Essential Cartan subalgebras of $C^*$-algebras

TL;DR

This paper introduces the concept of essential commutative Cartan pairs in $C^*$-algebras, linking them to effective twisted groupoid $C^*$-algebras and exploring their automorphism groups.

Contribution

It generalizes the notion of Cartan pairs, characterizes them via effective twisted groupoids, and establishes a correspondence between automorphisms of twists and the pairs.

Findings

Essential Cartan pairs correspond to effective twisted groupoid $C^*$-algebras.

The underlying twisted groupoid is unique up to isomorphism.

Automorphism groups of twists and Cartan pairs are isomorphic.

Abstract

We define essential commutative Cartan pairs of $C^*$-algebras generalising the definition of Renault and show that such pairs are given by essential twisted groupoid $C^*$-algebras as defined by Kwa\'sniewski and Meyer. We show that the underlying twisted groupoid is effective, and is unique up to isomorphism among twists over effective groupoids giving rise to the essential commutative Cartan pair. We also show that for twists over effective groupoids giving rise to such pairs, the automorphism group of the twist is isomorphic to the automorphism group of the induced essential Cartan pair via explicit constructions.