Table Of ContentLecture Notes ni
Mathematics
Edited by .A Dold and .B Eckmann
975
Radical Banach Algebras
dna Automatic
ytiunitnoC
Proceedings of a Conference Held at California
State University, Long Beach, July 17- ,13 1891
Edited by J.M. Bachar, W.G. Bade, RC. Curtis ,.rJ
H.G. Dales, and .M .P Thomas
galreV-regnirpS
nilreB Heidelberg New kroY 1983
Editors
John .M Bachar
Department of Mathematics
California State University, Long Beach
Long Beach, CA 90840, USA
William G. Bade
Department of Mathematics,
University of California, Berkeley
Berkeley, CA 94?20, USA
Philip C. Curtis
Department of Mathematics,
University of California, Los Angeles
Los Angeles, CA 90024, USA
.H Garth Dales
School of Mathematics, University of Leeds
Leeds LS2 9JT, England
Marc R Thomas
Department of Mathematics
California State College, Bakersfield
Bakersfield, CA 93309, USA
AMS Subject Classifications (1980): 03 E50, 04A30, 13-02, 13A10, 13G 05,
J31 05, 46-02, 46-06, 46 H 05, 46 H10, 46 H15, 46H 20, 46J 05, 46J10,
46J15, 46J 20, 46J25, 46J30, 46J35
ISBN 3-540-11985-XSpringer-Verlag Berlin Heidelberg New York
ISBN 0-38?-ll985-XSpringer-Verlag New York Heidelberg Berlin
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DEDICATION
This volume is dedicated to Charles Rickart, one of the first to consider
questions of automatic continuity. His contributions to the theory of Banach
algebras underlie many of the problems considered here.
PREFACE
This volume contains the contributions to the Conference on Radical Banach
Algebras and Automatic Continuity, held at the California State University, Long Beach,
July 13 - 17, 1981, and the following study period from July 18 - .13 The purpose of
the conference was to present recent develolEents in these two areas and to explore
the connections between them.
The articles given here represent expanded versions of conference talks, to-
gether with solutions of various problems that were presented and discussed. The
papers contain, in varying degrees, historical background, syntheses and expository
accounts, and the development of new ideas and results. Further details of the
papers are given in the Introduction. The volume concludes with a list of unsolved
problems.
The editors, who also served as the organizing committee, wish to thank the
adminstration of the California State University, Long Beach, and particularly
President Stephen Horn, for generous financial support, and for the excellent working
conditions that were provided for the Conference. We are also grateful for additional
financial support fr~n the National Science Foundation, and for the travel grant frcm
the North Atlantic Treaty Organization which enabled .E Albrecht, .G .R Allan,
.H .G Dales, and M. Neumann to come to California to discuss their work.
Finally we wish to thank Elaine Barth, who typed the entire manuscript of this
volume with its many corrections. Through her patience and outstanding skill she
has made a major contribution to this endeavor.
The editors
John Bachar, Long Beach
William Bade, Berkeley
Fnilip Curtis, Los Angeles
Garth Dales, Leeds
Marc Thomas, Bakersfield
VI
TABLE OF CONTENTS
O. ~TRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . I
I. GENERAL THEORY OF RADICAL BANACK ALGEBRAS . . . . . . . . . . . . 3
E.l J. Esterl% Elements for a classification of eorgnutative radical
Banach algebras . . . . . . . . . . . . . . . . . . . . . . 4
E.2 J. Esterle~ Qtlasimultipliers, representations of H~ and the
closed ideal problem for cormnutative Banach algebras. . .... 66
z F. Zouakia, The theory of Cohen elements . . . . . . . . . . . 163
II. EXAMPLES OF RADICAL BANACH ALGEBRAS . . . . . . . . . . . . . . . 179
Da.1 H. G. Dales, Convolution algebras on the real line. • . . . . . . 180
Do.l Y. Domar~ Bilaterally translation-invariant subspaees of weighted
LP (l~) . . . . . . . . . . . . . . . . . . . . . . . . . . 210
Do.2 Y. Domar~ A solution of the translation-invariant subspace problem
for weighted L p on ,ql ql + or ~. . . . . . . . . . . . . . 214
