
growth of
llal+iyll
Im
tends to infinity. Thus
lyl
O(at) c 1R
iYll and so
0,
t > O.
In Theorem 5.6 we shall see that a stronger restriction on the
has fundamental implications for the structure of the
Banach algebra. First we recall some definitions. If log+a = log a
and
a ? 1,
for
log+a = 0
a continuous increasing function gn for all
a,s > O.
An ideal
for
then
a > 0,
Then
0 < a <- 1.
BL(X)
is
in a Banach algebra is called a primitive
J
ideal if there is an algebraically irreducible representation into
log+
log+ (as) <- log+a + log+s
(O,°), and
for a Banach space
X
such that
from A
6
Note that in a
ker 6 = J.
Banach algebra each primitive ideal is closed. If A is a commutative Banach algebra, then an ideal
J
character (that is,
is a maximal modular ideal), because the characters
J
is primitive if and only if it is the kernel of a
are the only irreducible representations of the algebra. Note that the probability Lebesque measure 1
on the unit circle
d6
T = {z E C
:
lzl = 1}
27r
is mapped to the measure z F' z + 1
: ID -
H,
1
dy
n
1
where
ID
on
i]R
by the conformal mapping
is the open unit disc
{z e C :
lzl
< 1}
in
1 - z a.
5.6
THEOREM
Let A be a Banach algebra, and let
t [* at
analytic semigroup from the open right half plane (atA)
= A
for all
t e H.
If
H
(1 + y2)-1 log+
then each proper closed (two sided) ideal in
A
:
H i A be an
into A Ilal+i.yll
dy
such that is finite,
is contained in a primitive
ideal.
We begin with a lemma that ensures that by moving to the right in the complex plane we obtain functions of exponential type.
79
5.7
LEMMA If
f
t f' at :
is finite, then t J'
1I
1R exponential type and
is an analytic semigroup satisfying
H - A
(1 + y2) 1 log+ al+iy II dy
fm (1 + t2)-1 log+
Ila3+iy11
a3+t
dy
Proof. The second conclusion follows from the subadditivity of the finiteness of the integral
1R (1 + y2)-1 dy.
is of
H -> A
is finite. log
and
We consider the first
conclusion which is proved by using the ideas behind the result that the difference of a set of positive measure contains an interval. Let y2)-1 log+ IIal+iyj1
M = 1R(1 + e-M/m so that
dy and choose a positive real number m Let V = {y E R : Ilal+iyll < emlyl}, and let p
3/4.
be Lebesque measure on R. M >_
I
mIyl
Then
for
\ (V u [-1,1])
2 -1 : R+ + R +
[l,-).
is a closed subset of R,
Now the function
y L V.
is positive, increasing on
Using the symmetry of the integral about
worst position of V with respect to
M-2
mfa S
+1 where
and
a large positive real number. Note that
8
for all
log+I1al+iy11 >_ mlyl
y F* y(1 + y )
on
V
(1 + y2)-1 dy where the integral is evaluated over the set
y (1 +
0
and decreasing
and considering the
we have
dy
Hence
2a = u(V n [-8,8]).
M 2 m
y2)-1
[-8,8],
[0,1],
1Y-1
dy = m log(8(a + l)-1)
a+ 1
so that
eM/m - 8(a + 1)-1 and Let
W E R With
a ? 38/4 - 1. Iwl
? 5,
and let
8 = 21w1, and suppose that
w j (V n [-8,8]) + (V n E-8,61)-
Then
(V n E-8,61 - w)
n
(V n [-8, 8]) = 0,
so that
28 ? u((V n C-8,8] - w)
2a- Iwl +2a
?38-4- 8/2.
n [-8,8]) + u(V n [-8,8])
80
contrary to the choice of
4 ? 6/2
Therefore
yl,y2 E V
w = yl + y2,
such that
[-S 8]
rt
a2+iwII < II
II
w
and
S.
Hence there are
so
al+iy2II emly11. emly2I e 4mIwI
<
The function y I} IIal+iyII : C-5,5] ->R is continuous, and so there is a constant C such that Ila2+iwll <_ C e4mIwl for all w E R. Because
are constants
ml
and C1
is a continuous semigroup there
(0,-) + A
t [* at :
such that
for all
IIatII < Cl emit
t ? 1.
If
z = x + iy E H, then I l a2+iy I
I l al+x l l
I I a3+z I I
ml (l+x)
because
I
4mlyl
5C Cl e
.e
< C C1 eml.exp
(max {m1,4m).21/2 IzI)
x + IyI <_ 21/2 IzI
by Cauchy's inequality. Thus
z 1*
a3+z
:
H -* A
is analytic of exponential type.
Note that in the above lemma, and in Theorem 5.6 we do not need an assumption on the growth of
IIatII
t ± 0, t > 0.
as
Perhaps if there
is a semigroup satisfying the hypotheses of Theorem 5.6, then there is one that is bounded on 5.8
(0,1].
PROOF OF THEOREM 5.6
We begin by showing that A
is not a radical algebra from which
the result will follow by standard Banach algebra techniques. Using the Hahn-Banach Theorem we choose We let
f(z) = F(a3+z)
for all
analytic in a neighbourhood in
H,
F E A z
such that
3
IIFII = 1
and
F(a
in a neighbourhood of
H-.
Then
(H - 1, say)
of
H ,
)
x 0. f
is of exponential type
and satisfies
(1 +
R
f
y2)-1 log+
IF(iy)Idy <-
(1 +
is
y2)-1 log+II-3+iyll
)R
is finite by Lemma 5.7. By the Ahlfors-Heins Theorem (Al.l) there is a
dy
81
constant all
r 1 log IF(reie)I -* c cos a
such that
c
e E
However
(-ii/2,7f/2).
r - -
as
for almost
r 1 log IF (reie)I:- r-1 log Ila3II + r-1
ie log IIaYe
tends to minus infinity as
II
r
tends to infinity, because
ie r [* are
(O,-)
is a continuous homomorphism into a radical Banach
-+ A
A
algebra. This gives a contradiction, and so
is not a radical Banach
algebra.
Finally let that
be a proper closed ideal in
J
is contained in no primitive ideal in
J
Banach algebra, and
t [-+ at + J :
y2)-1 log+ Ilal+iy
(1 +
1
H i A/J
A.
and suppose
A,
Then
is a radical
A/J
is an analytic semigroup with
+ JII dy
12
finite. This contradicts what we have just proved, and so A/J is not radical. Thus there is a primitive ideal in
containing
J.
COROLLARY
5.9
If
w c R'
A
is a proper closed ideal in
J
such that
fn(w) = 0
for all
L1 URn),
then there is a
f E J.
Proof. The Gaussian semigroup (2.15) or Poisson semigroup (2.17) in L1 CR
satisfy the hypotheses of Theorem 5.6. A character on form
n g [+ g
(w)
for some
carrier space of
w e IIt
with
L'cRn)
Rn
n)
is of the
L1fRn)
because of the identification of the and of the Gelfand transform with the
Fourier transform.
PROBLEM
5.10
Let
be a locally compact group. The group
G
of polynomial growth, or set
W c G
u(Wn) <_ Cn
w j E W}.
r,
if and only if for each compact sub-
G E CPG1,
there is a positive integer where
p
r
and
is left Haar measure on
and W
G G
growth if and only if there is an analytic semigroup
(i)
(ii)
(iii)
(bt * L1(G)) IIbtIIl = 1
N
bt ? 0
for all
t > 0,
IIb1+iyII1
= O(IyIN)
n
=
have polynomial t F+ bt
:
H -+ L1(G)
such that
= L1(G)
and
such that
C > 0
a
Does a metrizable locally compact group
and a positive integer
is said to be
G
=
(L1(G)
* bt)
for all
and as
t E H,
as a function and in the *-algebra
IYI
-+
y E R?
L1(G)
82
See Hulanicki [1974]. and Dixmier [1960].
COROLLARY
5.11
Let
A with
algebra
H -> A be an analytic semigroup into a Banach
t F' at
= A
(atA)
for all
continuous homomorphism from 1
+ iyII
y2)-1 log+
(1 +
I
A
If there is a non-zero
t e H.
into a radical Banach algebra, then
dy = °°
]ft
Proof. Let
where
B = (9(A))
is a continuous homomorphism from A
9
into a radical Banach algebra. Then semigroup, and
t 1+ bt = 0(at)
y2)-1 log+ IlbI + iyll
(1 +
:
H + B
is an analytic
dy
)7R y2)-1
f (1 +
5 log+ II a1I
(1 +
y2)-1 log+ Ilal + iYll
J ]R
IR
Since
r
dy +
is a radical algebra the first integral diverges by Theorem 5.6,
B
and hence so does the required integral.
SEMIGROUPS OF EXPONENTIAL TYPE - NONSEPARABILITY OF THE
5.12
MULTIPLIER ALGEBRA The basic result in this section is that if there is an analytic semigroup (atA)
t F' at
= A
H -> A
:
for all
into a commutative Banach algebra and
t e H
then the multiplier algebra example is
C (R), 0
t F' at
of A
Mul(A)
such that H,
is nonseparable. The motivating
which has a bounded analytic semigroup defined in
and whose multiplier algebra, the algebra functions on
A
is of exponential type on
H
of continuous bounded
CbOR)
is non-separable. In this example the group of invertible
]R,
elements in the multiplier algebra
CbOR)
has
2 0 connected components.
To obtain these conclusions we require restrictions on the semigroup, and on the carrier space of
A#
to get the
2\O - components. Later in the section
we show that some restrictions are necessary by means of examples. Before proving the main theorem of this section we shall state and prove a technical
number theoretic lemma, and note some useful things about semi-
groups and the multiplier algebra.
5.13
LEMMA Let
there is a set
pair
a x 6 E W
V W
be an infinite subset of the positive integers N. of real numbers of cardinality
there is
n E V with
Then
2N0 such that for each
dy.
83
I exp (-27rina) - exp (-27rins) I ? 1. Proof. From mj+l ? 5mj
we choose an infinite subsequence
V
for all
is a subset of
T
If
j e 1N.
the characteristic function of
so that
T
is
XT
mj IN,
such that
we let
1 on T
and
XT
denote
0 off T,
and we let w(T) = 1
XT (j)
2 j=l
m. 3
Note that the series defining
w(T)
converges because
for all
m. ? 53 3
j.
Now we let
W =
: T c1N1.
{w(T)
T
Let
be subsets of N with
R
and
the least integer in the symmetric difference
Y=
T x R,
and let
(R\T) U (T\R).
n
be
If
n(XT(j) - XR(j)).
L
j>n
M. 3
then
IYI
<
n 2
E
j>n
m. 3 2n.5-3
X
j>n < n 2-15 n < 2-1
XT(j) = XR(j)
Since
for
1
<_
j
< n,
we have
Iexp (-27rni w(T)) - exp(-27rni w(R))I
= 11 - exp {t 7ri + 7iy) I with
+
n e T\R
if
11 + exp 7riy I
>-
1,
and
and so
-
if
n e R\T.
Since
IiYI
<_
7r/2,
exp (-27rni w (T) ) - exp (-27rni w (R)) I
? 1. This,
proves the lemma. We note that the ">- 1" in the lemma may be replaced by "? 2 - c"
if we replace the 5 in the proof by a large real number. We return to semigroups and multiplier algebras. Note that if
t 1' at :
H - A
is an analytic semigroup, then the semigroup is of
exponential type (that is, there are constants
C
and
K
such that
84
for all
IIatII <- C exp(Kltl)
t E H,
{IIatII
Iti
if and only if the set
t E H)
is bounded. If
<- 1)
A
is a commutative Banach algebra
with a bounded approximate identity bounded by embedding of a [* L
A
into the multiplier algebra where
: A -> Mul(A),
a
embedding. A semigroup
a
bt
t
for all
L (x) = ax
monomorphism. We shall regard A
then the natural
1,
defined by
Mul(A)
A
into
for all
via this
Mul(A)
from an additive subsemigroup of
is said to be strongly continuous if the map
Mul(A)
is an isometric
x E A,
as a closed ideal in
t E btx
into
C
is continuous
(See Chapter 6 for more on strongly continuous
x E A.
semigroups.)
THEOREM
5.14
let
t F' at
{IIatII
be an analytic semigroup with
H -> A
:
t E H,
:
be a commutative Banach algebra without identity, and
A
Let
Itl
= A.
(a1A)
If
then there is a one parameter group
is bounded,
<- 1}
is the identity of
Mul(A),
y
aly : R - Mul(A)
such that
t
at
is a strongly continuous norm discontinuous semigroup.
+ Mul(A)
H
Further
ao
is non-separable. If the carrier space of A#
Mul(A)
and
has only
finitely many components, then in the norm topology the group of invertible elements in
2K0
has at least
Mul(A)
Proof. We first extend the semigroup t N at
H ; Mul(A)
:
components.
t 1+ at
to a semigroup
H i Mul(A)
:
by strong continuity. The boundedness hypothesis
implies that the semigroup is of exponential type, and so is bounded on each bounded subset of k E A
and
s, t E H,
and
Y E R, h e A,
Let
H.
For all
e > 0.
we have
Ilat.h - as.hhl < (IIatII + IIatII) Ilh - a1.kll+llal+t.k Since
(a1A)
and then for Ilal+t.k
-
and
s
al+s.kll
converges in operator
we may choose
= A,
aly
to
H
on
t E H
k
with
so that Iiy - sl
is very small. Hence iy
A by
for each
t F' at
:
as
t
A
because
as
t
We define the
h c A.
tends to
iy, t E H.
t F' at :
H - A,
is a strongly
H + Mul.(A)
continuous semigroup and is bounded on bounded subsets of is the identity operator on
A
converges in
From this definition and the properties of the semigroup direct calculations yield that
very small
Iiy - t)
and
y E R and each
aly(h) = lim at .h
is very small,
Ilh - a1.k1l
at .h
a1+s.kll.
1
ao(a1k) = a k
H_.
and
Also
(a1A)
ao = A.
