Proceedings of the Conference on Banach Algebras and Several Complex Variables (Contemporary Mathematics)

PROCEEDINGS OF THE CONFERENCE ON

Banach Algebras and Several Complex Variable

AmE ICAn mATH mATICAL SOCIETY

U0LU E32

PROCEEDINGS OF THE CONFERENCE ON

Banach Algebras and Several Complex Variable

AmE ICAn mATH mATICAL SOCIETY

U0LU E32

COIlTEMPORAn mATHEmATICS Titles in this Series Volume

1 Markov random fields and their applications, Ross Kindermann and J. Laurie Snell

2

3 4 5

6

7

8 9 10

11 12

13 14 15 16 17

18 19 20

Proceedings of the conference on Integration, topology, and geometry in linear spaces, William H. Graves. Editor The closed graph and P-closed graph properties In general topology, T. R. Hamlett and L. L. Herrington Problems of elastic stability and vibrations. Vadim Komkov. Editor Rational constructions of modules for simple Lie allebras, George B. Seligman Umbral calculus and Hopf algebras, Robert Morris. Editor Complex contour Integral representation of cardinal spline functions, Walter Schempp Ordered fields and real algebraic geometry, D. W. Dubois and T. Recio. Editors Papers In algebra, analysis and statistics. R. Lidl. Editor Operator allebras and K-theory, Ronald G. Douglas and Claude Schochet. Editors Plane ellipticity and related problems. Robert P. Gilbert. Editor Symposium on algebraic topology In honor of Jose Adem, Samuel Gitler, Editor Algebraists' homage: Papers in ring theory and related topics, ·S. A. Amitsur. D. J. Saltman and G. B. Seligman, Editors Lectures on Nielsen fixed point theory, Boju Jiang Advanced analytic number theory. Part I: Ramification theoretic methods. Carlos J. Moreno Complex rep.r esentations of GL(2, K) for finite fields K, lIya Piatetski-Shapiro Nonlinear partial differential equations, Joel A. Smaller. Editor Fixed points and nonexpansive mappings, Robert C. Sine. Editor Proceedings of the Northwestern homotopy theory conference, Haynes R. Miller and Stewart B. Priddy. Editors Low dimensional topology, Samuel J. lomonaco. Jr.. Editor

Titles in this Series Volume 21

Topological methods in nonlinear functional analysis, S. P. Singh. S. Thomeier. and B. Watson. Editors 22 Factorizations of b" ± 1. b = 2, 3. 5,6, 7, 10, 11. 12 up to high powers. John Brillhart. D. H. Lehmer. J. L. Selfridge. Bryant Tuckerman. and S. S. Wagstaff. Jr. 23 Chapter 9 of Ramanujan's second notebook-Infinite series identities, transformations, and evaluations, Bruce C. Berndt and Padmini T. Joshi 24 Central extensions, Galois groups, and ideal class groups of number fields, A. Frohlich 25 Value distribution theory and its applications. Chung-Chun Yang. Editor 26 Conference in modern analysis and probability, Richard Beals. Anatole Beck. Alexandra Bellow and Arshag Hajian. Editors

27

Microlocal analysis, M. Salah Baouendi. Richard Beals and Linda Preiss Rothschild. Editors

28 29

Fluids and plasmas: geometry and dynamics, Jerrold E. Marsden. Editor Automated theorem proving, W. W. Bledsoe and Donald Loveland. Editors

30 Mathematical applications of category theory, J. W . Gray. Editor

31 32

33

Axiomatic set theory, James E. Baumgartner. Donald A. Martin and Saharon Shelah. Editors Proceedings of the conference on Banach algebras and several complex variables, F. Greenleaf and D. Gulick. Editors Contributions to group theory. Kenneth I. Appel. John G. Ratcliffe and Paul E. Schupp. Editors

conTEMPORARY MATHEMATICS Volume 32

PROCEEDINGS OF THE CONFERENCE ON

Banach Algebras and Several Complex Variables F. Greenleaf and D. Gulick. Editors

AMERICAn MATHEMATICAL SOCIETY providence • RhOda Island

EDITORIAL BOARD Kenneth Kunen James I. Lepowsky Johannes C. C. Nitsche Irving Reiner

R. O. Wells, Jr., managing editor Jeff Cheeger Adriano M. Garsia

PROCEEDINGS OF THE CONFERENCE ON BANACH ALGEBRAS AND SEVERAL COMPLEX VARIABLES HELD AT YALE UNIVERSITY NEW HAVEN, CONNECTICUT JUNE 21-24, 1983

These proceedings were prepared by the American Mathematical Society with partial support from the National Science Foundation Grant MCS 8218075. 1980 Mathematics Subject Classification. Primary 32Axx, 32Bxx, 32Exx, 32Fxx, 46Hxx, 46Jxx.

Library of Cong,.. Cataloging in Publication Data Confere.,ce on Banach Algebras and Several Complex Variables (1983: Vale University) Proceeding. of the Conference on Banach Algebras and Several Complex Varlablesp (Contemporary mathematics; v. 32) Held to honor Prof. Charles E. Rlckart. Bibliography: p.

1. Banach algebras-Congresses. 3. III.

2.

Rlckart, C. E. (Charles Earl), 1913) Alckart, C. E. (Charles Earl), 1913-

Functions of several complex varlables-Congre.ses. .

I. Greenleaf, Frederick P. II. Gulick. Denny. . IV. Title. V. Series: Contemporary mathematics

(American Mathematical Society); v. 32.

0A326.C65

1983

512'.55

84-18443

ISBN 0-8218-5034-2 (alk. paper)

Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such a5 to copy an article for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews provided the customary acknowledgement of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication (including abstracts) is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Executive Director, American Mathematical Society, P. O. Box 6248, Providence, Rhode Island 02940. The appearance of the code on the first page of an article in this volume indicates the copyright owner's consent for copying beyond that permitted by Sections 107 or 108 of the U. S. Copyright Law. provided that the fee of $1.00 plus $.25 per page for each copy be paid directly to Copyright Clearance Center, Inc., 21 Congress Street, Salem, Massachusetts 01970. This consent does not extend to other kinds of copying, such as copying for general distribution, for advertising or promotion purposes, for creating new collective works or for resale.

Copyright © 1984 by the American Mathematical Society Printed in the United States of America All rights reserved except those granted to the United States Government This volume was printed directly from author prepared copy. The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability.

A conference in honor of CHARLES E. RICKART

upon his retirement from Yale University.

CONTENTS

xi

Introduction

xiii

Signatures of Participants H. Alexander

Capacities in

¢n

G. Allan

Holomorphic Left-Inverse Functions

B. Aupetit

Geometry of Pseudoconvex Open Sets and Distribution of Values of Analytic Multivalued Functions

15

J. Bachar

Some Results on Range Transformations Between Function Spaces

35

w.

Bade

Recent Results in the Ideal Theory of Radical Convolution Algebras

63

w.

Bade & P. Curtis

Module Derivations from Commutative Banach Algebras

71

F. Bonsall

Criteria for Boundedness and Compactness of Hankel Opera tors

83

H. Dales

Algebra and Topology in Banach Algebras

97

J. Esterle

Mittag-Leffler Methods in the Theory of Banach Algebras and a New Approach to Mlchael's Problem

107

I. Glicksberg

Orthogonal and Representing Measures

131

B. Kramm.

Nuclearity (resp. Schwartzity) Helps to Embed Holomorphic Structure into Spectra

143

D. Kumagai

Maximum Modulus Algebr~s and Multi-Dimensional Analytic Structure

163

K. Laursen

Central Factorization in C*-Algebras and its Use in Automatic Continuity

169

J. McClure

Nonstandard Ideals and Approximations in Primary Weighted tl-Algebras

177

A. O'Farrell, K. Preskenis & D. Walsh

Holomorphic Approximation in Lipschitz Norms

187

1

7

x

CONTENTS

M. Rajagopalan & P. Ramakrishnan

Uses of ~S in Invariant Means and Extremely Left Amenable Semigroups

195

R. Rochberg

Deformation Theory for Uniform Algebras: Introduction

An

209

W. Rudin

Nevanlinna's Interpolation Theorem Revisited

217

S. Sakai

Unbounded Derivations in C*-Algebras and Statistical Mechanics

223

D. Sarason

Remotely Almost Periodic Functions

237

Z. Slodkowski

Analytic Hultifunctions, q-Plurisubharmonic Functions and Uniform Algebras

243

E. Stout

Algebraic Domains in Stein Manifolds

259

J. Wada

Sets of Best Approximations to Elements in Certain Function Spaces

267

J. Wenner

Green's Functions and Polynomial Hulls

B. Yood

Continuous Homomorphisms and Derivations on Banach Algebras

273 279

w.

The Maximal Ideal Space of a Commutative Banach Algebra

Zame

285

INTRODUCTION A Conference on Banach Algebras and Several Complex Variables was held on June 21-24, 1983, to honor Professor Charles E. Rickart as he retired after 40 years at Yale University. *

These Proceedings contain articles submitted

for the Conference. The topics, both at the Conference and in these Proceedings, represent recent advances in a wide spectrum of topics related to Banach algebras, function algebras and infinite dimensional holomorphy.

Professor Rickart has

profoundly affected these areas, and the many participants who have been either associates or graduate students of Professor Rickart would like to join in thanking him for his inspiration. The preparatory work for the conference rested primarily with R. R. Coifman and Phil Curtis, as well as Phyllis Stevens of the Department of Mathematics at Yale University, who lent an expert hand that was invaluable to the conference.

The painstaking job of retyping each manuscript to appear

in these Proceedings fell onto the shoulders of Donna Belli, Caroline Curtis, Mary Ellen Del Vecchio and Bernadette Highsmith ef the Department of Mathematics, under the careful and patient supervision of Regina Hoffman.

To

all those who helped make the conference successful and memorable, as well as to the National Science Foundation ** for its generous support, we wish to express our gratitude. The editors would also like to thank the following conference speakers whose results are appearing elsewhere:

B. Cole, E. Effros, T. Gamelin,

T. Lyons, M. Sibony. With this volume we extend our thanks to Professor Rickart, who has for such a long time served as teacher, researcher, administrator and friend. Fred Greenleaf Denny Gulick

* After

completing his undergraduate study at the University of Kansas, Professor Rickart began his move east, earning his Ph.D. at the University of Michigan in 1941 and serving as Benjamin Peirce Instructor at Harvard University from 1941 to 1943. He joined the Yale faculty in 1943, and became one of the pioneers in the study of Banach algebras. In 1960, Professor Rickart published his classical treatise General Theory .2!. Banach Algebra. More recently his interests have turned to the study of infinite dimensional holomorphy, culminating in his 1979 book Natural Function Algebras, an important contribution to this subject.

** NSF

Grant:

MCS-82l7l28 xi

SIGNATURES OF PARTICIPANTS

'Yl~,.q-~

,/1 /~ .~. .1~

~

t>r.....J-r

~ ~~ ()..-. l-k,~

f}s~ HcL~-.. J ohVl Do/(;;:;,..J ~'j.OJ./ Y

}/tSS I'M

AtJv- .&;~~ [JUt'

9~-J.RJ

k h..()"L/iTl S017

Rtch~, K.lk-~b.~

'6·«1\ __ . .

xiii

P\u

f E.-c~T

Contemporary Mathematics Volume 32, 1984

CAPACITIES IN H. Alexander

1.

INTRODUCTION.

en

*

Various notions of capacity in higher dimensional complex

spaces have been studied during the last few years.

In the classical case of

logarithmic capacity in the complex plane, it turns out that several different possible definitions of a capacity do in fact yield the logarithmic capacity. In the higher dimensional case one obtains different capacities and the problem of relating them to each other. There is naturally a close connection between capacities, pluripolar sets and plurisubharmonic functions.

In particular, Bedford and Taylor have applied

their "local" capacity to obtain results which add considerably to our knowledge about plurisubharmonic functions. The object of this paper is to give a survey of some of these recent developments.

We begin with a list of several equivalent definitions of loga-

rithmic capacity; in higher dimensions these lead to different capacities.

Then

we consider a connection between equilibrium measure and Jensen measure; the local capacity of Bedford and Taylor; Siciak's capacity; the capacities defined from Tchebychef polynomials; and some relationships among the various capacities.

2.

CAPACITY IN

~

I

• We shall recall some of the several equivalent definitions

of logarithmic capacity. 2.1.

Potential theory.

bability measure on

Put

V

measure

= inf ~

IJ.

I(~).

on

defined to be

*Supported

K,

If

The book of Tsuji [12] is a good reference for this. Let

K be a compact subset of

Cl •

If

~ is a pro-

the energy integral is defined by

V
exists and is equal to the capacity of K. This limit is also equal to 2.2.

Tchebycheff polynomials.

11k

i~f ~

2.3.

;

this follows from the relation

Robin's constant.

singularity at

Let

g(z)

JENSEN MEASURE.

inf {llpIlK:

mj +k

~

mj

p

~.

be the Green's function for

i\K

with

K; 0).

nlen near ~ we have One shows that e- Y is also the capacity of

(assume cap

w

g(z) = loglzl + y + o( Izl). The y is Robin's constant.

3.

Put

K.

Bishop [6] has introduced the notion of Jensen measure

for uniform algebras.

His argument works in the more general context [2] of a

multiplicative semigroup (MSG) of continuous functions; namely, for a compact Hausdorff space ~

X we say that

(a)

ftg E G

(b)

G contains the constants.

f

A closed subset f

E G.

(i)

r of

Suppose that

~(fg)

I

~

~ /If /IX'

C,

boundary for

is an MSG

C C(X)

if

and

X is a boundary for

= ~(f)~(g)

I~(f)

(iii)

• g EG

G

G if

IIf IIr

=

is a complex valued functional on f,g E G,

for all

(ii)

~(1)

IIf "X

for all

G satisfying

= 1 and

Then there exists a probability measure

1.1.

on

r

(a

satisfying logl~(f) I ~ J loglfldl.l.

r

for all f E G. We say that ~ is a Jensen measure for ~. In Bishop's case, G is a uniform algebra on X and ~ is a homomorphism of G. For an arbitrary MSG ~(f)

G,

= exp(J

a functional

~

satisfying

(i) - (iii)

is defined by

X. We shall show that the equilibrium measure of (2.1) can be obtained as a Jensen measure in the following way. For K e t a compact set of positive capacity c, we let P be the MSG consisting of the set of all polynomials n restricted to K. Define a functional ~ on P by ~(p) = an c where n n-l p ( z ) = anz + an_lz + .•. + a O is a polynomial of degree nand c = capacity of K. It is clear that ~ satisfies (i) and (ii). To verify that

10glf/d V),

where

v

is any probability measure on

I~(p) I ~ IIpllK we need only observe that

~

is monic and therefore by n

(2.2) , > m

-

n

> _ cn •

CAPACITIES Now since 0

Aul

n B(zO,e» =


2 +

Sketch of proof. The

n

functions

,

tjJ (n-l)

r/J(n-l) 2

1

D

cP n

2 2

...

cP'n

=

1

cP(n-l) n

D is not

ANALYTIC MULTIVALUEO FUNCTIONS

19

We introduce also: 1

1

cp (n-l)

¢

1

q, (n-l)

(n-l) 2

1

n

:1>1 Let

D1

be the minor determinant corresponding to the

first line of

~i

and let

0,

consequently

6

m(r,~) < N(r,~)

1

N(r,~)

-

1

vanishing, because 1

N(r ,0) - N(r '0)

,

6

+m(r,6 l ) +m(r,6) +0(1).

= 0/CP2"'¢ 11 ,

we get

N(r,~)

~.

m(r'¢l).

If all the

1

- N(r,X) •

so:

denotes the greatest of the

T(r)

element in the

be the corresponding minor determinant in

We have

If

i-th

T(r,cp ), p

for

p. l, ••• ,n,

applying

the previous inequalities we get T(r)
0 " > 0 for o
0 be such that B(ai,e) n B(aO,e) = 0 for i ~ j. By the localization theorem (see for instance Theorem 3.14 in [4], or [7], [13]), A ~ K(A) n B(ai,e) are analytic multivalued on a disc B(AO'~l) with ~l

is included in

exceptional values in the sense of Picard - i.e., for every

exists of

Ai(A)

IIK(A)

K(X)

E~.

h.

First we suppose that

ul ,u2 , •.• ,un

such that

F(X,u i )

F(X,u)

= ki'

has for all

The identities

u

n n

+ Al p . ) u

n-1 n

+. .. + An (X)

define a Cramer system for the unknowns Vandermonde determinant:

Ai(X)

=

k

n

because its determinant is a

n

21

ANALYTIC MULTI VALUED FUNCTIONS n-l

...

0-2

u1

ul

u1

1

u2

1

n-2 u2

n-l

u2

~

u

n-l

un-2

Consequently diction. which

u n

0

0

is constant for all

Ai

0

- Log e,

be such that

so by the analog of the

Liouville theorem for subharmonic functions we conclude that is constant. Xl

such that

Let

zl

zl

be in the boundary of

is not in the boundary of

and suppose that there exists

E,

K(X I ).

It is obvious that

Zl i K(X l ) because K(A l ) c E and Zl E boundary E. such that B(zl,r) n K(~l) = ~. Let ~2 be such that z2 E 8(zl,~)\E ~~.

There exists r

B(zl'S)

r > 0

n K(X2) = ~

and

We have

dist(z2,K(X 2»

~ ;r and dist(z2.K(A 1 » ~ ~r

and this contradicts

dist(z2,K(X2»

part of the tbeorem.

If

= dist(Z2,K(~*».

sOAwe get the first

E c K(X)

K is bounded we have

K()JA ~ EA

part, and moreover

- Log dist(zO,K(l»

by definition, so

by the first is constant.

K(A)A

This result obviously extends tbe classical theorem of Liouville for entire functions case.

h

K(~)

because

= {heAl}

is analytic multivalued in that

In the second part of theorem we cannot conclude, in general, that

is constant.

To see that let us consider K(~)

.. {z E I:

K(O)

=

K(X)

K defined by

Izi = I},

~

for

1 0

{z E I:

which is analytic multivalued.

E.

THEOREM 1.7. Let K be an analytic mu1tivalued function on K(~)A is constant or U K(~)A is dense in t.

Then either

AEI:

PROOF.

Zo E E and

Suppose there exist

for all

K(i~)A

r > 0

such that

B(zO,r)

n K(~) A = ~

E E. Then u(z) = I/(z-zO) is holomorphic in a neighborhood of for all ~ E t. So by Theorem 3.9 in [4] (see also [1], [13] and [21] ~

for easier proofs), L(~)

is analytic multivalued on constant, so

L(A)A

=

L(O)A.

nomially compact subset of L(~)

A

~

A

u(K(A) )

A

= u(K(O»,

= I:

{u(z) : z E K(A)A}

and

L(A)

C

B(O,l). r

By Theorem 1.6,

It is easy to verify that 1:\8(zO,r) hence

u

L(A)A is

transforms a poly-

into a polynomially compact set. K(A)

A

=

K(O)

A

because

u

So

is one-to-one.

25

ANALYTIC MULTI VALUED FUNCTIONS

From this theorem we can use a very elegant argument due to T. J. Ransford

[13] in order to obtain a nice result previously obtained by the author with the help of more complicated arguments explained in §2. THEOREM 1.8.

K be an analytic multivalued function on

Let

there exists a constant number

C

E.

Suppose that

such that

IRe u-Re vi ~ C

Max

u,vEK(\) for all set

\ E t.

Then there exist an entire function

h

and a fixed compact

E such that

for all

\ E t.

In particular this happens if the diameter of

E.

formly bounded on PROOF.

Let

u(A)

= Max{Re u : u E K(\)} and v(\) = Min{Re v ; v E K(\)}.

The two functions A E t.

So

u-v

is uni-

K(\)

u

and

-v

are subharmonic and

is a constant

u.

But

u

= v~

u(\) - v(\) implies that

~

C for all

u

is simul-

taneously subharmonic and superharmonic, so it is harmonic on all the complex plane.

Hence there exists an entire function \ E t.

for all

Let

h

L(A) • {z-h(\) : z E K(\)}.

such that Then

L

u(\)

=

Re b(\)

is analytic multi-

t. But L(A) is always included in the half-plane {z : Re z ~ By Theorem 1.7, L(\)A is a fixed compact set E, so K(\)A = {h(\)} + E.

valued on

OJ.

A. Zraibi and I obtained in [7] the following generalization of the Picard theorem for analytic multivalued functions: E,

then either

K(A)A

is a

K(\)A

theorem and is rather complicated. that if

K is analytic multivalued on

is constant or the complement of the union of the sets

Go-set having zero capacity.

metric proof.

if

The original proof uses the Frostman

I now intend to give an easy and more geo-

As explained in the introduction, Z. Slodkowski [16] noticed

K is analytic multi valued then the complement of its graph is a

pseudoconvex open subset of

[2.

Moreover, in [16] he discovered the striking

fact that the theory of analytic multivalued functions with values in

E, the

theory of fibres for uniform algebras and spectral theory are locally equivalent. More precisely, if

K is analytic multivalued on an open subset

with values in

then for every relatively compact subdomain

E,

exists an analytic function from

V into

L(l2)

U of V of

E, U there

(the algebra of bounded

AUPETIT

e2 )

operators on the standard Hilbert space all

~

E V,

A on a compact set

and there exist a uniform algebra

V c ~\f(K)

f, g E A such that

elements

K(~) = Sp f(~),

such that

K(~) = g(fl(~»

and

for

K and two

~ E V.

for all

The following lemma will show that it is always possible to associate a lot of analytic multivalued functions to a pseudoconvex open subset of in fact, the four theories of pseudoconvex open subsets of

E,

valued functions with values in

[2,

£2.

So,

analytic multi-

spectra of analytic families of operators

and fibres associated to uniform algebras are "metaphysically" equivalent in the sense that any result in one of these theories will give new information for the others.

This is one of the reasons why I believe that a deeper knowledge

of the geometry of pseudoconvex open subsets of

will have a great impact

[2

on spectral theory. LEMMA 1.9. (AO,a)

Let

E~.

~ be a non-void pseudoconvex open subset of E2 and let D is the open set of

Suppose that

Then the multivalued function

D

K defined on

+ a

K(A)

AE£

such that

(~,a)

E ~.

K(~)

is

~

U {a}

: (~,z) ~ ~}

is analytic multivalued.

PROOF.

It is obvious that

closed because if or

u;Ea

and

u

K(A)

E K(X),

n

lim l/(u -a) n

1

is non-void for

lim u

n

= u,

= l/(u-a).

~

E D.

then either (~,a

But

The set u=a

1

+-) t u -a n

Q

(so

u E K(A»,

implies

(A,a + --) ~ 5"l, so u E K(X). Moreover K(~) is a compact subset of E u-a because if u E K(}..) with limlunl = + 00, it follows from un = zl_a' with n (~,z

n

) ~ 52,

that

us now show that Because

a E

K(~)

lim z =a n

'

so

(X,a)

t

and this is a contradigtion.

S-~.

Let

21 = {(A,Z) ~ E D, z E E, z t K(A)} is pseudoconvex. for all A E D we have (A,a) i Ql for ~ E D, so A E D,

z E a:\{a},

(X 1

'M

+

a) E Q}

= u -l(sa) n D x (1:\ {a})

where

u(z)

because

= z-a 1 +

u-l (52)

a

and

is analytic on D x (E\{a})

D

x

(a:\{a}).

Thus

Q1

is pseudoconvex

are pseudoconvex.

We now need a technical but very important result in the theory of analytic multivalued functions.

ANALYTIC MULTIVALUED FUNCTIONS LEMMA 1.10.

