JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 204, 522]544Ž. 1996 ARTICLE NO. 0452 Complex Strongly Extreme Points in Quasi-Normed Spaces* Zhibao Hu Miami Uni¨ersity, Oxford, Ohio 45056-16411 View metadata, citation and similar papers at core.ac.uk brought to you by CORE and provided by Elsevier - Publisher Connector Douglas Mupasiri Uni¨ersity of Northern Iowa, Cedar Falls, Iowa 50614-0506 Submitted by Richard M. Aron Received April 13, 1994 We study the complex strongly extreme points ofŽ. bounded subsets of continu- ously quasi-normed vector spaces X over C. When X is a complex normed linear space, these points are the complex analogues of the familiarŽ. real strongly extreme points. We show that if X is a complex Banach space then the complex strongly extreme points of BX admit several equivalent formulations some of which are in terms of ``pointwise'' versions of well known moduli of complex convexity. We use this result to obtain a characterization of the complex extreme points of B Ž . and B Ž m . where 0 - p - `, X and each X , j I, are complex l pjjX gIpL , X j g Banach spaces. Q 1996 Academic Press, Inc. 1. INTRODUCTION The study of complex geometry dates back to a 1969 paper of E. Thorp and R. Whitleywx 14 . In that paper the authors characterized the complex * The results in this paper constitute part of a Ph.D. thesis submitted by the second author in 1992 at Northern Illinois University, Dekalb, Illinois. The authors thank Professors Patrick N. Dowling and Mark A. Smith for their advice and helpful comments. 522 0022-247Xr96 $18.00 Copyright Q 1996 by Academic Press, Inc. All rights of reproduction in any form reserved. COMPLEX STRONGLY EXTREME POINTS 523 Banach spaces X for which the strong maximum modulus theorem holds for X-valued analytic functions defined on an open connected subset of the complex plane in terms of a geometric condition called ``complex strict convexity.'' In 1975 J. Globevnikwx 3 brought complex geometry into sharper focus by introducing complex uniform convexity, a notion which is stronger than complex strict convexity, as a natural generalization ofŽ. real uniform convexity. Globevnik then proceeded to show that for a measure space Ž.V,S,m, the complex space L1 Ž.m, X is not only complex strictly convex as Thorp and Whitley had shown inwx 14 , but is in fact complex uniformly convex. Almost a decade later, a fundamental paper on complex convexity of quasi-normed linear spaces was published by W. J. Davis, D. J. H. Garling, and N. Tomczak-Jaegermannwx 1 . The paper considered a number of topics related to complex convexity. The topics discussed included various moduli of complex convexityŽ which give a measure of the plurisub- harmonicity of the quasi-norm unlike theŽ. real moduli of convexity which give a measure of the convexity of the unit ball of normed spaces. ; the connection between the behavior of suitable martingales taking values in complex quasi-normed spaces and the existence of equivalent complex uniformly convex quasi-norms; the relation of complex convexity with cotype; and results concerning complex Banach lattices. In this paper we study the complex strongly extreme points of the closed unit ball of continuously quasi-normed spaces X over C. When X is a complex normed linear space, these points are the complex analogues of the familiarŽ. real strongly extreme points. We show that if X is a complex Banach space then the complex strongly extreme points of B admit several equivalent formulations some of which are in terms of ``pointwise'' ver- sions of well known moduli of complex convexity. We use this result to obtain a characterization of the complex extreme points of B Ž . and l pjjX gI B where 0 - p - `, X and each X , j I, are complex Banach L pŽ m, X . j g space. We remark that in the real case it is well known that a norm one function f in LpLŽ.m, X is a strongly extreme point of B Žm,X.if and only Ž. Ž. if f v r5f v 5is a strongly extreme point of BX for m-almost all v in the support of f. The survey articlewx 12 gives a more detailed account of this and related issues. Our results for the case of the complex function spaces LpŽ.m,Xparallel the known results in the real case. The paper is organized as follows. In Section 2 we review some basic definitions and relevant preliminaries. In Section 3 we