Pseudo-Cartan Inclusions

Abstract

A pseudo-Cartan inclusion is a regular inclusion having a Cartan envelope. Unital pseudo-Cartan inclusions were classified by Pitts; we extend this classification to include the non-unital case. The class of pseudo-Cartan inclusions coincides with the class of regular inclusions having the faithful unique pseudo-expectation property and can also be described using the ideal intersection property. We describe the twisted groupoid associated with the Cartan envelope of a pseudo-Cartan inclusion. These results significantly extend previous results obtained for the unital setting. We explore properties of pseudo-Cartan inclusions and the relationship between a pseudo-Cartan inclusion and its Cartan envelope. For example, if $\mathcal D\subseteq \mathcal C$ is a pseudo-Cartan inclusion with Cartan envelope $\mathcal B\subseteq \mathcal A$, then $\mathcal C$ is simple if and only if $\mathcal A$ is simple. Also every regular $*$-automorphism of $\mathcal C$ uniquely extends to a $*$-automorphism of $\mathcal A$. We show that the inductive limit of pseudo-Cartan inclusions with suitable connecting maps is a pseudo-Cartan inclusion, and the minimal tensor product of pseudo-Cartan inclusions is a pseudo-Cartan inclusion. Further, we describe the Cartan envelope of pseudo-Cartan inclusions arising from these constructions. We conclude with some applications and a few open questions.