27

K be analytic multivalued on a domain

Let

U of

E.

Then we

have the following properties: i)

either the set of capacity zero or

ii)

X E U such that K(X)

either the set of of outer capacity

= to}

A E U such that zero~E

that case there exists the

Ai

K(X) .. {OJ for all X E U,

holomorphic on

U,

GS-set of outer

is finite is a

Go-set

is finite for all X E U. and in n n-1 u + A1 (A)u + •.. + An(A). with

K{X)

F(A.u)

K(\)

is a

=

such that

K(X)

=

{z : F(A,Z)

= O}

(see Theorem 1.5), iii)

either the set of

A E U such that

Go-set of outer capacity zero or

K(A)

K(A)

is countable is a

is countable for all

A E U.

PROOF. Part i) comes easily from subharmonicity of A ~ Log Max{lzl : z ~ K(X)} and H. Cartan's theorem on polar sets. Part ii) was given in general for the first time by H. Yamaguchi f20] but his argument is not completely convincing. With the help of holomorphic sections of smooth analytic multivalued functions, it was partly corrected in [4]. Now this result is perfectly proved, and even simplified in [7]. [21] or [14].

Part iii)

in 112], but the proof contains fallacies.

was given by T. Nishino

With ideas contained in [6], this

result is now proved correctly and more easily in [13], [14] and (5]. THEOREM 1.11. domain of i)

[

Let

Q

such that

U

either the set of

x

{OJ c Q.

either the set of

{A} x [

C Q

is a

G5 -set of

U x [ c Q,

A E U such that

number of points, is a

U be a

Then we have the following properties:

A E U such that

outer capacity zero or ii)

E2, and let

be a pseudoconvex open subset of

{X} x E c

Q,

except for a finite

Go-set of outer capacity zero or

Ux t

nQ

is the complement of an analytic variety. This 1s obvious by applying Lemma 1.9 with

PROOF.

and ii) if we remark that {A} x

[

c

Q,

{A} x [ c

Q

a

=0

is equivalent to

and Lemma 1.10 i) KOJ .. {OJ

except for a finite number of points, is equivalent to

and that K(X)

being finite.

We are now able to give a generalization of the Picard theorem to analytic multiva1ued functions.

28

AUPETIT We shall say that the analytic multivalued function

there exists an integer

K(jJ.)

n

~

and

1

G{A,Z)

implies

n

K is algebroid if

AI, ... ,An

entire functions

= 0,

such that

G{~.z) = ~n + Al(z)~n-l + •.. +

where

An (z). THEOREM 1.12.

K be an analytic multivalued function on

Let

E.

U is a

If

E\K{AO)' for__~ ~O E E, then either U is a component of E\K{~), for all A E t, or U\ U K{A) is ~ Go-set of outer capacity zero. Ma: . II ~1l particular i f we consider the analyt1c multivalued function K. then either

component of

is constant or C\ U K{~)" is a Go-set of outer capacity zero. MoreII ~E~ over. if K is not cons taut and is not algebroid, then the set F of z for

K(A)II

{~: z E K(A)"}

which

PROOF.

is finite, is a

We may suppose that

~O

= O.

convex, so the same is true for

Go-set of outer capacity zero.

Then Q

=

Q' = {(z,A) :

{(~,z)

(~,z)

E Q}

So we get the first part by applying Theorem 1.11 i). We now prove the last part.

we get the result. the intersection of Moreover

E\K(A)"

FeU.

t\K(jJ.)A

and F

= U.

proof of Theorem 1.5, we conclude that

Is this result the best one?

U x {O} c Q'.

Considering

Let

U

= a:\

n

IJ.EI: is always non-empty,

K(A)II,

the

K{jJ.)A.

Because

U is connecF

is a

But using the argument given in the

"

K

is algebroid.

Given a compact set

C of capacity zero, is

it possible to construct an analytic multivRlued function

f

and

is pseudo-

So by Theorem 1.11 ii) we conclude that either

Go-set having zero capacity or

11:\

K{~)}

U is the unbounded one, so by Theorem 1.6 and Theorem 1.11 i}

only component

ted.

: z f

K on

a:

such that

U K(A)A = C? Is it even possible to do that for K(A) = Sp f(A), where AU: is an analytic function from E into the algebra of compact operators on A. Zralbi [21] obtained the following particular cases:

some Banach space? If

C is a subset of

Ie

not containing

0

and having at most

limit point, there exists an analytic multivalued function

t\

U K{A)"

0

as a

K such that

= C.

ME

If

C is a compact subset of

analytic multi valued function is that

K(A)

t

of capacity zero then there exists an

has holes and the

K(~)A

t\ U

K(~) = C (but the problem ME cover all the plane!).

K such that

ANALYTIC MULTIVALUED FUNCTIONS

29

Almost simultaneously, F. V. Atkinson, B. SZ-Nagy and J. Smultjan (see the references in [6]) obtained the fallowing result: analytic function from a domain

D of

z·E Sp f

(~)

be an

into the algebra of compact oper-

t

ators on a Banach space, and suppose that A E D for which

~ ~ f(~)

let

t

z

Sp f(O).

Then the set of all

is closed and discrete in

with K compact, it is the classical result of F. Riesz.

f (~)

For

D.

0::

For a general

B. Sz.-Nagy believed that this result is deeper than the previous one.

I+>..K,

f, In fact

the three given proofs used extensively the important fact that the projections associated to isolated spectral values of a compact operator are finite-dimensional.

In [6J, using complex function theory, we proved that this result does

not depend really on functional analysis - i.e., that the operator is compact but only an the geometry of the spectra.

K be analytic multivalued on an open subset

Let I-L

E K(~O)'

exist K(A)

a disc

n6

~O

for some

K n K(~O)

is finite for

IX - Aol

< r.

{~}

0::

and an

E.

We say that

K(~O)

is a good isolated point of

E D,

such that

6

D of r > 0

i f there

such that the set

By the scarcity theorem for analytic

exactly

n

points for all

subset.

This. condition of "good isolated point" means geometrically that around

I~

-

~ol
")

such that

dist(O,K(A»

K.

It is a purely geometrical proof; see [21, pp. 41-42).

All of this suggests the following problems: i)

If

K is a continuous and convex analytic multivalued function on

does it have entire selections?

E,

ANALYTIC MULTIVALUED FUNCTIONS ii)

33

Let K be analytic multivalued on D. Given Zo E aK(AO)' when does there exist (at least locally) a finite analytic multivalued function small?

Zo E L(AO)

L such that

and

L(A) c K(A)

for

IA - Aol

These very difficult problems are, of course, intimately related to the problem of holomorphic support functions for pseudoconvex domains (see [11, pp. 113-114]) and the two problems of H. Alexander and J. Wermer given in [1] and [2].

Anyhow, any solution of one of these questions would have important applications in spectral theory, giving precise information on the distribution and the growth of spectral values.

The only result obtained until now is Theorem 2.6. To state it, we let K be a continuous and convex analytic multivalued function on ~, and let B(K(A» denote the diameter of K(A). We define

OO(A)

6(K(~»

Max

=

I~I

= IAI

which is a continuous and non-decreasing function of nmoREM 2.6. on t.

Let

IAI.

K be a continuous and convex analytic multivalued function

Given an increasing sequence

sequence of polynomials

fR

(Rn)

going to infinity, there e;dsts a

such that

n

MaxlfR (A)-zi ~ 3 oo(Rn ) zEK(A.) n for all

IA.I

~

Rn

and all

n.

r be a circle of radius R, and let r ~ oo(R). Applying the arguments of [1] to r, r and the continuous function a given by Lemma 2.5, PROOF.

Let

we conclude that there exists

fR

continuous for

IAI ~ Rand holomorphic for

IAI < R, such that IAI· R implies IfR(A.) - a(A) I < 2r. By Taylor's theorem applied to {A. IAI ~ R}, we may suppose that fR is a polynomial. Moreover IfR(A.) - zl ~ IfR(z) - a(A) But so

(A,z) ~ IfR(A) - zl IfR(A) - zl < 3r

for

I

+ la(A) - zl




-+>

B(X,Z)

has been called "the functions that operate in

A)

= C(X,Z),

Op(Ay

When

= A,

A. "

the algebra of all Z-valued continuous functions on

X.

has been called "the functions that operate weakly in

C)

A. "

On p. 167 of Rickart's treatise [26], brief mention is made about the problem of which functions operate in a Banach algebra. The main emphasis in this paper will be on conditions ensuring that Op

B) c Op(Ay

-+

or that C)

Op(Ay>- C) c C(Y ,Z).

is trivially true whenever

will be on conditions ensuring assumptions in (1.2) with

Op(Ay

-+

Since the inclusion B c C,

C) c C(Y,Z).

We will work under the

X compact; sometimes we will give results when

is only assumed to be a set of continuous functions on algebra of continuous functions; mostly, we will assume in

~,

most of the focus

though some results hold for

Z

=R

as well as

A

X rather than a Banach Z

= I:

and

Y open

~.

In Section 2, some examples of earlier results on range transformations will be given.

In Section 3, the main results will be stated.

Section 4 con-

tains proofs of these as well as other theorems of relevance and interest. Section 5 contains a brief survey of other results regarding discontinuity, non-measurability, measurability, and analyticity of functions that operate. full proofs of these and other related topics will appear in a forthcoming research monograph, "Range Transformations Between Function Spaces," by the author. It should be noted in passing that one can define "domain transformations," and even "domain-range transformation pairs," between function spaces. In the most general setting, the definition is as follows. be sets, X'

to

A(X,Y) yl,

functions, (1.3)

and

A'(X',Y')

respectively.

f

X to

X, Y, X', yl Y and from

A domain-range transformation pair is a pair of

to, p) , where 6 :X' For every

sets of functions from

Let

-+

X, p:Y -. Y',

E A(X,Y),

satisfying

pofo(S E A'(X',Y').

RANGE TRANSFORMATIONS

37

f Y

~

X

Xl

The class of all pairs A'(X',Y'».

t6, p)

In the case

X

Figure 2

lp

01

• yl

~fo6

satisfying (1.3) is denoted

= X'

and

= the

6

Op(A(X. Y)

-+

identity map, one sees that we

are dealing with a "pure" range transformation,

p.

In the case

Y

III

and

y'

p - the identity map, we are dealing with a "pure" domain transformation, 6. The subject of domain transformations will not be taken up here.

For an

extensive bibliography and survey of results, see E.A. Nordgren's "Composition Operators in Hilbert Spaces" in {3]. 2.

EXAMPLES OF EARLIER RESULTS ON RANGE TRANSFORMATIONS First. we introduce some notation and state some basic facts. When X is a compact Hausdorff space and Z is ~ or ~. a function

algebra

A on

X means an algebra

A(X,Z)

over

Z of Z-valued continuous

functions on

X whose algebra operations are the pointwise operations of func-

tions on

and which separates the points and contains all constant functions.

X.

A is inverse-closed on on X,

(l/f)

is in

X means that for every

A,

where

a Hanach function algebra

~

II· II

lex)

=1

for all

any non-empty open set Y.

x

For any function algebra

Y as explained above. on X (where

~,

Y in When

A,

X (i.e., H(Y)

Ay

A that never vanishes

for all

x

in

X.

A is

A is a function algebra on

such that

in

in

= l/f(x)

X means that

having a Banach algebra norm fined by

(l/f) (x)

f

11111 ]

= 1,

where

1

EA

is the identity of

X

is deA).

For

denotes the holomorphic functions on

is the set of

f

in

A having range in

A is an inverse-closed Banach function algebra

X is infinite), the A-valued integral.

(2.1)

can be defined (see [26], [13], or [8]).

Here

ber of closed rectifiable curves in Y enclosing its "interior." over Y, that

f (X)

f(~A)'

is identical with

A,

where

and is independent of ~A

inverse-closed on Ff(X)

X;

= Fof(x)

for all

uous complex homomorphism of algebra

B,

A,

because

this fact makes the construction of x E X,

is because "point evaluation at ~anach

in

f)

A.

x"

hence

Ff

y.

Note

is the set of all (neces-

sarily continuous) non-zero complex homomorphisms of over,

(the range of

The integral is defined as the limit of finite Riemann sums

the limit exists in the norm of

f(X)

is a union of a finite num-

y

= Fo£

is a member of

cP A'

Ff

for all

A is

possible. f E

Ay.

MoreThis

and so is a contin-

For any semi-simple commutative unital

Gelfand proved that

B is isometrically isomorphic to a

BACHAR

38

Banach function algebra isomorphism defined on

~

9:B X

= ~A

X·,

A on a compact Hausdorff space,

A is ,. defined by by f(~) = ~(f)

a(f) = f

(for all

(for all,.

~

with the weakest topology under which every

f

..f

the isometric

fEB),

where

is

E ~A)' where ~A is endowed is continuous, where A is

the algebra,. (necessarily a ,. function algebra that ,. is contained in all such f, and where ilfll;;; IIfli (for all f) is the norm on

C(X,t»

of

A.

Two early results on range transformations are as follows. THEOREM 2.1.

Let

A be an inverse-closed complex Banach function algebra on

the infinite compact Hausdorff space

X,

set in

that is, every holomorphic function

y

11:.

Then

algebra

B,

-I>

A),

Let

01;1.

B be any semi-simple commutative unital complex Banach

and let the Banach function algebra

ation.

Then A is inverse-closed on

in

H (Y) c

Q; ,

Y be any non-empty open

A.

operates in

THEOREM 2.2.

H(Y) c. Op(Ay

and let

A be lts Gelfand represent-

and for any non-empty open set

~A'

Y

Op(Ay .... A) •

'rhe latter theorem, often called the Gelfand-Silov Theorem, is thus a corollary of Theorem 2.1; the proofs of both theorems ar.e essentially those found in [26], [13] or IS]. The question of when

OrAAy

-+

= H(Y)

B)

naturally follows from these

two theorems, that is, when it is true that the set of functions operating from one Banach function algebra of holomorphic functions on

Y.

A to another one,

B,

is precisely the set

The following two theorems provide information

in the case where

Y 1s the open unit disc.

THEOREM 2.3 [24].

Let

A be a non-self-adjoint Banach function algebra on the

infinite compact Hausdorff space

X,

whose complete algebra norm is the uniform

X. Let Y = {zl Izl < I}, the open unit disc. Then every complex continuous function, F, on Y that is also in O~Ay .... A) is holomorphic

norm on

.2a. Y. THEOREM 2.4 [1].

Let

A be a Banach funct ion algebra under the uniform norm

9n the infinite compact Hausdorff space

X,

closed function algebra on of complex conjugates unit disc.

Then

(~

X)

ORAy .... B)

=

X

"'

B:> A be another uniformlx B does not contain the se! CA let

and suppose of the functions in

A.

Let

..

Y be the open

H(Y).

It is thus seen that Theorem 2.3 is a corollary of Theorem 2.4, and that the restriction that

F be continuous on Y can be removed.

A result on real-analyticity of functions that operate is the THEOREM 2.5 [27].

Let

fol~owing.

A be the Banach function algebra of all absolutely

39

RANGE TRANSFORMATIONS

T-

{z

E

[llzl = I}

open set in

n=--

1)

on the compact space 00

(the unit circle), with norm

containing

~

0

in

{e\}\EA

such that for all

A

= x.

lim e\x ">..EA

(3.3.5)2 There is a K> 0 such that for every finite set {xl" .• ,xn } CA and every e > 0 there is an element e E A (with liell :: K) such that IIxi - exill


on

X

that are constant on some open neighborhood (depending

Though it is known that this function algebra has no

on

complete algebra

norm, this will also follow from the result (proven below) that

any Banach

f)

RANGE TRANSFORMATIONS function algebra

43

on the infinite compact Hausdorff space

is a non-isolated point that there is an

f

x E X with

locally zero, satisfies the condition

M

x

such that

E M

is infinite.

Xf

Because of Proposition 4.1. we shall hereafter assume (equivalently,

dim A

=~

such that there

X

X is infinite

X has a non-isolated point) whenever

or

A is a

X. This is because of the obvious fact that when X is finite, every f E A has finite range, and hence every function F on Y (continuous or not) operates weakly 1n A, and so this case 1s completely function algebra

~

OP(Ay

settled, 1. e.,

C)

-to

We next present

Y.

consists of every function on

an elementary result concerning continuity of

functions

that operate weakly.

PROPOSITION 4.3. (1)

{cn }

C

ni

=

)

there is a sUbseguence

y,

II

X.

!!!.

A

(i)

tion 3.3). and ak

-to

{ck }

-+

O.

C).

WI,

l~m

sup(ak+x f )

¢~.

{aki },

= a ki

with

Choose

Applying

> 0

-to

y

C)

-+

C

WI.

Moreover, for any

C (Y)

Y

a

sequence

lim a ki

there are

X is compact, the closure

is continuous, we must have

ox ,

of

let

15 > 0

for all

x'

in

X,

x

We have

is

then

WI

be arbitrary.

{Xi}

= 0,

Op(Ay

-+

Put

ak

C)

Xf

C

Thus

above).

xi {XI} f (x')

there is an

y

in

X

IIfllX

there is an

such that for all

N


I(Foh)(x") - (Foh)(x')1

such that

ox I

so

and since

such that Also,

e

x,

g E A and

is compact in

- x.

Thus, let

to the boundary of

f € A such that

lim sup,

C(Y).

C

- c k -y (k E N),

this implies

f.

be arbitrary and pick

x" EO,.

A

By definition of

Since

Xf = f(X),

As for (ii), it

from the definitions of each (Defini-

to be half the distance of

x E X f • Put g = x h = g+y E Ay (see choice of e

cause

WI

we find that there is

and

+ xi'

g

ck

compact,

N.

Op(Ay

-

A is

We shall verify the second condition in (i).

Z

C

Y.

quence

implies

51

Thus we will prove that i f

Let F E Op(Ay

~

such that

}

only if the second condition in (i) holds.

is straightforward that

i

ni

is just the basic fact that a function between twu metric spaces is

continuous if and

x

{c

Z.

PROOF.

that

and every sequence

Y E Y

then

51

either of these two properties fmp1ies that

y € Y

Z any function.

-to

be any affine-like set of Z-va1ued continuous functions on

A

the compact Hausdorff space open in

F:Y

Z,

F(y).

.1!!

(ii)

Y open in

~,

is continuous if and only if for every

F

Y converging to

lim F(c 1

Z = R 2!

Let

N such that

for all

f

44

5

BACHAR

>

l(Foh)(xi) - (Foh) (x') I ~ IF(h(xi»

=

i

and so

=

IF(x-f(xi)+Y) - F(x-f(x')+y) I IF(a k +y) - F(y)

1=

- F(h(x'»

I = IF(C k

IF(g(xi)+y) - F(g(x')+y)I

I

IF(X-Xi+y) - F(y)

) - F(y)

I

for all

i

~

N,

i

ts continuous.

F

We next show that the measure condition in (ii) of the Main Theorem 3.4 implies that

is

A

SI.

~

PROPOSITION 4.4.

A be any affine-like set of Z-valued continuous func-

X.

tions on the compact Hausdorff space m(X f )

0,

where

Let

{c k }

:->

o

PROOF.

If there is an

Z,

is Lebesgue measure in

m

m(X f )

iance of Lebesgue measure,

m(c k + Xf )

=

o

O.

for all

m(aX f ) = m(c k + ax f )

o

k,

for all

()

small as you please by proper choice of

and

IlafOllx

O.

>

o

o

also have

Moreover, it

k.

m(aX f ) = lalm(Xf )

Z,

tn

S1.

By translation invar-

o

a "I: 0

is well-known that for every

A is

then

Z be any sequence converging to

C

.=,su;:;:,;c::;.::h~..:t;.:.:h:::a~t;

fOE A

Thus we

can be made

as

Further, the sets

a "I: O.

00

0< m(aX f ) = m(c +aX f ) U (c k + aX f ) (n E~) satisfy 5n+1 C Sn n 0 k=n 0 o :: m(Sn)' and m(Sn+l) O. In partiw n=l n n n 0 cular, this implies n 5 = lim sup(ck+aX f ) ~ ¢, as was to be proved.

Sn -

n=l n

k

0

We now turn to the properties mentioned in (iii) of Main Theorem 3.4. PROPOSITION 4.5.

(i)

Hausdorff space,

X.

is

Moreover,

WHOI.

~

A be any complex function algebra on a compact

For any non-isolated

M is x X is metric, then the sequence {x}

if

if

If, in addition,

~

p(x , x n )

0'

where

P

LZ

in

is the metric on

X.

3.4(iii)

=$

is

M x

(iii)

II

A

or is

B.A.I.

PROOF. that

5IF, wrF plus the condition

1

E M x

neighborhood

1/2).

Since

such that Nl(x)

M

x

is

in Theorem

on

in

f

-+ C) C C(Y).

For (i), first select P (x,x 1 )


is a complex Banach function algebra and

op(Ay

-

~

M has a B. A. I. => M is SlY x x or WIY plus the condition on f

WMOI.

LZ,

.2!. l4MOT, then

an

Mx

and either

M x

A is a complex Banach function algebra, then

the following implications hold: WIF,

then

C X\{x}

Ll

(3.3.2) can be chosen so that (ii)

x E X,

Xl

t

A separates points and contains

fl (Xl) = 1.

such that

We suppose, inductively, that

(in the metric case, choose

x

f1

Since

M

vanishes on

x

is

Nl(x).

m open neighborhoods,

=X

LZ,

~"'J

Clearly,

Xl ~ Nl(x).

1

points

have been selected such that

{x1, .•. ,Xm } c X\{x}

open

N. (x) (i=l, ••• ,m),

NO(X)

Nl(x)

there is

there is an

have been selected such that

J

lI,

so

Nm(x)'

that

m distinct

RANGE TRANSFORMATIONS Xi E Ni _ I (x)\N i (x) for i=I, .•• ,m,

(4.5.1) that

45

{fI, ••• ,f } eM

m functions

m

have been selected such

x

that

(4.5.2) and fi is zero on Ni (x) for i E U, ... ,m}.

(4.5.3) In case

X is metric, we

Now since

x

is non-isolated, there is an

metric, choose

LZ,

xm+l

* fm+1(xm+1)

such that

such that

= 1.

Since

xm+1 ~ Nm+l (x)

we have Nm+I(x)

for

0

=

N~l(x)

for

p=l, .•• ,m

Nm(X) c ••• c N1 (x».

for

i=l, ••. ,m.

xm+l E Nm(x)\{x};

N~l (x) n Nm(x) ,

Nm(x) , fm+I(X j ) - 0

C

fp(xm+I)

==

i

fm+I

such that

(use the definition of

1

=

1/2

in case

* C-f 1) ••• (D-fm) • fm+1 ;: fm+I

Define

fm+1(xm+l)


1 =

(_1}n+1(~), n=l,2,3, ••••

2

n-.L

so

BACHAR ""

aa

1 + La I ~)' = 1 + L. (_1)n+1(~), 0 < a < 1. To show the n=l n=l "" latter converges, we apply Raabe's test: given Z a n with all a n > 0, if a n +1 n-1 there 1s a p > 1 such that n(---- - 1) ~ -p for all sufficiently large n, an aa then converges. Thus, La a n n=l a(a- 1 )··.(a-n ) n! 1- 1) n (l (n+1)! a (a- 1 ) ..• (a-[n-1]) - n , - 1) - ~(la-nl-n-1) - n(l an+1 .. ~(n-a-n-1) = ~(-1-a) n+1 n+l n+1

II (I_Z)Cl Il

Thus,

-(1 + ~)


0, there is an with

a

0 < a ~ ao'

l!ea (l-7J)1I < 5.