85
t f at
If
Mul(A) were continuous, then there would be a small
H-
:
t > 0 such that II at - a°II = I I at - 1 I < 1. Mul(A),
in
at would be invertible
Thus
I
A would contain the identity of
and so
contrary to
Mul(A)
hypothesis.
and let
r
Let
G
denote the group of invertible elements in
and
s
be distinct real numbers. Suppose that
are in the same component of principal component of using Let
b 27ri
:
H
and b t e A
= 1
t F' Obt)
H - C
:
_ (alA)
(b1A)
t
(b ) = exp(-nt)
an identity the spectrum n e U}
for all
is
0 e o(b1)
and
in the unital Banach subalgebra
e
¢(e) = 0
is a character on if
4(b1)
b1
is a character on
and
= 0.
then
B,
analytic semigroup. Since bounded subsets of
{11(l - e)b 1
and so
Either
and
4(e) = 1.
so
k
such that
Hence
If
(1 - e)A
is an
are bounded on Ilbiyll <- k
y 1* biy,
for all
it follows that
o(b1)
which is a contradiction.
is infinite or our supposition that G
air
and
is wrong. If the carrier space of
has at most a finite number of components, then 2
is the same as
T \ a(b1).
in
is bounded. By Theorem 5.6 it follows that e = 1,
are in the same component of
has at least
A
t F exp(tc)
2n - periodicity of
y c ]R}
II
= 0,
4(e) = 1
t N (1 - e)bt : H i (1 - e)A
at
there is a
H-,
Using the l+iy
y e [O,2,r].
t
bl
of A
B
Note that since the spectrum
for all
4(b1) a 0
then
A,
al)-l
(b1 -
is a radical Banach algebra, and
G
does not have
A
is finite the unital Banach algebra generated by b1
that generated by
ais
A
t o H.
such that if
b1
4(b1) x 0 and
(1 - e)b
in
is
U
By the single variable analytic functional calculus for
generated by
¢
bl
Then
t e H.
Because
- log IIb1II.
and
<_ Ilblll
such that there is a
n
is finite. Then
o(b1)
there is an idempotent
of b1
then
A,
such that
for all
of the element
a(b1)
Assume that
if
Then
exp R (27ri + 1) _ f (b2fri + 1) = fl b l
Now
non-empty and is bounded below by
a(bl) x {O}.
aiy = exp c.
is a character on
¢
be the set of integers
U
A with
:
If
is an integer. Also exp a = I¢(bl)I
s
on
{O} u{exp(-nt)
is the
bt = ayt/21exp(itc/2n).
Hence there is a complex number
for all t e H. Let
<- log llblll.
character
t e H.
aiy
ais
is a continuous semigroup and is non-zero because
= A.
(bt) = exp (Rt) = exp S so that S
for all
such that
c e Mul(A)
be defined by
-> Mul(A)
and
is commutative (as we may check
Mul(A)
there is an element
A2 = A)
t N bt
and
y = r - s x 0
Then
G.
Since
G.
Mul(A),
air
a(b1)
A#
is finite and so
80 components. We now show that in this case
Mul(A)
is
86
nonseparable. There is a constant with all
Since
<- 1.
Itl
with
r, s c 3R
<-
Iri
such that
a
ai(r-s)ll>
III -
1, 1,
at least the cardinality of
Isl
<-
1.
Thus a dense subset of
To complete the proof we show that
is infinite. Because
t e H)
V
to
U = {n E IN
:
for
Mul(A)
has
is bounded below by
U
is nonseparable if
Mul(A)
exp(-nt) e a(bt)
a,8 e W with
given by Lemma 5.13. Let
for all
the set
-logllblll,
W be the set of cardinality
is infinite. Let
V = U n 3N
t e iIR
Ilair - aisll > a-1
[-1,1].
is infinite. In this case
a(b1)
for all
Ilatll <- a
we have
corresponding
2110
Then there is
a x S.
such that
n e V
lexp(-27rnia) - exp(-27rnis)I ? 1,
and there is a character
A#
on
process. For each
d e Mul(A)
z e A
implies that
x F (dx)
the functional
this constant by and
from A
and that
Mul(A),
m
Mul(A)
$
because
: A
x F* +(dx)
is a constant multiple of
(d) 4'(x) = (dx)
restricted to
A
4,
I
and (z) x 0
= 0
fi(x)
for all
by a standard
f(dx) (z) _ ¢(dxz) = (dz) 4(x) = 0.
so that
4(d)
to
From this equation it follows that
x e A.
= exp(-nt)
¢(bt)
the linear functional
has kernel containing the kernel of with
such that
We now extend the character
t e H.
Hence
and we denote
for all
d e Mul(A)
is a character on
is equal to
0.
Therefore
0(bl+iy)
O(biy) O(b1) exp(-niy)
= exp(-n(l+iy)) = exp(-niy)
_
for all
IIb27ria
y e7R.
-
b(bl),
and O(bly) _
Hence
b27TiSlI
(e2Tria)
(e2nis)
Il = Iexp(-27rnia) - exp(-2Trnis) I >-
So
Mul(A)
5.15
1k
is nonseparable and the proof is complete.
EXAMPLE
Note that we cannot deduce that
G
has uncountably many
components without the hypothesis that the carrier space of
A#
A = c0
so that
Mul(A) = £'.
Then there is a semigroup in
the hypotheses of the theorem. For example, let
at = (n-t)
has only a Take
finite number of components. Here is an example to show this. A
satisfying
for all
t e H.
87
However the group of invertible elements in then
a G,
(an)
£7
is connected because if
(an) = exp((log knI + i Arg an))
where
Arg is the
principal value of the argument. We are using the observation that if is invertible, then
5.16
:
{Ia nI
(an)
is bounded away from zero.
n EIN)
COROLLARY
Let A be a commutative Banach algebra without identity and with only a finite number of components in the carrier space of A A
If
.
is a quotient of a uniform algebra with a countable bounded approximate
identity, then the group of invertible elements in
has A
Mul(A)
components.
Proof. Let
and
= B
(btB)
A
be the uniform algebra of which
B
Theorem 3.1 there is an analytic semigroup v(bt) < exp(irIIm tI/2)
spectral radius. Since the norm in
is the quotient. By
t [' bt :
for all
B
such that
H -> B
where
t e H,
hypotheses of Theorem 5.14 are satisfied by the semigroup H -> A = B/J.
v
is the
is equal to the spectral radius, the bt + J
t
This completes the proof.
EXAMPLE
5.17
We shall now give an example of a Banach algebra satisfying the hypotheses of the above corollary. Let continuous in the closed right half plane plane as
be the algebra of functions
21
analytic in the open half
H ,
and tending to zero at infinity in the sense that
H,
With the uniform norm on
IzI - -, z e H-.
H-,
If(z)I
-> 0
is a separable
2C
uniform algebra isometrically isomorphic to the maximal ideal {g a A(D)
:
g(l) = O}
in the disc algebra
theorem on large semidiscs
I I f I I = sup { I f (iy) I z e H-,
let
{z e H
:
IzI
Using the maximal modulus
A(D).
we see that
<_ n},
for all f e 2 . For all
y e ]R}
en(z) = n(n+z)-1,
then
en e 2C
and
is easily seen to be an approximate identity in 21
to 1 uniformly in compact subsets of
A of IN
quotient
]R
as
n
n E IN
IIenII <_
Further
1.
bec-use
,
and all a
n
(iy)
en
tends
tends to infinity. A
with at most a finite number of components in the
carrier space of A# will satisfy the hypotheses of Corollary 5.16. Let e_z
g(z) =
on
21
for all
and II gfIL = II fIL
ideal in 21.
teH
z e H-.
A character
Then multiplication by
for all q
on
f E91. 2C
Hence
has the form
g
is a multiplier
g2( is a proper closed 4(f) = f(t)
for some
and all f e 21. Since g(t) x 0 for all t e H, no character on A = 2[/g2C is a radical Banach algebra and
2C annihilates g2[ . Thus
88
satisfies the hypotheses of Corollary 3.16 because the carrier space of A#
is a single point.
EXAMPLE
5.18
We shall construct an example of a separable Banach algebra A with a countable bounded approximate identity with multiplier algebra equal to
Mul(A)
This shows that some hypothesis on the semigroup
A ® T1.
by
is essential for the conclusions of Theorem 5.14. Let complex sequences
(a
n
be the set of
such that
)
II(an)ii = sup lanl + L lan+l - anI 1
by
is finite. With this norm and coordinatewise algebraic operations, is a Banach space. An elementary calculation using the inequality
m Ian+l Sn+l - ansnl 1
nnC
(Isn+ll.lan+l
1
by
shows that by
Let n
Snl)
- anI + lanl'Isn+l -
is a unital Banach algebra with the coordinatewise product.
denote the set of
(a
n
)
in
such that
by
bvo ®C 1
by = by0 ® C 1
and
by,
is only equivalent to that on
the definition of the norm on
by by
identity. Let
f
n
by,
for all
n.
then the norms match here
then
,
is isomorphic
Mul(A)
A has a countable bounded approximate
and that
a
in the n-th place and
1
C
e
n
= Ln f.. 1
Then
f
3
A direct calculation shows that
approximate identity in
in
is a Banach algebra.
by
denote the sequence with
zeros elsewhere, and let
sup
If we replace the
lim sup,
We shall show that, if A = by0 (bicontinuously) to
is a closed
though the usual norm on by.
but partial summation is needed to show
tends to zero as
n
by, bvo
tends to infinity. Since lim is a character on
maximal ideal in
a
For each
A.
n
,
e
{en
(an) E by
n :
E bvo
and
n E]N}
L(a)
define
lie
n
11 = 2
is a bounded on
bvo
n
by
L(a )(Sn)
n from
by
into
=
(tn$n). Then
is a bicontinuous monomorphism
(an) F L(a )
n Mul(bvo).
If
T E Mul(bvo),
so that there is a complex sequence
(Yn)
then
such that
T(f ) m
= T(f
m
2)
T(fm) = Ymfm
= T(f ) m
for all
f
in
89
in. Hence
m-1
n < m) +
max { IYnI for all
so that
m E IN
(yn)
Iyn+l-ynI + IYml = IIT(em) II < 2IITII Further
E by.
Tem = L(y )em
for all
n
Since
m E IN.
T
and
L(Y )
are continuous multipliers, and
is a bounded approximate identity, we have
T = L(y ).
by0
is not a radical algebra. Does there exist a
separable commutative radical Banach algebra identity such that
This completes
n
the properties of the example.
The algebra
{e m : m E IN)
A
with a bounded approximate
Mul(A) = A ®T 1?
NOTES AND REMARKS ON CHAPTER 5
5.19
Remarks on 5.2 - growth on rays The results in this section are from Esterle [1980e].
Theorems
5.3 and 5.4 are Theorems 3.1 and 3.6 of Esterle [1980e]. However our version of Theorem 5.4 is weaker than Esterle's in that his semigroup is infinitely differentiable. The infinite differentiability of the semigroup comes by choosing an infinitely differentiable semigroup such that
at
at
t [' 6t * at
in
and all its derivatives are in fl Dom (Dn), where
Ll OR+)
D
1
(= differentiation) is the infinitesimal generator of the strongly continuous semigroup
t f' dt
on
L1 (R+).
For example the semigroup
of 2.9 has the required properties. See Esterle [1980e] for details.
t f+ Ct
We shall now mention related results. If
t F' at
:
[O,-) } A
is a semigroup
that has an extension to an analytic semigroup in an open neighbourhood of (O,-)
in
T,
then there exists a
A > 0
such that
lim exp(-Ar).log IIaril = 0 (see Esterle [1980e] Theorem 3.3). r IlanIIl/n
Esterle [1980e] also investigates the rates of decrease of for
a
in a radical Banach algebra using various general methods.
Bade and Dales [1981] study similar problems for the radical algebras L1(R+,w)
providing specific rates of growth depending on
further results on the rates of growth of
nIIl/n IIa
w.
There are
in Esterle [1980d].
Remarks on 5.5 - growth on vertical lines The results in this section are in Esterle [1980f] though sometimes the minor details are a little different. Lemma 5.7 was suggested to me by A.M.Davie as a way of eliminating the hypothesis of exponential type. The following references are related to problem 5.10: Leptin [1973], [1976], Hulanicki [1974], and Dixmier [1960]. See also Dales and Hayman [1981].
90
Remarks on 5.12 - semigroups of exponential type
The results in this section are all from Esterle [1980c], and we have not discussed all the theorems in that paper. The discontinuity of the one parameter group
constructed in the proof of
y E aly : ]R -> Mul(A)
Theorem 5.14 leads quickly to the fact that
is non-separable by
Mul(A)
the following result of Esterle's [1980c, Theorem 3.1]. THEOREM. Let
X
be a Banach space, and let
strongly continuous semigroup. If the set the norm topology on
BL(X),
bt
t
{bt
:
:
(O,°°) - BL(X)
t > O}
be a
is separable in
then the semigroup is continuous in the norm
topology.
The proof of this uses the separation of Borel sets by analytic sets (see Hoffman-JOrgensen [1970] Theorem 5, Section 2, Chapter 3) to show that the semigroup
t N bt
is measurable. From this the result follows by
a standard theorem in the theory of one parameter semigroups (see Hille and Phillips [1974]). If the semigroup in the above theorem is actually a one
parameter group, then the continuity of the semigroup may be proved by versions of the closed graph theorem for metric groups thereby avoiding the separation theorem and the result from semigroup theory. Example 5.18 is due to S.Grabiner [1980].
91
6
6.1
NILPOTENT SEMIGROUPS AND PROPER CLOSED IDEALS
INTRODUCTION
We know that an analytic semigroup
Banach algebra A has the property that
t N at =
(atA)
into a
H -> A
:
for all
(a1A)
t e H.
In this Chapter we shall be concerned with continuous semigroups t [* at
:
(O,°°)
satisfying
-> A
(
u
atA)
= A
and
for each
(arA)_ x A
t>O r > O.
Clearly these semigroups are not analytic. However analyticity will
play an important role later in this Chapter. In the first section the standard Hille-Yoshida Theorem is proved for strongly continuous contraction semigroups on a Banach space. There are excellent accounts of this theorem and some of its applications in Dunford and Schwartz [1958], Reed and Simon [1972], and Hille and Phillips [1974]. we-have included it for completeness. From Corollary 6.9 on the results are less standard and involve Banach algebra conditions or the nilpotency of the semigroups. In the process we prove a hyperinvariant subspace theorem for a suitable quasinilpotent operator on a Banach space, and investigate when there is a norm reducing
monomorphism from
L* [O,1] into a Banach algebra. These results are due to
J.Esterle and were given in detail in his 1979 U.C.L.A. lectures.
6.2
STRONGLY CONTINUOUS CONTRACTION SEMIGROUPS ON BANACH SPACES In this section we introduce the notation and definitions
required later in this chapter, and give a proof of the Hille-Yoshida Theorem. The simplest example underlying this theorem is that, if is a continuous semigroup with R E T
such that
Re R <- 0
and
for all
latl 5 1
at = exp(tR)
to be thought of as a linear operator on semigroup at T
by
BL(X)
t = O. for
X
T,
t ' at
t > 0,
for all
:
(O,.) +
then there is an
t > O.
