A pair (M,D) consisting of a von Neumann algebra M containing a MASA D is a Cartan pair if the span of the unitaries in M which normalize D has weak-* dense span in M and if there is a normal conditional expectation of M onto D. In the 1970s, Feldman and Moore used measured equivalence relations and a cohomology theory for equivalence relations to classify the family of all separably acting Cartan pairs.

In this talk, I will describe joint work with Allan Donsig and Adam Fuller. We use extensions of inverse semigroups to classify all Cartan pairs. Our approach is more algebraic than the approach of Feldman-Moore. Also, our approach does not explicitly use much measure theory, instead the measure theory is encoded into the meet semi-lattice structure of a fundamental inverse semigroup constructed from the Cartan pair.