By choosing

a o> 0

a

=

II e

such that for all

2I1Qn&_111'

the last

&

estimate above reduces to

a
0,

0 < a < 0/3, ~

° < a < 1, aa'" a/3

choosing

we have yields

and we are done.

K be any infinite compact set in

!E&

(the closed unit disk), 1 E K,

•

+ ••• }] = a[l + (l+a)]

1

n(n-l)

Since

~ n~:~l)

1

such that

~

is an accumulation point of

KeD ~

K.

A be as in Proposition 4.7, let ~ = {g E C(K) Ig is the restriction to K of some f E A}, and let MI = {g E C(K) Ig is the restriction to K of K

~ f E MI }.

tion to

K),

Then under the map

9:A'" C(K)

defined by

is algebraically homomorphic onto

A

9:f'" f IK

AK' Ml

(restric-

is algebraically

homomorphic onto

Ml ; and under the quotient norm, !I!f+IKili E inf 'If+hli, K hEIK is the closed ideal of functions in A vanishing on K, A/IK ~

~

IK

MI/IK

are Banach algebras.

MlK'

resp.,

under

and

A/TK

and

MlK = {f E AKlf(l)

= o}

is a maximal ideal in

approximate identity in a < 1,

Ml/IK'

{(J-~)

K,

~

are isomorphic with

the natural transference of the quotient norm,

both are Banach function algebras on

o
0,

contains at least one point of

0

centered and also

¢\U

U}.

at least one point of

For any infinite compact set

LEMMA 4.11.

and ending at

is an open and connected subset of

for every

of radius

z

Fr C

K c ~,

GO

is contained in

K

and contains infinitely many points. PROOF.

x E Fr Cco

Let

By compactness.of With

1

open disk

f

.

x

Since

z E: Caa'

C...,

and joining z

x

*

there is an

Caa

since

is open.

Suppose

x

+ K.

z

of

and

C",,'

z :I: x

Clearly, the closed segment

[x,z] c Sf)(x) i..: S5(x) c t\K. there is a finite polygonal line P(z,..,) contained in t'\K (which is not in K) to Attach [x,z] to p (z,ao) at

z,

.10 •

and obtain a finite polygonal line,

...

Q(x,oc),

contained in

and joining

~\K

*

Since by assumption x K, we conclude by definition of Cw that x E: C... , which contradicts the fact that x Ceo' Thus the supposition x K can not hold, so x E K and Fr Cco t; K. x

to

C00

must contain at least one point

S5(x)

since

Clearly,

Xo E: K such that dist(x,K) = Ix-xol :;:. O. the closed disk S''i (x) lies in the complement of K. The

K,

'21 x-x 0 I,

5 =

.

*

*

Next, let Zo E K, and let R(ZO;0) = {zO} U {w E ~Iw -;: zO' arg(w-wO) - O}, 0 ~ e < 2n (Figure 3). TIIUS, R(ZO;8) is the ray beginning

K

R(zO;S) Figure 3

at

zo'

zo and making angle

real axis.

Since

i.e., there is R(zO;9)}.

we

e

relative to a line through

zo

K is compact, there is a "last" point EK

n R(zO;9)

such that

It is easy to verify that

Iwe-zol

Le;:; {w E

clw

the disjOint union of the two "open ll line segments From this it clearly follows that

we

E Fr

e....

we

parallel to the in

K

= sup{lw-zol:w

n

R(zo;9),

EK

n

E R(ZO;S), Iw-wel > O} (zO,we)

and

(We ,00) •

is

S3

RANGE TRANSFORMATIONS There must exist

90 , 0 S 00 < 211,

K = {zO}' contradicting the fact that S :; {a E [0,211) lwei: z} is non-empty. {wa}OES

such that waO~ zO' otherwise K is infinite. Thus the set If S is infinite, then the set

e

is a set of distinct points (Le.,

'I- a'

in

S

wa i: wa'~

implies

K

and {we}eES C Fr Coo. On the other hand, if S is finite, then, since is infinite, there must exist at least one 80 , 0 ~ eO < 211, such that K

n (zO'w e ]

K

n (zO,w e )

is infinite.

It is clear (since

is finite) that

5

o

Fr Coo'

C

and so

Fr Coo

is infinite.

o

DEFINITION 4.12.

Let

as in Remark 4.10. if there is

z

x

to

x EK and there is

E C00

with

IX> ,

P(z

,~)

P (z ,co)

C

x [\K,

-Then

K.

5

C00

{x}.

-

Coo'

S5 (zx)\{x}

that

and

Q(x,oo)

z

Q(x,~)\{x}

c

x

c.

x

of

00

Finally, it is easy to

00

in

K is in

Fr Cao •

(K) be the set of all circularly accessible points of ca is dense in Fr C .

S

Let ca

be

00

x is the finite polygonal line connecting

then the union

(K)

OIl

0/2 > 0, x E Fr C • Then there is x' E S5/2(x) x' K. Since K is compact, there is x" ~ K

Let

PROOF.

So (zx) n K

such that

'\ > 0

see that every circularly accessible point LEMMA 4.14.

C

[x,z] with x is a finite polygonal line connecting x to

z, x Q(x ,ao)\ {x} C G:\K

such that

and let

x

x joined at

P(Z ,(0), x

[

is called circularly accessible from

A point

Furthermore, if

C00 •

be an infinite compact set in

It is easy to see, hy the definition of

REMARK 4.13. L

K

DO

nC

00

,

and

*

clearly,

Figure 4 K

Ix'-x"l = dist(x' ,K) ;:: 5' > 0

such that x E Fr Cao

C

+ lx'-x" I

< 5/2

SS,(x')

nK -

radius

5'

K

on

we have

K,

+ 0/2

~

x'

and

X"'

E (x', x") ;

XII,

dist(x' ,K) ~ Ix-x'i < 6/2.

o. We claim that x"

by the definition of

about

Ca' (x').

=

6' •

ES

If

Clearly, since Also,

ca (K).

C6 ,(x ' )

lx-x" I ~ lx-x' I

Observe that is the circle of

x" E C5 ,(x'), but there may be other points of However, if we let (x' ,x") be the open line segment between

then

x',

(Figure 4).

then

(X',XIl) C

SR'(x').

then SIx '" -x " I (x''')

Clearly.

nK =

{x"}.

(x I ,x") c C.

"" Hence x" E S

ca

Let

(K),

as

claimed. PROPOSITION 4.15. ~ K be an infinite compact set in C, and let P(K,G:) • be the uniform closure of all complex polYnomials in z (see Definition 3.1). ~

of

~A

be the compact Hausdorff space of all non-zero complex homomorphisms... P(K,~), !!!h K imbedded homeomorphica11y in ~A via the standard ;

54

BACHAR

_".. po~i:.n~t:;...;e:;;.v~a::.;l:;.;u_a:;.;t:o.:i:;.:o:;.:no:.'_'.::m:::a:.l;p;..:(..:;s:.;;e_e;...:S_e_c_t_i;.;:::0~n:...;;;2.).:.._.;;:;E.;.:x,;:,t,:e.:;.n,:d-.:ea::.;;.ch... ~A

uous function on C'00 co {Z * P(~A) I z

~

~,

in the usual way.

P(z,~) C t - P(~A)}.

and such that

Suppose there is

Fr C:,

each open disk

K.

entirely outside

1\1. -\0 I
zEK 1 P(K,¢). In any commutative Banach algebra we have

and

and

C

Also,

C

.5 = dist(~O,K).

is invertible in

vCf) ~.: I'fll

Fr C~

-

which connects

P(z.~),

~

z, z E K,

E C"" hv definition of Coo' By definition of \.0 E Fr e.." contrary to assumption. Thus \0 f K.

\.1 E C!.,

e/2

=

p(z)

Xo

Thus,

we conclude

such that

!z

to a contin-

\0 E Fr C~\Fr Coo. SUEl20se \0 E K. By definition of 56 P"O) contains a point Xc E C!,. By definition of

C!., xl) t p(

0,

there exists

je

».

) - f. +k(x (x _x)-ll < e.. nO Je nO nO is a Cauchy sequence. Let y E K.

Then

such

RANGE TRANSFORMATIONS

57

I (fj-f·+k)(y)I = Iy-xll gx " (y)···gx" (y)-gx" (y)···gx (y)1 l nO+1 nO+j nO+1 nO+j+k ::: Iy-xlllgx" ···gx" - gx" ···gx" 11K nO+1 nO+j nO+1 nO+j+k ::: Iy-xl [llgx " 11 K'. '1Igx " 11K + IIg x " 11 K" 'lIgx " 11 K] nO+1 nO+j nO+1 nO+j+k
0,

there is

The properties of the

n

g" x n

derived in the proof of (i) enable one to demonstrate this, and the straightforward details are omitted. PROPOSITION 4.18. Hausdorff space

Let X.

A be a Banach function algebra on the compact

~:

)

58

BACHAR

dim A

(i) •

(ii)

let

x E X that is non-isolated.

X is infinite.

(iv)

PROOF.

(linear space dimension).

There is an

- (iii)

..

=~

There is an

f E A such that

Xf

is infinite •

-

By Proposition 4.1, it suffices to show only that (ii) - (iv).

x E X be non-isolated.

Thus,

M is LZ, then M is WHO I by Propox x sition 4.5, and by the construction used in the next to last paragraph there, one obtains an element H

If

in

A

with infinite range.

If, on the other hand,

M is not LZ, then (se"e 3.3.1) there is an f E M such that for every x x open neighborhood 0 of x, f does not vanish on O. Using the Hausdorff x

x

property, it is easy to prove that

f

has infinite range.

COROLLARY 4.19.

If the Banach function algebra

Hausdorff space

X has the uniform norm on

there is an element

f

~

A such that

A on the infinite compact

X as its complete norm, then.

Xf

is infinite.

We can use the previous results to prove the Main Theorem now.

However,

we first will prove the Reduction Theorem. PROOF OF REDUCTION THEOREM 3.2.

The proofs of each of the four parts are

essentially the same, so only part (iv) will be done in detail. F E Op(Ay(X,£)

First, let ~

f E i. 1I

II'

F E Op(AOy (Xf,t:)

-I-

~ C(X,~».

We must show that for every

T~

C(Xf'G:».

this end, let n

g E AO (X£, t:)

...

be

Y

Z Ia I < and n-O n n=O We must sh2w Fog E C(Xf'~)' i.e., Fog is continuous on Xf' n ~ n ~ Now the element G E Z a fn E A since " Z anf II ==: z lanillfli S Z laIlI A'

where

q>

x

(f)

= f(x),

via the

~A

f E A.

A be a Banach function algebra on the infinite compact Hausdorff i=1,2,

f-1 = ±I,

If

(5.3.1)

there

f l ,f 2 E A such that

are functions

o E int(elX f

pair

and if

1

+ EZX f )

~ ~

for some specific

Z

(el,e Z)'

then (5.3.2)

every

F E Op(Ay

~

C)

is of Baire class

1

(or equivalently, is the pointwise limit of some

{Fn } of continuous functions on V). (ii) If there is an f E A such that Xf has an infinite connected component, then 0 E int(X f + iXf } ~~, which, in turn, implies (5.3.2). (iii) If A is natural and if there is f E A such that Xf contains a countably infinite sequence of pairwise disjoint closed subsets, E (n E ~), sequence

n

with dist (X f \En ,En ) > 0 for n E N, then Op(~~ -I> C) C C(Y}. If there exists --y no such f, then (5.3.2) holds, even when A is not necessarily natural. In [6] the following results are proved:

(5.4) X,

If

A is a Banach function algebra on the infinite compact metric space

and if

is natural, then either Op(Ay

A

F E OpCAy ~ A)

of

~ C) C C(Y),

or else every

is locally Lipschitz on some dense open subset (depending on F)

Y.

(5.5)

There is a Banach function algebra

is an

F E

Op(Ay

-I>

single point of

A)

(F

A on

X

= [0,1]

such that there

operates "stronglyll) which is discontinuous at a

Y.

Since it is possible that a function that is locally Lipschitz on a dense open subset of

Y may actually be non-Lebesgue measurable (such an example is

easy to construct), we see that (5.3) (iii), together with the fact that Op(Ay

~

A)

C

Op(Ay

-I>

C),

actually strengthens the conclusion of (5.4) to the

fact that (5.3.2) holds also. In view of the above results, the following unsolved problems arise. Ql.

For every complex Banach function algebra Hausdorff space F E

O~Ay

-I>

Baire class Q2.

C}

a

Same as Ql, with

X,

and for

Y open in

A on an infinite compact t,

is it true that every

is Lebesgue measurable (or Borel measurable, or of

a)?

for some Op(Ay

-I>

C)

replaced by

Op(Ay

-I>

A).

61

RANGE TRANSFORMATIONS

Q3.

Does there exist points in

F E Op(Ay

~

A)

with infinitely many discontinuity

Y?

In view of the above results and those of Section 4, a negative answer to Q1 could obtain only when (2)

f E Ay, m(X f ) = 0, condition (3.3.6) fails for every

(3)

conditions (5.3) (i) [i.e., (5.3.1)] and (5.3)

(4)

A is not natural,

(5)

every candidate

(1)

for all

f E A,

(ii) fail for all

f E A,

F E Op(Ay

~

able must be such that int DF = ~. In [2J we prove that an F E Op(Ay

C) ~

that might be non-Lebesgue measurC)

exists which has infinitely many

discontinuity points on Y in the case where A is the Banach function algebra of absolutely convergent power series restricted to a very rapidly convergent sequence,

X - {xn } c {zl Izi < I},

converging to

O.

REFERENCES 1.

J.M. Bachar, Jr., Composition mappings between function spaces, Thesis, UCLA, June 1970.

Ph.D.

2.

J.M. Bachar, Jr., Range Transformations Between Function Spaces, research monograph, to appear.

3.

J.M. Bachar, Jr., Hilbert Space Operators, Lecture Notes in Mathematics 693, edited by D.W. Hadwin and J.M. Bachar, Jr., Springer-Verlag, 1978.

4.

P.C. Curtis, Jr., Topics in Banach spaces of continuous functions, Lecture Note Series No. 25, Matematisk Institut, Aarhus Universitet, December 1970.

5.

P.C. Curtis, Jr. and H. Stetkaer, A factorization theorem for analytic functions operating in a Banach algebra, Pac. J. Math. 37 (1971), 337343.

6.

H.G. Dales and A.M. Davie, Quasi-analytic Banach function algebras, J. Functional Analysis 13 (1973), 28-50.

7.

R.S. Doran and J. Wichmann, Approximate identities and factorization in Banach modules, Lecture Notes in Mathematics 768, Springer-Verlag, 1979.

8.

N. Dunford and J.T. Schwartz, Linear Operators, Part I: General Theorv. Interscience, New York, 1958.

9.

o.

Hatori, Functions which operate on the real part of a function algebra, Proc. A.M.S., 83 (1981), 565-568.

10.

H. Helson and J.P. Kahane, Sur 1es fonctions operant dans les alg~bres de transformees de Fourier de suites ou de fonctions sommab1es, C.R. Acad. Sci. Paris 247 (1958), 626-628.

11.

E. Hewitt and K. Stromberg, Real and Abstract Analysis. Springer-Verlag, 1965.

12.

H. Helson, J.P. Kahane, Y. Katznelson, W. Rudin, The functions which operate on Fourier transforms, Ac~a Math. 102 (1959), 135-157.

62

13. 14.

BACHAR

E. Hille and R.S. Phillips, Functional Analvsis and Semi-Groups. Am. Math. Soc. Colloquium Publ. 31, Providence, 1957. , , , J.P. Kahane, Sur un theoreme de Wiener-Levy, C.R. Acad. Sci. Paris 246 (1958), 1949-1951.

15.

J.P. Kahane, Sur un theor~me de Paul Ma11iavin, C.R. Acad. Sci. Paris 248 (1959), 2943-2944.

16.

J.P. Kahane and Y. Katzne!son, Sur le reciproque du theoreme de WienerLevy, C.R. Acad. Sci. Paris 248 (1959), 1279-1281.

17.

J.P •. Kahane and W. Rudin, Caracterisation des fonctions qui operent sur les coefficients de Fourier-Stie1tjes, C.R. Acad. Sci. Paris 247 (1958), 773-775. I , , Y. Katznelson, Sur les fonctions operant sur l'a1gebre des series de Fourier absolument convergentes, C.R. Acad. Sci. Paris 247 (1958), 404 ... 406.

18.

,

,

~

19.

Y. Katznelson, A1gebres caracterisees par les fonctions qui operant sur elles, C.R. Acad. Sci. Paris 247 (1958), 903-905.

20.

Y. Katzne1son, Sur Ie calcu! symbolique dans quelques a1gebres de Banach,

,

Ann. Sci. Ecole Norm. Sup. 76 (1959), 83-124. 21.

Y. Katzne1son, A characterization of the algebra of all continuous functions on a compact Hausdorff space, Bull. Am. Math. Soc. 66 (1960), 313315.

22.

, , ' . Y. Katznelson, Sur les algehres dont les elements nonnegatifs admettent des racines carres, Ann. Sci. tcole Norm. Sup. 77 (1960), 167-174.

23.

Y. Katznelson and W. Rudin, The Stone-Weierstrass property in Banach

algebras, Pac. J. Math. 11 (1961), 253-265. 24.

K. de Leeuw and Y. Katznelson, Functions that operate on non-self adjoint algebras, J. Analyse. Math. 11 (1963), 207-219.

25.

p. Malliavin, Calcu1 symbolique et sous-algebres Math. France 87 (1959), 181-190.

26.

C. Rickart, General Theory of Banach

27.

W. Rudin, Fourier Analysis on GrouEs, Interscience, 1962.

28.

W. Rudin, Real and Complex Analysis, McGraw Hill,

29.

S. Sidney, Functions which operate on the real part of a uniform algebra, Pac. J. Math. 80 (1979). ~65-272.

30.

W. Sprag1in, Partial interpolation and the operational calculus in Banach algebras, Ph.D. Thesis, UCLA, 1966.

31.

J. Wermer, The space of real parts of a function algebra, Pac. J. Math. 13 (1963), 1423-1426.

de Ll(G), Bull. Soc.

Algebras, Van Nostrand, 1960.

DEPARTMENT OF MATHEMATICS CALIFORNIA STATE UNIVERSITY AT LONG BEACH LONG BEACH, CA 90840

1966.

Contemporary Mathematics Volume 32, 1984

RECENT RESULTS IN THE IDEAL THEORY OF RADICAL CONVOLUTION ALGEBRAS William G. Bade In this survey I will discuss some problems concerning the structure of the family of closed ideals in certain radical convolution algebras on the

positive integers.

I

shall give background to these problems and describe

exciting results that have been found in the past two years.

In conclusion,

I shall briefly discuss the corresponding problems on the half-line.

A real-valued function w defined on ~+ = {n E ~:n ~ a} + and if function if wen) > a for all n E ~, w(m+n) 5 w(m)w(n)

for

m,n

w is radical i f

We say that the weight function

is a weight

E7/. lim w(n)l/n n-+OO

= 0.

For con-

venience we assume that w is non-increasing and tnat w(O) = 1. An example _n 2 for n E ~+. Denote by el(w) of a radical weight is given by wen) - e the set of all complex-valued functions x on '11+ for which l!x!1 = i Ix(n) Iw(n) < Then .e l (w) is a Banach algebra for the convolution n=O multiplication IXI.

(x*y)(n)

n

~ x(j)y(n-j)

=

for

n E ~+,

j=O withunit

e- [1,0,0, ••• ]

and generator

we can write

=

[0,1,0,0, ••• ].

If

1

x E t (w),

00

x -

and regard

z

tl(w)

volution of

as a Banach algebra of formal power series.

with

x

Z x(n)z n , naO

z

yields the right shift of

(z*x)(n)

= x(n+l)

n

=

{x:x(i)

These ideals, together with

1. QUESTION.

If

= 0, (0)

for and

i < n}, t

1

(w),

for

by one coordinate place: n E ~+.

x E t 1 (w),

for

There are certain obvious closed ideals in

M

x

Note that con-

t 1 (w), n

namely the ideals

= 1,2, ••••

are called the standard ideals.

w is a radical weight, is every clesed ideal in

a

standard ideal? © 1984 American Mathematical Society 0271-4132/84 SI.OO + S.25 per page 63

64

BADE

This question, and its analogue in the continuous case, are our main 1 concerns in this paper. The question for "(w) is attributed to Silov in 1941. If x E ,lew) and x ¢ 0, write «(x) - inf{i:x(i) ~ O}. For a radical weight w the following are equivalent: (a) all closed ideals in "l(w) are standard, 1 1 (b) for each x E "(w) with x ~ 0, x*" (w) - M ( )' _~_ 1 a x 1 (c) for each x E t (w) with a(x) ~ 1, the ideal x*t (w) contains (d)

some power of z, the closed subspaces of

"l(w)

z

invariant under convolution by

are totally ordered by inclusion. In view of (d) we call a radical weight w unicellular if any of the conditions (a)-(d) hold. The ideal question for

I t (w)

is part of a larger problem for weighted

1 + "(Z) by setting S(e) - Xnen+1' where ~n > 0 and en - {6kn :k E ~+}, for n n - 0,1,2,.... Then S is equivalent to the operator of convolution by z 1 1 in the space t (w), where wen) - ~0.~r .. ~n-1. In general, t (w) will not be an algebra unless conditions are placed on the ~n's. However, when w is a radical weight, the closed ideals of "l(w) are the closed subspaces invariant under cODVolution by z, and correspond to the closed subspaces of tl(~+) which are invariant for S. The early results concerning the ideal question for radical weights appeared between 1968 and 1974 in papers by Nikolskii [11]-[13], Grabiner [5]I8J, and Belson [9]. We describe a few of these below.

shift operators.

Let

S be the weighted shift defined on the space

DEFINITION. A radical weight w is a basis weight if for each r 2. there exists a constant Cr such that w(m+n+r)

~

C w(m+r)w(n+r), r

~

1

for m,n E ~+•

The condition says that every left shift of w is essentially submulti2 plicative. An example of a basis weight is wen) - e-n • The following elementary but important result is due to Niko1skil [11]. We give a proof by Belson [9].

3.

THEOREM.

1

Suppose 1 £' (w) is a non-standard closed ideal. define a multiplication by

PROOF.

M1

Every basis weight is unicellular.

(xty)(n) - (x*y)(n+l),

for

Then

On

n E J'+ and x,y.1ft.

Then (M1,.) is s Banach algebra with unit z- [0,1,0 •••• ]. Its unique maximal ideal is M2 , and I is a closed ideal of (M2 ,G). Hence I ~ !f2" An inductive argument shows that

RADICAL CONVOLUTION ALGEBRAS

65

00

len M n

=

(0),

n=l

so that

I

is unicellular.

For

w to be a basis weight it is sufficient that In w be concave. w(n+k) decrease to o This condition is equivalent to the condition that wen) Grabiner [6] has shown that for a basis weight every (non-closed) principal ideal x*t1 (w)

as n

~ ~

for

~

k

1.

contains a power of

Basis weights are quite special.

z.

In the negative direction, Niko1skii [13] constructed a class of weight sequences

t 1 (w)

w for which there were non-standard closed subspaces in

invariant under the right shift.

It was believed that his method yielded

algebra weights of the type we are considering for which there were non-standard closed ideals.

However, M.P. Thomas showed in 1979 that Nikolski! 's

argument did not work for algebras.

Thus the question of whether or not there

existed radical algebra weights yielding non-standard ideals became of great interest. In the past two years there have been two major results concerning this question, both of which are due to M.P. Thomas.

