Continuous Semigroups in Banach Algebras (London Mathematical Society Lecture Note Series)

LONDON MATHEMATICAL SOCIETY LECTURE NOTE SERIES Managing Editor:

Professor I.M.James, Mathematical Institute, 24-29 St Giles, Oxford 1.

General cohomology theory and K-theory, P.HILTON

4.

Algebraic topology:

5.

Commutative algebra, J.T.KNIGHT

8.

Integration and harmonic analysis on compact groups, R.E.EDWARDS

9.

Elliptic functions and elliptic curves, P.DU VAL

a student's guide, J.F.ADAMS

10.

Numerical ranges II, F.F.BONSALL & J.DUNCAN

11.

New developments in topology, G.SEGAL (ed.)

12.

Symposium on complex analysis, Canterbury, 1973, J.CLUNIE & W.K.HAYMAN (eds.)

13.

Combinatorics: Proceedings of the British combinatorial conference 1973, T.P.MCDONOUGH & V.C.MAVRON (eds.)

14. 15.

Analytic theory of abelian varieties, H.P.F.SWINNERTON-DYER An introduction to topological groups, P.J.HIGGINS

16.

Topics in finite groups, T.M.GAGEN

17.

Differentiable germs and catastrophes, Th.BROCKER & L.LANDER

18.

A geometric approach to homology theory, S.BUONCRISTIANO, C.P.ROURKE & B.J.SANDERSON

20.

Sheaf theory, B.R.TENNISON

21.

Automatic continuity of linear operators, A.M.SINCLAIR

23.

Parallelisms of complete designs, P.J.CAMERON

24.

The topology of Stiefel manifolds, I.M.JAMES

25.

Lie groups and compact groups, J.F.PRICE Transformation groups: Proceedings of the conference in the University of

26.

Newcastle upon Tyne, August 1976, C.KOSNIOWSKI 27.

Skew field constructions, P.M.COHN

28.

Brownian motion, Hardy spaces and bounded mean oscillation, K.E.PETERSEN

29.

Pontryagin duality and the structure of locally compact abelian groups, S.A.MORRIS

30.

Interaction models, N.L.BIGGS

31.

Continuous crossed products and type III von Neumann algebras, A.VAN DAELE

32. 33.

Uniform algebras and Jensen measures, T.W.GAMELIN Permutation groups and combinatorial structures, N.L.BIGGS & A.T.WHITE

34.

Representation theory of Lie groups, M.F.ATIYAH et al.

35.

Trace ideals and their applications, B.SIMON

36.

Homological group theory, C.T.C.WALL (ed.) Partially ordered rings and semi-algebraic geometry, G.W.BRUMFIEL

37. 38.

Surveys in combinatorics, B.BOLLOBAS (ed.)

39.

Affine sets and affine groups, D.G.NORTHCOTT

40.

Introduction to Hp spaces, P.J.KOOSIS

41.

Theory and applications of Hopf bifurcation, B.D.HASSARD, N.D.KAZARINOFF & Y-H.WAN

42.

Topics in the theory of group presentations, D.L.JOHNSON

43.

Graphs, codes and designs, P.J.CAMERON & J.H.VAN LINT

44.

Z/2-homotopy theory, M.C.CRABB

45.

Recursion theory: its generalisations and applications, F.R.DRAKE & S.S.WAINER (eds.)

46.

p-adic analysis:

47. 48.

Coding the Universe, A. SELLER, R. JENSEN & P. WELCH Low-dimensional topology, R. BROWN & T.L. THICKSTUN (eds.)

49.

Finite geometries and designs, P. CAMERON, J.W.P. HIRSCHFELD & D.R. HUGHES (eds.)

50.

Commutator Calculus and groups of homotopy classes, H.J. BAUES

51. 52.

Synthetic differential geometry, A. KOCK Combinatorics, H.N.V. TEMPERLEY (ed.)

53.

Singularity theory, V.I. ARNOLD

54.

Martov processes and related problems of analysis, E.B. DYNKIN

55.

Ordered permutation groups, A.M.W. GLASS

56.

Journees arithmetiques 1980, J.V. ARMITAGE (ed.)

57.

Techniques of geometric topology, R.A. FENN

58.

Singularities of differentiable functions, J. MARTINET

59.

Applicable differential geometry, F.A.E. PIRANI and M. CRAMPIN Integrable systems, S.P. NOVIKOV et al.

60.

a short course on recent work, N.KOBLITZ

61.

The core model, A. DODD

62.

Economics for mathematicians, J.W.S. CASSELS

63.

Continuous semigroups in Banach algebras, A.M. SINCLAIR

64.

Basic concepts of enriched category theory, G.M. KELLY

65.

Several complex variables and complex manifolds I, M.J. FIELD

66.

Several complex variables and complex manifolds II, M.J. FIELD

67.

Classification problems in ergodic theory, W. PARRY & S. TUNCEL

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

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 0,

The integral is a 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

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 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 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 _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) -.

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)

_ 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 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 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
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 0,

y2

b

for all t> 0.

1

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 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

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) 0.

following argument shows. Let

t N xt A

I xt I -

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 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

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
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
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
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

L1OR +,w) such that llatll 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 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


R is continuous, and so there is a constant C such that Ila2+iwll n

M. 3

then

IYI


n

m. 3 2n.5-3

X

j>n < n 2-15 n < 2-1

XT(j) = XR(j)

Since

for

1

-

1,

and

and so

-

if

n e R\T.

Since

IiYI

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,

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 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 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 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 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 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 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 0.

PROBLEM Can the hypothesis

IIT(A - T)-lII 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 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.

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 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 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
O.

w

and

for all

satisfying

Recall that a function

log+ w = log w

w > O.

Note that

log+ (uw) 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

rky. 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.



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