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Central decompositions for compact convex sets

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Central decompositions for compact convex sets COMPOSITIO MATHEMATICA A. J. ELLIS Central decompositions for compact convex sets Compositio Mathematica, tome 30, no 3 (1975), p. 211-219. <> © Foundation Compositio Mathematica, 1975, tous droits réservés. L’accès aux archives de la revue « Compositio Mathematica » (http:// implique l’accord avec les conditions générales d’utilisation ( Toute utilisation commer- ciale ou impression systématique est constitutive d’une infraction pé- nale. Toute copie ou impression de ce fichier doit contenir la pré- sente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques 211 CENTRAL DECOMPOSITIONS FOR COMPACT CONVEX SETS A. J. Ellis COMPOSITIO MATHEMATICA, Vol. 30, Fasc. 3, 1975, pag. 211-219 Noordhoff International Publishing Printed in the Netherlands 1. Introduction In this paper we continue the investigation, begun in [8], into facial decompositions for compact convex sets K. In particular we study conditions on K under which the Bishop decomposition determines A(K), or at least determines the centre of A(K) ; the special case when K is the state space of a unital C*-algebra is’ investigated in this connection. In the final section we prove a result for function algebras which is related to facial decompositions, and we also give a simple geometrical proof of the Hoffman-Wermer theorem. We are indebted to several mathematicians for discussions concerning the contents of this paper, and in particular to E. G. Effros and E. St~rmer. 2. Terminology and preliminaries Let K be a compact convex subset of a locally convex Hausdorff space and let A(K) denote the Banach space of all continuous real-valued affine functions on K, endowed with the supremum norm. The set of extreme points of K will be denoted by ôK, and its closure by ôK. Th

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