The first of these gives a

new and important class of weights having only standard closed ideals. The second is the construction of a difficult pathological weight which has a nonstandard ideal. I shall try to explain both of these results in an intuitive way. For the first theorem we say that a radical weight the function

w(n)l/n

decreases monotonically to zero as

comes from the fact that the region below the graph of from the origin. if an element

y

w

is star-shaped if

n -+

IXI.

This name

In w is illuminated

Thomas shows that star-shaped weights have the property that =

00

L y(n)z

n=O cation by a power of z,

n

of

t

1

(w)

is shifted to the right by multipli-

one can get a sharp estimate for the tail of the

resulting series: (II)

if

k:!: 1,

then

1\

y(n)zn+k ll 5 w(m)k/mllyll,

Z

for

m E IN.

n=m+l One does not know whether all star-shaped weights are unicellular. However, this is true with a small additional assumption on the rate of decrease of wen) l/n. 4.

THEOREM (Thomas [16]). is star-shaped and nw(n) lIn

All closed ideals in -+

0

as

n

To give an idea of the proof, let and

x(l) = 1.

We wish to show that

~

tl(w)

are standard if

w

00.

1

x E .e (w),

x*tl(w) = MI'

and !=Ittppose that Let

c = {c{n)

the unique complex sequence (called the associated sequence to

x)

t11=0

x(O)

=0

be

such that

66

BADE

GO

Z c(n)z n=O

n

satisfies the equation

...

GO

( ~ c(n)zn)( ~ x(m)zm) n=O m-l

in the algebra

=z The sequence

of formal power series.

G:[[z]]

c

will not in

1

t (w).

The strategy is to show that there exists a sequence of GO n Z c(n) z partial sums of the series such that in t 1 (w) we have n n-O p n lim x*( L c(n)z ) = z. n=O p-n-l k We must estimate the distance from z to x* Z c(k)z. Actually, it is 3 k=O n-l k+2 to x* ~ c(k)z • This is more convenient to estimate the distance from z general be in

k=O

1

sufficient because x*t (w) is standard if it contains z3. Since n-1 k X1c To c(k)z agree on [0,1,2, ••• ,n], we have, using (N), that

z

and

k=O

liz

3

n-1 k+2 - x* l c(k)z II ~o

~

= IIQ

n

n-l k+2 +3(x* Z c(k)z )/1 ~O

n-1

( z Ic(k) Iw(n+l)

k+l n+.r

1

)w(n+l)

n+r

k=O

(where for any series

y

00

n

= Z y(n)z,

Z Ic(k)lw(n +1) k=O

GO

n=m enters.

It suffices

p

= O(n ) P

3

z.

Thomas proves the existence of this

He shows that if there is no such sequence,

then w cannot be a radical weight. esting argument.

~

such that

p

in order to approximate

sequence by an indirect argument.

n

n +1

p

~....

n

Z y(n)z ).

~

n -1

p

as

{n p };c1

to prove that there exists a sequence

as

...

we write

n=O 1/ n Now the condition that nw(n) ~ 0

IIxll

I

shall not try to describe this inter-

Clearly more investigation is needed to get an effective CD

construction of

{n} p p=l· Such a construction would seem to be the key to further positive results of this type. THEOREM (Thomas (181). There exists a radical weight w and an elemen"t x E t 1 (w) such that x*t 1 (w) is a non-standard ideal. 5.

I will try to give the flavor of Thomas' construction, but I cannot ade-

quately convey the difficulty of his argument.

The paper [10] of McClure

gives an illuminating discussion of the problems that must be surmounted • Let such that n(j)

...

{n(j)}j=l n(1)

partition

=1 ~.

be a strictly increasing sequence of positive integers and

n(j+1) > n(j)(n(j) + 1)

for

j

We may assign values of the function

~

1.

The integers

w on

110

{n(j)}j=l

RADICAL CONVOLUTION ALGEBRAS

so that

{w(n(j»};=l

is strictly decreasing and

67

0

wenCk»~ ~ 1.


0, there exists

1.

This

A is compact.

N such that 2

(n ~ N),

I¢(z n ) I < e and we choose

~

1¢1 2 (z) - I¢(z) 12

We show that this implies that will prove that

87

(8)

with

p

(9)

We prove that there exists z E K(at~)

whenever

with

~ lal
I (z) -

I¢ (z) I

2

~l30d,

~ i~{1¢12(Zn)

~

°

as

0

let

Izi be

A

Izi

as

zn E K(a,T).

Then

~

- I¢(zn) 12} < i!g e.

1, and so

-+

1, and so

cj:.

limllAv n--

complete.

is compact.

A

-cp

Since

compa~t.

are both compact, and so

R¢

~

with

Therefore, by (6) and (8),

1¢1 2 (a) - I¢(a) 12 Thus

n

is analytic,

E VMO. z

II

=

R¢

= 0.

Therefore 0, and the proof is

n

We end this section with an elementary proposition which tells us when the linear span

X of

{v

z

: n Em}

is dense in

H2.

If

X is dense in

n

H2, then, of course, every bounded (compact) linear mapping of

X into

HZ

extends by continuity to a bounded (compact) linear operator on

H2.

PROPOSITION 4.

D and let H2 if and

Let

{z} n

denote the linear span of only if

Zm

n=l

(l-Iz I) n

= ~.

be a sequence of distinct points of {v

z

: n E IN}. n

Then

X

is dense in

X

88

BONSALL

PROOF.

h E H2

Let

(h,vz ) = O.

with

By Cauchy's integral theorem,

n

(1-lzI 2 )

(h,v) = h(z) z

and so

h(z) n

=0

[2, p. 18]) and so X is dense in On the other hand, i f

,

~~ (1-lz I) = ~, we have n=1 n

If

EN).

(n

\

h

with

(l-Iz I)
0, and choose

n

i=n Then choose

o

I) 2 .

J

E N such that

o GO

(X,

j=O

a i . Ia j

]·-0 -

a 1 . Ia j J

I)

2

(12)

< e •

nl E N with 00

X,

a ij Ia.

.

j=n

J

1

1

I < "4

k

E.

\I

(n)

(1= O,l, ••• ,n -1).

(13)

Ibij-b~nj) II a.l) 2.

(14)

o

0

We have ) fl/ 2

II (B-B n

Since

Ibij-b~;) I ~

2a ij ,

~

X,...

i=O

(12)

(X,GO

J

j-O

and

(14)

show that (15)

is

90

BONSALL

Now choose

such that, for all

N EN e

n 1- 1 (n) Z Ibij-b ij

j-O

Witn

Ila j

n

~

N , &

1 & k2

I < 2:(n)

(1 :: O,l, ••• ,n -1). o

0

(13), this gives ~

< (.L) n o

II(B-B

and (lS) now gives

n

)fil 2

(i :: O,l, ••• ,n -1), o

< 5&

whenever

n

~ N&•

The following construction was suggested by the proof of Paley's theorem in Kwapie6 and Pe1czy~ski [3].

I am grateful to Chandler Davis for a

remark that has made the construction simpler than the one I had first adopted.

Given a matrix

let

E

o

..

to},

n

,e n

denote the finite rank bounded

1

and, for

defined as follows.

1J

n E IN, let

E

n

- {k E tl : 2 n- 1 =:: k < 2 n }.

n" 0,1,2, ••• , let

Then, for

(i+j E E )

(n) a ij

={ {

b (n) ij

=

::

1J

THEOREM 7.

(i)

(n)

a ij

n

(all other (1 :::

i,j)

j)

(i < j) (n) _ ben)

ij

a ij

If ,£110

(liB fIl2+IIC*fIl2)

( B e., B e k ) mJ n

b~;) = 0 unless

and

i=O

for all

b(.m.>-b(.n) k ~J

~

2m- I !: i + j < 2m with

m ~ n + 2, we have

With b(n) - 0 ik

=

... ~

Thus

k.

B

i:'! j, and therefore unless

bi~) = 0 unless 2n

f.l Bn g,

C*f m

and similarly

m

From the orthogonality relations

(18)

< _ .1.,

in

which case

1 Cn*g.

we now have for all

f E H,

(19)

and (20)

together with similar identities involving Suppose now that

..

00

B

1.

n=O A

n

(16)

= Bn

Z C* ncO n

and

n

holds.

~onverge

+ C , it follows that n

{B 2n+1 } and

Then, by

(19)

and

(20), the series

in the strong operator topology.

Z

A

Since

couverges in the weak operator topology

ncO n

to some bounded operator

A.

Given

n , and then

i,j E Z+, we have

_{a

o

(A e. ,e.) -

nJ

1.

ij

II

i

+

j

EE n

for some o

= no )

(n

(n :; n )

o

IX>

Thus

a 1J.

= ZncO (Ane.,e.) = J ~

ed operator

(Ae.,e.), and J

~

(a;j)

A.

If the inequality (17) holds, (19) gives P 1

B2n+1 •

similarly for

liz

p

cnll

*

p

= liz

ncO

converges to

is the matrix of the bound-

~

n-O

Therefore ~

liZ

n-O

IIZP

Bn ":: 2M"i.

Cn " ~ 2M , and it follows that

A In the weak operator topology.

B2nf1l2!: M//fIl2, and

ncO

Similarly,

IIAII ~

I

4MJi , since

p

Z An n-O

92

BONSALL (ii)

of a bounded operator 60 + 8 2 +••. + B2n •

and

Ill.

den) -~ a. ij iJ


q

n

n=q

N

H,

co.

8 are applicable to every Hankel matrix

>· Corollary 8 can be strengthened when the matrix elements are non-nega-

tive.

BONSALL

94

COROLLARY 9. (i)

(a ij )

is the matrix of a bounded operator if and only if 00

Z

/lBnf/l2
J

below. There is another positive result about homomorphisms that is an

a: A ~ B be a homomorphism with separating space G(e), (a) be a sequence in A. Set Gn - e(an ••• a l )G(9). Then clearly n

important tool. and let

Gn+l (N

C

Gn •

Let

A form of the stability lemma asserts that there exists

depends on the sequence

(a

n

»

such that

Gn = GN for n

~

N.

N ER This

102

DALES

principle was used by Johnson and Sinclair in [23]; see also [24] for a general form of the result. theorem

The result leads to the following prime ideal

([33,11.4J, [6,2.7]).

commutative Banach algebra ideal

K of

Let

9: A -,. B be a

A such that

9(A)

B such that the homomorphism

homomorphism from a

= B.

A4 B

Then there is a closed 8/K

4

has a prime kernel.

Combining this fact with the theorem of Bade and Curtis, we see that, i f

e

there is a discontinuous homomorphism is a maximal ideal Banach algebra

M of

from an algebraC(X), then there

C(X), a prime ideal

R, and an embedding

P with

P; M,

a radical

M/P -.. R.

Eventually, it was shown (see [11],[9,9.6]) that, given such an and such a an

P

em~edding

in M/p

C(X) 4

such that

Hlp

has cardinality

'~l'

R for certain radical Banach algebras

The question concerning homomorphisms from study of ideals and order structure in study of radical Banach algebras. ed the radical Banach algebras

e(X)

e(X)

M

there does exist

R.

thus led to a deep

and other algebras, and to a

Indeed, Esterle (see [14]) has characteriz-

R which can occur in the above result, and

he has also given a comprehensive classification scheme for commutative, radical Hanach algebras. The result shows that, if the eontinuum hypothesis be assumed, then there is a discontinuous homomorphism from each infinite, compact space

C(X)

into a Banach algebra for

X.

It is a remarkable fact that the assumption of the continuum hypothesis cannot be dropped from this result: from each algebra Solovay.

C(X)

models of

ZFC

in which each homomorphism

are continuous have been constructed by Woodin and

See [lO,§4] for a discussion of this aspect of the story from the

point of view of an analyst. Now let us turn to consideration of homomorphisms from possibly :l.oncommutative C*-a1gebras. 4.

~UESTION

For which C*-algebras

A is it true that each homomorphism from

A into a Banach algebra is automatically continuous? Let

A Je a C*-algebra,

homomorphism.

I(e)

nC

B be a Banach algebra, and

A.

A.

Thus, if

4

B a

It follows from Bade and Curtis's theorem, as above, that

has finite codimension in

of

9: A

C for each commutative C*-subalgebra

But this is sufficient to show that A is an infinite-dimensional

ideal of finite codimension, then I(a)

c

A.

I (6)

C

has Unite codimension in

C*-al~ebra

with no proper, closed

If, further,

A has an

ALGEBRA AND TOPOLOGY identity, then

e

I(9) - A, and so

of proving that each homomorphism

103

is continuous.

B(H)

-+

This is the easiest way

B is automatically continuous

([22],[33,12.4]).

A slight modification of the method shows that each

homomorphism

-+

K(H)

B is automatically continuous:

C*-algebra of compact operators on a Hilbert space that the multiplier algebra of

here,

K(H)

denotes the

H, and one uses the fact

K(H), B(H), has no proper closed, cofinite

ideals. A second technique for proving that each homomorphism from certain C*-algebras is continuous was also introduced by Johnson ([21]):

it applies

to C*-algebras with many projections, and. shows, for example, that each homomorphism from a von Neumann algebra which has no direct summand of type I is continuous. Let

A be a C*-algebra.

a *-homomorphism

IT

from

A representation of dimension M (t), the algebra of

A into

n

n x n

n

of

A is

matrices.

The representation is irreducible if {O} and

Mn (~)

are the only linear

subspaces of

n(A).

Two irreducible represent-

M «() n

ations of dimension

which are invariant for n

are equivalent if they have the same kernel.

It is proved in [1] that, if

A is a C*-algebra widch has infinitely

many non-equivalent irreducible representations of dimension n E N, then there is a discontinuous homomorphism from made that otherwise all homomorphisms from

n

for some

A, and the guess is

A are automatically continuous.

By combining the two techniques mentioned above, and by using technical decomposition theorems for C*-algebras, it can be proved that the guess is true for a rather large class of C*-algebras, the so-called the class includes

(i)

each

AW*, and hence each von Neumann, algebra;

(ii)

each closed ideal in an AW*-algebra;

(iv)

the algebra

A ® K(H)

AW*M-algebras:

(iii)

for each C*-algebra

each commutative C*-algebra; A.

However, many C*-algebras are not of this c:ass.

It seems likely that

a new idea ,,,rill be necessary to resolve the question for these algebras. is a particular challenge.

Let

Here

A be a (topologically) Simple, infinite-

dimensional C*-algebra without identity.

Is every homomorphism from

A

automatically continuous? We do have an analogue of the Bade-Curtis theorem and of the prime ideal theorem for non-commutative C*-algebraR: from

[32]~

[271, and unpublished work of Cusack.

tinuous homomorphism from a C*-algebra subspace algebra of

the following result is taken

F of A.

A such that Further,

F

~

e = '"" +

morphism which coincides with

9

A.

on

9: A ~ B be a discon-

Then there is a finite-dimensional

1(9) = A and

'J, where

Let

~:

F + 1(9),

A

F + I(G) -+

is a dense sub-

B is a continuous homo-

and

\/: A -" B is a linear

104

DALES

vII(9)

map such that ideal from

P

in

M such that

~

G(9) P

is a homomorphism.

Finally, there is a prime

is the kernel of a discontinuous homomorphism

M into a Banach algebra. We saw at the beginning of this article that the uniqueness of norm

problem is closely related to the question of the automatic continuity of epimorphisms onto Banach algebras. QUESTION S.

Consider the following question.

Is each epimorphism from a C*-algebra onto a Banach algebra

automatically continuous? It was first proved by &ster1e ([13]) that each epimorphism from a COlmnutative C*-algebra is indeed continuous: proof.

see [14] for a very elegant

In [1], the same result is proved for AW*M-algebras, and it has been

proved by Laursen for epimorphisms with commutative range (see [26]): proofs rely heavily on &sterle's result.

these

However, the question is open for

general C*-algebras. Let me conclude by mentioning some sources where more detailed information can be found.

The theory of the automatic continuity of linear operators

between Banach spaces is given in Sinclair's book [33].

The survey [9] con-

centrates on the theory of homomorphisms between Banach algebras and of derivations into modules, and the article [251 discusses the automatic continuity of "intertwining operators", a class of linear maps which includes both ilOmomorphism and derivations: Banach algebra on [0,1].

c(n)[O,l]

the main applications are to maps from the

of n-times continuously differentiable functions

More details of the theory of homomorphisms from C*-algebras are

given in [10].

Finally, let me mention the volume [4], the proceedings of a

conference held in 1981, where many relevant articles can be found.

A longer

list of open questions can be found at the end of that volume.

REFERENCES 1.

E. Albrecht and H. G. Dales, Continuity of homomorphisms from C*-a1gebras and other Banach algebras, in [4], 375-396.

2.

B. Aupetit, Propriet's spectrales des alg~br~s de Banach, Lecture Notes in Maths., 735, Springer-Verlag, 1979.

3.

B. Aupetit, The uniqueness of the complete norm topology in Banach algebras and Banach Jordan algebras, J. Functional Analysis 47 (1982), 1-6.

4.

J. M. Bachar et al (ed.), Ra~ica1 Banach algebras and automatic continuity, Proceedings, Long Beach, 1981, Lecture Notes in Maths. Springer-Verlag, 1983.

975.

105

ALGEB RA AND TOPOLOGY ~

W. G. dade and P. C. Curtis, Jr., Homomorphisms of commutative Banach algebras, Amer. J. Math. 82 (1960), 851-866.

6.

W. G. Bade and P. C. Curtis, Jr., Prime ideals and automatic continuity problems for Banach algebras, J. Functional Analysis 29 (1978), 88-103.

7.

W. G. Bade and P. C. Curtis, Jr., Module derivations from conmutative J:Sanacll algebras, these proceedings, 7l-8l. J. Cusack, Automatic continuity and topologically simple radical Banach algebras, J. London Math. Soc. (2) 16 (1977), 493-500.

8. 9.

H. G. Dales, Automatic continuity: 10 (1978), 129-183.

a survey, Bull. London Math. Soc.

10.

H. G. Dales, Automatic continuity of homomorphisms from C*-algebras, in "Functional Analysis: surveys and related results III" (ed. K.-D. Bierstedt and B. Fuchsteiner), North Holland, 1984.

11.

H. G. Dales and J. Esterle, Discontinuous homomorphisms from Bull. Amer. Math. Soc. (2) 83 (1977), 257-258.

12.

M. Eidelheit, On isomorphisms of rings of linear operators, Studia Math., 9 (1940), 97-105.

13.

J. Esterle, Theorems of Gelfand-Mazur type and continuity of epimorphisms

from

C(K) , J. Functional Analysis

36

(l9~O),

C(X) ,

273-286.

14.

J. Esterle, Elements for a classification of commutative radical Banach algebras, in [4], 4-65.

15.

J. Esterle, Quasimultipliers, representations of H, and the closed ideal problem for commutative Banach algebras, in [4]', 66-162.

16.

C. Feldman, The Wedderburn principal theorem in Banach algebras, Proc. Amer. Math. Soc. 2 (1951), 771-777.

17.

I. M. Gelfand, Normierte Ringe, Mat. Sbornik

18.

F. Hausdorff, Zur Theorie der linearen metrischen Rlume, J. Reine Angew. Math. 167 (1932), 294-311.

19.

N. p. Jewell and A. M. Sinclair, Epimorphisms and derivations on

...

Ll(O,l)

9 (1941), 3-24.

are continuous, Bull. London Math. Soc.

8 (1976), 135-139.

20.

B. E. Johnson, The uniqueness of the (complete) norm topology, Bull. Amer. Math. Soc. 73 (1967), 537-541.

21.

B. E. Johnson, Continuity of homomor~hisms of algebras of operators, J. London Math. Soc. 42 (1967), 537-541.

22.

B. E. Johnson, Continuity of homomorphisms of algebras of operators (II), J. London Math. Soc. (2) 1 (1969), 81-84.

23.

B. E. Johnson, and A. M. Sinclair, Continuity of linear operators conunuting with continuous linear operators II, Trans. Amer. Math. Soc. 146 (1969),533-540.

24.

K. B. Laursen, Some remarks on automatic continuity, Lecture Notes in Mathematics 572 (1976), Springer-Verlag, 96-108.

25.

K. B. Laursen, Automatic continuity 01 generalized intertwining operators, Diss. Mat. (Rozprawy Mat.) 189 (1981).

26.

K. B. Laursen, Epimorphisms of C*-algebras, in "Functional Analysis: surveys and related results III" (ed. K.-D. Bierstedt and B. Fuchsteiner), North Holland, 1984.

2~

K. B. Laursen and A. M. Sinclair, Lifting matrix units in C*-a1gebras II, Math. Scand. 37 (1975), 167-172.

106

DALES

28.

J. A. Lindberg, A class of commutative Banach algebras with unique complete norm topology and continuous derivations, Proc. Amer. Math. Soc. 29 (1911), 516-520.

29.

R. J. Loy, Uniqueness of the complete norm topology and continuity of derivations on Banach algebras, ~hoku Math. J. 22 (1970), 371-378.

30.

C. E. Rickart, The uniqueness of norm problem in Banach algebras, Ann. of Math. 51 (1950), 615-628.

31.

C. E. Rickart, General theory of Banach algebras, van Nostrand, Princeton, 1960.

32.

A. M. Sinclair, Homomorphisms from C*-algebras, Proc. London Math. Soc. (3) 29 (1~74), 435-452; Corrigendum, 32 (1976), 322.

33.

A. M. Sinclair, Automatic continuity of linear operators, London Math. Soc. Lecture Note Series 21, Cambridge University Press, 1976.

SCHOOL OF MATHEMATICS UNIVERSITY OF LEEDS LEEDS, LS2 9JT ENGLAND

Contemporary Mathematics Volume 32, 1984

t-UTl'AG-LEFFLER METHODS IN THE THEORY OF BANACH ALGEBRAS

AND A NEW APPROACH TO MICHAEL'S PROBLEM Jean Esterle 1.

INTRODUCTION The aim of this p;lper is to describe two useful tools in the theory of

Banach algebras (the Mittag-Leffler Theorem on inverse limits and the notion of bounded approximate identity), and to apply these ideas to operator theory and to tne theory of entire functions of several complex vari abIes. This paper does not intend to present a survey of recent progress ill thi:! theory of Banach algebras.

Works as important as Dale's construction of

discontinuous homomorphisms from C(K) [14J, Domar's characterization [17] of closed ideals of Ll(IR+, e- t2 ), and Thomas's construci:ion [47] of a weighted radical Banaell algebra of power series with nonstandard ideals (see Bade's report in this volume [5]) will not he discussed.

We just lvish to

presl~nt

a

more-or-less connected set of ideas appearing in works of Arens [3], Allan (1). Sinclair L45j, Dixon [16], and in various papers [18]-[22J of the autill)r, which lead to one of the known constructions of discontinuous homomorphisms

C(K)

frolll

and yield some progress in the general theory of Banach algebras.

We also show how a slight reformulation of the Mittag-Leffler Theurem on inverse limits, given in Theorem 2.1, gives a tool which can be used to prove Cohen's Factorizatiun Theorem [12J (see Theorem 4.3). Sec::ions 2-5 are essentially expository.

In Section 2 we present the

Mittag-Leffler Theorem on inverse limits in a slightly more general form than usual, and show how this theorem includes the Baire Category Theorem.

We

!illOW

also the connection between the abstract version and the classical MittagLeffler Theorem about meromorphic functions with prescribed poles and Singular parts. Section 3 gives some applications to the theory of Banach algebras. first one is a step of G. R. Allan's construction of an embedding of into Banach algebras [11.

The

t[[X]]

The second one is a part of the author's classifica-

tion of commutative radical Banach algebras [21].

