# Compact convex sets and their affine function spaces

Abstract (Summary)

(Uncorrected OCR) Abstract of thesis entitled "Compact convex sets and their affine function spaces" submitted by Chan Jor-Ting for the degree of Doctor of Philosophy, May 1987 Throughout E will be a real Banach space, X a compact Hausdorff space and K a compact convex subset of a locally convex Hausdorff space. We shall denote by C(X,E) and A(K,E) the vector spaces of continuous E-valued functions on X and the continuous E-valued affine functions on K respectively. Both of them are Banach spaces under the supremum norm. Let deK be the set of extreme points of K. If K is a 'Bauer simplex', then deK is closed and A(K,E) ?C{deK,E). Therefore Banach spaces of the type A(K, E) are more general than those C(X,E) spaces. The main objective of this thesis is to extend some of the results for C(X,E) to A(K, E). Altogether there are five chapters. The first chapter focus on vector measure theory. We prove that if A is a subspace of C(X,E), then.every bounded linear functional on A admits an '^-boundary' ?-valued measure representation. This result generalizes the Choquet-Bishop-de Leeuw theorem to vector-valued function spaces. It has been obtained by the author independently of Saab and Talagrand. We also employ vector measure techniques to give a simple and transparent proof of a characterization theorem for extreme functional on A(K,E). The second chapter deals with the 'facial topology' defined on deK. Let K(E) denote the vector space of all bounded linear operators on E under the strong operator topology. The main theorem of this chapter asserts that $ : deK ? t (E) is continuous for the respective topologies above if and only if 3> 'acts' on A(K,E). This result will be needed in subsequent chapters. As an application we deduce a theorem of Andersen and Atkinson concerning facially continuous functions from deK into a Banach algebra. The aim of the third chapter is the study of the M-structure of A(K, E). In brief the Af-structure theory examines the operators in the centralizer, the M-summands and the Af-ideals. The centralizer of E will be denoted by 2(E). We characterize the centralizer of A(K,E) as facially continuous functions on deK into Z(E). We also prove that it is the strong operator closure of Z{A(K)) 2(E). Similar results are obtained for the biafline functions spaces. The above characterization is then used to yield a complete description for M-summands of A(K, E). Some special cases are discussed. If E is reflexive and has no non-trivial Af-ideal, we prove that J is an Af-ideal in A(K) <8>c E if and only if there is a closed 'split face' F of K such that / = {/ e A(K) ? E : f\F = 0}. In the fourth chapter we try to extend Jerion's generalized Banach-Stone theorem to aflSne function spaces. Specifically we prove that if K\ and if2 are Choquet simplexes and if E satisfies a certain condition (**) then every isometric isomorphism on A(Ki,E) onto A(K*Bibliographical Information:*