$C^*$-extreme maps and $*$-homomorphisms from $C(X)$ to finite von Neumann algebras

Abstract

Given a unital inclusion of $C^*$-algebras $\mathcal C \subset \mathcal A$ and a unital inclusion $\mathcal C \subset \mathcal B$ into a von Neumann algebra $\mathcal B$, we investigate the extreme points of unital completely positive maps from $\mathcal A$ to $\mathcal B$ that fix $\mathcal C$ denoted by $UCP_\mathcal C(\mathcal A, \mathcal B)$. This space is obviously convex and $C^*$-convex with respect to the $C^*$-algebra $\mathcal C' \cap \mathcal B$. In this article we show that the $\mathcal C' \cap \mathcal B$-extreme points are exactly the $*$-homomorphisms that fix $\mathcal C$ when $\mathcal A$ is commutative and $\mathcal B$ has a normal faithful center valued trace. This generalizes a result due to Farenick and Morenz where $\mathcal C = \mathbb C 1$ and $\mathcal B = M_n(\mathbb C)$.