107

(We show that a rauical

© 1984 American Mathematical Society 027]·4132/84 $1.00 + S.25 per page

ESTERLE

108 Banach algebra

R possesses a nonzero element

x

such that

2 x E [x R]

if

R possesses a nonzero rational semigroup.)

and only if

In Section 4 we present a new "Mittag-Leffler approach" to Cohen's Factorization Theorem, and some standard consequences of this theorem which seem like the theorem itself

to have been ignored by many distinguished

analysts and specialists of Lie groups. Sectlon 5 gives some "spectral mapping theorems" which do not use the Mittag-Leffler Tlleorem but instead are based upon a game involving topological divisors of 7.eros related to bounded approximate identities. SeLl ion 6 gives anew, unpublished approach to Michael's problem l34] about continuity of characters on Fr/ehet algebr-as. to P. G. Dixon and the author.) l~rechet

commutative

n

VI

We show in particular that all characters on

algebnlfi would be continuous i[ it were possible

to const ruct a sequence such that

(F n> n::l ?

0

(This approach is due

•••

1'n(q:-)

0

of entire functionfi of

= 0.

(:2

into itself

The existence of such a sequence is

n

unclear. and it even seems to be an open problem to decide whether for each entire function

from

2

n Fn (( 2 )

~ ~

Examples due to Fatou [26] and Bieberbach [6J show that some one-to-one entire functions F: a: 2 -+ «;2 F

t

into i.tself.

1, have nondense ranges.

whose Jacobians are identically equal to

The usual

approach consists of considering solutions of some functional equations (references, which go baCK to Poincar~'s tilesiti, are given in Section 6), but we present here

Cl

new appraoch which consistl:i tlf constructing sequences of

«;2

analytic automorphisms of

with respect to the a-topology.

of Jacobian 1 which have a nonsurjective limit (This idea is intimately related to the nOLion We construct in particular a

of Cohen clements l19] for Banach algebras.) one-to-one entire function

{( x , y) E (

2

IRe

F: ¢2

~ a:2 l:;uch that

x ::- 1, Re y > I}

Jacobian, such that lar, G

(D 2 )

avoidl:i the sets

')

an d

{ (x, y) E

allows us to obtain a strange entire function -1

2 F(a:)

inf

(I u I, Iv I>

== 1

('" IRe Co: 0;2

for each

This

x .;: 1, Re y < l}. -+

(;2

with nonvanishing

(u, v) E G(a;2).

In particu-

meets the ranges of all nonconstant entire functions (where

denotes the closed polydisc of radius for the construction of

l~,

2).

D2

Theorem 2.1 is usad as a basis

and the Mittag-Leffler Theorem (applied to a rather

unusual system of complete metric spaces) is used to establish the connection between

~1ichael' s

problem and entire functions of several variables.

Michael's

problem seems in [act to be the starting poInt of this circle of ideas, since it was used as an essenti:ll tool in Arens' approach [3] to continuity of characters on Frechet algebras. "

MITTAG-LEFFLER METHODS

109

In this paper we have tried to show how the theory of Banach algebras can still bring results or ideas to other branches of mathematics, and how some general structure results can otill appear inside the theory. indebted to C. E. Rickart.

I am deeply

The results, the methods, the conception of

mathematics given in his treatise [41] were and are a constant reference for my research in mathematics.

2.

THE MITTAG-LEFFLER THEOREM ON INVERSE LIMITS Let

(E ) n rel

be a countable family of sets, and assume that for each

elements

x

= for each

projection from if

x E F, F

~

11 :::

n E

Also, if

E • m

d(x,F) .. inf

1T

(E,d)

is the set of all

n

satisfying

>1 n n_

We will denote by

1.

n E onto >1 n n_ E, then we set

n

+-

of the cartesian product

en (xn +1)

xn =

11m (E ,6)

Its projective limit

is a projective system.

m

the mth coordinate

is a metric space and

d(x,z).

zEF We have the following theorem.

,e ) be a projective system, where En is a complete n n metric space with respect to a metric dn for each n ~ 1. Assume that the

THEOREM 2.1.

Let

(E

following conditions hold:

dn (9 n (x),9 n (y»

(1)

for

S dn +1(x,y)

x,y E En +1 , n

1

~

00

(2 )

)..

Z

n

n=l


E

n

Then for

0 and let

x EEL.

By induction we define an element

satisfying the following conditions:

n::: k+1

we have d (y , e (y +1» n n n n

< ).. n

ESTERLE

110

This shows that the sequence

Ek for each k ~L for each k ::: I, we have

= lim n--

d l (x, 91

o en-l(Yn»n~+l

0

Denote by

in

dl (x,x l )

(9k

its limit.

xk

(E ,9 ). (xk)k~l E lim ofn 0

is a Cauchy sequence

Since

is continuous

9k

Also

9n (Y1\+1»

0 ••• 0

n

..:; d l (x,a l (Y2» + lim sup n-~

~

m=2

dl (91

tim_1 (Y m) ,91

0 ••• 0

0 ••• 0

9m(y m+l

»

o

+ 11m sup

dl(x, 8l (Y2»


1 n n_

>' n n:;...&.

subset of

V n

•

=

Then the sequence

m

Denote by

~

(W ,~)

n

~

1

set

d.

as

A routine well-known verification shows

we can define the topology of

with respect to which

is complete.

V

VI' hence dense in n, so that

Xl E

E.

But if

n

V , and

n

n~l

V ~

Denote by

n It follows from Corollary 2.2 that

identity map. each

W is any open

If

is a complete metric space ([11], Chapter V, Section 3), so

that for each

in

E.

is well defined, and it is clearly a distance which on

W defines the same topology that

is decreasing, and

the given metric on

d

x,y E W,

E, then for

W is open,

Since

E.

E.

f1 u • m~

E be a complete metric

be a countable family of dense open subsets of

is dense in

n~l n

Let

III

by a distance

n

Vn +1

n

.• V

n

n l (lim (V ,9

d

n

the

»

is dense n 11 (V ,e ) then Xl = x for (xn)n~l E lim -+n n n n Un = n Vn is dense in E. n~l n:::l -+-

We now give the usual Mittag-Leffler Theorem about meromorphic functions. COROLLARY 2.4.

U be an open subset of the complex plane, let

Let

be a discrete sequence of elements of of rational functions, where

=

S (z) n

each

S n

m n

A-

~

i 2n

i=1 (z - a ) n

(an ) n_ >1' such that the singular part of Denote by

s

1

positive reals with ~n

set Then

= {z

f

at

A

n

and

~ ~

E Uld(z, ~\U)

V

= WU

an

5n

is

for each

u ;.

If

f,

n

and

as

~O

Izl

< A-n }

is a relatively compact subset of

that either

U whose set of poles is n

~

1.

I, choose

and a decreasing sequence

and consider a component

v

on

f

the Riemann sphere

an increasing sequence

be a sequence

has the form

Then there exists a meromorphic function

PROOF:

(Sn)n~l

U, and let

(an)n:::l

{~}

is a bounded subset of

V

S\S~.

of

(where

c.

U

n

-~

GO

,

(en)n:::l of such that the open

is nonempty for each and

n ~ 1.

u = U Q.

Fix n ~ 1 reIn is bounded, it follows

Since ~ n n W is the unbounded component of

In the first case we see that

~\~),

V n (5\U)

or is

112

ESTERLE

nonempty. for each

V, and there exists

intersects

n

and

~

z

D(u,e )

Iz 1
for n::: 1 and Xl E

(E ,a» n

n

We have

n un(E).

n::1

This

is

113

MITTAG-LEFFLER METHODS

to[[X]] the algebra of all formal power series in one variable with zero constant term. The following theorem is a slightly weaker Denote by

version of a basic result of G. R. Allan [I, Lemma 3).

I

n

u:::l th.ere exists a unique algebra homomorphism

= tr(x).

q>(X)

Let

PROOF:

the map n

x E A.

A be a commutative Banach algebra, and let = n x A, and den~te by tr: A -I> A/r the natural surjection. Let

THEOREM 3.2.

~

1.

u

cP

is one-to-one i f

A is not unital.

= ~ ~

xn

be an element of

'O[[X]].

f -+ ~

n

n:!l. n x + xu.

Since

= A,

[xA]

n

- A,

such that

an :

Denote by

[a (A)]-

we have

[xA1

If

cp: 'O[ [X]] -+ A/I

The map

Set

=

A -I> A

A for each

It follows from Corollary 2.2 that there exists a sequence

of elements of

A such that

un

=

9n (un +l )

n

for

~

1.

(u) n n~l A routine induction

shows that

n

Now let Then

be another element of

u

u - u1 E

such that

A

n xn A, so that

~ ~ xm E xnA for n ~ 1. m=l m So the map cp: f -+ u is well-

u -

n~l

defined, and it is clearly an algebra homomorphism from Also f -

cp(X) = tr(x).

Z ~nxn.

If CPI 0

€.

A

lIunll == I e

to

A.

~ on AU such that A = Ker~. We extend to

Then there exists a character

An = A e

of elements of

n~l

n

such that

H

HI = [span AM]-.

PROOF:

(a)

A be a Banach

1, and let

algebra with bounded approximate identity bounded by Banach A-module.

Let

the module action of A on M in the obvious way. Denote by o II and set the closed unit ball of A, u = {y E Mlilly-x/l < €}. Denote by B the closure in 0 x U of the set of elements of the form {(a,a

-1

(e

.

1/-1

x) 1a E D II Inv A ,a

We can define the given topology of

x E U}.

by a distance with respect to which x,y E U let (8,5) Then ~ E

that

8

is a closed subset of

P

is complete.

lIa-bll + d(y,z).

is a complete metric space.

Now set for

8

Bp

Now for

Since Pick

B( (b n , b:lX) , (a,y»

-+

0

II (e n -e)(b n ~ (b n )e) II

-+

0

lIe-A(e-e) 1/ ::: 1 n

(e-;\.(e-e» -1 n

=

m

2: A (e-e)

m~u

n

verification shows that

-+0 00

(e )

,and

and fix

p

sup 1'P(b n ) 1

lim

of elements in

n ~1

n

-+

00,

n:: 1. m

for

and sllch that

A"

n

n::

Hence

lib -1 [e-X.(e-e ) ]-lx_yll ..... 0 n n

-1

-1

IIbn as

We

is invertible in

A",

and

n

x-xli:: kllenx-x!I

n -)

00.

-1

as

such that

1

-+....

1. Since IIX.m(e-e ) mlII

Le-~(e-en)]

~

n

--+

n

0

such

1'1> (a) 1== (~)p.

O

(or even of analytic semi-

t

groups (a )Re t>O~ in commutative Banach algebras with bounded approximate identities. We refer to Sinclair's monograph [45] for these topics. A formalization of these constructions along the lines of the present proof of Theorem 4.1 will hopefully be available in the near future in a forthcoming paper by Zouakia. An important example of a Banach algebra with left bounded approximate

identity is given by the group algebra group.

Ll(G), where

In this case we may choose for each compact neighborhood

a nonnegative-valued function

e V with support in

Jvev(~)ds = 1, and the family

(e V)

LlCG)

G is a locally compact

1.

bounded by

some other consequences.

f

1f

that for every

f E Ll(G) g

gives a bounded approximate identity for

= g*h,

... E U (G)

where

and

G.

00

U (G)

the set of all bounded,

It is a standard result that

g E Lm(G).

and every

f E LICG)

g,h E L1 (G), but there are also

For example, denote by

uniformly continuous functions on f*g E Uoo(G)

V such that

The factorization theorem shows that each

can be written in the form

V of unity

e. > 0

But a routine verification shows there exists a neighborhood

Ve

I:ev*g-glloo < Eo, so that Ll(G) * Uoo(G) is dense in U (G). e. We tllUS have the following consequence of Cohen's Factorization Theorem. 00

of unity such that

118

ESTERLE

CUROLLARY 4.4. 1£ G is Ll(l;) it L'l 0 there exists a bounded operator u g on E

of finite rank such that

x E K.

If

Ilu;;,,:::

.1

and

,Iu f; (x)-xll
1 n_

y

-+

~x

>1 n n_

®t, n

!lilt II


(u) c Sp u for

9: l\.

is a homomorphism, we always

B

-+

\-Je give here an example for which the

u E A.

existence of a bounded approximate identity in some subalgebra of that some elements of

Sp u

belong

Sp means that

variables with complex coefficients such that

(a i ) - X(a.)

for

(al, ••• ,a p )

A is polynomially generated by

x E A there exists a sequence

-to co •.

X{X i )

A

A are continuous.

a continuous character ~

~(xi)·

on

~

A be a commutative Frichet algebra.

A is polynomially generated by a finite family

PROOF:

the

J

,

R > 0, so for every

elements of a unital Frechet algebra, the series

122

ESTERLE

ip

i l

r.

Ai

(ll, ••• ,i p )

1··· P

A.

converges in

a l ••• a p

i

We can denote by

the sum of this series, and the map (al, ••• ,a p ) ~ f(al,···,a p } is a continuous map from AP into A. Also, if F = (fl, ••• ,f q ) is an f(al, ••• ,a p )

entire function from

[p

into

¢q, then the map

F(al, ••• ,a p > = (fl(al, ••• ,ap), •.• ,fq(al, ••• ,ap» is a continuous map from AP into Aq • Finally, if X is any character on a Fr~chet (al, ••• ,a p >

al~ebra

-~

Xp (al, ••• ,ap )

=

the map defined by the formula

(X{al}, ••• ,X(a p

».

We obtain the following proposition.

Let

PROPOSITION 6 .l••

r E H«(P,C q ). Then character X on A. PROOF:

xp

A, denote by

A be a commutative Freehet algebra, and let X (F(u» = F(X (u» for each u E AP and for each q

p

for each

f

E H(CP,C).

generated by

Denote by

(al, ••• ,a p )'

= X{f(al, ••• ,a p »

f(X(al), ••• ,X(ap »

We just have to prove that

B the closed unital subalgebra of

Since

XIB

A

is continuous by Corollary 6.3,

the result follows from the definition of

(a 1 , ••• ,a p )'

f

We now present the basic step in a fairly new approach to Michael's problem. THEOREM 6.5.

If there exists a discvntinuous character on a commutative

~

Freehet algebra

A, then for every projective system

p (a:

II

, F) n ' where

F

n

E

H«(

p

p

n+l,a: n)

n ~ 1, the projective limit

for

p

lim (a: n,F) n +PROOf':

is nonempty.

We may assume that

character on

At and set

A is unitol.

r.1 = Ker

for ~quipped

with the discrete topology.

Now consider the map

e l\

(al, •••• a

Pn+l

en

,xl'.'.'x

qn+l

En +l ) =

Then Let

n:! 2.

complete topological space, so that n.

X.

E

n

En

X be a discontinuous

M is dense in

E

n

Then ~

Let

Pn

=A

A.

qn

x M ,where

Put ql M is

=0

M is homeomorphic to a metrizable

is metrizable and complete for defined by the formula

MITTAG-LEFFLER METHODS Since n

M

~ 1,

n ~

M

is dense in

on (E n +l )

A,

It follows from Corollary 2.2 that of lim (E ~

Pn

n

;(.

Pn

EA , (u)

n

x

n

,en ).

For each

is dense in n

III

n

we can write

for each

E n

Pick an element

.

= (un ,xn ), where

U n

qn

EM, and

n

= XP

(F (u +1»

n

n

n

=

belongs to

Then

F (X n

Pn+l

lim ~

(u +1». n

Pn

«(

Let

z

n

=X

n

into itself such that

Pn

(u)

for

n

n

1.

:.>0

,F), and the theorem is proved. n sequence

If there exists a

COROLLARY 6.6. from

n

is continuous for each

lim(E ,0 ) ; -t-

u

en

is equipped with the discrete topology,

and since

1.

123

of entire

= Ill,

Flo •••

function~

then all

n~l

characters on all commutative Fr~chet algebras are continuous.

n

PROOF:

Fl

Fn «(2)C lim (t 2 ' Fn ) •

0 ••• 0

n~l

~

Most of the credit for thit; approach to

~lichael' s

problem belongs

to P. G. Dixon, who mentioned a result similar to Corollary 6.b during a

discust;lon with the author over a cup of coffee during a NBFAS :;eminar at Edinburgh in June 1978.

The above formulation and the Mittag-Leffler proof

of Theorem 6.5 were obt ained by the author in November 1982, just after obtaining with rio G. Dales a short Hittag-Leffler type proof of Shah's theorem on continuity of positive linear forms on So-algebras with continuous involution [43].

Proposition 6.4, and hence Theorem 6.5, can be extended to

nonco~nutative rrechet algebras by using some algebras of formal power series

of noncommuting variables, but we will no do this here. Despite the simplicity of its statement, the question of the existence of

n

a

sequence

Flo •••

0

n~l

(F) n

F

II

of elements of

~l

(~2)

-

III

H«2,C 2 )

such that

seems to be a difficult problem.

Note that it

follows immediately from the big Picard theorem that the complement of o f «() contains at most one point for any sequence ( f ) of il f1 0 n n n~l n:::l nonconstant entire functions on t. But it follows from constructions made

by Fatou [26J and Bieberbach [6] in the twenties that there exists an entire one-to-one function dense in

~2.

F: t'

'}

-+

to.

of Jacobian One such that

2

F(t)

The literature about these functions is rather sparse,

known constructions are based on the following idea.

is not but all

124

ESTERLE

e

Take an analytic automorphism at

O.

and

of

,p

with a repulsive fixed point

Then there exists an analytic function ~ll

invertible, where

F' (0)

S~2

and

F:

wi.th

S~1 -+ Q 2

F(O) = 0

are open neighborhoods of

0

OaF = FoB. where B is a suitable analytic automorphism of B-n(y) -+ u as n -+..., for y E ,p. In the case where the

such that satisfying

F'(O)

eigenvalues of relation

• ••

do not satisfy any

m ""I\. p = 1

with

P

(ml, •••• Rl p )

:1

,p

and

(0, ••• ,0), one can just take

B

9'(0), and the treatment of

this case is related to the 1878 thesis of H. Poin(,.llr~ [37], in the context of partial differential equations. and to another paper of Poincare' L38] when

e

When p = 2 one can always take B to be of the ('1.X,AZY + ux q ), as shown in 1911 by Latt~s L321. A discussion

is a polynomial map.

form

(x.y)

-+

of the general case

was given by Reich in 1969 [391, [40J, but Reich's work

might overlap some results d'le to

,

.

Po~ncare

and Dulac, mentioned in Arnold's

book [4).

Anyway, \-1hat happens is that the solution of the equation

e~

can be extended to an entire function from

=

FoB

is one-to-one.

F(t 2 ) = {z E ,2

Also

Ie-n(z)

-+

,2

O}.

into itself which

But i f

has another

F

n~'"

P.

repulsive fixed point

then

e-n(z)

r-

n>GO .

some open neighborhood of

(:3, so that

F(G: 2 )

V is

z E V. where

for all

-+ ::l

is not dense in

,-. ?

Bieberbach's original example corresponds to the automorphism

for which both

and

(0,0)

(1,1)

are repu.l.sive fixed points.

Other con-

structions USing different automorphisms can be found in Sadu11aev [42], Kodaira [31], and Nishimura

lJ5J~

and a very clear exposition of Bierberbach's /

) original constru«.:tion is given by Stehle [46]. ::Ii.ow that, [or each

IF f. (z) I = 0 (exp(

> 0

E.

Izl~».

there exists a

Sibony and Pit Mann Wong [44)

Bierberbach function

F

such that

F-

We will not try to give comprehensive references

here, but just present another approach. Denote by

Aut1(t P)

the set of all analytic automorphisms of

,p

of

Jacobian identically equal to i, and denote by B(~P) the closure of Aut l (f.:p) in H(t P .t P) with respect to the topology cr of uniform convergence on compact subsets of J(F)(z) = 1

for every

fCP. It follows from Cauchy's inequalities that F E 8(fC P ) antl every z E (p, and it follows from

[7, Chapter 8, Theorem 9J that all elements of

B(~P)

idea just consists of finding a convergent sequence 2

Autl(C)

whose limit

F

avoids a suitable set.

elements of

B(¢P)

can be found in

elements of

B(G: P )

is always a Runge domain.)

[23].

are one-to-one.

(en)n~1

Our

of elements of

(Further properties of

In particular, the range of

MITTAG-LEFFLER METHODS

n~ 0, let 6n1 =

Fix

=

lI!

{z E

II Re z I !:

t

Riemann sphere

U {~}.

0:

Then

n

= {z

A = Al U A2 U 63•

and

n+ ;},

IRe z ~ n+l}, lI2

{z E 0:

125

n

S\F

n

z

E eiRe

Denote by

n

-n-l},

!:

the

S

is connected and locally connected at

infinity, and it follows from a deep theorem of Arakelian (see [2], or Theorem 1 on p. 11 of [27 J) that each function ~

analytic on

can be uniformly approximated on (fp)p~1

particular, there exists a sequence as

f (z) .... 0

p

(z) .... -2

P ....

00

uniformly on

2

((x,y) E re 2 Isup(l x l, For each

LEMMA 6.7. 2

Iyl)

!:

n

0

~

uniformly on

6,1

and

n'

n ~

~

-a, Re y

-a},

~}. We have the following lemma. there exists a sequence

(ep)p~l

of elements in

satisfying the following conditions:

Autl(C)

-1

(1)

en (U n+1 U Vn+l ) c Un+2 IJ Vn+2 '

(2)

ep (z)

PROOF:

In

of entire functions such that

p .... ~ uniformly on ~o Now for a E R, let p 2 2 u = {(z,y) E C IRe x ~ a. Re y ~ a} and V = {(x,y) E £ IRe x a a and for P > 0 denote by D~ the closed polydisc f

F and

F by entire functions.

{j,3, f (z) .... 2 n p

as

continuous on

f

-+

z

(fp)p~l

Let

a-pl ( z )

and

.... z

as

p ....

00

uniformly on Dn°

be the sequence of entire functions described above.

Taking away some terms of the sequence if necessary, we may assume that for p

~

1, Re fp(z)

Next we let

~

23

for

9p (x,y)

automorphism of

n+l, and

,2, and 6;1 is the map ~

Re(x+fp(Y»

and

n +

5 2'

Similarly we have

Po

Now there exists !Re (Y-fp(x)

that

~

p

-+

~

J(e) p

I ~ 21

~

1

such that

for each

uniformly for

=1

Re fp(z)

(x-fp(y-fp(x», y-fp(x».

=

If (x,y) E un+1 ' then

as

Re z

IRe

P ~ Po

zl

~

~

Then

for

9p

~

Re z

-n-l.

is an analytic

(x,y) .... (X+fp(Y), y+fp(x+fp(Y»). so that

Re(y+fp(Y»

-1

8p (Vn+l ) c Vn+2 •

IRe (x+fp(Y» I ~ and each

21

1

follows from a routine computation.

~

n +

25

Thus (1)

and

holds.

and

(x, y) ED. n

n +2' we see that

We now obtain the following theorem.

3 - 2

(2)

Since holds.

f

P

(z)

-+

0

The fact

126

ESTERLE F · There exists an ent i re one-co-one f unet10n

l'HEORE:-t 6.8.

J(F) :: L and such that F(G: 2 ) U1 = {(K,y) ~ C2 /Re x ";> 1, Re y ;;> I} that

,2

-+ ... ... 2

sue h

avoids the sets and

= ({x,y)

V1

,2 IRe x

E

< -1,

Re y < -I}. PROOF:

= Un

Wn

Let

=

d(C,H)

n::: 1. Also, if C, H E H(G: 2 .~ 2 ), set

for

UV n

1: 2-n inf (l, p (G-ll» n

n=1

= sup IG(z)-H(z) I (here we use the notation

p (G-H)

where

n

I (x,y) I

Izl~

(Ix I, Iy I».

