Maps with the unique extension property and \({C}^\ast\)-extreme points (Q1623866)

The author gives an unexpected example of a weak\(^*\) compact set \(K\) of Hilbert space operators that is \(B\)-convex over an abelian von Neumann algebra \(B\), but does not have any \(B\)-extreme points. He investigates \(C^*\)-extreme points for singly generated weak\(^*\) compact \(C^*\)-convex sets of compact operators. He extends the theory of maps with the unique extension property to the context when Hilbert spaces are replaced by self-dual Hilbert \(C^*\)-modules. He then proves a Krein-Milman-type theorem for \(\ell^\infty(A)\) in the case when \(A\) has separable predual, under the assumption that the finite part in the central decomposition of \(A\) is injective.