Abstract
Let K be a simplex and let A0(K) denote the space of continuous affine functions on K vanishing at a fixed extreme point, denoted by 0. We prove that if any extreme operator T from a Banach space X to A0(K) is a nice operator (that is, T⁎, the adjoint of T, preserves extreme points), then the facial topology of the set of extreme points different from 0 is discrete, and so A0(K) is isometrically isomorphic to c0(I) for some set I. From here we derive the corresponding result for A(K), namely, if K is a simplex such that each extreme operator from any Banach space to A(K) is a nice operator, then the set of extreme points of K is finite.
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