Noncommutative Cartan C*-subalgebras

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.

This version is accepted for publication in the Transactions of the American Mathematical Society. Proposition 4.8 is reformulated (due to a previous gap in the proof). As a consequence, we replaced almost faithful by faithful in a few places

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