= sup

Then

d

H(1E 2 ,re 2 ) ,

is a distance on

which 2

is complete with respect to this distance.

2

H«( ,re )

defines the topology of uniform convergence on compact se"s, and Next, let

E = {G ~ 0(re 2 ) iG(~2\W

1) c [2\W l }. Then El ~ 0, and each En is a closed n+ subset of 8([). Now let. G E En and set Gp = GOS p ' where (9 p )p:!1 is the sequence given by the lemma. Since 8p (z) -+ Z as p uniformly for Iz! ::: n, we have lim sup d(C,C p ) ::: 2- n . Also e;1(wn+1) C Wn+2 ' so that 2'

n

-)0

2

p-"'"

2

\1 p tIE ~.Jn+2) C II: \W n +1

and

n

Theorem 2.1 that

E

n:!l

£

Gp E En+1

0.

'"

n

Pick

F E

z

n

E.

n~l n 2

rE.

COROLLARY 6.9.

There exists an entire function

element

(x,y)

PROOF:

Denote by

G = HoF.

of

Wl

2

z E: q: , and such that

H the map

(::,y)

REMARK 6.10.

H«(2,(2) with

n

n::1 Dl Z 2

G: [2

case where

z E ,2.

If

(2

-+

is

such that for each

(u,v) E C(,2)

(x,y) E F(e 2 ).

2

0

then

Since

(F) .... l

=

0

F2

of el...'ments

2

n

(Flo F2)(C ) D2 = 0, etc •• 2 -1 i f and only if F2 (e ) n Fl (02) = 0, and

is the function

nonconstant function

n~

n

0, and then F2 with

n DZ

n

F ([ ) ~ 0, on~ could try to construct

02( at least i f the Jacobian of

Fl

F

(exp(x-l), exp(y-l», and set

is certainly unbounded if the boundary of interior of

J(F) s 1,

1, the corollary follows.

n Flo •••

o F2 ) (t )

Then

inf (!x!,ly!)::: 1

To construct the desired sequence

with 2

Fl «()

~

-+

for every

(u,v) = (exp(x-l), exp(y-l», where ini (Re x, Re y)

It follows then from

G([2).

J(G)(z) ~ 0

Then

~

p.

This prnves the tht!orem.

r'(z)

for each

for each

for each

one-to-one and

J(G)(z) ~ 0

...

F1

F1 (t 2 ) meets the never vanishes). In the

G g:l.ven in Corollary 6.9, then no such

call exist.

Indeed, i f

(GDH)(,2)

n 02

-

0, then

MITTAG-LEFFLER METHG)DS

2

(GOH)(a:)

~G2

c :~l

U g2' where

Ql = {(x,y) E

127

a: 2 Ilxl:!

2,

iyl ::

I}

and

{(x,y) E c2 11x I :: 1, Iy I ::: 2}. Since (GoH)(a: 2 ) is connected, then either (GoH) (a: 2 ) is contained in Ql or (GoH) (a: 2 ) is contained in 2 We have GoR = (f l ,f 2), where fl and H(G; ,IE). But f2 belong to =

and and

are not dense in

H is constant since

a:,

so that

and

£2

are constant,

G is locally one-to-one.

Some other aspects of the theory of Bieberbach functions can be found in [23], and a full discussion of this new approach to Michael's problem, with a comprehensive presentation of the theory of Bierberbach functions, will be given in a forthcoming joint paper by P. G. Dixon and the author.

REFERENCES 1.

G. R. Allan, Embedding the algebra of all formal power series in a Banach algebra, Proc. London Math. Soc. (3) 2S (1972), 129-340.

2.

N. Arakelian, Uniform approximation on closed sets by entire functions, lzv. Akad. Nauk. SSSR 28 (1964), 1187-1206 (Russian).

3.

R. Arens, Dense inverse limit rings,

,

~lichigan

:-tath. J. 5 (1958), 169-182.

...

4.

V. Arnold, Chapitres supplementaires a 1a theorie des equations differentielles ordinaires, Editions de Moscou, 1980.

5.

W. G. Bade, Recent results in the ideal theory of radical convolution algebras, these proceedings, 63-69.

6.

7.

L. Bieberbach, Beispiel zweier ganzen Functionen zweier komplexer Variablen, welche eine sch1ichte volumtreue Abbi1dung des Rn auf einem Teil seiner selbst vermitten, S. B. Preuss. Akad. Wis9 (1933), 476-479. S. Bochner and W. Martin, Several Complex Variables, Princeton University Press, 1948.

8.

F. F. Bonsall and J. Duncan, Complete Normed Algebras, Springer-Verlag, 1973.

9.

N. Bourbaki, Iopologie Generale, Chapitre II, Hermann, 1960.

10.

B. Chevreau and J. Esterle, Banach algebras methods in operator theory, Proceedings of the 7th Conference in Operator Theory, Timisoara (June, 1983), to appear.

11.

G. Choquet, Cours d' Analyse - Topologie, Masson, 1964.

12.

p. J. Cohen, Factorization in group algebras, Duke Math. J. 26 (1959), 199-206.

13.

J. B. Conway, Functions of One Complex Variable, 2nd Edition, SpringerVerlag, 1978.

14.

H. G. Dales, A discontinuous homomorphism from (1979), 647-734.

15.

H. G. Dales, Automatic continuity: (1978), 129-183.

16.

P. G. Dixon, private discussion.

C(X), Amer. J. Math. 101

a survey, Bull. London Math. Soc. 10

128

ESTERLE

17.

Y. OOl11ar, A solution of the translation invariant subspace problem for weighted LP on R,R+ or Z, Radic

a,

~

"",

I

s II 611sup I g(X\V) In.

we have

~ sup I g(XW)

I=

a < 1.

i. e., on a neighborhood of

M.

P in

In the context of function theory, this immediately implies the following

K c« with dense connected interior

fact for a compact

u: 6

on

U and vanishing on a non-void relatively open subset

on

K.

(One has only to choose a

assumes its maximum modulus over

v

20 I. K near

aK only in

v,

E C(K)

V of

analytic

aK vanishes

z ..... -1 ~-zO A = A(K) of course.

for which with

(For a proof from the more standard function theory viewpoint, see [34, p.l20].) Wermer's maximality theorem and (1) combine to yield a simple proof [l4,P.40] of Rado's theorem (that

6

E C(K)

and analytic on

U\{lCO)

is in

A(K). To further emphasize the notion of a peak set we note (2)

if

Indeed with

r

is a peak set then Iz.P =

{6

E A: 6(P)

= o}

AI P

and

is closed in 9

C( j.") •

peaking on

P we can observe

that for the quotient norm,

so

116 + kP11 ~

1I(6Ip)ll oo ;

for the reverse inequality, note that

vides a mul tipliC'.ative linear functional on

AlkP,

so

16(1/» I

ra

t0pology of c.)mp:lct convergent:e all A .....

f, f«(p)

induces the homeomorphism

aA)

endowed with the Gelfand topol'Jgy). !IlUY lJe

interpreted as a function

(m its "natural" l:,

Y 1n some domain of holomorphy

a: n ,

tuere exists a homeomorphism

is holomorphic for all

f

in

~

Y ~ U, such that in the diagram

A.

REMARKS:

1)

Formally, this seems to be a rather complicated notion.

In (2.2) we shall

transform this notion into a very simple but equivalent one. The price to be paid for this is the introduction of the involved notion of a Stein algebra. 2)

If definition

(2.0) is satisfied then it follows easi:y (e.g. from (2.2»

that all the components of (2.1).

-1

cp

: U -+ Y satisfy

The classical case occurs, of course, when

algebra.

-1-1

{9

)l'···'(cI>

)n E

Au.

A is a uniform Banach

We want to consider: Problem I.

Give necessary and sufficient conditions for

A to exhibit holomorphic structure at a point of

aA.

There is a related classical problem which seems to have lost some of its attraction.

Gleason introduced his notion of parts because it looked

likely at that time that the parts would be the most natural pieces of exhibiting holomorphic structure. case, too.

aA

We shall contribute two solutions to this

More precisely, we will examine:

148

KRAMM

Problem II:

Let

n

C

a A be a Gleason part. this time endowed

with the metric topolOgy of conditions that

A'.

Give necessary and sufficient

exhibit holomorphic structure at all points

An

\pErr. In order to make this meaningful, carry Definitivn 2.1 over to obvious way.

(It doesn't matter that

U

= n,

in the

need not be Gelfand open.)

TI

It Is possible to reformulate the results below for Problem II

RE~~RK.

a local version for

into

rr, but you will see that the version just posed will be

more convenient. (2.2).

The following theorem will be fundamental for the sequel.

THEOREM.

Let

~

B be a (uB)-algebra and let

E aB

be given.

The following conditions are equivalent: (i) (ii)

B has holomorphic structure at There is

Au

that

~

cp E aB;

open (A-convex) neighborhood

is

~

U c aB

of

Ij)

such

Stein algebra.

To the best of my knowledge, this theorem does not appear in the literature. So we shall give a full proof here.

But first we fix some notation and provide

some preparation. (2.3). For the notion of Stein analytic space we refer to

[5].

We'll use the

following nice function algebraic characterizat:ion of Stein spaces.

This is

the famous Igusa-Remmert-Forster theorem [4]:

An analytic space j

with

Ij)

x

(2.4) DEFINITION.

REMARK.

x

•

A is called a Stein algebra if there

such that

A

If you aLe given a Stein algebra

space associated with spectrum

(X,O)

-+ ~

is a homeomorphism.

A (uF)-algebra

exists a Stein space

is a Stein space iff the natural

: X ... C"'J(X), x

= f(x),

(f)

(X,O)

aA

A.

~

O(X)

A, then there must be some

How to find it? aA

Stein

Well, by (2.3) we know that the

must be the carrier space (up to a

obtain the "right" sheaf on

as topological algebras.

~.omeomorphism).

In order to

let us momentarily forget our situation.

We introduce a most natural and simple sheaf

Ax

for quite general function

HOLOMORPHIC STRUCTURE algebras (A,X). Consider the family subsets of X; this family is a presheaf on

X.

associated with this presheaf.

(In the case

A=

U

That's it.

149

running through all open Let Ax be the sheaf X .. CIA we set

Ax.) Now let's go back to the above.

structure sheaf for

A, triat is,

(CIA,A)

A (unique up to biholomorphisms). [3].

In [11] we show that

A is the

is the Stein space associated to

A different approach is given by Forster

It is more complicated, but includes also the non-reduced case, which

we exclude. (2.5).

We need the following permanence property of Stein algebras.

THEOREM.

Let

A be a Stein algebra and

Ao

C

A a closed suba1gebra such

that the adjoint spectral map aA ~ CIAo is proper. Then Ao is a Stein algebra, too. Its spectrum is obtained by identifying those points in A

Ao •

which cannot be separated by For a proof see [9, p. 202]. (2.6)

Proof of "(1) - (ii)" of Theorem (2.2) :

and

P, for which

w

D

wz

z E Zp'

and hence

= ywz = xz. The Pedersen ideal

where

0

~

x EP

and

compactly supported in

f

Po

is generated by elements of the form

is a non-negative, real-valued continuous function,

]0,-( [5, p. 175].

ization claim for all suchelements.

If

and compactly supported real functions elements

Yl""'Yn in

f(x),

P so that

0

Suppose we have proved the factor~

x E Po

then tbere are continuous

f 1 , ••• ,f n : ]o,~[ ~ R+

and positive

174

LAURSEN

(where

and

qlz

vanishing on some neighborhood of i

E Zp with qlz vanishing on a neighborhood of P such that zi = tiz (suitable t i ) and thus x ~ (lxiti)z. The generalized polar decompositi.on [5, 1.4.5] can then be used to complete the argument. So it remains to prove that for an element f(x) (with x and f as specified before) we can find a central element z for which

O. If also and r · m(k) + 1. Now (4) and V2m (k)

So we can assume

+1

j < k.

j - k, then necessarily (6) give

p

= 1, q. 0,

=

Then

= by (4). S1nce j ~ I, P < r, and therefore p - 1 < m(k). The required inequality now follows, either from the induction hypothesis and (3), if r - 1 < m(k) ,

or from one of cases (ii) and (iii), below.

(i1) Suppose 1 = j = k - 1. Then (1) implies p + q - r + 2. Therefore, p =: m(k-l) + 1 and q!: m(k-l) + 1 imply r!: 2m(k-1) - m(k) - 2. Note also that p> 0, since m(k-l) + m(k-l) < m(k). So

Vm(k-l) + P vm(k_l) + q

=

~

m(k)-l

vr-l v v p-l q-l

182

MCCLURE

•

v

\n(k) -1

v

r-1 Vr

r

v p _1 v q _l

\n(k)-l (Ar_1) -1 (Ar)-l uSing (5), the induction hypothesis and the fact that Since r < m(k) , holds. (iii)

Suppose

i

j


Am(k)-l (Ar )

(9) implies

O.

~

1,

p - 1 + q - 1 • r < m(k).

and the required inequality

Now m(k-l) + p + m(j) +

k - 1.

a

-1

m(j) + q + P

= m(k-l) +

+ q + P < 2m(k-l) + 1,

2 + r.

q =

Also,

and therefore

m(k) + r m(j) + q
r}.

Berndtsson refers to Ovrelid [16] for

functions on

~.

C1

dependence of the various

However, Ovrelid refers to Hormander and Bungart.

There

190

O'FARRELL, PRESKENIS AND WALSH

are (at least) three published proofs of the desired facts (solubility of Cousin and related problems with smooth dependence on a parameter) - by Bishop [3], Bungart [4], and Weinstock [21]. is the most elementary. product theory. (3)

Of the three, Bishop's method

The others use the powerful Grothendieck tensor

Berndtsson's function

H has Weinstock's "omitted sector property", Le.

for each ~ there exists 6 such that H~ .z) takes no value in the sector {w E I: : 0 < I wi < 6. I Imwl + 6 Rew < O}. (He also needs this fact, to establish the relation O

be a given integer.

elements with right zero multiplication. ~X

EXAMPLE 2.7. ax

2c

have that

a E S

X be a set with

That is,

ab

=b

so

n

for all a,b E X.

a

Let

=x

Let

is

Z.

X, and each element of X is a right zero if multiplicative left invariant means.

multiplication. that

there is an

This obviously is not satisfied by

Let

Then clearly has exactly n

x,y E S

S

X.

So

X

N be the set of positive integers with right zero Thus

ab

for all

=

b

for all

a,b E N.

x E ~N.

a E N and

Then it is easily checked Since I~NI = 2c (see [18]) we

is a possible value of the cardinality set of all multiplicative

left invariant means on a semigroup. We remark that if

n > 1

even right amenable though

in Example 2.6 then that semigroup

X is extremely left amenable.

X is not

So it is interesting

that this cannot happen if the multiplicative left invariant mean is unique, as the following theorem shows: THEOREM 2.8.

Let

S be a semigroup.

left invariant mean. PROOF.

Let

bead) - (ba)d

a E

a · ad.

zero of

~S.

S

be a right zero of

~S

S have a unique multiplicative

is also extremely right amenable.

by Lemma 2.2, so

right zero of that

Then

Let

=

bead)

Let

~S.

(ba)d

=

b, dES.

(ad).

Thus

Then

ad

is also a

The uniqueness of multiplicative left invariant mean implies

~S.

Since Then

dES ea

is arbitrary it follows that

a

is also a left

is also a multiplicative right invariant mean

S.

o~

Thus we have proved the theorem. The above theorem does not imply that if

S has a unique multiplicative

left invariant mean then it also has a unique multiplicative right invariant mean.

It would be interesting to know whether a semigroup is uniquely

extremely left amenable if and only if it is uniquely extremely right amenable. We can settle this problem in the affirmative in one particular case.

But

then we have to consider continuous extensions of the semigroup operation of S

~S

to

and their relation to extreme amenability.

THEOREM 2.9. on

~S

Let

S be a semigroup.

G>t

so that the following holds:

(a)

~S

(b)

S

is a semigroup under is a subsemigroup of

given operation on (c)

Then there is a binary operation

x

at

y

at . ~S

at

under

and

G>t

agrees with the

S.

is continuous in

y

~

~S

for any fixed

x

of

~S.

EXTREMELY LEFT AMENABLE SUBGROUPS

er

Similarly, an extension defined in

~S

PROOF.

xES

r

continuous extension of

r

~

of the semigroup operation on

S

can be

which is continuous in the left variable only. let

x

For

203

be the map

x

~S.

to x xES.

y

~

If

yx

a E

on ~S

S, let

and let

r

x be the map

t

be the

a r (a) - ax for all Then t is a continuous map from S into x a and hence has a unique continuous extension ta to ~S. Finally if

'V

'V

~S

a, b E

a 0 b

that

Let

ba

be a net in

bc~ = lat(bac~).

5!

... l~t[(lat(aba»C~]

that

~t

b

=

0t

~

converging to

c~

Now

c~ E S

=

~S

50

lt ~

and

Gt

ba

a net in bc~.

50

~

50

~t(bac~)]

b)c~]

b.

5

~S

ta

~S.

S

l~t[~t(abac~)]

=

Gt

It is clear

•

So we get the theorem. It is not true

is a semi group then the multiplication in

to a semigroup operation in

'"

from definition of

is a semigroup under

a, b E 5.

for all

0t

=

a,b.c E

Let c~

and

E S,

l~t[a

l~t[(a

~S.

= tb(c)

(b (!)t c)

(b 0 t c) =

c.

a E

for all

in the theory of numbers is given in [4].

in general that if separately.

b

et(bc~).

0t

ab

~

A use of

a

Then it is clear from the definition of

Now

since

= (a Gt b) a

S

~5.

in

c

St (b 0 t c) = lt a

a t b

'V

= ta(b).

G)t b

is continuous in

converging to a

a

put

~5,

S

extends

which is continuous in each variable

Needless to say, the multiplication need not necessarily extend to

a jointly continuous multiplication in multiplications from

~S

to

5

~S.

Continuous extensions of

is a very fascinating and difficult subject,

and some partial results have been obtained by H. Mankowitz [15], T. Macri [16], R.P. Hunter and L. W. Anderson [1], Aravamudan [2], and others.

The following

gives us an interesting class of semigroups: DEFINITION 2.10. in

5

A semigroup

S

is called R-semigroup if the multiplication

extends jointly continuously to a semigroup operation on

semigroup

S

is called a V-semigroup if the multiplication in

semigroup operation on

~S

~5.

S

The

extends to a

which is continuous in each variable separately.

Now we are ready to improve our Theorem 2.9 for the class of V-semigroups. THEOREM 2.11.

Let

S be a semigroup.

Then the following are equivalent:

(i)

S

has a unique multiplicative left invariant mean.

(ii)

The collection of all left thick subsets of

S

in

S.

(iii)

~S

has a unique right zero under the operation

(iv)

Given

f E m(S)

is an ultrafilter

or .

there exists a unique constant function in

is the weak * -closure of the set

{r f I a E S}. a A similar theorem holds if left is interchanged with right throughout. where

kef)

kef),

204

RAJAGOPALAN AND RAMAKRISHNAN

PROOF.

The equivalence (i) - (ii)

follows from Theorem 1.12.

The

equivalence of (ii) and (iii) follows from Theorem 1.12 and Lemmas 1.7 and 1.10.

We now show (iv)

right zero of in

~S,

in

then

r

a

f

~

(iii).

and a r a converges in weak a.e. on To every g2 E A(B), g2 ~ 0, corresponds then an h E H(B) such that (a) Re h ~ P[q>] in B, (b) Re h * ... q> a.e. on S, (1)

(c)

g2

divides

h;

i.e.,

h E g2H(B).

Conclusions (a) and (b) show that function in

P[q>] - Re h

B whose radial limits are

integral of a positive measure

S.

on

~

0

S

a.e.

[cr];

is a nonnegative harmonic it is thus the Poisson

that is singular with respect to

cr.

It is in this form (involving measures) that (a) and (p) are stated in [1], as well as in [7]. The next theorem is not stated in [1] but, as we shall see, it is an immediate corollary of Theorem 2, and it contains many interesting special cases that do occur in [1]; see also [3]. Smirnov class

N*(B).

the functions

10g+lf

every

Recall that r

I,

N*(B)

0 < r < 1,

e > 0 should correspond a

The proper setting for it seems to be the consists of all

YeS

with

cr(Y) < 0

for which

form a uniformly integrable family: 0 > 0

such that

fylog+lf(r~) Idcr(~) for every

f E H(B)

< e

and for every

r E (0,1).

to

NEVANLINNA'S INTERPOLATION THEOREM THEOREM 3.

219

Assume that

(i)

f

E N* (B), f

(ii)

t

~ If * I

'=

0;

s, t/ If * I

a.e. on

agrees a.e. with some lower semicont-

inuous function, and

f slog 'lr da < (iii)

00;

gl E A(B), gl ,0,

there exists

such that

Ig~ I ~ lo~ a.e. on To every (a) (b)

F 0,

g2 E A(B) , g2

S.

If I corresponds then a function

F such that

F E N*(B) , \F

* I = 'it

on S , have the same zeros 'in

a •e •

(c)

F and

f

(d)

F E f + g2·H(B).

Note that (d) implies that

F (z)

;;0

f Cz}

B,

wherever

g2 (z)

matches (interpolates) f on the zero-variety of g2' If 0 < p ~ ~ and the data f and t are in HP(B) pectively, then

F

is also in

I'l

and

Thus

O.

LP(a),

F

res-

HP(B).

To prove Theorem 3, apply Theorem 2 to ep

Note that

ep E L1 (a)

since

= log

--\- • * If \ fsloglf Ida> _00 [6, p.8S].

Let

h

be given by

Theorem 2, and put h F = fe •

The assumption

f E N*(B)

implies 10glfl ~ p[logff * I].

Hence 10giFI ~ p[loglf * I] + prep] so that

10g+IFI

= P[log t]

is dominated by the Poisson integral of

+

1

log 'it E L (a).

This

gives the required uniform integrability, and proves (a). (b) follows from Theorem 2(b); (c) is obvious, and (d) holds because g2 divides hand h divides eh - 1, hence F - f. Q,E.D. Here are some special cases of interest:

1 (1) Take 'it = 1, choose f E H00 (B), f ~ 0, If\ < 2' so that If * I agrees a.e. with some upper semicontinuous function, Theorem 3 furnishes inner functions F in B with the same zeros as f.

(2)

= zn'

Take

f E HOO (Bn _1 ), If I ~ 1,

Conclusion:

'" - 1.

There is an inner function F(zl'····zn_l'O)

Apply Theorem 3 with F

in

Bn

= f(zl,···,zn_l)·

so that

81 • 82

220

RUDIN

To see this, one has to verify (iii), but this is an easy consequence of the Schwarz lemma. (3)

V

Take

= 1.

f

1s then seen to be

Every bounded, lower semicontinuous, strictly positive

IF * I

F E H~ (8).

a.e. for some

This last application shows, incidentally, that the lower semicontinuity hypothesis cannot be dropped from Theorems 2 and 3 when

n > 1,

has then what I have called the LSC property; see [7]. ~

examples, that

"g2 E A"

cannot be replaced by

do not know whether the same is true of ~

keep

0

sufficiently far away from

gl.

when

mai~ pu~pose

,/(f I

and to keep

H (B)

One can also show, by m

"g2 E H"

The

~

because n > 1,

of

gl

away from

but I

1s to

1.

The question arises now whether these various continuity assumptions can n = 1.

be dropped when

The answer is affirmative, and the proof turns out to

be a surprisingly simple application of Theorem 1. Here is Aleksandrov's theorem: Assume that

THEOREM 4.

~ E Ll(T), ~ ~ 0,

(i)

m

g E H (U), g

To every

~

(a)

Re h

(b)

Re h *

(c)

h E g.H(U}.

