Lecture 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 This work si tcejbus ot .thgirypoc llA rights era ,devreser rehtehw eht whole or trap fo eht lairetam si ,denrecnoc yllacificeps those fo ,noitalsnart ,gnitnirper esu-er fo ,snoitartsulli ,gnitsacdaorb noitcudorper yb gniypocotohp enihcam or ralimis ,snaem dna egarots ni data .sknab rednU § 54 fo eht namreG thgirypoC waL erehw copies era edam rof rehto naht etavirp ,esu a eef si elbayap ot tfahcsllesegsgnutrewreV" ,"troW .hcinuM © yb galreV-regnirpS Berlin grebledieH 3891 detnirP ni ynamreG gnitnirP dna :gnidnib Beltz ,kcurdtesffO .rtsgreB/hcabsmeH 012345-0413/6412 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