Convex sets associated to C⁎-algebras

Abstract

For a separable unital C⁎-algebra A and a separable McDuff II1-factor M, we show that the space Homw(A,M) of weak approximate unitary equivalence classes of unital ⁎-homomorphisms A→M may be considered as a closed, bounded, convex subset of a separable Banach space – a variation on N. Brown's convex structure Hom(N,RU). Many separable unital C⁎-algebras, including all (separable unital) nuclear C⁎-algebras, have the property that for any McDuff II1-factor M, Homw(A,M) is affinely homeomorphic to the trace space of A. In general Homw(A,M) and the trace space of A do not share the same data (several examples are provided). We characterize extreme points of Homw(A,M) in many cases, and we give two different conditions – one necessary and the other sufficient – for extremality in general. The universality of C⁎(F∞) is reflected in the fact that for any unital separable A,Homw(A,M) may be embedded as a face in Homw(C⁎(F∞),M). We also extend Brown's construction to apply more generally to Hom(A,MU).