Abstract Renault proved in 2008 [22, Theorem 5.2] that if G is a topologically principal groupoid, then C 0 ( G ( 0 ) ) is a Cartan subalgebra in C r ⁎ ( G , Σ ) for any twist Σ over G . However, there are many groupoids which are not topologically principal, yet their (twisted) C ⁎ -algebras admit Cartan subalgebras. This paper gives a dynamical description of a class of such Cartan subalgebras, by identifying conditions on a 2-cocycle c on G and a subgroupoid S ⊆ G under which C r ⁎ ( S , c ) is Cartan in C r ⁎ ( G , c ) . When G is a discrete group, we also describe the Weyl groupoid and twist associated to these Cartan pairs, under mild additional hypotheses.

中文翻译：

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非主扭曲群群 C⁎-代数的嘉当子代数

摘要 Renault 在 2008 年证明了 [22, Theorem 5.2] 如果 G 是拓扑主群群，则 C 0 ( G ( 0 ) ) 是 C r ⁎ ( G , Σ ) 中的 Cartan 子代数，对于 G 上的任何扭曲 Σ。然而，有许多群群不是拓扑主体，但它们的（扭曲的）C ⁎ -代数承认嘉当子代数。本文通过确定 G 上的 2-cocycle c 和子群 S ⊆ G 上的条件，其中 C r ⁎ ( S , c ) 是 C r ⁎ ( G , C ） 。当 G 是一个离散群时，我们还描述了与这些 Cartan 对相关的 Weyl groupoid 和扭曲，在温和的附加假设下。