daB W. G. Bade, Multipliers of weighted ~l-algebras. • . . . . . . . 227
La K. B. Laursen, Ideal structure in radical sequence algebras. . 248
Th M. P. Thomas, Approximation in the radical algebra Ii(~n) when
~n } is star-shaped . . . . . . . . . . . . . . . . . . . . 258
Gra-Th S. Grabiner and M. P. Thomas~ A class of unicellular shifts which
contains non-strictly cyclic shifts. . . . . . . . . . . . . . 273
AI.I G. R. Allan, An inequality involving product measures. . .... 277
W.l G. A. Willis~ The norms of powers of functions in the Volterra
algebra . . . . . . . . . . . . . . . . . . . . . . . . . 280
Gra S. Grabiner~ Weighted convolution algebras as analogues of Banach
algebras of power series. . . . . . . . . . . . . . . . . . 282
AI. 2 G. R. Allan~ Commutative Banach algebras with power-series
generators . . . . . . . . . . . . . . . . . . . . . . . . 290
N. Gr~nbaek~ Weighted discrete convolution algebras. . ..... 295
Bac G. F. Bache!is~ Some radical quotients in harmonic analysis. . 301
Do. 3 Y. Domar, A Banaeh algebra related to the disk algebra. . .... 309
III. AUTOMATIC CONTINUITY FOR HOMOMORPHISMS AND DERIVATIONS . . . . . . . 312
uA
B. Aupetit, Automatic continuity conditions for a linear mapping
from a Banach algebra onto a semi-simple Banach algebra. . . . . 313
Lo.1 R. J. Loy, The uniqueness of norm problem in Banaeh algebras with
finite dimensional radical. . . . . . . . . . . . . . . . . 317
c P. C. Curtis, Jr., Derivations in commutative Banach algebras. . 328
~
J. C. Tripp, Automatic continuity of homomorphisms into Banach
algebras. . . . . . . . . . . . . . . . . . . . . . . . 334
llV
Table of Contents - continued
Di P.G. Dixon, On the intersection of the principal ideals
generated by powers in a Banach algebra . . . . . . . . . . . . 340
A-N.I E. Albrecht and M. Neumann, Automatic continuity for operators
of local type . . . . . . . . . . . . . . . . . . . . . . 342
A-N.2 E. Albrecht and M. Neumann, Continuity properties of
k
C -homomorphisms . . . . . . . . . . . . . . . . . . . . . . 356
A-Da E. Albrecht and H. G. Dales, Continuity of homomorphisms from
C*-algebras and other Banach algebras . . . . . . . . . . . . . 375
Da.W H.G. Dales and G. A. Willis, Cofinite ideals in Banach algebras,
and finite-dimensional representations of group algebras . . . . . 397
W.2 G.A. Willis, The continuity of derivations from group algebras
and factorization in eofinite ideals . . . . . . . . . ~ . . . 408
IV. CONTINUITY OF LINEAR FUNCTIONALS . . . . . . . . . . . . . . . . 422
M G.H. Meisters, Some problems and results on translation-
invariant linear forms . . . . . . . . . . . . . . . . . . . 423
Lo.2 R.J. Loy, The uniqueness of Riemann integration . . . . . . . . 445
Da.2 H.G. Dales, The continuity of traces. '. . . . . . . . . . . . 451
V. OP~ QUESTIONS . . . . . . . . . . . . . . . . . . . . . . 459
PARTIC IPANTS
.E Albrecht (University of Saarland, Saarbr~cken, West Germany)
.G .R Allan (Cambridge University, Cambridge, England)
.R Arens (UCLA, Los Angeles, CA)
.B Aupetit (University of Laval, Quebec, Canada)
J. M. Bachar, Jr. (California State University Long Beach, Long Beach, CA)
.G .F Bachelis (Wayne State University, Detroit, MI)
W. .G Bade (UC Berkeley, Berkeley, CA)
.T .G Cho (Sogang University, Seoul, Korea)
.P .C Curtis, Jr. (UCLA, Los Angeles, CA)
.H .G Dales (University of Leeds, Leeds, England)
Y. Dcmar (University of Uppsala, Uppsala, Sweden)
J. Esterle (University of Bordeaux, Talence, France)
.T Gamelin (UCLA, Los Angeles, CA)
.S Grabiner (Pomona College, Claremont, CA)
N. Gr~nbaek (University of Copenhagen,Copenhagen, Denmark)
M. J. Hoffman (California State University t LOs Angeles, LOs Angeles, CA)
.D .L Johnson (Hughes Aircraft, Culver City, CA)
Ho Kamowitz (University of Massachusetts~ Boston, MA)
J. Koliha (University of Melbourne, Melbourne, Australia)
.K .B Laursen (University of Copenhagen, Copenhagen, Denmark)
J. A. Lindberg, Jr. (Syracuse University, Syracuse, NY)
.R J. Loy (Australian National University, Canberra, Australia)
