C *-uniqueness Results for Groupoids

Abstract

For a 2nd-countable locally compact Hausdorff étale groupoid ${\mathcal{G}}$ with a continuous $2$-cocycle $\sigma $ we find conditions that guarantee that $\ell ^1 ({\mathcal{G}},\sigma )$ has a unique $C^*$-norm.

Detecting ideals in reduced crossed product C*-algebras of topological dynamical systems

“…In this section we prove the L 1 -ideal intersection property for certain twisted groupoids, in the same spirit as [AO22] did for cocycle twisted groupoids. In view of applications in Section 6, our statements are established in slightly greater generality.…”

“…In the remainder of this section we aim to prove that if sufficiently many fibres of the isotropy bundle have the ℓ 1 -ideal intersection property, then the full isotropy bundle has the L 1 -ideal intersection property. Following the same strategy as in [AO22], we achieve this by decomposing any C * -completion of L 1 (I G , I E ) as a C * -bundle over G (0) . We will need the following lemma in order to describe the fibres of this bundle.…”

“…The proof of Theorem C employs groupoid techniques in order to set up an induction scheme. The assumptions on amenable subgroups are exploited to construct a twisted groupoid, which is analysed from the point of view of the L 1 -ideal intersection property for groupoids, previously studied in [AO22]. Ultimately our induction argument reduces the maximal possible Hirsch length of polycyclic subgroups.…”

“…Ultimately our induction argument reduces the maximal possible Hirsch length of polycyclic subgroups. The following generalisation of work from [AO22] allows us to analyse the twisted groupoids we obtain when applying our strategy. It might be of independent interest.…”

A uniqueness theorem for twisted groupoid C*-algebras

“…Renault's reconstruction theorem is of particular importance to the classification program for C*-algebras, given Li's recent article [29] showing that every simple classifiable C*-algebra has a Cartan subalgebra (and is therefore a twisted groupoid C*-algebra), and the work of Barlak and Li [7,8] describing the connections between the UCT problem and Cartan subalgebras in C*-algebras. The increasing interest in twisted groupoid C*-algebras (see, for instance, [3,6,10,11,15,16,17,23]) has also recently inspired the introduction of twisted Steinberg algebras, which are a purely algebraic analogue of twisted groupoid C*-algebras (see [4,5]).…”