Simplicity of algebras associated to non-Hausdorff groupoids

Abstract

We prove a uniqueness theorem and give a characterization of simplicity for Steinberg algebras associated to non-Hausdorff ample groupoids. We also prove a uniqueness theorem and give a characterization of simplicity for the C * -algebra associated to non-Hausdorffétale groupoids. Then we show how our results apply in the setting of tight representations of inverse semigroups, groups acting on graphs, and self-similar actions. In particular, we show that the C * -algebra and the complex Steinberg algebra of the self-similar action of the Grigorchuk group are simple but the Steinberg algebra with coefficients in Z 2 is not simple.2010 Mathematics Subject Classification. 16S99, 16S10, 22A22, 46L05, 46L55.

Quotients of Étale Groupoids and the Abelianizations of Groupoid C*-Algebras

“…for all α ∈ G, αs(α) = r(α)α = α holds, (3) for all (α, β) ∈ G (2) , s(αβ) = s(β) and r(αβ) = r(α) hold,…”

“…every γ ∈ G, there exists γ ∈ G which satisfies (γ , γ), (γ, γ ) ∈ G (2) and s(γ) = γ γ and r(γ) = γγ . Since the element γ in (5) is uniquely determined by γ, γ is called the inverse of γ and denoted by γ −1 .…”

“…By Lemma 1.2.3, the universal norm takes values in [0, ∞). Since the left regular representations of C(G) induces a faithful *-representation of C(G), the universal norm becomes a C*-norm (see [2,Section 4]). The completion of C(G) by universal norm is denoted by C * (G).…”

A von Neumann algebraic approach to self-similar group actions

“…In some cases with the assumption that G is amenable, it has been shown that O Gmax and O Gmin are isomorphic by proving the simplicity of the reduced groupoid C * -algebra. See [3,11]. It is one of a problems on the C * -algebraic aspect that several Cuntz-Pimsner algebras from a self-similar group action might not be isomorphic especially in non-amenable case.…”

“…If G is an amenable group, O Gmin and O Gmax are isomorphic in some cases. See[3,11] for example. To understand O Gmin we consider another algebra from self-similar actions.…”