Here
R
is
and is the derivative of the
This little result will be generalised by replacing a Banach space, and finding necessary and sufficient
conditions on a closed linear operator
R
on the Banach space
X
for it to
92
generate the semigroup in a suitable way. Throughout this section let
X
denote a Banach space.
DEFINITIONS
6.3
R on a Banach space
A closed (linear) operator operator
R
defined on a dense linear subspace
has closed graph
{(x,Rx)
x e D(R)}
:
D(R)
is a linear
X
X
such that
R
The closed graph property
X x X.
in
of
is equivalent to the condition that
E D(R), xn -+x a X, and
x n
x E D(R)
Rx
n
-+y E X imply that
Rx = Y.
and
Note that each continuous linear operator on
X
is a closed linear operator
and that, using Zorn's lemma, one can easily construct linear operators on dense subspaces of an infinite dimensional Banach space that are not closed.
The following method of constructing closed operators lies behind the definition of the resolvent of a closed operator. Let one-to-one linear operator on a Banach space Define
R : Tx f+ x
domain
D(R) = TX,
Then R
TX -+ X.
:
because
with
x E TX
and
Rx = T
lx
calculation we could check that the Laplacian on derivative on
L1OR)
The complement of A F+
R)-1
(A -
:
6.4
}-+
D(R)
p(R)
onto
C
in
and the
L2(Rn)
X
R on a Banach
A
such that
A - R
is a one-to-one
whose algebraic inverse is continuous.
is the spectrum of
R,
and the function
is called the resolvent of
p(R) -+ BL(X)
R.
R be a closed operator on a Banach space
empty resolvent set A
imply that
LEMMA Let
map
X.
By direct
= y.
of a closed operator
p(R)
is the set of complex numbers
X
linear operator from
dense in
are closed operators.
The resolvent set space
TX
is a closed linear operator with
xn c TX, xn -+ x E X, R xn + y E X
so that
xn = TRxn -+ x = Ty
X
be a continuous
T
(A -
p(R).
R)-1 :
commutative subset of
Then the resolvent set
p(R) -+ BL(X)
p(R)
with non-
X
is open, and the
is an analytic function into a
BL(X).
Proof. An elementary algebraic calculation shows that, if
A,
}i
e p(R),
then
93
R)-1
(A -
R)-1
- (p -
R)-1
= (p - 1) (A -
(u - R) (A - R)-1.
(p - A) (p - R) The openness of the resolvent set of
R
follows from the corresponding
result for bounded linear operators. Let
la- XI
and let
A E p(R)
with
a E C
Then
a - R = (A - R) + a - A
(1 + (a - A) (A - R)-1) (A - R) on
so that
V(R)
I
because
(a - R)-1 E BL(X)
(A - R)-1 E BL(X)
and
is an invertible bounded linear operator since
(1 + (a - A)(A - R)-1)
Using the geometric series expansion of
(a - A)(A - R)-111< 1.
(1 + (a - A)(A - R)we obtain 11(a
- R) -ll I s II( X - R)111 (1 - la - A I
.
I
I
A - R) -1 I I
(A - R)-1 - (a - R)-1 = (a - A)(A - R)-1(a - R)-1 A
(A - R)-1 :
(A - a)
p(R)
)-l.
From the equation
we deduce firstly that
is continuous, and then after dividing by
BL(X)
that this function is analytic. This proves the lemma.
6.5
DEFINITIONS
A semigroup
from
t F' bt
a contraction semigroup if
into a Banach algebra is called
(O,-)
for all
IIbtil <- 1
t > O.
Most of the semigroups we constructed in Chapters 2 and 3 are contraction semigroups if attention is restricted to the positive real numbers. A function
F
strongly continuous if
from a topological space t {* F(t).x
Clearly a (norm) continuous function from
U
into
BL(X)
is continuous for all
X
U -
:
U
into
BL(X)
is
x E X.
is strongly
continuous.
Here are a couple of examples of semigroups that satisfy the hypotheses of the following lemma, which is part of the Hille-Yoshida Theorem.
If we take f E L2 OR), t e H,
X = L2 OR)
and w E 7R,
(btf) (w) = f (w + t)
and
then t F' bt
continuous contraction (semi)group with R = D,
where
the space of
on
JR
and
b° = I.
for all is a strongly
The closed operator
is the differentiation operator, and the domain of
D
f E L2(R)
such that
f
R
is
is differentiable almost everywhere
Df E L2 OR). If
group
]R -> BL (L2 O2) )
t N at :
A
is a Banach algebra with a continuous contraction semi-
(O,-) + A
satisfying
(
u
t>O
atA)
= A,
then we take
94
X = A, b° = the identity operator on t > 0
and all
Then
x e A.
bt(x) = at.x
and
A,
t F' bt
The generator R
continuous contraction semigroup with b° = I.
A
semigroup is a closed operator on R(x.a) = R(x).a
6.6
for all
x e D(R)
t N bt
[o,-)
for all
is a strongly
[O,-) -> BL(A)
:
of the
satisfying the multiplier equation and
a E A.
LEMMA Let
tion semigroup with
b° = I.
-
be a strongly continuous contrac-
BL(X)
Let
D(R) _ {x e X : lim t-1(bt - 1).x exists in X}. t->O, t>O
Then
is a dense linear subspace of
D(R)
for all
of
p(R)
R
II (A - R)-11I <_ (Re A)-1
Proof. Clearly we average
is a closed operator on
for all
and
A E H.
To show that it is dense
X.
over small intervals of the form [0,s].
s (1)
x e X
If
and
f0
s r
rs
bw t-1 (bt - 1) x dw
bwx dw =
t-1(bt - 1) = t-1
J
=t -1
t - 0, t > 0,
0
(bt+w - bw)x dw
Jo
fs+t
ft r b x dr
bw xdw -t -1
I
s
1
o
because w f+ bwx :
is a continuous function
[0,-) - X
s
s
with
H,
then
s > 0,
as
Further the
X.
contains the open right half plane
is a linear subspace of
D(R)
bw.x
R
then
x e D(R),
resolvent set
Rx = lim t 1(bt - 1).x
and if
X,
b°x = x.
Thus
fo
bw x dw a D(R)
and
bw x dw )= (bs - 1)x
R (
0 s_ 1
for all
x e X
and all
s > O.
bw x dw + x
Since
as
s -> 0,
0
it follows that
s > 0,
D(R)
is dense in
X.
If
then we also
x e D(R),
(s
obtain from (1) that
bw R x dw = (bs - 1).x for all
s > O.
Jo
We shall use this equality to show that Let
x
n
be a sequence in
D(R)
such that
x
n
R has a closed graph.
- x e X
and
Rx
n
-> y e X
95
as
Then
n i -.
s-1
(bs - 1) x s-1
= lim
(bs - 1)x
n
s
(
= lim s-1
bw R x
I
1
0
n
dw
s
= s-1
bw y dw f-
-* y
s - O, s > O
as
because the function
w N bwy
and
Rx = y.
Hence
x e V(R)
[O,°)
:
is continuous with
-> X
eaw
The Laplace transform of the function -a e H
is the function
A F' (A - a)-1
the operator
H
- C.
[O,-)
for
-; C
This is the motivation
given below. For each A E H
is defined by
X
on
R(A)
:
R(A) _ (A - R)-1
behind the definition of
w 1*
b°y = y.
e-Xw bw x dw
R(A)x = J 0
for each x e X,
Clearly the integral is defined and convergent for
x c X.
and
is a linear operator on
R(A)
X.
all
Further
IIR(A)xII 5 fo - a w(Re a) jjbwjj jjxjj dw A)-1
5 (Re for all
x e X,
so that
i1xli R(A)
t 1(bt - 1) R(A)x e-aw (bt+w = t -l
- b w )
x c X
If
E BL(X).
and
A E H,
then
x dw
Jo = t -l
eat
e Xv by x dv - t 1
o
I
=t 1 (eat
e-aw bwx
I
it
- 1) Jo
e- avbvxdv -
eat tl I
dw
a awbwxdw
Jo
-> AR(X)x - x as
t - O, t >0.
(A - R) R(A)
Thus
R(A)x a D(R),
and R R(A)x = AR(A)x - x
is the identity operator on
X.
If
x E D(R)
and
so that A E H,
96
then
R(A) t 1 (bt - 1)x =
e-aw (bt+w
t-1 r
- bw)x dw
Jo
AR(A)x - x
converges to
by the definition of
the identity operator on linear operator, and
6.7
as in the calculation above,
t - 0, t > 0,
as
R(A)Rx
and converges to
D(R).
(A - R) -1
Thus
R.
R(A)(A - R) (A - R)-1
is equal to
R(A)
is
as a
BL(X), which completes the proof.
is in
THEOREM (HILLE-YOSHIDA THEOREM) Let
R be a closed linear operator on a Banach space
is a strongly continuous contraction semigroup bo = 1
Hence
satisfying
Rx = lim t 1 (bt - 1)x
bt
t
for all
There
X.
with
(O,-) + BL(X)
:
if and only
x E D(R)
t- O
if
(A - R)-1 E BL(X)
t f bt
for all
and II (A - R)-111:5 A-1
If
A > O.
satisfy the above conditions, then the open right half plane
contained in the resolvent set
p(R)
of
(A - R)-1x = 1
R,
e-aw
and
R
H
is
bwx dw
0
for all
x E X,
the function
and
A E H,
A)-1
II(A - R)-1II5(Re
A F' (A - R)-1 : H - BL(X)
for all
and
A E H,
is analytic. Further for each
x E X
Ilbtx - exp t(A2(A - R)-1- A)xII
tends to zero uniformly for
t
in compact subsets of [0,00)
as
tends
A
to infinity.
The operator R occurring in the Hille-Yoshida Theorem is called the generator or infinitesimal generator of the semigroup
Lemma
t [* bt.
6.6 gives half the above result. The heuristic motivation for the
construction of bt sense. Formally
from
This is why we expect A
is that we want
bt = exp(t R)
A2(A - R)-1 - A = R(l - R/a)-1
tends to infinity and each
as
R
bt
A2(A - R)-1 - A
converges to
in a suitable
R
as
A
is a continuous linear operator.
to be a suitable limit of
exp t(A2(A - R)-1 - A)
tends to infinity, where the exponential of a bounded linear operator
is defined by the power series for the exponential.
97
Proof of the Hille-Yoshida Theorem. Suppose that all
A > 0.
zero as
and
(A - R) -1 E BL(X)
x E D(R),
If
then
(A(A - R)
tends to infinity. Since
A
is dense in
D(R)
tends to zero as
X,
and
as
A
A > 0.
x E D(R),
If
for all
a standard argument shows that x E X.
convenience in the following calculations, we let for all
then
X-
I
for
-1)x = (A - R)Rx tends to
IIA(A - R)-111:5 1
tends to infinity for all
A
II(A - R)-111
A > 0
(A(A - R)-1 - 1)x
For notational
RA = A2(A - R)-1 - A
RAx = A(A - R)1 Rx
tends to
Rx
tends to infinity.
With these little preliminaries out of the way we turn to the semigroups. Since
RA E BL(X)
the semigroup
t + exp tAR
:
[O,-) - BL(X)
may be defined using the power series expansion for the exponential function. For each positive
A
and
t,
Ilexp t exp(-tA)
(ta)n IIA(A n=0
R)-1I In
n!
< 1
because
positive
IIA(A - R)-111:5 1.
and
A
and exp (wR
v
V.
By Lemma 6.4
RA
and
RV
commute for all
Differentiating the power series defining
)
we obtain
d
Iexp(wRA).exp((t - w)RV)
dw
= exp(w RA).(RA - RV).exp((t - w)RV)
for all
w, t, A,v > 0.
Integrating this, we have
II(exp (tRA) - exp(tRV)).xII
ft d
0 dw (t <_
J
l
) exp(wRA). exp((t - w)RV).x1 t
dw
)
exp(wRA).(RA - RV). exp((t - w)RV).x
0
<_ Jt1I(RA-RV)xIIdw 0
t 11 (R - RV) x II
dw
exp(wR
98
for all A
x E X.
and
x e D(R),
If
then
tend to infinity so that
v
zero uniformly in
in bounded subsets of
t
to infinity. From the density of
[O,-)
as
A
the operator bt
and
and
t
as
[O,-)
btx = lim
and
A
tend
v
it follows that
u,
tend to infinity for all
v
is defined by
tends to
and the observation that
X
in
tends to zero uniformly in
II(exp(tRX) - exp(tRij )).xll
of
D(R)
for all positive
1
Ilexp(tRP)II <-
tends to zero as
II(Rx - R)xII
II(exp(tRA) - exp(tRV)).xll
in bounded subsets
t
For each
x e X.
exp(tRX).x
for all
t > 0
x e X.
A-*O
Clearly bt e BL(X)
and
the above limits for t [* bt
[O,-) - BL(X)
bt
b° = I
and
for all
1
<-
Ilbtil
The uniformity of
t > O.
in bounded subsets of
t
[O,-)
implies that
is strongly continuous. The semigroup property of
follow from the corresponding properties of
Let
Tx = lim t-1 (b t->O
t
x e D(T) = {y e X
for all
- 1)x
exp(tRA).
To complete the proof we show that
lim t 1(bt - 1) y exists in X}. t->O
and
D(T) = D(R)
We do this by using a formula that occurred in
T = R.
Lemma 6.6. Either by integrating the power series for the exponential factor in the integrand term by term or from the proof of Lemma 6.6 (applied to the semigroup
we have
t F' exp tRA),
t
RAx dw
exp(wR
(exp (t Rx) - 1)x = J 0
x e X
for all
bwRx
and all
uniformly for 1
Ilexp(wRA)ll
and
t
for all
Because
A > O.
w e [O,t]
as
A
exp(w RA)Rx
tends to
tends to infinity, and because
the above equations converge to
A > 0,
(t
bwRx dw
(bt - 1)x = J
as
A
tends to infinity for all
x c X.
Dividing
0
t > 0
by
and letting
w ' bwRx then
t
tend to zero, we obtain
from the definition of
x e D(R) :
[O,-) - X.
(A - R) D(R) = X
D(R) = D(T)
because
Hence so
T
D(R) c D(T)
and
(A - T) D(R) = X,
(A - T)
Tx = Rx
for all
and the continuity of
is one-to-one on
R = T
on
D(R).
If A > 0,
and it follows that D(T).
This completes the
proof of the Hille-Yoshida Theorem.
