2025/02/04 by Pitts, David R.
#46L05 #FOS: Mathematics #Operator Algebras (math.OA)
paper · doi:10.48550/arxiv.2502.01975
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⊆ \mathcal C is a pseudo-Cartan inclusion with Cartan envelope \mathcal B⊆ \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.