characterize the complex strongly extreme points of BX . In Section 4 we characterize the complex strongly extreme points of B and B for 0 - p - `. L ppŽ m, X . l Ž X . Our arguments for the complex space LpŽ.m, X are direct and straightfor- ward and apply to the real case as well. 524 HU AND MUPASIRI 2. BASIC DEFINITIONS AND PRELIMINARIES Throughout this paper all vector spaces are assumed to be over the field of complex numbers. DEFINITION 2.1. Let X be a vector space. A map 55? : X ª w0, `.is called a quasi-seminorm if Ž.155<<55axsax;agC,xgX Ž.2 there exists a constant C s CX Ž ,55.G1 such that 5xqy 5FCxŽ. 55q 55y for all pairs x, y g X. We call 55? a p-seminorm where 0 - p F 1 if in addition p p p Ž.3 55xqy F55x q55y ,x,ygX. If also 55x s 0 implies x s 0, then 55? is a quasi-norm Žrespectively, p-norm. if conditionsŽ.Ž.Ž 1 ] 2 respectively, conditionsŽ.Ž.. 1 ] 3 hold. The pair Ž X,55. , or simply X if there is no ambiguity, is then called a quasi-seminormed, p-seminormed, quasi-normed, p-normed,ornormed space according to whether 55? is a quasi-seminorm, a p-seminorm, a quasi-norm, a p-norm, or a norm, respectively. We remark that 55? induces a metrizable vector topology on X. If this topology is complete, then the pair Ž X,55. is called a quasi-Banach space. In case 55? is a quasi-norm which is uniformly continuous on 55? -bounded subsets of X, then the pair Ž X,55. or simply X, if no confusion results, is called a continuously quasi-normed space. Note that by propertyŽ. 3 of Definition 2.1, a p-seminorm is uniformly continuous with respect to the uniformity it generates on X. We also note that according to the Aoki]Rolewicz theoremwx 9 , every quasi-norm is equivalent to a p-norm for some p,0-pF1. Consequently, it is customary to assume that all quasi-norms are p-norms for some p. We shall not need to make this assumption here. DEFINITION 2.2. Let X be a quasi-normed space. An upper-semicon- tinuous function w: X ª wy`, `. is said to be plurisubharmonic if for every x, y g X p du 2 iu w Ž.x F Hw Ž.x q ey . 0 2p COMPLEX STRONGLY EXTREME POINTS 525 DEFINITION 2.3. Davis, Garling, and Tomczak-Jaegermann call a con- tinuously quasi-normed space Ž X, 55.locally PL-con¨ex if for any x, y g X, there exists d s dŽ.x, y ) 0 such that p du 2 iu 55xFH 5xqre y 5 for all 0 - r - d . 0 2p Remark. By Proposition 2.2 ofwx 1 , the defining condition for PL-con- vexity is equivalent to requiring that log55? be plurisubharmonic. Since, by Jensen's inequality, 55? is plurisubharmonic if log 55? is plurisubharmonic, a continuously quasi-normed space is locally PL-convex if and only if it is equipped with a plurisubharmonic normŽ cf. Lemma 2.2 ofwx 8. It thus follows that in the setting of continuously quasi-normed spaces the terms locally PL-con¨ex and PL-con¨ex are equivalent. A superclass of the PL-convex spaces is the class of `` A-convex'' spaces introduced by N. Kalton inwx 8 . Kalton calls a complex quasi-normed space A-con¨ex if it admits an equivalent plurisubharmonic norm. These spaces have the characteristic ability to control the behavior of vector-valued analytic functions taking values in them in the following way. Let X be an A-convex space and let AX0Ž.denote the collection of all X-valued functions which are continuous on the closed unit disc of the complex plane and analytic in its interior. X is said to satisfy theŽ. weak Maximum Modulus PrincipleŽ. MMP if there is a constant M ) 0 so that for all Ž. fgAX0 we have f Ž.0 F M max fzŽ.. <<zs1 A result of Kaltonwx 8 says that the A-convex quasi-Banach spaces are precisely the spaces X satisfying the MMP. We conclude this section by recalling the definitions of two continuously quasi-normed quasi-Banach spaces whose complex geometry will be of particular interest to us in Section 4. EFINITION ` Ž. D 2.4. Let 0 - p - and let XjjgIbe a family of complex Ž. Banach spaces. The l pi-sum of the family X igIis the linear space Ž. Ä <Ž. 55 `4 lXpjjgIj[f:IªDgIjXfjgX jfor all j and f p - where the 55? 55 ŽÝ 5Ž.5p.1rp quasi-norm on X is given by f p s jg I fj and Ý 5 Ž.5p ÄÝ 5Ž.5p 4 jg Ijfj [sup gFfj :F;Iand F is finite . DEFINITION 2.5. Let X be a complex Banach space. For 0 - p F `,we denote by LpŽ.V, S, m; X the Lebesgue]Bochner space of m-equivalence 526 HU AND MUPASIRI classes of strongly measurable functions f: V ª X for which Ž.p Ž.