Noncommutative Cartan C*-subalgebras

TL;DR

This paper characterizes noncommutative Cartan subalgebras in C*-algebras through various algebraic and dynamical properties, extending classical results to the noncommutative setting.

Contribution

It provides multiple characterizations of noncommutative Cartan subalgebras and extends Renault's classical results to the noncommutative case.

Findings

Unique crossed product decomposition for noncommutative Cartan subalgebras

Relation between noncommutative Cartan subalgebras and aperiodic inclusions

Extension of Renault's characterization to noncommutative setting

Abstract

We characterise Exel's noncommutative Cartan subalgebras in several ways using uniqueness of conditional expectations, relative commutants, or purely outer inverse semigroup actions. We describe in which sense the crossed product decomposition for a noncommutative Cartan subalgebra is unique. We relate the property of being a noncommutative Cartan subalgebra to aperiodic inclusions and effectivity of dual groupoids. In particular, we extend Renault's characterisation of commutative Cartan subalgebras.