6.8
EXAMPLES
We shall now sketch two examples to illustrate the above theorem. These examples are discussed in detail in Hille and Phillips [1974] (see
99
Chapter 19). Let
X = L2 OR)
(btf)(x) = f(x+t)
for all
on
L2 R),
bt
on
by
L2OR)
Clearly bt
is an isometric operator
is a (semi)group. The strong
t F bt : R i BL(X)
and
continuity of
and define t E 3R.
follows easily from the density of
t J* bt
space of continuous functions with compact support, in
strong continuity of bt
on the normed space
infinitesimal generator equal to the set of
D(R)
everywhere on
(C cm), 11-IL).
such that
f e L2(R)
and is in
]R
L2 OR)
of the semigroup is the derivative
R
and the The d dx
with
is defined almost
df dx
on the space of
R = d
That
L2OR).
the
Cc OR),
(9 continuously differentiable functions with compact support is clear.
Properties of shift semigroups are intimately linked with the differential operator
d/dx.
In the second example we let
btf = Gt*f
for all
and all
t > 0
semigroup. By Theorem 2.15
t N bt
with
equal to the set of
D(R)
everywhere and (3
tat
R
is the Gaussian
of the semigroup is the Laplacian such that
Lf
exists almost
R = 0 follows from the relation
in Theorem 2.15.
/
COROLLARY
6.9
be a Banach algebra. There is an analytic semigroup
A
Let
such that
t [* at : H + A
H
there is a
such that
u e A
IIu(A - u)-lll <_
1
= A = (Aat)
(atA)
right half plane
and that
for all
= A = (Au)
,
9(f) =
a(u)
in the open if and only if
(O,-) = 0,
n
and
A > O. 9
:
Ll(R)+ + A by
Then
e
is a norm reducing homomorphism from
and we may extend
9
to a homomorphism from
f(t) at dt. 1
t
r > 0
for all
Ilan1 <_ 1 (uA)
for all
Proof. Suppose that the semigroup exists. We define
into
Gt
and
is a strongly continuous
CO,-) - BL(X)
f E L1ORn)
That
Af e L1(Rn).
= 0
- A\(Gt*f)
where
f e L1OR') :
contraction semigroup. The generator A
X = 1,1 00), b°f = f,
L1CIR+)
0
A,
by defining vn = (-1)n
9(1) = 1. I
semigroup in
n ,
where L1CR+).
I
Let t
If
into A#
for all w e 1R+. Then t-1 -1 is the fractional integral = w e -w r(t) v(w) = -e -w
(w)
L1 aR+)li
A e C
with
lA
> 1,
then
100 a-1 + a-1
(A - v)-1 (w) =
1 (-l)n wn-1 a-w n=l
An (n-1):
= A-1 - a-1 exp(-(l + 1/A)w)
Since
w 2 0.
for all A
all
A e H.
exp(-(l + 1/X)w)
so that
IIv(A -
v)-1II1
,
for
u
A
A
identity for
:
g
x e A
Theorem
A.
L1 OR+)# I A#.
;
A with the required
Au(A - u)-1,
-u(A - u)-1 y
n e ]N}
implies
follow from the correspond-
u
we have
since
-u(A - u)-Iux
IIu(1 - u)-111 :5 1.
converges to
tends to zero. Similar calculations with {-u(n 1 - u)-1
_
= A = (A at)
(atA)
exists in
tends to zero for all
as
This inequality also shows that
Further
from which we obtain
The other properties of
tends to
imply that
_
via the norm reducing homomorphism
u + u2(A - u)-1 =
ux
A > 0.
The condition
9(v).
Conversely suppose that
as
LCR+)#
(v*L1(R+))-
properties. Because
y e A
for
L1CR+)
exp(-w(l + 1/A))
= A = (A.0(L1 Ot+)))
= A = (Au)-. v
a-1
= (1 + A)-1 for all
u =
(0(L1 Dt+)).A)
ing ones for
=
v)-1(w)
We let
= L10R+).
is in
exists in
(A - v)-1
A similar calculation to the above shows that
VOL -
(u A)
a-1
in the open right half plane H,
all
that
A-1 -
w 1+
u
y
for all
on the right
is a countable bounded approximate
completes the proof.
3.1
6.10 NILPOTENT SEMIGROUPS In this section we find necessary and sufficient conditions on the resolvent of the generator of a strongly continuous contraction semigroup for the semigroup to be nilpotent. The same idea is used to investigate
hyperinvariant subspaces for suitable quasinilpotent operators on a Banach space.
THEOREM
6.11
Let
X
be a Banach space, and let
t f' bt
a strongly continuous contraction semigroup with infinitesimal generator Li)
(n. I (1 - R) -n I) l/n <_ M (ii)
There is an
M > 0
1:° = I
be
[0,-) -> BL(X)
and with
R.
The semigroup is nilpotent with I
:
such that
and only if there is a non-zero
if and only if
bM = 0
for all
n E IN.
(bMX)
F e X
is neither such that
{O}
nor
X
if
101
{[n:IF((l - R)-nx)I71/n Proof. We define
6
:
:
is bounded for each
n E 3N}
L1 pt+) i BL(X)
by
x e X.
f(t)bt.x dt
6(f)x = 1
for all
0
x e X
f e L1(R+).
and all
The integral exists since
is continuous and bounded for all 0
is a homomorphism from
into
L1CR+)
a bounded approximate identity in b° = I
that
x e X
for all for
we obtain
w
so that
t > 0
and
BL(X)
L1(R+)
CO,-) -+ X
such that
11611
!5
1.
Using
bounded by 1 and the observation
From Lemma 6.6,
11811 = 1.
t f+ bt.x :
A direct calculation shows that
x e X.
(1 - R)-1 = 6(I1),
where
(1 - R)-lx = Joe wbwx dw 0
It(w) = P(t)-1
is the fractional integral semigroup in
wt-1
e -w
L1 Ot+)(see
2.6).
Suppose that
M > 0 with
Theorem we choose a non-zero and each
F e X*
(bMX)
x X.
annihilating
Using the Hahn-Banach bM.X.
For each
n E IN
x e X,
IF((1 - R)-nx)I
IF(6(In)x) I w--1 e-w
= If
F(bwx) dwl
r (n) rM wn-l a-w Jo r (n) o
dw
Mn
n:
This proves the necessity in (ii) and similar working, using
in
proves it in case W.
place of
We consider the converses. In case (i) we consider all
with IIxiI s 1 the given
M > 0
and all F e
F e X
X*
with
and consider all
x e X
IIFII <_ 1, and in case (ii) we use x e X. Suppose that there is an
such that
In! F((1 - R)-nx)Il/n <- M
for all positive integers and
x,
n.
In case (i) the
M
is independent of
but in (ii) it may depend on both of them. For each
A e Q
F
102
I
R)-n-lx)I
IF(-2,!ia)n (1 -
na0 <
(2,r A ) n Mn+l
C
(n+1):
n==0
<- M exp (2wM IXI),
and hence we may define the function
G(1) _
by
G
F((-21TiX)n (1 - R) -n-l x)
1
n=O for all
If
AEG
Then
A E T.
is an entire function of exponential type
G
with II27rX(1 - R)-l II < 1,
1 (-2niX)n (1 - R)-n-1 n=o
27TM.
then
converges in
to
BL(X)
(1 + 2nia(1 - R)-1)-1 = (1 + 2nia - R)-1
so that
G(A) = F((1 + 27ria - R)-lx).
The functions
G
and
are analytic in a neighbourhood of the closed
A N F((1 + 21TiX - R)-lx)
lower half plane -iH ,
and hence these functions are equal on
-iH
By
.
the Hille-Yoshida Theorem
(1 + 27ria - R) 1x= r e (1+2nia)w bw x dw Jo
for each
A em,
and so f27riXw eF(e-wbwx)
G(A) =
dw.
0
We define the function K(w) = 0
for
w < 0.
K e L1(R) n L2CR).
K
on
Then
]R
by
IK(w)I s
Further
G
-w
G
fnr all
w > 0
is the Fourier transform
by Plancherel's Theorem the restriction of Rudin [1966]). Because
for w a 0
K(w) = F(e wbwx)
G
has exponential type
to
IR 2irM
Note that the
2n
KA
is in
of
K,
and
L2(R)(see
the Paley-Wiener
Theorem (see Rudin [1966]) implies that the support of K [-M,M].
and
and
is contained in
occurring in the exponent of our definition of
the Fourier transform appears
in the relationship between the support of
103
K
and the type of
are done because
F(bwx) = 0
Thus
G.
M was independent of
Theorem ensures that bM = 0. for all
for each
w >_ j}
space of
w >- M.
Then each
B
(bNX)_
F(bNX) = {O}
0 < t < r,
then
2 (brX)
(btX)
{x e X : F(bwx) = 0
B.
is a closed linear sub-
B.
then
(btX)
= X
for all
t > 0
(btX)
x X
for all
t > 0,
6.12
The
However
since
= X
(brX)
for some
= (bs(brX)
(br+s X)
r > 0,
_ (bsX)
)
.
and the strong continuity of
and b° = I
[O,-) - BL(X)
u B. = X. j=l
for some positive integer N, N If b X could be {O}.
= X
Also if
.
Thus
small enough
N
* X.
and
In case (i) we
x - the Hahn-Banach
In case (ii) we let
j e 3N.
Baire Category Theorem implies that
t [+ bt :
and
and what we have just proved shows that
X,
and so
for all F
ensure that
is nonzero for
(btX)
This completes the proof.
t.
REMARKS
Note that by using Stirling's Formula, nl/2 nn a n+0(1/n)
nr(n)
as
tends to infinity, the condition
n
Theorem 6.11(i) may be replaced by
(1 - R) -nlll/n 5 M
IIn:
{n 11(1 - R) -nlll/n
n E 3N}
but with the loss of the nice relationship between bM = 0
in
is bounded,
and the bound
M.
T
If
is a continuous linear operator on a Banach space
then a hyperinvariant subspace space
Y
X
of
such that
Y
for
SY c Y
The second commutant of an operator of all
with
R E BL(X)
R
is a proper closed linear sub-
T
for all
T
S E BL(X)
commuting with
on a Banach space
X
S E BL(X)
commutes with all
T.
is the set commuting
T.
THEOREM
6.13
Let space
such that
X,
X
T
be a non-zero continuous linear operator on a Banach
satisfying
(0,") n a(T) = 0,
and
IIT(A - t)-111:5 1
for all
A > 0. ((i)
If (TX)
= X
and
{n
IITnIIl/n
:
n E I }
nilpotent strongly continuous semigroup b° = I (ii)
and bt
is bounded, then there is a t f* bt : [0,-)
in the second commutant of
If there is a non-zero
F E X*
such that
T
for all
-* BL(X)
with
t > 0.
{n IF(Tnx)I1/n : n e 3N)
104
is bounded for all
inequality
tends to y Therefore
tends to
as
T)-1
T + T2(A -
Tx
as
and the
it follows that
y e X
because
dense linear subspace
TX
of
closed operator satisfying
II ()`R - 1)_111 5 1 5 II Tn IIl/n II l - T II
R = T 1
and so
X,
T)-1
T(A -
:
l)-1
= (AR -
= X.
(TX)
is a one-to-one continuous linear operator from
T
y
-T(A - T)
tends to zero. Hence
A
tends to zero for all
A
T)-1
= AT(A -
A > 0,
for all
IIT(A - T)-lII 5 1
- T(X - T)-1Tx
has a hyperinvariant subspace.
T
then
x e X,
Proof.(i) From the equation
onto the
X
D(R) = TX -
is a
X
and
E BL(X)
for all A > 0 (see 6.3). Further 11(l - R) -njIl/n for all n E 3N so that R satisfies the hypotheses
of Theorem 6.11(i) as modified by Stirling's Formula in Remark 6.12. Hence the strongly continuous semigroup R of
exp t(A2(A
-
R)-1
:
[0,-)
in the second commutant of
t I-+ bt
:
T
T
is the strong limit
-A) as
A
tends to infinity
t > 0.
is in the second commutant of
bt
is neither
(brX) t,
is
The strong continuity of
and bN = 0 for some positive
such that
r > 0
generated by
f+ BL(X)
is closed under strong limits,
for all
[0,=) -+ BL(X), b° = I,
that there is an
l)-1
-A) = exp t(A2 TOT -
Because the second commutant of
br
t f+ bt
is nilpotent. From the Hille-Yoshida Theorem bt
{0}
nor
ensure
N X.
Since
is a hyperinvariant subspace
(brX)
of X. (ii).
We may suppose that
invariant subspace of
X.
= X.
(TX)
for otherwise
is a hyper-
(TX)
By the first part of the proof of (i),
R = T-1
is the infinitesimal generator of a strongly continuous semigroup t f+ bt :
[0,m) -+ BL(X)
in the second commutant of
T.
By Stirling's
Formula applied as in Remark 6.12, the boundedness of the set {nIF(Tnx)Il/n
n 63N)
:
for each
of the set
{(n!IF(Tnx)I)1/n
n! IF(Tnx)I
55 Mn
n!
for all
:
x c X
n c ]N}
where
n c 1N
is equivalent to the boundedness for each
M
x E X.
depends on
IF (1 - R) -nx)I
= n! IF(Tn(T - 1)x)I IF(T"+jx)I
5 n!
I
j=0 < Mn eM
(n + j - 1)(n + j - 2)...n.Mn+j j!
(n + j):
Suppose that x.
Then
105
for all {O}
By Theorem 6.1 it follows that
n E I.
nor
for some
X
6.14
is neither
(brX)
This completes the proof.
r > 0.
PROBLEM Can the hypothesis
IIT(A - T)-lII <_ 1
for all
A > 0 be
omitted from the hypotheses of Theorem 6.13?
6.15
EXAMPLE
Theorem 6.13 may be used to obtain the obvious hyperinvariant subspaces of the Volterra operator
f(w)dw : L2[0,1] - L2[0,1]
T : f 1+
fo
T
by showing that
satisfies the hypotheses of Theorem 6.13. The strongly
continuous semigroup
t f+ bt :
is the shift semigroup on
-> BL(L2[0,1])
[O,-)
L2[O,1] defined by
(btf)(x) =l0 f(t - x) for all
and
0
t
given by the theorem
f E L2[0,1].
O<_x
this example is that the closed hyperinvariant subspaces arise from L 2[O,1]
being a Banach module over
L* [O,1].
We shall discuss this in
more detail later in the chapter, but we illustrate it briefly here. We have a norm reducing homomorphism into
BL(L2[0,1])
g E L2[0,1],
and
properties of T
u
0(u) = T where in
L[O,1]
except for the condition
6.16
6
from the Volterra algebra
6(f)g = f*g
given by
for all
u(w) = 1
f E L1[0,1]
for all
L1*[O,1] and
w E [0,1].
The
give the hypotheses of Theorem 6.13(i) for (TX)
= X
which must be checked directly.