.G .H Meisters (University of Nebraska-Lincoln, Lincoln, NE)
M. Neumann (University of Essen, Essen, West Germany)
.B Rentzsch (TRW, Redondo Beach, CA)
.C Rickart (Yale University, New Haven, CT)
M. .P Thomas (California State College Bakersfield, Bakersfield, CA)
J. .C Tripp (Southeast Missouri State University, Cape Girardeau, MO)
N. N .P Viet (UC Berkeley, Berkeley, CA)
.S Walsh (UC Berkeley, Berkeley, CA)
°G .A Willis (University of New South Wales, Kensington, New South Wales,
Australia )
W. Zame (SUNY, Buffalo, NY)
INTRODUCTION
The basic problem of automatic continuity theory is to give algebraic conditions
which ensure that a linear operator between, say~ two Banach spaces is necessarily
continuous. This problem is of particular interest in the case of a homomorphism
between two Banach algebras. Other automatic continuity questions arise in the
study of derivations from Banach algebras to suitable modules and in the study of
translation invariant functionals on function spaces.
There is a fundamental connection between questions of automatic continuity
and the structure of radical Banaeh algebras. For example, the recent construction
of a discontinuous homomorphism from C(X), the algebra of all continuous, complex-
valued functions on an infinite compact space X, depends on important structural
properties of certain commutative radical Banach algebras.
The 30 papers in this volume present the latest developments in these two
theories and explore the connections between them. Section I is devoted to the
general theory of commutative radical Banach algebras. In E.I, Jean Esterle
gives a comprehensive classification of commutative radical Banach algebras based
on the types of semigroups which these algebras contain. Esterle shows in this
paper precisely which commutative radical Banach algebras R with unit adjoined
can serve as the range of a discontinuous homomorphism from C(X). Assuming the
continuum hypothesis, such a discontinuous homomorphism w from C(X) into
R • ~e exists if and only if R contains a rational semigroup over Q+. An
equivalent condition is that R contains a non-nilpotent element of finite closed
descent. This paper also contains some new short proofs of earlier theorems of the
author, e.g., that each epimorphism from C(X) onto a Banach algebra is automatically
continuous.
In E.2 Esterle investigates the question of whether or not a commutative
radical Banach algebra must contain a non-trivial closed ideal. Substantial partial
results are obtained on this fundamental open problem, which in turn is related to
the invariant subspace problem for Banach spaces. Improvements of some recent
results on invariant subspaces are given as well.
Section II is concerned with particular examples of radical Banach algebras.
In IDa. l, H. .G Dales gives a survey of radical convolution algebras on the line
and half line. The algebra Ll(~) on ~+, where ~ is a rapidly decreasing
weight function, has been much studied in recent years. Particular interest has
centered on the problem of determining for which radical weights, ,~ every closed
ideal of Lm(~) is a standard ideal, that is, an ideal consisting of those functions
with support in an interval ~,~). In Do.1, Y. Domar gives the first results on
this problem, showing that for a wide class of radical weights w on ~+ each
closed ideal is indeed standard. In Do.2, Domar gives the final details of a
solution to the problem of when spectral analysis holds for the analogous Beurling
algebras on B.
If ~ is a rapidly decreasing weight sequence on ~+, then ~l(w) is a
radical Banach algebra of power series. At present there are no known examples of
weight sequences for which ~l(w) contains non-standard ideals. A construction
by N. .K Nikolskii around 1970 of a weight sequence with this property has been shown
to be incorrect. The problem of characterizing those weight sequences for which each
ideal is standard motivates several of the papers in Section .II The multiplier
algebras for the power series algebras ~i(~) are discussed in Bad and La.
