Unique expectations for discrete crossed products

TL;DR

This paper characterizes when the inclusion of a unital $C^*$-algebra into its crossed product by a discrete group has unique conditional or pseudo-expectations, linking these properties to the underlying dynamics and strengthening existing simplicity results.

Contribution

It provides a dynamical characterization of unique expectations in crossed products and enhances previous results on $C^*$-simplicity of such algebras.

Findings

Characterization of unique conditional expectations based on dynamics

Conditions for unique pseudo-expectations in crossed products

Strengthened results on $C^*$-simplicity of crossed products

Abstract

Let $G$ be a discrete group acting on a unital $C^*$-algebra $\mathcal{A}$ by $*$-automorphisms. We characterize (in terms of the dynamics) when the inclusion $\mathcal{A} \subseteq \mathcal{A} \rtimes_r G$ has a unique conditional expectation, and when it has a unique pseudo-expectation (in the sense of Pitts). Likewise for the inclusion $\mathcal{A} \subseteq \mathcal{A} \rtimes G$. As an application, we (slightly) strengthen results of Kishimoto and Archbold-Spielberg concerning $C^*$-simplicity of $\mathcal{A} \rtimes_r G$.