PROOF:

-....

~(eie)de >

I1T log -1T

(ii)

~

P[q']

=

~

corresEonds "then an

0,

in

U,

a.e.

on

There is an F E HeU)

T,

with

in

U,

fEW, f

~

1.

= P[~].

Re F F

for some

such that

h E HeU)

Since

Re F > 0,

l+f

= 1-f

Consequently,

so that 10g(1 - If * I2 )

T.

=

log ~ + 2 10gll - f * I ~

E. Let B be the Blaschke product with the same zeros as g. By Theorem 1 there is an inner" function u in U such that B divides f - u. Thus g divides f - u. Put

a.e. on

Our first proposition shows therefore that

h = l+f _

l-f Then

f - u

divides

h.

This proves (c). Re h

Since

u

is inner,

f

.!±!!. . l-u

Also,

= P[~] _ Rel+U •

Re{(l+u)/(l-u)}

1-u

is positive in

U and has boundary values

NEVANLINNA'S INTERPOLATION THEOREM

O' a.e.

on

T.

221

Q.E.D.

This gives (a) and (b).

We now come to the announced stronger form of Theorem 1: THEOREM 5.

Assume that f E N.(U), f

(i)

If• I

t ~

t

0,

a.e. ~ T, n ~ i9 I_nlog 10&lf.l(e )de >

(ii) (iii)

g E Hm(U), g ~ 0,

To every

_me

corresponds then a function

(a)

F E N.(U) ,

(b) (c)

IF I = t a.e. on T, F and f have the same zeros in

(d)

F

F

such that

• E

U,

f + g·HCU).

This follows from Theorem 4 in precisely the way in which Theorem 3 was proved from Theorem 2. When t

=

1

and

g

is a Blaschke product, this is Theorem 1, but with

(c) as an added conclusion. 1 has no zeros in

U,

In particular, if the given function

in Theorem

then the interpolation can be done by a zero-free (i.e.,

singular) inner function. One final remark: The desired function outer function whose absolute value is formed with the zeros of satisfy (a), (b), (c).

f

f,

F must be the product of (1) the

'it on T,

(2) the Blaschke product

and (3) some singular inner function, in order to

The point of Theorem 5 is simply that the singular inner

factor can be so chosen that the interpolation property (d) holds as well. REFERENCES 1.

A.B. Aleksandrov, Existence of inner functions in the unit ball, Mat. Sb. 118 (160), N2(6) (1982), 147-163.

2.

John B. Garnett, Bounded Analytic Functions, Academic Press, 1981.

3.

Monique Hakim and Nessim Sibony, Va leurs au bord des modules de fonctions holomorphes, Math. Ann. 264 (1983), 197-210.

4.

Kenneth Hoffman, Banach Spaces of Analytic Functions, Prentice Hall, 1962.

5.

Karel de Leeuw and Walter Rudin, Extreme points and extremum problems in HI, Pacific J. Math. 8 (1958), 467-485. Walter Rudin, Function Theory in the Unit Ball of ,n , Springer Verlag, 1980. Walter Rudin, Inner functions in the unit ball of Cn , J. Functional Analysis 50 (1983), 100-126.

6. 7.

DEPARTMENT OF MATHEMATICS UNIVERSITY OF WISCONSIN-MADISON MADISON, WISCONSIN 53706

Contemporary Mathematics Volume 32, 1984

UNBOUNDED DERIVATIONS IN C*-ALGEBRAS AND STATISTICAL MECHANICS (KMS states, bounded perturbations and phase transition)

1.

INTRODUCTION There is a good possibility that the theory of quantum lattice systems

in statistical mechanics may be well-developed within the theory of unbounded derivations in C*-algebras.

In fact, many theorems in the theory of quantum

lattice systems have been formulated for

l~ormal

hyperfinite C*-algebras (called UHF algebras).

*-derivations in uniformly One of the most ambitious pro-

grams in the theory of unbounded derivations is to develop statistical mechanics within the C*-frame work.

Especially the abstraction and generaliza-

tion of the phase transition theory in classical lattice systems to the C*theory, including quantum lattice systems is one of the most important subjects. This program is not so easy, because the phase transition has not been established even for the three-dimensional Heisenberg ferromagnet with nearest interaction (for the anti-ferrogmagnet it has been proved by Dyson. Lieb and it Simon). In this paper, as a step to bring the phase transition into the C theory, we shall study KMS states in detail, and as an application, we shall show the absence theorem of phase transition in lattice systems with bounded surface energy in the most general form.

This was previously done for normal

it-derivations in UHF algebras ([2], [7], [11]). eliminate the assumption of UHF algebras.

In this paper, we shall

This becomes possible, because the

set of all KMS states obtained after bounded perturbations with bound less than a fixed number is relatively weakly compact in the set of all normal states defined by a starting KMS state (Theorem 2.2, (8».

Because of the restriction

of space, most of the theorems will be stated without proof.

More details on

the matters discussed here will appear in my forthcoming book [12]. it C -DYNAMICAL SYSTEMS AND KMS STATES it Let A be a C -algebra with identity and t -+ at be a strongly continuous one-parameter group of *-autoNorphisms on A. The system {A,a} is called

2.

a C*-dynamics. generator of Let

0(5)

Let a;

a

then

t

= exp to (t E R), where 0 is the infinitesimal 5

is a well-behaved closed *-derivation in

be the domain of

5;

then

V(5)

is a dense

A.

it-subalgebra

© 1984 American Mathematical Society 0271·4132/84 $1.00 + $.25 per page

223

224

SAKAI

of

A and

~i)

e(ab)

e

satisfies the following properties:

= e(a)b +

ae(b)

(a,b E

Vee»~;

* e(a)

(ii)

= e(a) *

(a E

Vee»~.

GO

An element

a

n V(6 n )

in

n=l

is said to be analytic if there is a positive

'"'

such that

~ (a E A). Let n=O A(6) be the set of all analytic elements in A with respect to e. a(EA(e» : /Ie n (a)/1 n is said to be entire analytic if u , r < + '"' for all positive numbers

number

r

n=O

r.

n.

The set of all entire analytic elements with respect to

Al (c).

a(EAl(e»

A with

a *-subalgebra of

A2 (e). A and it is dense in A.

2.1.

Lo:!t

13(a t (b)a) evide~ce

is

if

{A,a} at inverse temperature (resp. 0

which is analytic on

A2 (e)

\3, a state

For a real number

a,b E A, there is a bounded continuous function

Sp = {z E tlO ~ Im(z) ~ 13}

Ma

The set of all geometric elements in

a

r~spect

is denoted by

is said to be 6eometric if there is a positive number

/Ion(a)1I ~ Mnllall (n=1,2, ••• ).

such that

e

for

t E R.

of being the abstract formulation

of the condition for equilibrium of states (cf. [3]). Let

4>

be a KMS state for {A,a} at

\3;

then

4>

is invariant under

a - i.e., 4>(a t (a» = 4>(a) (t E R, a E A). Let {TT! be ti.le GNS representation of A constructed via 4>. Put uq,(t)a

(a t (a*a» = q,(a*a) = la4>"' U4>(t) can be uniquely extended to a unitary operator on H

exp-t(Hcp+rrcp(k»exp tHcp = ~ (-l)P!rrcp(a is (k»rrcp(a iS (k» ••• p-O 0;;asl=s2=' < 1 •• :is p:it 2

Hence

(t E R), where

••• rr (a. (k»ds 1 ds 2 ••• ds E rr~(A) cp 15 p P ~ of ( • ).

(

)

is the closure

00

lIexp-t(H~+rr~(k»exp tH~1I

Moreover

~

~

~

~

~

p=O

flla t

51

(k)lIl1a. (k)1I 15 2

O;;a51;;a52;;a···;;a5p~t

••• lIa.1S (k) IIdslds2' •• ds P • p

Suppose that lIon(k)1I ~ Mnllkll (n=0,1,2, ••• ); then on(k) (is.) n lIa 1S .(k)1I = lI(exp is j o)(k)1I = II ~O nl J 1/ J n n n00

00

::::

~

M

n=O

Is I ,j

n.

Hls.1

IIkll = e

J

Ilk II

(j=I,2, ... p).

Hence

= i[H~+rrcp(k),rr~(a)] = rr~«O+6ik)(a» and so 1T~ ( exp

t ( o+Oik )() a)

in the strong operator top01cgy of

__ eit(H~+rr~(k»1T~(a) e-it(H~+rr~(k». ~

~

~

~

~

B(Hcp)'

226

SAKAI ~ ~

Let

k

(x)

=

(H +rr (k» tk ck

~Htk

2

2

(rr (x)e

e

tjJ

k

~(H +IT' (k»

_

1

ck

e

cp'

k

E A). Then cp (x)/cp (1) is a KMS state for k at (3. In fact, for a,b t: A2 (O+6 ik ), cp (a exp (x

-p (llcp +11 cp (k» ( 1f () a e

rr

tjJ

(b)

e

cp

e 2

~H

( e 2

e

2

[3H

:J. 2

= (e ~

~(H

-

ck

2

e

~H

cp

e

'jH

:J 2

e

e

2

1

e

2

cp ) _ ?(H~+11!l!(k»

pHck

~Hcp

-

_ PH~

2

Tlcp(b) e

e

2

2

e

p(H!l!+rrcke K»)

~Hp' 2

e

2

e

TTcp(a).

PHck

2

e 2

1

cp'

1) ;p

= elf cp (b)rr cp (a)

e

(use that

is a KMS state for

tjJ

=

~

2

e

p{H2+11 ~ (k»

e

i~(o+Oik)(b»

lq"lcp)

+11 (k» !l!

ellck +n !l! (k»

e

PH!l! e

cp

cp'

2

rrcp(a)e

r~ (Hcp +rr ~)

rr (b)e

1

P (H +rr (k» !l! P.

2

{A,exp t(o+Oik) (t E R)}

[:' (Hck+rr ck (k»

pliq,

J _ P(Hck+1T~(k»

=

tcp)

I

2

e

2

e

i3 (Hcp +rr cp (k) ) •

~) (Irq, +rrip (k) )

(

~Htk

ck·

2

e

Now we shall show that

{A, exp to}

for

cp

00

cp

~).

at

eiz(Hcp +rr cp (k» e -izH E rr (A)

'" e' )p e iz(H cp +rr cp ek» e -izHcp =11 cf> ( u 1Z

1 )

z E t.

In fact,

f

p=o

Hence e(e

e

Mizi

)

izeH +IT (k»

cp

cp

e

II k lllzl.

For

-izH

cp E rr (A)

cp

and

III. e iz(Hcp+rr cp (k» e -izHcp " ~

b E A 2 (o),

P(Hck +Tr q, (k» 1 )

cp

_

e

~eH

+11 (k»

CP!P 2

=

(e

2

~ (H

+rr (k»

ck ck 2

rr (b)e

cp

~(H

d>

cp

_ ~ (Hck +rr ck (k) )

+rr (k»

2

1 ,

e

cb

lcp' e

2

lq)

UNBOUNDED DERIVATIONS ~ (H

(a)

of KMS states).

+rr (k»

cp

cp

lcp' lcp) (a E A).

+rr (k»

cp cp

lcp' lcp)

-~ (Hcp +Tf cp (k) ) ~Hcp

(rrcp(a)e

e -~(H

=

(Tlcp (a)e

~H

+rr (k»

cp

cp

e

-~(H +rr (k»

where

cp

e

2.2. THEOREM. for

141 , lcp> CPlq,' 141 )

(a E A)

~H

41

e

41 E rrq,(A).

Let

cp

be a KMS state for

~ ~ 0 (resp.

Then we can show the following theorem.

~ ~ Im(z) ~ 0

adjoint portion of the weak closure (z,h) ~ f(z,h)

a mapping

~ < 0)

for

M of

S~ x MS

of

= {ziO

at~, s~

{A,a}

and let

rrq,(a)

in

~

Im(z)

~ ~}

M b~ _~I.!..~self­ S

Then there is

Hq,

into the predual

M.

of

M satisfy-

ing the following conditions: for (2) s~

x E j'.I, hEMs, f (z, h) (x)

For

If a directed set

number) converges to

h

{f(z,ha )} converges to compact subset of S~. (4) f(i~,h)

For

hEMs,

{ha.}

f(z,h)

f(O,h)

IIhali ~ M

with

f(i~,h){eit(Hcp+h)e-itHq, x)

in the norm of

=~

,where

p03i~ive

is a faithful normal

(t E R)

M.

¢(x)

h E

tl,

and

=

(xlcp,lcp)(x EM),

f(t + i~,h)(x) f(t,h){x)

2

)

and

lq,' e

2

1 41 )

M,

and

and

=

= f(O,h)(xeit(Hcp+h)e-itllcp)

f(i~,h)

~(Hp +h)

~(Hp+h)

e

2

lcp E Vee

B(Hcp)' tilen

uniformly on every

~(Hp+h)

For

(M, Uixed

linear functional on

and moreover,

t E R.

= (x

s~.

in the strong operator topology of

eit(Hcp+h)e-itHcp E M (t E R)

(5)

of s M

in

s~.

is a bounded continuous func tion on

s~

and is analytic in the interior (3)

for

z E

(x EM).

(x)

..

228

SAKAI Im(z)(H!/? +h)

f(z,h)(lH)

=

( e iRe(z) e

2

1) 4>

cJ>

for

E S~.

7.

And, if {h} converges strongly to a Im(z)(H +h ) -

cp

M, then r.,1( z) (Hp+h)

y.

2

{e

h with

2

converges to {e

Im(z) - ~)

in the norm of Let 0 be a bounded *-derivation on A and let TI (0 (a» = -- 0 4> 4> i[h,n (a)] (a E A) with hEMs (cf. [9]); then tea) = 4> -f(lp,h)(nep(a»/f(ip,h)(lH ) (a E A) is a KMS state for {A,exp t(o+OO) (6)

0, let

ry

relatively a(M:,:,M)-compact in

ry

closure of

in

= {f(ip,h) I M*.

(i)

and is analytic

~

IFt,;(Z)(X) I F~(O)(xx*)

k 2

for each

F~(Z)

elPIYllxll,

IF~(t+iP)(x) I

(x EM); (ii)

and

l~tF~(t+iP)(X) I ~

and

~(x)

= F~(O)(x)

F~

Sp

be the

is

a(M*,,\I)-

is a faithful normal ~

E r , there is a bounded Y

satisfying the following Sp

Sp' and

F~(iP)(x*X)~

(iii)

and

F~(t)(X)

x E M,

yelP IYllxlla.e.;

IF~(t)(x) I ~ F~(t+iP)(x)

and

d

IdtF~(t)(x) I ~ yllxll

~(x)

=

F~(iP)(x)

a.e.,

(x E M),

(x EM).

and

a(M*,M)-closure of on

ry

1'y

is bounded continuous on

t E R, and

>

0

S2

let

f(0,h)(h2)~ ~ Y

a(M*,M)-relatively compact subset in (the

of

:::

for each

More generally, for y

f(ip,h)(h2)~ ~ Y

Sp

F~(z)(x)

x E M,

are differentiable for almost all

(9)

on

o the interior Sp

1n

ry

in -

~

M and for each

M*-yalued continuous function ~F~erties:

Moreover, let

M*; then each

positive linear functional on

Ilhl! ~ y, hEMs}; then

M*.

y

= {f(ip,h)

hEMs}; then

f(iP ,h) (LH S2

Y

) ~ y,

ep is again a

Furthermore, for each

~

-

E S2 y

S2y in M*) there is a bounded continuous function satisfying the same properties occurring in (8).

UNBOUNDED DERIVATIONS REMARK.

229

The assertions (1), (2), (3), (5) and (6) were proved by H. Araki

[1] in slightly different forms. The assertions, (8) and (9) are new. These assertions are the key lemmas to show a generalized absence theorem of phase transition. In mathematical physics, it is important to study the strong convergence of the one-parameter groups of *-automorphisms. 2.3. DEFINITION. Let an t ~ an,t (n=1,2, ••• ) and a : t ~ at be a family of strongly continuous one-parameter groups of *-automorphisms on a C*-algebra

A.

a

is said to be a strong limit of

{a} n

= strong

(denoted by

lim a or at = strong lim a ) if lI a n ,t(a) - a t (a)1I ~ 0 n n n n,t uniformly on every compact subset of R for each fixed a E A. (By using

a

the Baire's category theorem, one can easily see that (si.mple convergence) for every !la.n,t(a) - at(a)/I ~ 2.4. PROPOSITI0N.

110.n, tea)

- at(a)1I ~ 0

a E A implies the uniform convergence

0 on every compact subset of

R.)

at"" exp to and at = exp to ; then n, -1 n-=l at = strong- lim a n,t-iff (1-0) ~ (1-0) strongly in B(A), where n -is the algebra of all bounded operators on A. PROOF.

Let

By the Kato-Trotter theorem ([13]) in semi-group theory,

(1-0 )-1 ~ (1-0)-1 (strongly) is equivalent to

110.n, tea) - a t (a)1I ~

n

t ~ O.

for

For

t
r.:t... ,n (a n, t(b)a)dt, for 0 0 * Z E S~, where S~ is the interior of S~. Let AO be a C -subalgebra of A generated by {p(t)(b)}; then AO is separable; hence there is a subsequence

{nj } of

ct>~,nj

{n} such that

(aat(b»

~ ct>~(aat(b»

(t E R).

~ lIaanj,t(b) - aat(b)1I + Ict>~,nj(aat(b» - ct>~(aat(b»1 ~ O(n j ~ 00),

and

t(b» I ~ ilallllbil

Ict>r.:t n (aan

and

Ict>j:l n (an

j'

""j

""j

t(b)a) I

:iii

lIalillbll •

j'

Hence by the dominated convergence theorem, there is a bounded continuous function

on the strip which is analytic in the interior of the strip

F

a,b

such that

lim y-+o

3.

q.e.u.

PHASE TRANS IT ION Let us begin with the definition of phase transition.

3.1. DEFINITION. Suppose that Then

{A,a}

{A,a.}

110

be a C*-dynamics and let

{A,a}

be a real number.

~

has at least one KMS state at the inverse temperature

is said to have phase transition at

KMS states at to have

Let

{A,a.} If phase transition at

~,

then it is said

~.

If a-general normal *-derivation then by Proposition 2.11, 6

if it has at least two

~

has only one KMS state at

~.

~.

6

has bounded surface energy,

is a pregenerator and

exp tB = strong lim exp tO ih • n

Now we shall show the following theorem. 3.2. THEOREM.

Suppose that

n

has a unique tracial state

A has a unique tracial state

(consequently, *-derivation

A (n=1,2, ••• )

0

"t).

"tn

If a general normal

in A has the bounded surface energy, then the C*-dynamics

{A, exp tB (t E R)}

has a unique KMS state at

(namely it has no phase transition at

~

~

for each real number

for each real number

~).

~

SAKAI

234 PROOF.

Since

exp t6

by Proposition 2.12

strong lim exp tO ih

=

it has a KMS state forneach

{A, exp to (t E R)}

factorial KMS states for {n~ ,u~ ,H~}

A has a tracial state,

and

~.

at

~

be the covariant representation of

Let ~1'~2 , and let

be two

{A, exp to (t E R)}

con-

III

structed via for

a

A •

~

n

~l·

(exp to )(a) n has a unique tracial state,

On = 5 + 0i(k -h ); then

Let

n

Since f (i~, n

A

(k -h »(n,j, (a» n

't(ae

'f'1

n

.. f(i~,TT

Since

4>1

'tee

(k -h »(lH ) n n ~ 1

cr(M*,M)-compact in

M*(M

= TT~

n) (a E A ) • n

-'--~~k:--­

n)

IIn~l (kn-h n )II ~ 0(1), by Theorem 2.2 (8),

relatively

to' k )(a) ~ n

-~k

n ~1

= (exp

n

{f(~,n,j,(k

'f'

n

-h »} n

is.

(A)"), so that by Eberlein's 1

theorem there is a subsequence

{f(iA n

h»} of {f(i~,n,j, (k -h »} n.- n. 'f'l n n J J which converges to a normal faithful state t in cr(M*,M). Hence 1'"

4>1

(k

t(l14> (a» 1

(a

Quite similarly, we start with 11

~2

(A)"

~2;

E A).

then there is a normal state

S

on

such that

(a

E A).

)

E;.(1H

q,2 t(TT~

Hence

(a» 1

E;.(nq, (a»

2

--'~--=----

t(lH) q, 1

(a E A), and so

is quasi-equivalent

E;.(lH)

q,2

q.e.d.

UNBOUNDED DERIVATIONS

235

REFERENCES 1.

H. Araki, Relative Hamiltonian for faithful normal states, pub1. RMS, Kyoto Univ. Vol. 9 (1973), 165-209.

2.

, On the uniqueness of KMS states of one-dimensional quantum lattice system, Corom. Math. Phys. 44 (1975), 1-7.

3.

R. Haag, N. Hugenholtz and M. Winnink, On the equilibrium states in quantum statistical mechanics, Comm. Math. Phys. 5 (1967), 215-236.

4.

E. Hille, Analytic Function Theory, Vols. I, II. 1959, 1962.

5.

E. Hille and R. Phillips, Functional Analysis and Semi-groups, Amer. Math. Soc. Colloquium publ. Vol. 31, Providence, 1957.

6.

A. Kishimoto, Dissipations and derivations, Comm. Math. Phys. 47 (1976), 25-32.

7.

, On uniqueness of KMS states of one-dimensional quantum lattice systems. Comm. Math. Phys. 47 (1976), 167-170.

8.

R. T. Powers and S. Sakai, Existence of ground states and KMS states for approximately inner dynamics, Comm. Math. Phys. 39 (1975), 273-288.

9.

S. Sakai, C*-algebras and W*-algebras, Springer-Verlag, New York, 1971.

Ginn & Company, Boston.

10.

, On one-parameter subgroups of *-automorphisms on operator algebras and the corresponding unbounded derivations, Amer. J. Math. 98 (1976), 427-440.

11.

, On co~autative normal *-derivations II, J. Functional Analysis 21 (1976), 203-208.

12.

, Operator algebras in dynamical systems, to appear in the series of Encyclopedia of Mathematics.

13.

T. Kato. Perturbation Theory for Linear Operators, Springer-Verlag, New York, 1966.

14.

P. J~rgensen, Trace states and KMS states for approximately inner dynamical one-parameter groups of *-automorphisms, Corom. Math. Phys. 53 (1977), 135-142.

DEPARTMENT OF MATHE¥ATICS FACULTY OF HUMANITIES p~ SCIENCES NIHON UNIVERSITY TOKYO, JAPAN

Contemporary Mathematics Volume 32, 1984

REMOTELY ALMOST PERIODIC FUNCTIONS Donald Sarason This paper concerns a generalization of the notion of almost periodicity which, to my knowledge, has not appeared previously in the literature.

I call

functions which are almost periodic in this generalized sense remotely almost periodic functions.

The term "asymptotically almost periodic" would perhaps be

preferable had it not already been used by M. Frechet [4] to refer to a related but rather more restricted generalization of almost periodicity. almost periodic functions form a closed subalgebra,

RAP.

of

of bounded, uniformly continuous, complex valued functions on The main result to be established here is that algebra, by

AP,

RAP

The remotely

BUC, R,

the algebra the real line.

is generated, as a Banach

the algebra of Bohr almost periodic functions, and another

algebra, called

SO, consisting of functions which oscillate slowly at

One can define

RAP

by slightly modifying the definition of

~.

AP.

The

discussion here will be limited to the real line, although it will be clear that a similar development is possible in a more general context. number, we let T t

t+a.

