Measured inverse semigroups and their actions on von Neumann algebras and equivalence relations

TL;DR

This paper provides a unified and rigorous proof that certain regular inclusions of von Neumann algebras and equivalence relations can be characterized as crossed products with cocycle actions of quotient groupoids, filling a gap in the literature.

Contribution

It offers a comprehensive proof connecting inverse semigroups, groupoids, and their actions on von Neumann algebras and equivalence relations, generalizing known results.

Findings

Regular inclusions arise as crossed products with quotient groupoids

Unified approach using inverse semigroup and groupoid correspondence

Fills gaps in the literature with rigorous proofs

Abstract

It is known to experts that certain regular inclusions of von Neumann algebras arise as crossed products with cocycle actions of the canonical quotient groupoids associated with the inclusions. Similarly, `strongly normal' inclusions of standard equivalence relations arise as semi-direct products with cocycle actions of the quotient groupoids. However, to the author's knowledge, rigorous proofs of these results in full generality are absent in the literature. In this article, we exploit the usual correspondence between inverse semigroups and groupoids, and give a unified approach to proving these `folklore' results and fill this gap in the literature.