PROPER CLOSED IDEALS IN RADICAL BANACH ALGEBRAS
We now turn to continuous semigroups in a Banach algebra, and relate these to proper closed ideals in the algebra. The main problem in
this area of research is, does a commutative radical Banach algebra A have a proper closed ideal? x f+ bx : A -+ A
If there is a non-zero
b E A
such that
is a compact linear operator, then Lomonosov's Theorem
(see Radjavi and Rosenthal [1973]) implies that this operator has a hyperinvariant subspace, and so
A has a proper closed ideal. In a similar way
we shall convert the operator theory results of the previous section into Banach algebra results. Before turning to our proper closed ideals and
106
nilpotent semigroups, we briefly consider quasinilpotent semigroups.
THEOREM
6.17
Let A be a Banach algebra, let t continuous contraction semigroup, and let The semigroup
u =
F+ at
(0,-) -+ A be a
:
u
is quasinilpotent if and only if
t F+ at
be in
-0 a-t at dt J0
A.
is quasi-
nilpotent.
Proof. Suppose 0 <_ II (at)njI'n at
t > 0,
then
t > 1,
since to - n > 0, so the spectral radius of
'-Ilanlllln,
is zero. If
and
is quasinilpotent. If
a1
there is an
such that
m E IN
quasinilpotent elements is a radical algebra. Hence
Conversely let {a
t
there is a character
Q
semigroup so there is a t f+ at _ e-t
0(u) -_
e
0
nilpotencce of
6.18
on
and
B,
t F+ 0(at)
:
[O,-)
4(at) = e$t
is a contraction semigroup
<_ 0.
8 e C
dt = (-1 +
S)- 1
Re (B)
is a continuous
-+ T
for all
t > 0.
Hence
is non-zero, contrary to the quasi-
u.
PROBLEM
Let
A
be a Banach algebra, and let t F+ at
continuous contraction semigroup such that If the semigroup is quasinilpotent, is
A
:
(0,00)
-+ A be a
= A = (U A at) t>O
(U at A)
t>O
6.19
t > O}.
:
is not a radical algebra. Then
B
such that
Bt
{at
be the commutative Banach algebra generated
B
and suppose that
t > O},
:
Since
amt
is quasinilpotent
u
as it is in the commutative Banach algebra generated by
by
thus
mt > 1
are quasinilpotent. A commutative Banach algebra generated by
at
.
a radical Banach algebra?
THEOREM
Let A be a commutative Banach algebra. There is a continuous contraction nilpotent semigroup
t * at
:
-+ A with
(0,0D)
u at A)
(
= A
t>O
if and only if there is a non-zero
u e A
such that
(--,0) = 0, Ilu(A + u)-lII < 1 for all A > 0,
and
(uA)
= A, a(u) n
{nllunll11n
:
n e N}
is bounded.
Proof. To prove this result we shall combine the operator theory results of this chapter with the existence of a suitable continuous semigroup given by Theorem 3.1. Suppose that a continuous nilpotent contraction semigroup t F+ at
:
(O,0D)
-
A
exists satisfying
A)
= A.
We define
107
b t E BL(A)
Then
for all
bt (x) = a t .x
by
t N bt
(tiOat
continuous since
A)
of this semigroup, and let
t > 0
and
and let
x E A,
b° = I.
is a contraction semigroup, which is strongly
[O,-) - BL(A)
:
Let
= A.(
u = TO
R
be the infinitesimal generator Note that this integral
t at dt.
0
A
converges in lle-tatll < e-t.
for all Lu
by
because
and so the operator
x E A,
)-1
U
= U + (1 -
R)-1)-1
= (1 - R)-1((X + 1)l-1 -
R)-1 A-1
Since the spectra of
A > 0.
a(u) n (--,0) _ 0.
u
in
((1 + A)a-1 -
=
u
U
a-1
R)-1)-1
L (A + L)-1 = (1 - R)-1(a + (1 -
equal,
(1 - R)- l.x = ux
is left multiplication
(1 - R)-1
and
BL(A)
for all
is continuous and
(0,-) - A
Hence
u.
(A + L is in
t E e tat :
From the Hille-Yoshida Theorem we see that
A
and
in
L U
BL(A)
R)-
are
Further
llu(X + u)-11l=llLU(A + Lu)-111 A
because
has a bounded approximate identity
The estimate on 11(u - R)-111
by 1.
Ilu(X + u)-111 = < a-1
for all
bt
A > 0.
Also
nitl/n
n
in the Hille-Yoshida Theorem gives
- R)-111 = (A + 1)-1 < 1
uA = (1 - R)-lA = D(R)
is nilpotent, because
{n1l(1 - R)
at
45]N}
is dense in
A.
The semigroup
is nilpotent, and thus the set
is bounded by Theorem 6.11 and Remark 6.12.
This completes the proof of this implication because
all
n E 3N}, say, bounded
:
a-lll((l+A)a-1
A)a-1)-1
((1 +
{a 1/n
(1 - R)-n = un
for
n E N. Conversely suppose that there is a
properties. We define
T
:
x f -ux : A - A.
u
in
Then T
A
with the required
satisfies the hypo-
theses of Theorem 6.13, and there is a strongly continuous contraction semigroup
t F' bt :
[O,-) + BL(A)
u - u2(A + u)-1 = Au(A + ux
as
A
u)-1
with b° = I for A > 0,
tends to zero for each
x E A.
and
bN = 0.
Since
we have
u(A + u)-tux
Because
llu(A + u)-111:5 1
tends to and
108
= A, it follows that
(uA)
tends to zero.
A
approximate identity in
u)-1
for all
= A
The map
t > 0.
(0,") -A such that
t J* at
t N bt(at)
continuous contraction semigroup and
(0,-)
t N bt
for all
(btA)
and so
t > 0,
is a
-* A
(0,-) - A
t f' at
is a
is a strongly continuous
contraction semigroup into the multiplier algebra of =
as
bounded by 1. By Theorem 3.1 there is a
A
continuous contraction semigroup,because
(bt at A)
y e A
for each
y
is a countable bounded
n E IN}
continuous contraction semigroup (at A)
tends to
u(A + u)-1y
{u(n-1 +
Thus
Finally
A.
A
t 1* btat
is
the required nilpotent semigroup.
Theorem 6.19 may be used to give conditions on a Banach algebra that ensure that there is a continuous homomorphism from L1*[0,1] into the Banach algebra.
6.20
COROLLARY
Let A be a commutative Banach algebra. There is a continuous norm reducing monomorphism
A
such that
that
1
L1* [0,l]
if and only if there is
= A
u)-111<_
IIu(A -
= A,
(uA)
from the Volterra algebra
8
(8(L1*[0,1]).A)
for all
A > 0,
{i junlll/n
and
into
u e A
such :
n E 3N)
is bounded.
u with these properties exists in
Proof. If a
there is a continuous contraction semigroup (t>Oat
= A
A)
and
aN = 0
we may assume that at = 0 (1
6
:
f(t) at dt
f F' J
:
for some
N.
if and only if
L1*[O,1] - A.
then by Theorem 6.19
A,
t N at :
(0,°^) - A
such that
By a change of scale in
Clearly
t
We let
t ? 1. 6
is a norm reducing
0
homomorphism from follows from of
L
[0,1]
(j0at A)
into
= A.
If
is a proper closed ideal
8
J
A, 8
in
in the Volterra algebra is of the form for some
[O,a]}
a 2 0
J
satisfies
function of the interval to
a
so
0
a
as
n
(8(L1*[0,1]).A)
is not one-to-one, then the kernel
L[0,1].
Each closed ideal
{f e L1*[O,1]
:
f = 0
J
a.e. on
J
is assumed to be non-zero, the a < 1.
[a,a + 1/n],
tends to infinity. Thus
If
then
a
is the characteristic
fn
fn E J
as = 0,
and
n 8(fn)
tends
which gives a contradiction
is one-to-one.
Conversely if the norm reducing monomorphism exists, we let u = 8(v)
where
v(t) = 1
= A
(see Dales [1978], Dixmier [1949], or Radjavi
and Rosenthal [1973]). Since corresponding to
and the property
for
t E [0,1].
The properties of
u now
109
and from
v
follow from the corresponding ones for
(9(L1*1O,1]).A)
= A.
In the proof of Theorem 6.19 we used Theorem 6.13(i) with other techniques. By modifying the proof of Theorem 6.19 slightly and using Theorem 6.13(11) we obtain the following Theorem, whose proof we omit.
6.21
THEOREM
Let A be a commutative Banach algebra. There is a continuous contraction semigroup
A nor
is neither
(arA)
a non-zero
F e A
:
{0}
r > 0
for some
A > 0,
for all
such that
(0,W) - A
u e A with
and
IIu(A + u)-111:5 1
for all
t [* at
(tUOat A)
if and only if there is
= A, a(u) n (--,0)
(uA)
{nIF(unx)I1/n
and
= A and
:
n E w)
is bounded
x e A.
EXAMPLES
6.22
We shall now consider two examples of Banach algebras that satisfy Theorems 6.19 and 6.21.
In the Volterra algebra s e [0,1].
for all
be
of
corresponding to u t
for
0 <- t 5 1
is
t
6
t
* It,
t N It
where
L1*[0,1] could
is the fractional integral semigroup of
restricted as a function on IR+
to
[0,1].
be the Banach algebra of continuous functions in the
Let IU
closed right half plane
H ,
analytic in
H,
and tending to zero as
tends to infinity. This algebra was introduced in Example 5.17. Let
Izi
in
be defined by
2[
are in
2(
for all
u :
A > 0,
for all
z e H_.
Thus
z N (z + 1)-1 : H
- Q.
Then
u
u,2[-* `u
(bt91)
for all
(A + u)-1
and
IIu(A + u)-111 :5 1
for all
A > 0.
The strongly
continuous contraction semigroup of multipliers generated by
is bt(z) =
and
(1 + A + Az)-1
u(A + u)-1(z)
u e
t I+ 6t,
and is zero for
A corresponding continuous contraction semigroup in
L1cR+)
u
L1*[0,1J
is the unit point mass at
dt
t > 1.
u(s) = 1
The strongly continuous semigroup in the multiplier
algebra Mul(L1*[0,1]) where
we could take
L1*[0,1]
e-t.e-tz
t > 0.
for all z e H
and
R : of N -f
t > 0. Further
110
6.23
PROBLEM Is there a nice class of Banach algebras with countable bounded
approximate identities such that we can find all proper closed ideals in each algebra? If there is a continuous norm reducing monomorphism L 1 *[O,1]
into a commutative Banach algebra
(9(L1*[O,1]).A)
= A,
A
9
from
such that
what additional properties are required to ensure
that all proper closed ideals in
A
arise from proper closed ideals in
L 1*[0,1]?
6.24
REMARKS AND NOTES Our discussion of the Hille-Yoshida Theorem is standard and
as we have noted earlier there are accounts in several references (see Hille and Phillips [1974], Dunford and Schwartz [1958], and Reed and Simon [1975]). There is a nice account of the generators of analytic semigroups in Reed and Simon [1975]. From Theorem 6.11 onwards the results in this chapter are taken from the seminars and postgraduate lectures that Jean Esterle gave at U.C.L.A. in 1979. Though the discussion differs slightly from his, these results are due to Esterle. At present Esterle has not published these results, and I am grateful for his permission to include them in these notes.
111
APPENDIX 1
:
THE AHLFORS-HEINS THEOREM
In this appendix we shall prove a special case of the AhlforsHeins Theorem which will be strong enough for the applications in these notes. Our hypotheses are stronger than those of the full result in that we require
(
to + f(i )ldy < W
112
1 + Y
y-2 log+lf(iy) f(-iy) dyl <
rather than f_1
2
and analyticity in a neighbourhood of the closed half plane. For a discussion of this point see Boas
[1954, p.114]. The conclusions in the
theorem we prove are weaker in that the convergence holds except for a set
of measure zero rather than except in a set of outer capacity zero. Our proof will assume the complex analysis that is in Rudin [1966], Real and Complex Analysis, and so on the way we shall prove results of Carleman and Nevanlinna on analytic functions of exponential type in a half plane. We define w 2t
Further
1.
log+ w = 0
is an increasing function of for all
if
0 < w < 1
log w = log w - log+ w
u, w > O.
w
and
for all
satisfying
Recall that a function
log+ w = log w
w > O.
Note that
log+ (uw) <_ log+u + log+ w f
analytic in the open right
half plane is of exponential type, or of exponential type T, a non-negative real number lim sup R-1 log (sup{lf(z)l R -
-
If C
f
:
is of exponential type
such that
A1.1
If(z)I
H ,
if there is
such that z E H,
T,
IzI
S R})
<_
then for each
<- C exp((T + c)Izl)
for all
T.
there is a constant
c > 0 z E H.
THEOREM (AHLFORS-HEINS) Let
plane
T
if
log+
f
and let
be analytic in a neighbourhood of the closed right half f
be of exponential type in
H.
If
I
to + f(i )ldy
1+y2 is finite, then
112 2Y1Tr/2
cos 0 log If(reie)I d0
a = lim /2
exists in
]R,
and
lim r 1 log If(reie)I = a cos 0
r-for almost all
0 E (-n/2, II/2).
We shall firstly prove a classical theorem of Carleman on functions analytic in a half plane, and obtain a number of properties of f
from this theorem. These properties and the Poisson formula for a semi-
disc are combined to give a proof of the theorem of Nevanlinna which expresses
in terms of the zeros of
log If(z)I
a convolution with a Poisson kernel along
DR.
f
in
H
and in terms of
In the proof of the Ahlfors-
Heins Theorem it is only the logarithm of the Blaschke product of the zeros of
that requires a rather delicate argument. The proofs are broken up
f
into lemmas, and our first lemma helps in various calculations with the zeros of
f.
A1.2
LEMMA If
B E H
with
< R and if
IBI
(R2+Bz
h(z)
R
for all
z E H,
z)
Ih(z)I =1 for all
then
z E i ]R u{z E C
:
IzI =R1
and
-1
h' (0) = 2(Re B) r 1
R2
Proof. Expanding
2 _
h
IBI 2 in a power series in
h(z) = 1 + z r- 1 1\
- i
B
z
+ s R2
+ S \+ z2..., R2
Now
h(iy) = \ 1 - iy/B/
we obtain
\R2 + iys/
which gives
h'(O).
113
for all y c R, and
I\
R + Rei
a /I
(Re-i
The lemma follows from these equalities because
A1.3
and let
f(O) = 1,
the open right half plane If
R > 0
101
= I81.
be analytic in a neighbourhood of the closed right half
f
with
H
6 E R.
e
THEOREM (Carleman's Theorem) Let
plane
for all
h(Rei0) = 8 r S - Reiel. /Re-i8 + 81
and
R
I
=1
then
rk,
R rk
cos ek
\12-1
rk
cos P log
/2
f(Re')I
Lw12
R + 1 2n fO
in
f
dR
log If(iy).f(-iy)I dy - 1 Re f- (O).
