Cartan Triples

TL;DR

This paper introduces Cartan triples, a new generalization of Cartan MASAs in von Neumann algebras, establishing a correspondence with Clifford extensions and extending spectral theorems and Aoi's theorem within this framework.

Contribution

It defines Cartan triples, links them to Clifford extensions of inverse semigroups, and extends key spectral and structural theorems in the context of von Neumann algebras.

Findings

Established a one-to-one correspondence between Cartan triples and Clifford extensions.

Developed a spectral theorem for bimodules in this setting.

Extended Aoi's theorem to Cartan triples.

Abstract

We introduce the class of Cartan triples as a generalization of the notion of a Cartan MASA in a von Neumann algebra. We obtain a one-to-one correspondence between Cartan triples and certain Clifford extensions of inverse semigroups. Moreover, there is a spectral theorem describing bimodules in terms of their support sets in the fundamental inverse semigroup and, as a corollary, an extension of Aoi's theorem to this setting. This context contains that of Fulman's generalization of Cartan MASAs and we discuss his generalization in an appendix.