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 $\ast$-automorphism of $\mathcal{C}$ having $\mathcal{D}$ as an invariant subspace uniquely extends to a $\ast$-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.

This version fixes typos, contains changes made in response to comments and suggestions from referees and readers, and also includes corrections to several issues found by one of the referees. To appear in Canadian Mathematical Communications. 62 pages