Barycentric decompositions in the space of weak expectations

TL;DR

This paper characterizes the extreme points of the space of weak expectations for a representation of a separable C*-algebra, providing a clearer understanding of its convex structure using operator theory.

Contribution

It explicitly identifies the extreme points of the space of weak expectations, advancing the barycentric decomposition theory in operator algebras.

Findings

Explicit characterization of extreme points

Application of operator theoretic techniques

Enhanced understanding of convex structure in weak expectations

Abstract

The space of weak expectations for a given representation of a (unital) separable C*-algebra is a compact convex set of (unital) completely positive maps in the BW topology, when it is non-empty. An application of the classical Choquet theory gives a barycentric decomposition of a weak expectation in that set. However, to complete the barycentric picture, one needs to know the extreme points of the compact convex set in question. In this article, we explicitly identify the set of extreme points of the space of weak expectations for a given representation, using operator theoretic techniques.