A Generalization of Renault's Theorem for Cartan Subalgebras

TL;DR

This paper extends Renault's theorem for Cartan subalgebras by removing the need for second countability and separability, replacing topological principality with effectiveness in the underlying groupoid.

Contribution

It generalizes Renault's theorem by weakening key assumptions, broadening its applicability to more groupoids and Cartan subalgebras.

Findings

The theorem holds without second countability.

Effectiveness replaces topological principality.

Broader class of groupoids and subalgebras covered.

Abstract

We prove a generalized version of Renault's theorem for Cartan subalgebras. We show that the original assumptions of second countability and separability are not needed. This weakens the assumption of topological principality of the underlying groupoid to effectiveness.