Dense subalgebras of purely infinite simple groupoidC*-algebras

Abstract

A simple Steinberg algebra associated to an ample Hausdorff groupoid G is algebraically purely infinite if and only if the characteristic functions of compact open subsets of the unit space are infinite idempotents. If a simple Steinberg algebra is algebraically purely infinite, then the reduced groupoid C * -algebra C * r (G) is simple and purely infinite. But the Steinberg algebra seems to small for the converse to hold. For this purpose we introduce an intermediate * -algebra B(G) constructed using corners 1 U C * r (G)1 U for all compact open subsets U of the unit space of the groupoid. We then show that if G is minimal and effective, then B(G) is algebraically properly infinite if and only if C * r (G) is purely infinite simple. We apply our results to the algebras of higher-rank graphs.

Structure theory and stable rank for C*-algebras of finite higher-rank graphs

“…If contains a cycle, then another application of [14, Corollary 5.7] (see also [4, Remark 5.8]) gives that is purely infinite. Since is unital, simple and purely infinite, it contains two isometries with orthogonal ranges, so [34, Proposition 6.5] gives .…”

Structure theory and stable rank for C*-algebras of finite higher-rank graphs

“…In Section 3 we characterise which k-graphs yield C * -algebras with stable rank 1 (Theorem 3.1 and Corollary 3.4) and show how the dimension of the tori that form the components of their spectra can be read off from (the skeleton of) the k-graph, see Proposition 3. 3.…”

Tensor Products of Steinberg Algebras

“…For instance, the structure theory and ideal theory of Steinberg algebras are highly applicable to Leavitt path and inverse semigroup algebras [19,21,34,36]. Steinberg algebras have also led to new insights and progress in C * -algebras, especially on the topic of simple groupoid C * -algebras (see [10,11,22]). Besides those already mentioned, there are interesting examples of Steinberg algebras that arise out of higher-rank graphs, self-similar graphs, and various kinds of dynamical systems [9,10,16,20,22].…”