Facial topology and extreme operators

Let K   be a simplex and let A0(K)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)A0(K) is a nice operator (that is, T⁎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)A0(K) is isometrically isomorphic to c0(I)c0(I) for some set I  . From here we derive the corresponding result for A(K)A(K), namely, if K   is a simplex such that each extreme operator from any Banach space to A(K)A(K) is a nice operator, then the set of extreme points of K is finite.