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arXiv · 2403.15255

Non-traditional Cartan subalgebras in twisted groupoid C*-algebras

Anna Duwenig·Dana P. Williams·Joel Zimmerman
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Abstract

Well-known work of Renault shows that if $\mathcal{E}$ is a twist over a second countable, effective, \'etale groupoid $G$, then there is a naturally associated Cartan subalgebra of the reduced twisted groupoid C*-algebra $C^*_{r}(G; E)$, and that every Cartan subalgebra of a separable C*-algebra arises in this way. However twisted C*-algebras of non-effective groupoids $G$ can also possess Cartan subalgebras: In work by the first author together with Gillaspy, Norton, Reznikoff, and Wright, sufficient conditions on a subgroupoid $S$ of $G$ were found that ensure that $S$ gives rise to a Cartan subalgebra in the cocycle-twisted C*-algebra of $G$. In this paper, we extend these results to general twists $\mathcal{E}$, and we refine the conditions on the subgroupoid for $C^*_{r}(S;\mathcal{E}_S)$ to be a Cartan subalgebra of $C^*_{r}(G;\mathcal{E})$.

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Anna Duwenig, Dana P. Williams, Joel Zimmerman. 2024-03-22. Non-traditional Cartan subalgebras in twisted groupoid C*-algebras. https://doi.org/10.1093/imrn%2Frnaf011

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