- 1
(1y2
be the zeros of
repeated according to their multiplicities.
H,
is not an
rk
k
zk = rk.e
R2)
2
The last integral is convergent. Proof. p
and
Firstly suppose that r
be small and positive. Let
contour which follows the semicircles the y-axis between
iR
and
-iR
r
since
{z e H :
IzI
< R}
r
in
and
and
log f(z) = f (O) z +....
for
Let
IzI 5 R}.
IzI = R
in
H
and
except that where there is a zero of
H.
By Theorem 13.18 Rudin [1966, p.262]
z N log f(z) z
f
by a small semicircle with
is simply connected and
a logarithm function log such that
:
be the positively oriented closed
IzI = p
on the y-axis the contour detours round it centre the zero and radius
{z E H
has no zeros in
f
f(O) = 1,
we may choose
is analytic in this set
near zero in
H.
The integral
I
p'r
= 1 2ni
1
J P(z2
+ 1 log f(z) dz R2)
is zero, because the integrand is analytic in a neighbourhood of the region surrounded by radius
p
P.
For
p
is approximately
very small the integral round the semicircle of
114
-11/2 1
f' (O) pelf' (1
11/
J
21ri
2
I\
p
2
e
+ 1 ) ipel d>V = - f-(0)/2.
2i
R
The integral round the semicircle of radius 11/2 1
is bounded above by
depending on Because
+
i R elf
detouring a zero of
as
c
I(z-2 + R-2) log f(z)I
K
and
are constants f(a).
the integral round each little semicircle
r -> O,
r
tends to zero.
tends to zero the integral down the y-axis is
r
{-y-2 + R-2 Ii log f(iy) dy
1
RZ I Y 14
{k-2
_ - 1
J
y-2} log(f(iy) f(-iy)) dy
-
p
21r
R
1
where
on the y-axis tends to zero as
f
After taking the limit as
,ri
on the y-axis,
f
and the determination of the logarithm near
a,m,
- 1
of
m
C + K flog lz - all
r flog rl -> O
2
2)
-1r/2
of order
a
is
cos $ log
J
1TR
Near a zero
R
R_
(R-2 e-2i
f1
2
r
{y-2 - R
2} log (f(iy) f(-iy)) dy
271 f o
as
then
p -+ O. p
-> 0
Now taking the real part of the limit of
as r - 0 p,r we obtain zero is equal to the right hand side of Carleman's I
equation, that is,
O= 1
1r/2
cos >y. log If(Re)
I
d>(1
nRfoJ -11/2 + 1
R (R 2 - y-2) log If(iy) f(-iy)I dy - 1 Re f'(O)
2
WT
Finally suppose that {z E H :
Iz) < R}
and let
f(z) = g(z) h(z),
z1,...,zn
are the zeros of
f
in
with each zero repeated according to its multiplicity, where
n h(z) = jIIl (1 z/zj
R
zzj
115
Then
is analytic in a neighbourhood of
g(z)
has no zeros in
{z E H :
IzI
< R}, and
z E i1R u {w E C
for all
Ih(z)l = 1
{z E H
IzI
the result will follow once we have proved that 1/2 Re h"(O) hand side of the equation in the statement (because Since each factor in
h
< R},
g
by Lemma A1.2, so that
Iwl = R}
:
:
Further
g(O) = 1.
is the left
h(O) = g(O) = 1).
has value 1 at zero, from Lemma A1.2 we obtain
n
1/2 h'(0) _
-
(R-2
z7 -2) Re z7
1
as required.
COROLLARY
A1.4
Let and let
be analytic in a neighbourhood of
f
be of exponential type in
f
H
H
repeated according to their multiplicities.
log
If
with
f(O) x 0,
zl,z2.... in'H
with zeros
lf(iy)l dy is
1+y2
im finite, then
(i)
(ii)
I Re (zn 1) converges, 11 - z/z
log
L
converges for all
n
z E H
not a zero of
1 + z/znll
log
(iii)
(iv)
all
n.
<_
cos yllog If(Reie)IdP : R _>
f
for all
eMzl+k If
is finite, and the set
R
by a constant we may assume that n,
for all
and let
M and
k
z E H.
Choose
r > 0
is large and not equal to an
Theorem C
r
is bounded.
-n/2
Proof. By dividing
If(z)l
f(i )II dy
f71/2
1
nR
zn = rneien
I
1 +y2
fm
(rn-2
- R-2 )
rn cos en
f(O) = 1.
Let
be constants such that
rn,
such that
r < rn
then by Carleman's
for
f,
116 flT/2
= 1
(1)
.log If(Re)I#
cos
-7r/2 JR + 1
(
y-2
- R-2 )
log If(iy). f(-iy)I dy - 1/2 Re f' (0)
21T
R hR
where
R_2)
(MR + k) + 1
n
r
2Tr
is a constant depending on
C
log+ If(iy) f(-iy)Idy + C,
(y 2 -
and
r
The last integral is bounded above by
I
R F'
R.
y 2 log+ If(iy)I dy,
1
2ir which is finite. Hence there is a
but independent of
f
r<
YI
such that the increasing function
B
(r -2 - R 2) rn cos 0 n
r n
S
equal to any
r
R
as n.
tends to infinity through positive real values not
Thus
cos 0n = I Re(zn-1) r
n
partial sums are bounded above by w e T
If
with
converges because its
IwI
This proves (i).
S.
<_ 1/2,
then
Ilog 11 + wiI < 21w1.
Now
-+n-
1 - z/z = 1 - 2z Re(z -l) 1+ z/zn 1+z/z n so that
I1=z/znl
(2)
Ilog
for z e C
with
l+z/zn 11 - z/z
I
-1)
5 8IzI Re(zn-1)
11 + z/znI
Izi Re(z n-1)
< 1/2.
Hence if
z e H
is not a zero of
converges. This is (ii). Note that since
log I1 -- z/z
then
< 4Izl Re(
1 + z/zn
II
is less than 1 for z e H its logarithm is negative.
1 + z/zn If log = log + + log
R
is large and
R
is not an
rn,
then using
and part of inequality (1) we obtain
f,
117
-1
(y 2- R-2) log
J
2n
I f (iy) j dy
r
C + R 1(MR + k) +
I.
cos 6 + 1 rj
2n
rj
log+ If(iy)I dy. y2
r5IyI5R
The right hand side of this inequality is bounded above for all specified R
R
since the series and integral converge as
r <-
<- R/2,
IyI
then
3 y-2
(y-2 - R-2 )
tends to infinity. If
so that - 3 4
r
y-2 log- lf(iy)ldy
J
rSI I5R is bounded above for all - log
r.
It follows that
is finite, and this implies the convergence of
If(iy)I dy
1 +y2
f3R
because
Ilog If(iy)II dy
1+y2
fR
This proves (iii).
Ilogl = log+ - log-.
From Carleman's Theorem for
R not the modulus of a zero of
we have
f,
(n/2
- 1
cos 4' log
1
1Tr
If(R ei4')I d 4'
-n/2 n/2
cos e. + 1 j=1 r nR j
+ 1
IR (Y-2 -
2n
cos 4 If(R ei*)I d 4' J
-n/2
R-2 )
log+ If(fy) f(-iy)I dy + 1/2 If-(0)1'
JJJ o
R because
The right hand side is bounded as a function of 5 MR + k
log+ If(iy)I dy
and
1+y2
fR
log+ If(R e")I
is finite. Thus the set of
n/2
-1 nR
cos 4,. log
I f (R ei4')
-n/2
R not the modulus of a zero of
for
d*
f
is bounded above. Hence the set of
(n/2 1
nR
cos 4.Iloglf(Rel4)II di,
1
is bounded above for all
R ? 1.
This
-n/2
is seen to be true initially for
R
not a zero of
restriction is lifted because near a zero
z
n
f
f,
and then this
behaves like a constant
118
times (z - z
n
for some positive integer
) m
m
I1 IlogltII dt
and
1
is
l
finite. This completes the proof of Corollary Al.4.
Our next lemma is Poisson's formula for a semicircle, and it
will lead to the crucial theorem of Nevanlinna when we let
R
tend to
infinity. Though the proof of the lemma is similar to the proof of Carleman's Theorem we give it in detail.
A1.5
LEMMA Let
H
,
and let
be analytic in a neighbourhood of the closed halfplane
f
be the zeros of
zl,z2,...
in the open right half plane
f
repeated according to their multiplicities. If zero of
and
f
z = x + iy e H
if
with
IzI
R
H
is not the modulus of a
< R,
then
log If(z)I
log Izk
(1
I\l + z/zk
+ x
-R 1R
Iv -2yv+Izl2
+ 2Rx
(R2 ,, 2 2 IRel
Proof. Suppose that <- R}.
circle
Let
loglf(iv)I dv
z
2)
log
cos
-zI 2
has no zero in the half open semidisc
f
in
and along the y-axis between
H
that the contour detours round each zero of semicircle in
H
hood of the closed semidisc on
if_-R,R7.
j = 1 J 2ni r
iR
and
1
-iR
except
on the y-axis by a small
f
log f(z)
{z e H
:
IzI
is analytic in a neighbour-
<- R},
except for the zeros of
- z
w+z
+ z
wz-R2
log f(w) dw. Grouping the denominators
wz+R2
in suitable pairs we see that this is equal to
2ri
:
We consider the integral - 1
1
w-z
{z e H
with centre the zero. As in the proof of Theorem 1.3 we
choose and fix a logarithm so that
f
d$
IRe+zl 2
be the positively oriented closed contour round the semi-
r
IzI = R
2
v IzI2-2yvR2+R4
J -it
it
IzI
R2 - zzk/I
- R
1
it
R2 + zz
z/zki
fr 1 2x
lw2-2iyw-IzI2
log f(w) dw
- 2R2x
w2I.12-2iywR2-R4
I
119
and also to
1
-
1
1
Ti 2r
(w+z)(wz+R2)
Near one of the zeros likelw-alm,
m
where
I (Iz12 - R2) log f(w) dw.
1-
+
(w-Z) (wz-R2)
of
a
on the y-axis
f
If(w)I
behaves
is the order of the zero. Hence the integral along
the little semicircle round the zero will behave like a constant times mt log t,
where
t
is the readius of the little semicircle under considera-
tion, and this tends to zero as
tends to zero. We now let the radii of
t
the small semicircles tend to zero and we find the limit of two equivalent forms of
J
using the
on the y-axis and the semicircle of radius
J
R.
The limit is
f-R - 2R2x
2x
1
I-v2+2yv-IzI2
R
27r
fn/2
+ 1
log f(iv) dv I
-v2lzl2+2yvR2-R41
1
-
(1z12-R2) log f(Re) RieiodO.
1
2ni J- x/2
I
The real part of this is
Rv2-2yv+Izl2 2x
1 2rz
log If(iv)I dv
2R x
f_R v2IzI222yvR2+R4
+ 2Rx 1x/2 _n/2
(R2 -
log If(ReiO)I d
cos 0 IReiO+ZI2
IzI2)
on simplifying the term in brackets in the second integral. The only pole of the integrand in the integral defining term
(w-z)-1.
J
From the residue theorem
real part of this the result follows when
is at
w = z
J = log f(z), f
caused by the
and by taking the
has no zeros in
{zEH: IzI <_R}. Now suppose that {z E H
n 1- z/z, h(z) =
II
J=l 1+ z/z
f
has
n
zeros
z11...zn
in
with each zero repeated according to its multiplicity.Let
IzI
:
] ,
J
R2+z z k R2-zkz
.
Then
h
is analytic in a neighbourhood of
120
{z E H
:
IzI
and by Lemma A1.2 we have
<_ R},
z E i R u{w E C
Iwl = R}.
:
just proved applies to
Let
because
g
then what we have
has no zeros in
g
for
= 1
h(z)
g(z) = f(z)/h(z),
{z E H
:
< R}.
IzI
The lemma now follows.
A1.6
THEOREM (NEVANLINNA'S FORMULA) Let
plane
H
,
be analytic in a neighbourhood of the closed right half
f
and let
H with
be of exponential type in
f
y = lim inf r-1 log M(r),
where
M(r) = sup{If(z)I
:
z E H,
IzI
<_ r}.
If
rJ
to + f(iw)Idw
R
is finite, then
1 + w2 log If(z)I
log I-
= cx +
+ x
for
where
where
z = x + iy E H,
right half plane c = lim
r-
7Tr
dw
zl,z2,...
are the zeros of
f
in the open
repeated according to their multiplicities, and
H
2
log If(iw)I
n R (w - y) 2 + x2
'-1 + z/zn
f"n/2
cos 4.log
Further
d$.
Proof. This result is obtained by letting
c <- 2y.
R tend to infinity in the
Poisson formula for a semicircle (Lemma A1.5). We shall tackle the various terms on the right hand side of the equation in Lemma A1.5 one-by-one. The modulus of the difference between the first term
Log
1 - z/zk
.
R2 + zzk
Izkl
1 + z/zk
R2 - zzk
of the Poisson formula for a semicircle and
log
is
IzkI
log
1 - zR 2/zk
Izkl
< 8IzIR-2
C
Izkl
1 + zR2/zk
Re(zk 1)
121
by inequality (2) in the proof of Corollary Al.4 (see p.116) provided is large enough compared with
Since
Izi.
first term of Lemma A1.5 tends to the
I Re(zk 1)
I log
1 - z/zkl as 1 + Z/Z
R
converges, the
R
tends to
k
infinity.
The first part of the first integral equals
rR x J
log lf(iw)I. dw
1
-R w2-2yw+IzI2
r
JR r
which converges to
because
I
1 x r f3R x2 +(w-y)2
Ilog f(iw)Ildw
R
loglf(iw)ldw,
1
-R x2 +(w-Y)2
log If(iw)Idw
R
as
tends to infinity
is finite by Lemma Al.4(iii). Though we do not
1+w2
use it, we note that
x
where
log If(iw)Idw = (Px * log
1
J
R x2+(w-y)2
r
Px(y) =
x (x2+y2)
We turn to the second part of the first integral. If large that
< R/2,
lyl
G(w) =
and if
R2x
R
is so
then
w2x2+R4/4
R2x
=
w2lz12-2ywR2+R4
for
IwI 5 R.
