Convex sets associated to C⁎-algebras

For a separable unital C⁎C⁎-algebra AA and a separable McDuff II1-factor M  , we show that the space Homw(A,M)Homw(A,M) of weak approximate unitary equivalence classes of unital ⁎-homomorphisms A→MA→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)Hom(N,RU). Many separable unital C⁎C⁎-algebras, including all (separable unital) nuclear C⁎C⁎-algebras, have the property that for any McDuff II1-factor M  , Homw(A,M)Homw(A,M) is affinely homeomorphic to the trace space of AA. In general Homw(A,M)Homw(A,M) and the trace space of AA do not share the same data (several examples are provided). We characterize extreme points of Homw(A,M)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∞)C⁎(F∞) is reflected in the fact that for any unital separable A,Homw(A,M)A,Homw(A,M) may be embedded as a face in Homw(C⁎(F∞),M)Homw(C⁎(F∞),M). We also extend Brown's construction to apply more generally to Hom(A,MU)Hom(A,MU).