Noncommutative Cartan sub-algebras of C*-algebras

TL;DR

This paper extends Renault's characterization of C*-diagonals to noncommutative Cartan subalgebras using Fell bundles over inverse semigroups, providing a broader framework for understanding these structures.

Contribution

It introduces a noncommutative generalization of Renault's Cartan subalgebra characterization via Fell bundles over inverse semigroups.

Findings

Proves a theorem on the uniqueness of conditional expectations.

Reformulates Cartan subalgebras as twisted etale groupoids with noncommutative units.

Extends the theory to noncommutative Cartan subalgebras.

Abstract

J. Renault has recently found a generalization of the caracterization of C*-diagonals obtained by A. Kumjian in the eighties, which in turn is a C*-algebraic version of J. Feldman and C. Moore's well known Theorem on Cartan subalgebras of von Neumann algebras. Here we propose to give a version of Renault's result in which the Cartan subalgebra is not necessarily commutative [sic]. Instead of describing a Cartan pair as a twisted groupoid C*-algebra we use N. Sieben's notion of Fell bundles over inverse semigroups which we believe should be thought of as "twisted etale groupoids with noncommutative unit space". En passant we prove a theorem on uniqueness of conditional expectations.