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.

中文翻译：

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非交换 Cartan C*-子代数

我们使用条件期望的唯一性、相对对易或纯粹的外部逆半群作用以多种方式刻画 Exel 的非交换 Cartan 子代数。我们描述了非交换嘉当子代数的交叉积分解在哪种意义上是唯一的。我们将非交换 Cartan 子代数的性质与非周期包含和对偶群群的有效性联系起来。特别是，我们扩展了雷诺对可交换嘉当子代数的表征。