Unique Pseudo-Expectations for Dynamical $C^*$-Inclusions

Abstract

Let $(\mathcal{A},G,\alpha)$ be a $C^*$-dynamical system. Unique extension properties for the $C^*$-inclusion $\mathcal{A} \subseteq \mathcal{A} \rtimes_{\alpha,r} G$ have been studied extensively. In this paper, we investigate unique extension properties for some other natural ``dynamical'' $C^*$-inclusions, namely: $\bullet$ $C_r^*(G) \subseteq \mathcal{A} \rtimes_{\alpha,r} G$; $\bullet$ $\mathcal{A}^G \subseteq \mathcal{A}$, where $\mathcal{A}^G$ is the fixed-point subalgebra; $\bullet$ $Z(\mathcal{A})^c \subseteq \mathcal{A} \rtimes_{\alpha,r} G$, where $Z(\mathcal{A})^c = Z(\mathcal{A})' \cap (\mathcal{A} \rtimes_{\alpha,r} G)$ is the relative commutant of the center. Our hope is that a larger selection of examples will enable progress on some lingering open problems, in particular (1) the relationship between aperiodicity and the unique pseudo-expectation property and (2) the relationship between the faithful unique pseudo-expectation property and norming.