arXiv subjects

Joel Zimmerman

Publications and source records attributed to Joel Zimmerman.

3 recordsLinked to original sources

arXiv2025

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

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 $C^*_{r}(G; E)$, 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 $C^*_{r}(G;\mathcal{E})$.

math.OA↗

arXiv2022

Renault's $j$-map for Fell bundle $C^*$-algebras

If $p \colon \mathcal B\to G$ is a Fell bundle over an étale groupoid, then we show that there is an norm reducing injective linear map $j \colon C^*_r(G;\mathcal B)\to Γ_{0}(G;\mathcal B)$ generalizing the well know map $j \colon C^*_{r}(G)\to C_{0}(G)$ in the case of an étale groupoid.

math.OA↗

arXiv2022

A Steinberg algebra approach to étale groupoid C*-algebras

We construct the full and reduced C*-algebras of an ample groupoid from its complex Steinberg algebra. We also show that our construction gives the same C*-algebras as the standard constructions. In the last section, we consider an arbitrary locally compact, second-countable, étale groupoid, possibly non-Hausdorff. Using the techniques developed for Steinberg algebras, we show that every $*$-homomorphism from Connes' space of functions to $B(\mathcal{H})$ is automatically I-norm bounded. Previously, this was only known for Hausdorff groupoids.

math.OA↗