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

Abstract

Well-known work of Renault shows that if $\mathcal{E}$ is a twist over a second countable, effective, étale groupoid $G$, then there is a naturally associated Cartan subalgebra of the reduced twisted groupoid C*-algebra $C_r^{\ast }(G;E)$, 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 $C_r^{\ast }(S;{\mathcal{E}}_S)$ to be a Cartan subalgebra of $C_r^{\ast }(G;\mathcal{E})$.

v4: Fixed a gap in the proof of Prop 3.14 by adding Lemma 3.22. Added figures. [34p] Accepted for publication in International Mathematics Research Notices, published by Oxford University Press

References in corpus (3)

• Normalizers and Approximate Units for Inclusions of C*-Algebras
• Analyzing the Weyl construction for dynamical Cartan subalgebras
• The local bisection hypothesis for twisted groupoid C*-algebras