Convex Sets Associated to C*-Algebras

TL;DR

This paper studies the structure of the space of weak approximate unitary equivalence classes of *-homomorphisms from a separable unital C*-algebra to a McDuff II_1-factor, revealing its convex properties and characterizing extremal points.

Contribution

It introduces a convex structure on Hom_w(A,M), characterizes extreme points, and extends Brown's construction to a broader context, connecting these spaces to trace spaces and universality properties.

Findings

Hom_w(A,M) is a closed, convex subset of a Banach space.

When A is nuclear, Hom_w(A,M) is affinely homeomorphic to the trace space of A.

Conditions for extremality of elements in Hom_w(A,M) are established.

Abstract

For A a separable unital C*-algebra and M a separable McDuff II_1-factor, we show that the space Hom_w(A,M) of weak approximate unitary equivalence classes of unital *-homomorphisms A \rightarrow 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,R^U). When A is nuclear, Hom_w(A,M) is affinely homeomorphic to the trace space of A, but in general Hom_w(A,M) and the trace space of A do not share the same data (several examples are provided). We characterize extreme points of Hom_w(A,M) in the case where either A or M is amenable; and we give two different conditions -- one necessary and the other sufficient -- for extremality in general. The universality of C*(F_\infty) is reflected in the fact that for any unital separable A, Hom_w(A,M) may be embedded as a face in Hom_w(C*(F_\infty),M). We also extend Brown's construction to apply more generally to Hom(A,M^U).