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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. <http://www.numdam.org/item?id=CM_1975__30_3_211_0> © Foundation Compositio Mathematica, 1975, tous droits réservés. L’accès aux archives de la revue « Compositio Mathematica » (http:// http://www.compositio.nl/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). 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 http://www.numdam.org/ 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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