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.

Classification of regular subalgebras of the hyperfinite II1 factor

“…This bijective correspondence between regular inclusions B ⊂ M and cocycle crossed products can be viewed in different ways. In[DFP18], this is interpreted in the language of inverse semigroups, where cocycle actions of groupoids become extensions of inverse semigroups.…”