Noncommutative Cartan C*-subalgebras are exactly the reduced crossed products by closed, purely outer inverse semigroup actions, and this decomposition is unique up to a canonical refinement.
Math.316 (2017)
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Noncommutative Cartan C*-subalgebras
Noncommutative Cartan C*-subalgebras are exactly the reduced crossed products by closed, purely outer inverse semigroup actions, and this decomposition is unique up to a canonical refinement.