Now for
1
fwl?m
e > 0
log+I f (iw) l
R2x
5 G(w)
w2 x2+(R2-yw)2
given we choose
m
such that
dw 5 e,
1 + w2
Multiplying the integrand by
1+w2 and splitting the integral at ± m, l+w2
we
122
obtain
1
log If(iw)I dw
xR2
JR
n w2IzI2-2ywR2+R4
-R
<- max I G(w)(1 + w2)
Iwl
:
<_ m If to + f(iw)
R
n
+ e max
I
G(w).(l+w2)
m <_
5 Rl
IwI
n
11
R2x (m2+1)
m2x2+R4/4
since (R
w F' (1 + w2) G(w)
is large) and
to + f(iw)
.1. n
fIR
dw + c.8x it
1+w2
is an increasing function of
(1 + R2)G(R) <- 8x.
fR
1 n
dw
1+w2
R2x
Letting
R
w
R4 > 4x2
for
tend to infinity,
loglf(iw)I dw
-R
w2I.I2-2ywR2+R4
tends to zero.
We compare the last integral in
A1.5 with
f/2
2x
7r
log
cos
d$
though we drop the factor
2x
in the
-n/2
following calculations. Then n/2 1
cos t.loglf(Relo )I d
I
nR J-n/2 fn/2
- RI
(R2 -
cos $. log
Izi2)
n
do
(Rely-zI2IReio+ZI2
-n/2 fTr/2
d
cos 4.Ilog
1
nR
max
_n/2
1 -
R2(R2 - Iz12) IRelo-zI2IReio +zl2
for all large
R.
By Lemma
bounded as a function of
R.
A1.4(iv) the quantity before the maximum is Further the maximum tends to zero as
R
tends
123
to infinity.
From the above calculations and Lemma A1.5 it follows that the limit giving
c < 2y
exists and that
c
the inequality
If(Reio)I
log If(z)I for
<- M(R)
from the definitions of
c
and
has the required value. Using
< r/2
101
one easily obtains
This completes the proof of
y.
Theorem A1.6.
The Ahlfors-Heins Theorem is proved by replacing in Nevanlinna's formula and letting
R
by Rel+
z
tend to infinity. The limits of
the terms are easy to calculate except for the term I log
arising
from the zeros of
I log
f
essentially via the Blaschke product. The
I ...I
term we handle using the inequality in Lemma A1.7, which is due to A.M. Davie.
A1.7
LEMMA
on
i
then there are constants
0 < 1 < n/2,
If
A
and B
depending
such that
log u-w T- T
for all
< Re w (A + B Ilogle - 011)
+w
w = pelf E H \{O}
Iw12
and all
u = reie E H
with
7/2
and
lel <,4). Proof. We shall split the pairs
satisfying the hypotheses into two
(u,w)
sets
{(u,w) {(u,w)
:
lu - wI
>_ 1/2IwI cos},
Iu - wI
< 1/2IwI cos
and
}.
We consider these two cases separately. If
lu - wI
log (1 + x) < x
for
>_ 1/2 IwI cos
x > - 1
log lu + w u-w =1 2
log u + w lu - w
2
V,
we obtain
then using the inequality
124
=flog l1+4 Re wRe u t Iu - wI2 (1)
We now consider the more difficult case
and we use lu
wl
u = reie and
<_ 1/2 IwI cos l
Iu - wI
<_ 1,2IwI cos
Squaring the inequality
w = pelf. gives
r2 - 2rp cos (8 - ) + p 2 - 1/
p2
cos2',
4
which leads to
2r cos 8 cos , P
r2+2rsin 6sin +1-1 cost 2
r2 2
4
P
- 2r sin
AP
+ 1 - 1 cost 4
p
P
because
Isin 8 sin 4l <_ sin i,
(r - sin l 2)+ 3 cost , 4 P z 3 cos2 1). 4
Hence
8 sect p. 3
Using the inequality
Re w 2 1 IwI2
(2)
lul
t + t 1 > 2
for positive
for the product of two sines, we have
t = r/ , P
and the formula
125
u + w
= r
u - w
+ 2rp cos(8 + 4>) + p2
r2 - 2rp cos(O -
4>)
+ p2
= 1 + 4 cos 8 cos ¢ r/p + p/r - cos(8 - ¢)
<- 1 + 2 cos 8 cos
1 - cos(8 -
4>
4>)
=1+cos (0 +¢) 1-cos (8-4>) From the graphs of the functions 1 - cos
we observe that
1 - cos F >- 2E2/7r
and
2E2/n2
for
0 <
< n,
2
2
1
n/2
For
0
7/2
sin E 2 2E/7r for
n
the inequality may be obtained by integrating 0 <-
E
<- n/2,
which gives
1 - cos
2C2/7r2.
E2/n a
The remainder of the range may be checked using the convexity of and the convacity of
on
n/2 <- C 5 n.
Thus
1 - cos
126
u + w
log
u - w < log
2
(3)
G -cos < log (
<_
Using (1),
Tr
2
l
2 Ilog 16 - 011 + 2 log Tr.
(2), and (3) we see that
-
u - w
log
1 1u1
u + w
A = max
8 sec2 ry,
5 Re w (A + B
log I8 - I
I),
IwI2
where
16 sec2 i log Tr= 3
8 sec2
1
and
B = 16 sec2 3
This proves the lemma.
A1.8
PROOF OF THE AHLFORS-HEINS THEOREM Let
0 <
formula, and let
r
< 7/2.
We let
z = rei8with Iel
tend to infinity. The term
in Nevanlinna's
in Nevanlinna's
cx
formula gives us the limit required in the Ahlfors-Heins' Theorem, and we show that
r 1
times the other terms tend to zero for almost all
0
in
(-Tr/2, n/2) . We begin with the integral term
J = x IT
r
J] .
log f(iw)I dw
(w-y) 2+x2
in Nevanlinna's formula, and split the range of integration into and
Iwl 2 m
for some large
use the inequality
m.
Iwl
<_ m
In the latter range of integration we
127
cos 6
(w-rsin6)2 + r2cos28
which we obtain from
j < r
1
<-
1
<-
,
w2cos
w2cos6
For large
w2cos26 <- w2-2rwsin6 + r2.
dw +
to + f(iw)
1
m
We now choose, and fix,
log+lf(iw)I
I
dw
IwI?m
w2-2rwsin6 + r2
IwIsm
1
m,
w2
cos
so large that
to + f(iw)
1
dw
f
cosh
is very small. Then for
w2
IwI?m
sufficiently large the integral
r
I
also very small. This shows that I61
as
<- i
Jr 1
is
r
IZm
tends to zero uniformly in
tends to infinity.
r
We now tackle the sum of the logarithms in Nevanlinna's formula (that is, the logarithm of the Blaschke product of the zeros of f). Let c > 0,
A
and let
Lemma 1.7. Since of
f
in
and
be the constants corresponding to
B
converges (Al.4), where
I Re(z n-1)
we may choose
H,
Re(z
I
n?k
-1)
z
n
i
given by
are the zeros
so that
k
< E2.
n
From the equality
1 - rei6/zn
=
z
1 + reie/in
and Lemma 1.7 with
zn - re i6
u = reie
n
+ re 16
and w = zn = IznI 1 - re
_ - 1
F (reie)
r
log
we obtain
ie
/'n
1 + rei6/z
n
IA + B log le - nl
Rezn
n_k
1
n?k
elfin,
2
Izn
G(e) for all
r > 0
and
e e [-M1.
Let
U
be the set of all
e e
[-,
128
such that
lim sup - 1
r 4-
r
log
1
1 - reie/z
n=1 1 + reie/z
2E.
n n
Since
lim
1 - re
log
ie
n
1 + reie/z
for
1
n
6 e U
we have
:- n <_ k,
= 0
/z
r-
if and only if
lim sup F(reie) ? 2E.
r -> Thus
F(reie) ?
U c {e E {6 e If
p
G(e)
is Lebesgue measure on E.L (U)
<_
E
for some r}
E}.
>_
]R, then
G(e) ? E}
E.11 {e E
G(e)de
(A + B Ilog le - nlide
Rez
2J-
n?k <_ E2
(AnI 2B f
I log tldt)
0
since
< 7/2
Ie -
for all
nI
a constant depending on A
-.
log
lira sup - 1
r
r
1
for almost all 6 e 1 - rei6/znl
1+ reie zn 6 e (-it/2,a/2).
1,
n.
1 - reie/ z
where
u(U) s CE, E.
C is
Thus
< O
1 + rei6/z n reie
Because and hence
Therefore
but independent of
and B
zn
are in
log 11 - rei6/ I >_ 0
- 1 r
and
l
1+ re16 z
H,
we have
for all
n
Thus the lim sup above is equal to the limit almost every-
where and is zero almost everywhere. The proof of the Ahlfors-Heins' Theorem is complete.
129
The following corollary of Nevanlinna's formula will be used in Appendix 2 when we investigate closed ideals in
This corollary is a weaker
L1OR+,wti).
version of a theorem of Krein that an entire function which is the quotient of bounded analytic functions in the upper and lower half planes is of exponential type.
COROLLARY
A1.9
Let
be an entire function such that
f
left half plane
If there are functions
g
neighbourhood of the closed right half plane
H
-H.
and k bounded in Proof.
Since
If(iw)I
where
-< 1
H ,
analytic in a
k
and
with
fg = k
is of exponential type in
f
and
g
C.
is bounded in the left half plane, we may assume that
f
for all
We apply Nevanlinna's formula to
w E ]R.
zl,z2.... and
right half plane Since
then
is bounded in the
f
are the zeros of
wl,w2,...
g
k
and
and
g
k,
in the open
with each zero repeated according to its multiplicity.
H
f(z) = k(z)/g(z)
for
z
entire function, the zeros of
not a zero of
g
and since
are zeros of
k
of smaller multiplicity.
g
f
is an
Hence
1 - Z/w n
1+z/wn
1-z/z n
- log
1+z/z n
where the second summation is over those zeros of zeros of
g,
that is, over the zeros of
Nevanlinna's formula to all
g
and
k
in
f
k H.
not cancelled by
We now apply
obtaining, for some constant
z = x+iy E H,
log If(z)I
= log Ik(z)I - log Ig(z)I = Cx +
log 1
+ x
1-z/w
- log
1 + z/wn
J
1
z/i
1 + z/z n
loglk(iw)I -
log
dw (w - Y)2+x2
5 Cx,
C
and
130
because
If(iw)I
<-
1
for all
w E IR
and
<_ 1
ll----z/w
for all
j.
1+z/w n. J
Hence
f
is of exponential type in the right half plane, and so in
T.
A1.10 NOTES AND REMARKS The Ahlfors-Heins Theorem is in Ahlfors and Heins [1949], and Boas [1954] has a section devoted to it (Section 7.2) - see also Hayman [1956]. However there are two minor points in Boas's account that make
matters difficult for the innocent reader. Boas omits the factor (R2-znz)/(R2+znz) 2 (R
2
- znz)/(R -znz)
in many of the calculations - the factor in his case is
because he is working in the upper half plane (see
the list of errata in Boas [1974]). Also he works sometimes in the upper half plane and sometimes in the lower half plane, and though there is nothing wrong it adds further confusion. The account here is influenced by Boas [1954] (Section 7.2). Lemma A1.7 and this variation of the proof of the Ahlfors-Heins Theorem are due to A.M. Davie, and I am grateful for his permission to use these ideas here. Krein's Theorem is in Krein [1947], and I do not know of a simple published proof.
131
APPENDIX 2: ALLAN'S THEOREM - CLOSED IDEALS IN L1 OR+,w)
A major open problem in radical Banach algebra theory is, are all the proper closed ideals in L1OR ,w) equal to the standard ideals IS = {g E L1(R+,w) w?
:
g = 0
a.e. on [M]} for suitable radical weights
We shall prove a theorem of Allan [19791 that certain closed ideals in are of the standard form. We use this theorem in Chapter 4. We
L1(R+,w)
firstly recall some notation and ideas from 2.12. A radical weight
R
is a continuous positive valued function
satisfying
The Banach space
r J0-Ig(t)I
w(t) dt =
L1 R+, w)
IIghi < -}
= {g
:
g
on
w(r)l/r . O
as
R = CO,-) - (O,-)
w
s, t > 0
for all
w(s + t) <- w(s) w(t)
w
and
is measurable,
becomes a radical Banach algebra under the
0
convolution product
ft
(f * g)(t) =
f(t - w) g(w) dw
for
a.e.
t E Il2
0
THEOREM
A2.1
Let
ideal in
w
L1CR+,w).
1- If(t)I e-kt dt
be a radical weight on R+,
and let
f E J
If there is a non-zero
is finite, then there is a
y ? 0
J
and a
be a closed k > O
such that
such that
0
J = {g E L10R+,w)
Proof. If
0 5
g = 0
:
L1+,w) ;
g = 0
A direct calculation shows that each 1(0) = L1OR+, W).
I(y) = n {I(S)
on
[O,y7}.
let
I(g) _ {g E
and clearly
a.e.
:
B < y}
Let
so that
I(8)
is a closed ideal in
y = sup {g I(y) 2 J.
We shall prove the reverse inclusion.
on [O,S]}.
a.e.
If
:
1($) 2 J}. y = 0,
If
clearly
L1 CR+, w),
Y > 0,
I(y) 2 J.
132
+
Let L OR
= {h
)
< -} . Then
h measurable, I I h/w IL = II h I I
:
is a Banach space, and there is a natural duality between
L'OR+,w-l)
and
L1 OR +,w)
w- 1
,
= fo h(w) g(w) dw
given by
L'OR+,w-1)
for
0
LOR+,w)
space
h e LOR+,w-1)
and
g e L1(R+,w)
of
R+
Let
by
on the space of
a
is the infimum of the support of
a(g)
and let
fI e J
with the dual
L'OR+,w-1)
We define the function
L OR+,w).
measurable functions on
identifying
such that
K e 1R
g.
Ifl(t)I e(-K+1)t dt
J
0
is finite.
F(t) =
e-Kt
1)
with
w(t)1/t
ti=
as
-+ 0
on
G e L1 OR)
by
L1OR)
for all
L1OR)
Since
w e]R.
for all
for almost all t > O,
we
0
and the Fourier transform
n L2 (R)
We identify
t
of functions that vanish on
h(w) = h(-w)
Ig(t) I S IIgII w(t)
and
We let
* L1OR+,w)> = {O}.
1
G(t) = eKt g(t + 1)
We define
(--,O).
e]R+.
and
f1(t)
with the Banach subalgebra of
L1OR+)
have
g e LOR+,w
Let
G(t) e-lXtdt
=
Gn(A/2ir)
is an entire function that is bounded in the closed upper half plane R = {z e T
n
a neighbourhood of and
(-°°,0)
n
and
-II
For each
e
n
t E+ fl (t - x)
J
t f+ f1(t - x)
J
is in
fl * en,
[x - 1/n,x + 1/n].
for all
x > 0
so that
0
F(t) G(t + w) dt
F * G(-w) = J
=
is zero on
[O,-) -+ T
:
is the norm limit of
f1(t - x) g(t) dt = 0
0
F
[O,-).
