Decomposable Weak Expectations

TL;DR

This paper introduces decomposable weak expectations for representations of separable unital C*-algebras, establishing conditions for their existence and linking them to decomposable measures on the state space.

Contribution

It defines decomposable weak expectations, provides necessary and sufficient conditions for their existence, and connects them to decomposable measures on the state space.

Findings

Decomposable weak expectations exist under specific conditions.

Existence of decomposable weak expectations is equivalent to the presence of decomposable measures.

An example of a decomposable weak expectation is provided.

Abstract

In this article we define a special class of weak expectations for a representation of a separable unital C*-algebra, called decomposable weak expectation. We give necessary and sufficient conditions for such kind of weak expectations to exist for a given representation. Then we define decomposable measures on the state space of a C*-algebra and show that the GNS representation of a state admits a decomposable weak expectation if and only if there is a decomposable measure on the state space. Further we give an example of a decomposable weak expectation.