$C^{\ast }$-extreme maps and $\ast$-homomorphisms from $C(X)$ to finite von Neumann algebras

Abstract

Given a unital inclusion of $C^{\ast }$-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 $UC{P}_{\mathcal{C}}(\mathcal{A},\mathcal{B})$. This space is obviously convex and $C^{\ast }$-convex with respect to the $C^{\ast }$-algebra ${\mathcal{C}}^{\prime }\cap \mathcal{B}$. In this article we show that the ${\mathcal{C}}^{\prime }\cap \mathcal{B}$-extreme points are exactly the $\ast$-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 $\mathcal{B}=M_n(\mathbb{C})$.

arXiv comment7 pages