T

If

f

whose value at

t

a

BUC.

=

stand for the transformation on

a

is a function on is

f(T t). a

R,

then

* T f a

a

is a real

R of translation by

a:

will stand for the function

The functions we shall deal with all belong to

We shall measure the distance between two functions

by means of the supremum norm:

If

dist(f,g)

dist~(f,g)

= lim

=

IIf-gli. ~

f

and

g

in

BUC

We also define

suplf(t)-g(t)I.

It I""""

e is a positive number, the real number a is called an e-translation number of the function f provided dist(f,T *f) ~ e. We shall call a a a * remote e-translation number of f provided dist (f,Ta f) ~ e. The function f If

~

e > 0, its e-translation numbers form a relatively dense set. (A subset of R is said to be relatively dense if there is a bounded interval each of whose translates contains at least belongs to

AP

if it is in

one point of the set.) for every Like

e > 0, AP,

BUC

and, for every

We shall say that

f

is in

RAP

if it is in

BUC

and,

its remote e-translation numbers form a relatively dense set. the class

RAP

is a closed suba1gebra of

BUC.

The proof of

this statement is for the most part immediate, the only difficulty being the verification that

RAP

is closed under the formation of sums.

The same

© 1984 American Mathematical Society 0271-4132/84 51.00 + 5.25 per page

237

SARASON

238

di.fficulty arises with one does for

AP, and one can overcome it for

in the same way

[1, p. 36].

AP

The algebra

RAP

obviously contains

AP.

It also obviously contains

R that

the algebra of continuous functions on AP + GO

RAP

vanish at

GO'

The linear span

00.

is easily seen to be a closed algebra; it consists of the functions

,

that Frechet termed asymptotically almost periodic in the paper cited above. Another sUbclass of f

in

BUG

such that

RAP

is

* f-Taf

SO,

which by definition consists of all functions

is in

obviously a closed subalgebra of

Co

for every

BUC;

a.

The class

SO

is

AP,

it is nearly disjoint from

the

only functions common to both algebras being the constant functions. The considerations that led me to the paper.

RAP

will be mentioned at the end of

The bulk of the paper will be devoted to the proof of the following

assertion. THEOREM.

RAP

is the closed subalgebra of

BUC

generated by

AP

The proof will be indirect and will involve an analysis of Gelfand space (space of multiplicative linear functionals) of sis will reveal how

M(RAP)

can be built from

identify the functions in the algebras

AP, SO

transforms on the appropriate Gelfand spaces.

M(AP) and

and RAP

and

SO.

M(RAP) ,

RAP.

M(SO).

the

The analyWe shall

with their Gelfand

Each of these algebras is a

C*-algebra, so each i.s identified with the algebra of all continuous functions on its Gelfand space. We shall regard the real line, each point of

R,

as a subset of

R becomes a dense open subset of

*

M(RAP).

M(RAP)

the transformation T a

T extends to a homeomorphism of a acts as an isomorphism of the algebra RAP

and that homeomorphism is the desired extension of T

RAP. For

Under a

R,

in

onto itself.

In

onto itself; that isomor-

phism is induced by a homeomorphism of the Gelfand space by

by identifying

R with the corresponding evaluation functional on

this identification, fact,

M(RAP)

T . a

M(RAP)

onto itself,

We denote the extension

also.

a

The space M~(RAP).

M(RAP) - R

(the "fringe" of

M(RAP»

will be denoted by

It consists of two connected components, which can be thought of as

the fibers of

M(RAP)

denote the closure of

above

{T x:x E R}; a

the transfonnation group closed subset of

and

M(RAP)

ing transformation group.

-""

.

For

x

in

Mco (RAP)

we let

the latter set is the orbit of

x

is the smalles t {T:a E R}. The orbit closure S~ x a which contains x and is invariant under the precedClearly, a function in

RAP

belongs to

SO

if and

only if it is constant on

Qx for each x in M00 (RAP). The first main step in the proof of the theorem will be to show that each

of the orbit closures

Q x

is a replica of

M(AP).

Next, we shall show that

two orbit closures which are not identical are actually disjoint.

Finally, we

239

REMOTELY AUtOST PERIODIC FUNCTIONS

shall show that two distinct orbit closures can be separated by a function in Once that has been done, the theorem will be almost immediate.

SO.

M (RAP) and, for f "" whose value at t is f(TtX).

Let us fix an function on

R

in

x

because any remote e-translation number of T.* f.

number of

The function

x

g.

ishes on

T* f

RAP,

denote by

T*f

T*f is in AP, x is an ordinary e-translation

vanishes identically if and only if

x

the

x

The function

f

T* as a map of

Hence, we may as well regard

x

in

C(Q)

x

van-

f

into

x

AP,

and when so regarded it obviously preserves norms and is an algebraic isomorphism. For

s

R,

in

let

*

T (e ) = e (x)e , x s *8 S

Then

T

the range of

e

denote the exponential function

s

so the range of

T

is therefore der.c;e in

x

*

contains

x

AP.

mum norm), so it equals

AP.

Hence,

x

s

= e ist

By Bohr's theorem,

s

From the observation at the end of T*

the preceding paragraph we know that the range of

* T

e.

e (t)

is closed (in the supre-

x

gives an isomorphism of

C(Q) x

onto

AP.

*

The preceding discussion shows the

is dense in

T AP

x

AP,

so

AP IS2 x

is

C(Q ). Actually, as we shall now see, APIQ = C(Q). To establish x x x this we need only to show that APIS2 x is closed, which we can do by showing

dense in

that the restrict ion map from proving

IIfll

00

=

liTx*fll

last equality when

for all

00

f

to

AP

in

f

preserves norms.

We actually need only to prove the

AP.

is an exponential polynomial in

As is well known, the space

M(AP)

as addition.

phically) in in

M(AP).

M(AP)

w.

Now suppose that

(Here,

a finite set.)

R with its natural image

as a dense subgroup; we identify

The restriction of the functional AP.

We write the group operation on

The real line is embedded (continuously but not homeomor-

which we denote by nomial in

AP.

can be identified with the Bohr group,

the dual of the discrete real line [5, p.33l]. M(AP)

That amounts to

x

to

f = Z C(s)e s

C stands for a function on

AP s R

is an element of

M(AP)

is an exponential polywhich is

0

except on

From the equality T* f (t) x

'Ie

= Zs C(s)T xs e (t) = l C(s)e (x)e (t) s s s = Zs C(s)e s (w)e s (t) = Zs C(s)e s (w+t)

= f(w+t) , ,,:

we see that the values taken by by

f

on the coset

equality

=

IIflloo

w + R of

T f on R are the same as the values taken x M(AP). As that coset is dense in M(AP), the

* IITxfll""

follows, and so the equality

and

are in

AP1Qx

= C(Qx)

is

established.

LEMMA 1.

If

x

tical or disjoint.

y

Moo (RAP) ,

then

Q x

and

Q

y

are either iden-

240

SARASON To prove this we need only to show that, if

in

~.

on

C(~)

y

~x'

is in

then

x

is

Suppose y is in ~. Because API~ = C(~) and the restriction y x y y map is an isometry, the functional f ~ f(x) on AP can be regarded as acting point

in

Since

~.

y

f(x) for all f Because f (z) x it must be that z = x, so x is in

is in

z

= C(Q

Apls~

z

and so is represented by a point

y

in

~.

),

(x j )

Let

(Yj)

and

the same directed set)

as desired.

be two convergent nets in M"., 0

and covering

U, K cUe X,

Assume {U t } of

such that

is proper and each fiber is

U •

t'

there are polynomials

PO,Pl, ••• ,Pr

such that

(f l ,··· ,fm) 11K < B;

(iii) Then

C

(X,A)

fl, ••• ,fm E A and an open subset

the map

(ii)

K

C t

Ute

X be locally compact and

that for every compact set

(i)

(fllu, ••• ,fmlu}:u ~ F(U)

(X,A)

is a k-maximum set.

F(U)

has the k-maximum property.

PROOF OF THEOREM 3.2 (Sketch).

According to the

3.6 we have (roughly .,v m (f l ' ... , fm) E (A ~ B) , such

speaking) to find sufficiently many m-tuples that the map

F

=

(fll u, ••• ,fmlu)

is proper, has small fibers, and. F(U)

a (k+$.H)-maximum set for suitably chosen functions

Lemm~

U

C

X x Y.

We approximate given

by polynomials in elementary functions

and using Lemma 3.5 we choose tuples

is

g1 x gi, g1 E A, g'1 E B), , " (fi, ... ,f~,)E Am, (f ,f;,,) E Bm

1, ...

ANALYTIC MULTIFUNCTIONS

containing gI and g1 respectively. If we choose take F' = (filu, ••• ,f~,lu) and F" alike, and set

1, ...

251

U as the product (fl, ••• ,fm) -

U' xU",

(fi ® l, .•• ,f~ ® 1, 1 ® f ,1 ~ f;), F· (fllu, ••• ,fmlu), then F(U) .. F'(U') x F"(U") is a (k+t·H)-maximum set by Proposition 3.4 and Theorem 3.3.

We omit further details.

REMARK. If X and Yare complex varieties of pure dimension, then Theorem 3.3 is equivalent to the well-known assertion that diml(Xxy) = dimCX+dimeY. PROBLEM.

Is Theorem 3.2 still true without the assumption that

X is locally

compact? 4.

DUALITY BETWEEN k-MAXIMUM SETS AND

q-PSEUDOCONVEX DOMAINS

The following two results are crucial to our method of proving Theorem 3.3. The Duality Theorem 4.1, which generalizes Th. 1.3, makes it possible to translate statements on k-maximum sets into assertions on q-pseudoconvex domains, while Proposition 4.2 allows us to reduce problems concerning q-pseudoconvex domains to questions on q-plurisubharmonic functions -- more amenable to analytic techniques. DUALITY THEOREM 4.1. is q-pseudoconvex in set.

Let U, V be open in en, U c V and X = v\p. Then U V (0 ~ q ~ n-2) if and only if X is an (n-k-2)-maximum

PROPOSITION 4.2. Let U,V be open in following conditions are equivalent: (i)

U is q-pseudoconvex in

en, U c V, 0 ~ q ~ n-2.

Then the

V;

(ii) the canonical exhaustion function ·.z -+ -log dist(z ,aU) subharmonic near V n au;

is q-pluri-

(iii) there is a neighborhood W of V n au and a q-plurisubharmonic function u:W n U -+ [--,~), such that lim u(z) = m, for every z E V n au. z '-+z Let us see how these results work.

If

X,Y

are as in Th. 3.2, take open

sets VI c t n , V2 c ~ ~, such that X C VI' Y c V2 and X,Y are closed in VI' V2 respectively. Set Xl = XxV 2 , YI = VlxY. Then Xl and YI are kl and t1-maximum sets respectively, where kl .. k+m, tl .. n+t. (We check this for Xl' If to show that

U .. VlXX, U x V2 is

then VI x V2\X x V2 = U x V2 • By Th. 4.1 it suffices «n+m)-k l -2) = (n-k-2)-pseudoconvex in VI x V2 • It

is indeed so because the canonical exhaustion function of

U,

composed with the

252

SLODKOWSKI

projection of

U x V 2 onto U, gives an (n-k-2)-plurisubharmonic function satisfying Proposition 4.2 (iii». By these observations Theorem 3.3 is implied by the following assertion. INTERSECTION THEOREM 4.3.

V.

sobsets of

Let

Assume that

perty of orders

kl

and

Xl

Vc (N

n

Yl

be open and

;~,

Xl' YI relatively closed Xl' Yl have the maximum pro-

and

t 1 respectively

(0::: kl ,t 1 ::: N-I).

Then

Xl

n YI

is an (N+l-kl-tl)-maximum set. This result can be viewed as a generalization of the classical estimate of the dimension of the intersection of complex submanifolds. By the Duality Theorem 4.1, the Intersection Theorem 4.3 is equivalent to the following statement about the relative complements 4.4. in

Ul

= V\XI'

U2 = V\X2 •

UI , U2 eVe (n be open. If Ul and U2 are q- and r-pseudoconvex then Ul U U2 is (q+r+l)-pseudoconvex (in V) •

Let

V, Let

be continuous exhaustion functions for

u l ' u2

r-plurisubharmonic near

V

n aU l

and

V

n au 2 ,

UI

and U2 ' respectively. Set

q- and

u(z) ..

Then

u

is a continuous exhaustion function; by Proposition 4.2 it is enough

to prove that

u

is (q+r+l)-plurisubharmonic near

V

n a(Ul

U U2 ).

The next

theorem suffices to yield this. THEOREM 4.5. ulB

and

vlB

Let

B be an open ball in

(n

and

u,v E C(B).

are respectively q- and r-plurisubharmonic in

is (q+r+l)-plurisubharmonic in

Assume that B.

Then

min(u,v)

B.

Of course, because of its local nature, the theorem is true for an arbitrary open set

B.

Also the continuity assumption can be omitted.

Our proof of Theorem 4.5 is closely connected with the generalized Dirichlet problem studied by Hunt and Murray [6]. 5.

OPERATIONS ON q-PLURISUBHARMONIC FUNCTIONS AND THE GENERALIZED DIRICHLET PROBLEM It is easy to prove Th. 4.5 in case one of the functions is smooth:

253

ANALYTIC MULTIFUNCTIONS

5.1. A smooth q-plurisubharmonic function has the property (P q,r ): For every r-plurisubharmonic function v, the function min(u,v) is (q+r+l)-plurisubharmonic. If we knew that continuous q-plurisubharmonic functions could be approximated (locally) by smooth ones of this class (cf. Hunt and Murray [6]), the last observation would imply Th. 4.5.

Since we do not, we take a longer way:

we

prove that a continuous q-plurisubharmonic function can be obtained (locally) from smooth q-plurisubharmonic functions by some simple operations, repeated (infinitely) many times; moreover these operations preserve property More specifically, we let

AP

(P

) •

q,r denote the smallest class of upper semi-

q

which contains the class

continuous functions defined on open subsets of

of all smooth q-plurisubharmonic functions and is closed with respect to the operations:

(a)

functions; (d)

upper semi-continuous envelope of the supremum of a family of

(b)

restriction to a subset;

local correction:

such that

having given

lim sup ul (z')

~

u(z)

u

(c)

translation by a vector;

D and

in

for every

ul

in

n ~Dl'

zED

z '-+z

v(z.) z

to be

u(z)

for

z E D\Dl ~

max(u(z), u l (z»

and

for

E1\. It is easy to see that the class

(a) -(d)

) is preserved by operations q,r AP q • Therefore Theorem 4.5 is a consequence

by 5.1 - contains

and

(P

of the following result.

nmOREM 5.2.

If

o

where

~

q !: n-l,

!, and q-plurisubharmonic in B is an open ball in Cn , then ulB E AP (B). is continuous on

u

--

By properties of the class continuous function

v

in

AP

B,

q

open in

B.

such that

,n,

We claim that

v

==

u.

vlB E AP (B). q

vlaB

If not, then

Using methods of

= ulaB

we have to show that

u-v ~ 0

in

H.

-vIH,

it follows that

u+V

Let

u

and

v

is con-

(u-v) laH ulH

(u-v) IH ~ 0

=0

and

(u-v) IH

is

and (n-q-l)-plurisubharmonic by the next theorem (and the ,n).

be q- and r-plurisubharmonic respectively.

is (q+r)-plurisubharmonic.

is

is both q-pluri-

local maximum property of (n-l)-plurisubharmonic functions in

THEOREM 5.3.

v

To complete the proof of Theorem 5.2

Since

the sum of the q-plurisubharmonic function

and

H= {z E B:v(z) < u(z)}

and by arguments of Hunt and Murray [6], ulH

subharmonic and (n-q-l)-plurisubharmonic.

function

q

there exists a greatest upper semi-

Bremermann [5] and Walsh [22], one checks that tinuous on

B,

Then

254

SLODKOWSKI This result, as was hinted

the uniqueness of the

sol~tion

at by Hunt and Murray [6], is equivalent to to the generalized Dirichlet problem studied by

these authors. Cf. [18, Secs. 5 and 6] for more details. in our opinion incorrect, of Th. 5.3 in case 6.

(M. Kalka [7] gives a proof, q=r=(n/2)-1.)

REGULARIZATION OF q-PLURISUBHARMONIC FUNCTIONS BY MEANS OF CONVEX FUNCTIONS To prove Theorem 5.3 we approximate q-plurisubharmonic functions by func-

tions which, although not smooth, exhibit some regularity, and then prove the theorem for the approximations. The standard way of smoothing up a function

u

is to consider the con-

B(O,e), and /g=l. Since the class of q-plurisubharmonic functions is not closed with respect to the summation, this method is useless in this context. The fact that the supremum of a family of q-plurisubharmonic functions is q-plurisubharmonic suggested to us to introduce a new type of convolution. volution

u*g,

DEFINITION.

where

Let

convolution of

g

u,g u

is smooth,

and

Let

u

g

(i)

Set

(n.

The supremum-

Cn •

Let

is the function

=sup{ u(y) g(z-y):y

E

tn} •

be a bounded nonnegative function in

g (0) = 1 n un • u* s g. Then

~

~

0

smooth functions such that n·l,2, ••••

C

be bounded nonnegative functions on

u* g(z) s THEOREM 6.1.

supp g

there are constants

L(n)

g n

~

0

and supp g -n

g

be

n

B(O,l/n),

C

such that the functions

are convex on (ii)

if

u

is continuous and q-plurisubharmonic near

q-plurisubharmonic near A function

v

K and converge to

such that

v(z) +

u

uniformly on

K,

then

u

n

are

K.

~lzl2 is convex for some L ~ 0, will

be said to have lower bounded real Hessian.

In other words, it is a function

whose real Hessian in the sense of distribution theory is a vector measure with values in the convex set of symmetric matrices with lowest value bounded from below by -L. By some results in convex analysis, such a function has at almost every point a second order differential in the local (namely Peano) sense.

In such points real and complex Hessians can be defined.

possible the following characterization.

This makes

255

ANALYTIC MULTIFUNCTIONS THEOREM 6.2. Then

u

Let

u

be a function with lower bounded real Hessian in (0 ~

is q-plurisubharmonic

x E U the complex Hessian of

point

q ~ n-l) u

at

n

Uc C •

if and only if at almost each x

has at most

q

negative eigen-

values. The necessity of this condition is rather easy; as for the sufficiency, it can be reduced to the next theorem, whose proof, based on ideas from geometric measure theory, is omitted. THEOREM 6. 3. let

L

~

Let

be convex in

u

N

B(O,r) c R , r > 0, u

Ixl

Assume that for almost every

0.

the real Hessian of

u

at

is greater than

x


!

is harmonic in

A on IAI < 1,

as was to be

The answer to Question 2 is No, as is seen by the following two examples of admissible sets which admit no representation (3). EXAMPLE 1:

Let

x = X+

= {( A, w)

X

=

{( A, w)

and

X-

X+

We claim that

{(A,w) IIAI ::; 1,

I ::; 1},

jr.r2 - (A - 10)

11m w E X 11m w EX

> 0 }, < 0 }.

are connected components of

with

X,

Fix A.

Each point in

= "IIc +

w

with

Ic I :::

Since

1.

or

A - 10

v'c + A - 10

ASSERTION: PROOF:

X+

lies in

P

n

A - 10 ,

1m w > 0

It follows that

for each x+

x+

is

x+

n

{IAI

=

I}.

So

it

is a compact polyno-

X is

x+

A, c,

X

n {IAI =

I},

is admissible.

has no representation (3).

We pu t Q+

Let

= -Vc +

w

Also, since the Silov boundary of

the Silov boundary of

UX •

XA has the form

follows that our claim is correct. mially convex set.

= X+

Re ~ < 0 with

Let yr- denote the branch of the square root defined on ~ = 1.

X

=

[IAI < l]\X+, Q~

=

{w I(A'w) E Q+}.

g+ (A,·)

denote the Green' s function for Q+A with pole at 00. Suppose that X+ has a representation (3). Thus there are polynomials

as above with GO

n n=l For each

n,

we put

Green's function of

Q = {IAI < 1 }\K , n Pn (Qn) A with pole at

K . Pn

and we write 00.

Then, on

~ (A,·)

for the

~,

g (A,w) ... (d 1 P )log IF (A,w) I. n eg n n It follows tnat As

n

g

n

(\,w)

+

4

g (A,w)

{IAI < I} x {Iwl > R}

g+(A,w)

= loglwl

+ peA) +

Q n

as a function of

+

A, g (A,·) t g (A,·)

for each fixed

400,

that the function. large that

is pluriharmonic on

C

n

is pluriharmonic on Q+,

Iwl > R.

on

Q,.

~.

"-

X and w. It tollows

Choosing

R so

we have the expansion

al(A) altA) a 2 (A) a 2 (A) w + W + 2 + _2 + ••• ,

w valid for

+

w

Since g+ is pluriharmonic, we can choose a conjugate

function h+ such that

GREEN'S FUNCTIONS g+ (~.w) + ih+ (~,w) where

F

=

'2.77 F(~,w),

log w +

fI ~ I
R}, on

we see that

and vanishes for

f

w =

I}

x

Then,

is single-valued analytic on g+

Since

00.

is pluriharmonic 52+ U {w

can be analytically continued along each path in

f

{Iw I _~ R}.

=

oo}.

Since that domain is simply-connected, the resulting function, again written f,

is single-valued analytic on

is the ~reen's function of

g+(A,.)

Q~U

malone-one map of

~,aQ~

For fixed f(\,·)

If I

=1

+ aQ~

If(~,w)

and ~,

I=

c

with aQ+.

f (~, Vc

10)

A -

a constant

lei = 1. Hence

=

going to

00

O. f(~,·)

Hence

+ aQ\"

w in

to some neighborhood of

w

is a confor-

It follows

Since this is true for each

Then the variety

Vc +

w

If(~,Vc + ~ - 10) I

A-

1, IAI < 1.

is an analytic function of

aQ+.

Also

10,

< 1,

is

Also

I~I < 1.

on

IAI

Hence there is

with f(~,Vc + ~ - 10)

(4)

I~I < i.

= Yc'

~,

Differentiating (4) with respect to

we get

f\(~,Vc + A - 10) + fw(~,Ve + ~ - 10) .

(5)

w

for

1

+ aQ~.

extends analytically across

a subset of

+

00, f(A,·)

\,

+ aQ.

on

Fix

Q~ with pole at

to the unit disk, with

{oo}

extends analytically in

f

Also, since for fixed

= oo}.

is a real analytic simple closed curve.

extends continuously to that

Q+ U {w

Equation (5) holds for each (~,Vc + ~ - 10)

with

c

on

I~I ~ 1

Ic

and

+ A - 10

1\1

< 1.

The totality of points

1 = 1.

Icl

= 0,

1 2'1/c

=1

is precisely

+ aQ.

So we have

(6)

at each on

rl

~"2~.

(~,w)

+ E aQ.

For fixed

Since (6) holds on

+

aQ~,

f ~ CX., .)

(7)

at each point of

+

Q~.

~,

now,

and

f\(~,·)

f (\,.) w

are analytic

we have 1 + f w ( .\, • ) 2w

But for fixed + Hence at Q~.

~,

w

=

0

-+ f(~,w)

is a conformal map and so

never vanishes on w = 0, (7) gives a contradiction. We w conclude that x+ has no representation (3) • Since X+ is admissible, this f

gives a negative answer to Question 2. NOTE:

We did not appeal to Theorem 1 because we lacked enough information

about

cap(x~)

+

1n order to apply that result.

In the next example, we are

able to appeal to Theorem 1. EXAMPLE 2:

Put

x

{t~, w) II ~ lSI /4,

Iw(1 - ~w) I