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

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.