Ideal structure and pure infiniteness of ample groupoid -algebras

Abstract

In this paper, we study the ideal structure of reduced C * -algebras C * r (G) associated to étale groupoids G. In particular, we characterize when there is a one-to-one correspondence between the closed, two-sided ideals in C * r (G) and the open invariant subsets of the unit space G (0) of G. As a consequence, we show that if G is an inner exact, essentially principal, ample groupoid, then C * r (G) is (strongly) purely infinite if and only if every non-zero projection in C0(G (0) ) is properly infinite in C * r (G). We also establish a sufficient condition on the ample groupoid G that ensures pure infiniteness of C * r (G) in terms of paradoxicality of compact open subsets of the unit space G (0) .Finally, we introduce the type semigroup for ample groupoids and also obtain a dichotomy result: Let G be an ample groupoid with compact unit space which is minimal and topologically principal. If the type semigroup is almost unperforated, then C * r (G) is a simple C * -algebra which is either stably finite or strongly purely infinite.

The Groupoids of Adaptable Separated Graphs and Their Type Semigroups

“…Now, we will show that Typ(G) coincides with some recently defined type semigroups [12,33,34]. To this end, we recall the definition in [12] (which is equivalent to the ones given in [33] and [34], as shown in [34]).…”

A dichotomy for groupoid -algebras

“…Remark 3.4. A related definition of paradoxicality appears as [3,Definition 4.4]; our definitions is slightly more flexible in that we do not require that the sets s(E i ) in (2) cover A, and we do not insist on any orthogonality amongst the sets r(E i ). It is clear that if G is (E, k, l)-paradoxical in the sense of Bönicke and Li for some E, k, l, then it is (k, l)-paradoxical in our sense.…”

“…Our definition of the type semigroup of G is related to the definition given by Bönicke and Li in [3, Definition 5.3], though our definition is somewhat more algebraic and does not involve amplifying and passing to levels of G (0) ×N. The apparent discrepancy can be resolved using Proposition 5.7 below: Write R := N × N regarded as a discrete principal groupoid, and let KG denote the groupoid G × R. Identifying R (0) with N in the obvious way, it is straightforward to see that the type semigroup of G as defined in [3,Definition 5.3] is precisely the type semigroup of KG as defined by Definition 5.4. Proposition 5.7 shows that the type semigroups, in the sense of our definition, of KG and of G coincide, so we see that our definition of the type semigroup of G agrees with [3, Definition 5.3] up to canonical isomorphism.…”

“…As we were preparing the final version of this article, the article [3] was posted on the arXiv. In Sections 4 and 5 of that paper, Bönicke and Li have independently obtained a number of the results in this paper, for groupoids with compact unit spaces.…”

Stone duality and quasi‐orbit spaces for generalised C∗‐inclusions

“…And they are interesting objects in their own right (compare [26]). Moreover, under some freeness assumptions, the quasi‐orbit map is a homeomorphism (see [17, 27, 28, 41, 55, 58, 60] for the classical case of group actions, [25, Theorem 3.2] for partial group actions, [40, Theorem 6.8] for Fell bundles over discrete groups or [6, Theorem 3.17] for a recent result for groupoid ‐algebras of étale groupoids).…”