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arXiv · 2001.08270

Cartan subalgebras for non-principal twisted groupoid $C^*$-algebras

Anna Duwenig·Elizabeth Gillaspy·Rachael Norton·Sarah Reznikoff·Sarah Wright
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Abstract

Renault proved in 2008 that if $G$ is a topologically principal groupoid, then $C_0(G^{(0)})$ is a Cartan subalgebra in $C^*_r(G, \Sigma)$ for any twist $\Sigma$ 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 $S \subseteq G$ 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.

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Anna Duwenig, Elizabeth Gillaspy, Rachael Norton, Sarah Reznikoff, Sarah Wright. 2020-01-22. Cartan subalgebras for non-principal twisted groupoid $C^*$-algebras. https://doi.org/10.1016/j.jfa.2020.108611

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