2n times the characteristic function of
is
is analytic in
F
because
-II
is integrable on
the function
x > 0
because
f1(t) g(t + x)dt =
Thus
is bounded on
F
e(-K+1)t f(t)
t f+
(fl * L1OR+,w))
where
Further the Fourier transform
Im z ? O}.
:
0
I
e-' f(t) eKt.eKw g(t + w + 1) dt = 0
Jo
for all Let
w >_ -1.
L = F * G
transform
L
in
A
L1OR).
is zero on
-II,
and is bounded in
(-°,l]. {z a Q: -II.
:
Hence the Fourier Im z < 1}
of the
The entire function
satisfies the hypotheses of Corollary A1.9 if we rotate the complex plane
through
n/2
and identify the closed lower half plane
right half plane
G A,
L
is analytic in a neighbourhood
closed lower half plane G GA
Then
is in
L2
H
Thus
Gn
is of exponential type. Using
G,
and so
we may apply the Paley-wiener Theorem (see Rudin [1966])
OR)
and deduce that
.
-II with the closed
G,
and so
g,
has compact support.
Let
O
be the
133
supremum of the support of it follows that
From the definition of
g.
The Titchmarsh Convolution Theorem (see
a(G) = -a + 1.
Mikusinski [1959]) states that
a(F * G) = a(F) + a(G).
zero on
Thus
(-°°,l], a(F * G)
support of
is an arbitrary function in
g
it follows that
(fl * L1CR+,w))
isomorphism between Using
L1OR+,w)*
y = inf {a(h) : h e J},
Therefore
we obtain
fl
e > 0
and thus the restriction of
J,
J ? (h * L1(R+,w))
by the
= I(a(fl))
there is an h
to
y < a(f).
h e J
is in
[a(f),-)
is in
= I(a(h)) 2 I(y + c).
Since
to
Since
such that
[y,a(f)]
h
because
J 2 I(a(f))
we are done so we suppose that for
orthogonal
L'CR+,w 1)
(fl * L1 CR+,w))
Now the restriction of
a(h) <_ y + e.
in
is
and therefore the
and L(R+,a 1).
in place of
f
a(f) = y,
If
f e J.
F * G
Since
1 5 a(f1) - $ + 1,
is contained in the interval [O,a(f1)].
g
Since to
? 1.
Kt
G(t) = g(-t + 1)e
I(a(f)),
J n L1
so
OR+).
I(y + c)
U
e>O
is dense in
A2.2
I(y),
we have
J 2 I(y).
The proof is complete.
NOTES AND REMARKS
This is one of the two results in Allan [1979] showing that if there is a nice
f
in a closed ideal
J
of
Ll(R+, w),
then
J
is
standard. The other condition is a local one near zero. See Dales [1978] for a [1981].
general discussion. These results have been strengthened by Domar
134
APPENDIX 3: QUASICENTRAL BOUNDED APPROXIMATE IDENTITIES
In Chapters 3 and 4 we used bounded approximate identities,
which had nice properties with respect to derivations, multipliers, or automorphisms, to obtain corresponding properties for an analytic semigroup near
t [* at
0
in
The problem is when do these nice bounded
]R+.
In this appendix we show that in an Arens
approximate identities exist?
regular Banach algebra with a bounded approximate identity, there is a new bounded approximate identity that behaves well with respect to derivations, multipliers, and automorphisms. The idea is to lift the problem from the
algebra A
to the second dual
with the Arens product and exploit
A
**
the identity in
A3.1
A
given by the hypotheses.
DEFINITION Let
A
be a Banach algebra, and define the Arens product
F.G
**
on
A
as follows
=
= = **
*
for all
x,y a A, f e A
and
,
F,G E A
,
where
is the pairing
between dual Banach spaces. The reversed product by
x o y = yx
for all
x,y e A.
on the Banach algebra
o
The Banach algebra A
A
is defined
is said to be
**
Arens regular if the reverse of the Arens product in product of the reversed product in
A.
A
is the Arens
With a little calculation we could
write down the definition of Arens regularity symbolically and relate it to *
the relationship between A
and
be the natural embedding g from
A ,
A
but it is the idea we require. Let into
A**
given by
<x ,f> =
135
x e A
for all
A3.2
and
F e A.
LEMMA
A be a Banach algebra with a bounded approximate identity
Let
**
is Arens regular, then
A
If
A.
a(A**,A*)-closure of
the
has an identity
A
and
E,
**
**
Proof. Since the unit sphere of A topology, there is a net
fey
is compact in the
Y e r}
:
a bounded approximate identity for **
a(A
is in
E
An.
in
n
A )
fey
tends to
eY
and
A,
such that
A
*
a(A
E
:
-
y E r}
is
in the
*
,A
f e A%
If
- topology.
)
<E.f,x> = <E,f.x> = lim <e
then
,f.x> y
= lim = lim _ **
for all
x E A
so that
E.f = f.
F.E = F
Hence
F E A
for all
,
and
**
is a right identity for
E
A
Since
bounded approximate identity for
A,
Y E r}
fey
is also a left
F 0 E = F
we have
for all
F E A
**
where o
When
in A. **
A
.
is the Arens product in A
is Arens regular
A
arising from the reversed product
F 0 E = E.F,
so
A
Recall that a derivation D on an algebra operator on
A
let
a,b E A.
denote the group of continuous automorphisms on
Aut(A)
Der(A)
denote the Banach space of continuous derivations on
is Arens regular, then for each
A
for all
A be a Banach algebra with a bounded approximate identity
Let
let
D(ab) = aD(b) + D(a)b
satisfying
is a linear
THEOREM
A3.3
A,
is an identity for
E
This proves the lemma.
G c Aut(A),
D c Der(A),
and
e > 0
there is an
and all finite subsets e e co(A),
such that Ilex - xll + llx.e - xll < e , lla(e) - ell < c IID(e) II < e for all x E F, a E G, and D E D. ey
n
{e
:
y e r}
be the net in A
Y
tends to
E
**
in the
a(A
,
G = fall ...,am}
and
and If
F c A,
and
given by Lemma A3.2 such that
*
A )-topology. Choose
such that
S E r
II-1.x - xll + Ilx.ey - xll < c for all x E F and all y E r If
A.
the convex hull of
A,
Proof. Let
A,
D = {Dl,...,Dn},
let
W
with y >
B.
be the convex hull of
the set
{(1 - a1)eY,...,(l -
in the Banach space
Am+n
with norm
am)eY,D1(ey),...,Dn(ey))
l l (xj) 11 = max {I I xj 11
:
:
y E r, y > S}
1 < j < m + n}.
136
The norm closure of W
and the weak closure of
W
coincide by the Hahn-
Banach Theorem, and we shall show that (0) is in this closure. A straightforward calculation using the definition of the Arens product on
and of the second dual of an operator shows that
A**
an automorphism on A**
for
A**
D E Der(A).
E
f E A*
is
is a derivation on
D**
is the identity in
For each
(a** - 1)(E) = 0 = D**(E).
and
a E Aut(A),
for
Since
a**
A**,
we have
we have
O = <(a** - 1)E,f> (a* - 1)f>
<E,
= Jim <(a* - 1)f, e> = Jim
,
and
0 = = lim .
Hence
(0)
is in the weak closure of
combination IID(e)II < e
e
from
for all
{eY
:
Am+n.
W
in
y E r, y > $} such that
a E G
and
Thus there is a convex 11(o, - 1)eII < e and
The proof is complete.
D E D.
If the hypotheses of the above theorem are satisfied, then
A
has a quasicentral bounded approximate identity for each enveloping Banach algebra containing
A
as a two-sided ideal (see 4.8, proof of property 17).
Since C*-algebras are Arens regular (see Civin and Yood [1961]) this gives another way of writing the proof that a C*-algebra has a quasicentral
bounded approximate identity (see Arveson [1977], Akemann and Pedersen
[1977], and Sinclair [1979a]). Suppose that a Banach algebra A bounded approximate identity. Then
A
has a
has a quasicentral bounded approximate
identity if
(i)
(ii)
(iii)
If
A* = A.A* + A*.A
(see Sinclair [1979a]), or
A has a central bounded approximate identity (trivial),
A
or
is Arens regular
B = (A + C)-,
where
C
is a commutative Banach algebra generated as
a Banach algebra by the set U = {u E C
:
Hull = 11--111= 1},
then
A
has
137
a quasicentral bounded approximate identity. The proof is similar to A3.3 except that the element
E
theorem not from A3.2, and
being an identity for
A3.4
comes from the Markov-Kakutani fixed point E E A**
commutes with all
u E U
rather than
A**.
PROBLEM Let
G
be a locally compact group. When does
L1(G)
have a
quasicentral bounded approximate identity?
A3.5
NOTES AND REMARKS Lemma A3.2 is in Civin and Yood [1961]. Quasicentral bounded
approximate identities play a fundamental role in C*-algebra theory. See Arveson [1977], Akemann and Pedersen [1977], and Elliott [1977].
138
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143
INDEX
NOTE.
The terms "analytic semigroup", "Banach algebra",
"bounded approximate identity", and "semigroup" occur so frequently they are not covered in the index except for the first occurrence. The most important page number is underlined.
4 A# 33, 49 Aarnes and Kadison Ahlfors-Heins theorem 3, 10, 73, 111, 130 49, 136, Akemann and Pedersen
137
algebraically irreducible 78 representation Allan l0, 33, 40, 131, 133 Allan and Sinclair 49 analytic function into a Banach space 5 analytic semigroup 7 Arens product 134 Arens regular Banach algebra 10, 134 Arveson 47-,49, 136, 137 48, 135 automorphism Aut A = continuous automorphisms 135 on A backwards heat semigroup 8, 33 33, 49, 89 Bade and Dales Baire Category Theorem 103 4 Banach algebra Berge and Forst 2 19 Beta function BL(X)
by
88
5 C = complex numbers Ct 20, 29 C0(O) 12 8, 12, 22, 33 C*-algebra CL(X) = algebra of compact operators 41, 42, 43 112 Carleman's Theorem carrier space 84 5, 16 Cauchy's integral formula Cauchy semigroup = Poisson semigroup
29
Choi and Effros 43 Civin and Yood 136, 137 closed graph 92 closed linear operator 92 50 Cohen Cohen's factorization theorem 9, 35, 39, 50 completely positive operator 43 7, 93 contraction semigroup convex bounded approximate identity 36
co(A) = convex hull of A Curtis and Figa-Talamanca
135 69
4
111, 130 Boas Bochner 14 Bochner integral Bonsall and Duncan
3
2, 12, 48,
62
bounded approximate identity Butzer and Berhens 2
4
D(R) = domain of R 92 Dales 133 Davie 89, 123, 130 Der(A) = continuous derivations on A 135 48, 135 derivation
144
41, 87 disc algebra Dixmier 82, 89, 108 37, 38, 49, 69 Dixon 2, 33, 48, Doran and Wichman
Hulanicki 34, 82, 89 Hulanicki and Pytlik 49 Hunt 49 hyperinvarient subspace 10, 100, 103, 104, 105
69
Dunford and Schwartz 91, 110
1, 2, 10, It = fractional integral semigroup infinitesimal generator = generator
Elliott 47, 49, 137 equivalent norm 37 15, 19, 23 Erdelyi 2, 33, 48, 89, 90, Esterle 110 49 T11-factorization Evans and Lewis 49 exponential type 79, 82, 111
Jacobson semisimple algebra = semisimple algebra 44, 49, 69 Johnson
129, 130
Krein
FL(X) = algebra of finite rank operators 42 24, 102, 132 Fourier transform fractional integral semigroup 8, 16, 70, 101 functional calculus 37
F = Gamma function 15 8, 24, 25, Gaussian semigroup 35, 81, 99 15, 19 Gelfand map 8, 12, Gelfand-Naimark Theorem
L = Laplace transform 14, 22, 33, 38, 99 -17,-22, 24, 34 23, 33, 131 L1(Rw) L*[0,l] = Volterra algebra L1CR+) LlCRn)
23, 39
108 Lance 43
Laplace transform 15, 95 1, 25, 92, 99 Laplacian Leptin 89 Lie group 34 locally compact group 41, 81 105 Lomonosov's Theorem
13
Gelfand, Raikov, and Shilov generator = infinitesimal generator 1, 96 33 Ghahramani 90 Grabiner Gr$nbaek 49 group of invertible elements
33
Markov-Kakutani Fixed Point Theorem 137
Mikusidski 2, 133 Mul(A) = multiplier algebra of A 15, 82
4,
84
Gulick, Liu and van Rooij
H = {z E C
:
Rez > 0}
69
5
Hardy and Littlewood 33 49 Herbert and Lacey Hewitt 69 Hewitt and Ross 39, 41, 42, 48, 69 1, 2, 10, Hille and Phillips 33, 90, 91, 98, 110 Hille-Yoshida Theorem 1, 2, 3, 9, 10, 50, 91, 96, 110 Hoffman-JOrgensen 90 Holder's inequality 17
5 IN = positive integers Nevanlinna's formula 120 nilpotent semigroup 100, 106 nonseparable multiplier algebra 4 v(x) = spectral radius of x 43, 44 nuclear C -algebra
B2
Olver 15 one parameter semigroup = semigroup 1
Paley-Wiener Theorem 2, 33 Pedersen
102, 132
145
Plancherel's Theorem 102 Poisson's formula for a semicircle 118 Poisson semigroup = Pt 8, 24, 29, 35, 81 polynomial growth (in a group)
Varopoulos 69 Volterra algebra = L1*[0,l] Volterra operator 105
Weierstrass semigroup = Gaussian semigroup Widder 33 9, 78 Wiener Tauberian Theorem
81 78 primitive ideal proper closed ideal 49 Pytlik
10
2Z
quasicentral bounded approximate identity 136, 137 2, quasinilpotent semigroup 106
R = real numbers 5 radical convolution algebra = L1(R+, w ) radical weight 23, 40, 75, 131
Radjavi and Rosenthal
105,
108
Reed and Simon 1, 2, 91, 110 resolvent 1, 92 resolvent set 92 2, 62 Rickart Rudin 2, 111
semisimple algebra 2 12 a-compact a(x) = spectrum of x 4 9, 48, 49, 60, 65, Sinclair 69, 136 standard ideal 131 Stein 34, 49 Stein and Weiss 34 Stinespring 43 Stirling's Formula 103 Stone-tech compactification strongly continuous semigroup 1, 84, 93 subordinate 8, 14
Tauberian Theorem 77 Titchmarsh convolution theorem 15, 23, 133
uniform algebra
87
4
integers
5
