$C^\ast$-extreme maps and nests

Abstract

The generalized state space $ S_{\mathcal{H}}(\mathcal{\mathcal{A}})$ of all unital completely positive (UCP) maps on a unital $C^*$-algebra $\mathcal{A}$ taking values in the algebra $\mathcal{B}(\mathcal{H})$ of all bounded operators on a Hilbert space $\mathcal{H}$, is a $C^\ast$-convex set. In this paper, we establish a connection between $C^\ast$-extreme points of $S_{\mathcal{H}}(\mathcal{A})$ and a factorization property of certain algebras associated to the UCP map. In particular, this factorization property of some nest algebras is used to give a complete characterization of those $C^\ast$-extreme maps which are direct sums of pure UCP maps. This significantly extends a result of Farenick and Zhou [Proc. Amer. Math. Soc. 126 (1998)] from finite to infinite dimensional Hilbert spaces. Also it is shown that normal $C^\ast$-extreme maps on type $I$ factors are direct sums of normal pure UCP maps if and only if an associated algebra is reflexive. Further, a Krein-Milman type theorem is established for $C^\ast$-convexity of the set $ S_{\mathcal{H}}(\mathcal{A})$ equipped with bounded weak topology, whenever $\mathcal{A}$ is a separable $C^\ast$-algebra or it is a type $I$ factor. As an application, we provide a new proof of a classical factorization result on operator valued Hardy algebras.