Certain inequalities which may be relevant to the closed ideal problem are discussed
in AI.I and W.I. In Th, M. Thomas describes a class of weights for which each
closed ideal in ~i(~) is standard; this is the first new result of this type in
about i0 years.
The papers in Section III concern the automatic continuity of homomorphisms
and derivations. In Au, .B Aupetit gives a new proof of the well known theorem
of .B Johnson that a semi-simple Banach algebra necessarily has a unique Banach
algebra topology. This proof involves techniques from the theory of subharmonic
functions, and the ideas should be applicable to other automatic continuity
problems. Uniqueness of norm for nonsemisimple algebras is considered by .R .J Loy
in Lo.l. Automatic continuity for local algebras is surveyed by .E Albrecht and
M. Neumann in A-N 1,2.
The problem of whether or not a homomorphism between C* algebras must be
continuous has resisted solution for some time. Partial results on this question
and related problems are given by Albrecht and Dales in A-Da. Automatic con-
tinuity questions for derivations on group algebras are discussed in W.2 by
.G Willis. The more difficult problems here involve nonamenable groups~ particu-
larly the free group on two generators.
The automatic continuity of certain linear functionals on Banach algebras is
discussed in Section .VI Translation invariant functionals are surveyed by
.G Meisters in M. The most intractable problem which remains in this area is
whether or not each translation invariant linear functional on the space of
continuous functions on ~ with compact support is necessarily continuous. This
question is discussed by .R J. Loy in Lo.2. The theory of positive linear
functionals and traces is considered in Da.2.
The volume concludes with a list of open problems~ some well known and others
posed and discussed at the conference.
ELEMENTS FOR A CLASSIFICATION OF COMMUTATIVE RADICAL
BANACH ALGEBRAS
J. Esterle
1. Introduction
The purpose of this paper is the investigation of the relationships between
some natural algebraic and topological properties for commutative radical Banach
algebras. We shall see that a lot of apparently unrelated properties are in fact
equivalent, or stronger or weaker. This enables us to give a classification of
infinite-dimensional, commutative radical Banach algebras into nine classes. The
first one, the class of such algebras g for which the set of principal ideals of
• @e is not linearly ordered by inclusion~ is the biggest. In fact it is so
big that it contains all infinite-dimensional, commutative radical Banach algebras.
Then the classes get smaller and smaller, and the ninth one is the class of com-
mutative radical Banach algebras which possess a nonzero analytic semigroup
t
(a)Re t>0 over the right-hand half-plane such that su~ iitaH"" < + ~. This
0< .tI ~l
Re t > 0
ninth class does not contain all commutative radical Banach algebras with bounded
approximate identities (it is shown in Section 7 that Ll(~ +, e -t2) does not be-
long to the ninth class). This suggests that we introduce a tenth one, the class of
commutative radical Banach algebras which contain a nonzero analytic semigroup
t
(a)Re t>0 over the right-hand half-plane such that sup iltaJJ < + % but
Ret>0
this tenth class is unfortunately empty (see Section 7). This fact is an easy con-
sequence of the Ahlfors-Heins theorem for bounded analytic functions over the half-
plane.
The three main themes of our investigations are (i) partial approximate identi-
ties, (ii) semigroups, and (iii) factorization or division properties. By a partial
approximate identity (p.a.i.) in a Banach algebra g we mean a sequence (en) of
elements of g such that b = lim e b for some nonzero b c g. Many radical
n~ n
Banach algebras do not have any p.a.i.; for example, radical Banach algebras of power
series, or trivial radical Banach algebras in which the product of any two elements
equals zero. But the non-existence of p.a.i.'s in a radical Banach algebra
implies a nice algebraic property. We show in Section 3 that every nonzero element
of such an algebra g is equal to a finite product of irreducible elements of g.
This fact is of course obvious in the examples mentioned above, but is not immediate
in the case of the algebras g4' g5 and g6 of Section 7.
There are several natural ways to try to get a p.a.i. The nicest ones would
