Unique Pseudo-Expectations for Dynamical $C^{\ast }$-Inclusions

Let $(\mathcal{A},G,\alpha )$ be a $C^{\ast }$-dynamical system. Unique extension properties for the $C^{\ast }$-inclusion $\mathcal{A}\subseteq \mathcal{A}{⋊}_{\alpha ,r}G$ have been studied extensively. In this paper, we investigate unique extension properties for some other natural ``dynamical’’ $C^{\ast }$-inclusions, namely: $\bullet$ $C_r^{\ast }(G)\subseteq \mathcal{A}{⋊}_{\alpha ,r}G$; $\bullet$ ${\mathcal{A}}^G\subseteq \mathcal{A}$, where ${\mathcal{A}}^G$ is the fixed-point subalgebra; $\bullet$ $Z(\mathcal{A}{)}^c\subseteq \mathcal{A}{⋊}_{\alpha ,r}G$, where $Z(\mathcal{A}{)}^c=Z(\mathcal{A}{)}^{\mathrm{\prime }}\cap (\mathcal{A}{⋊}_{\alpha ,r}G)$ 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.