Unique conditional expectations for abelian $C^*$-inclusions

TL;DR

This paper characterizes when a unital abelian $C^*$-subalgebra inclusion admits a unique conditional expectation, especially in separable cases, and provides an example with unique expectation but multiple pseudo-expectations.

Contribution

It offers a topological characterization of unique conditional expectations for abelian $C^*$-inclusions and presents the first example with this property but multiple pseudo-expectations.

Findings

Characterization of unique conditional expectations in topological terms

First example of an inclusion with a unique expectation but multiple pseudo-expectations

Applicable to separable abelian $C^*$-algebra inclusions

Abstract

Let $D \subseteq A$ be an inclusion of unital abelian $C^*$-algebras. In this note we characterize (in topological terms) when there is a unique conditional expectation $E:A \to D$, at least when $A$ is separable. As an application, we provide the first example of an inclusion with a unique conditional expectation, but multiple pseudo-expectations (in the sense of Pitts).