Simplicity of inverse semigroup and étale groupoid algebras

Abstract

In this paper, we prove that the algebra of an étale groupoid with totally disconnected unit space has a simple algebra over a field if and only if the groupoid is minimal and effective and the only function of the algebra that vanishes on every open subset is the null function. Previous work on the subject required the groupoid to be also topologically principal, but we do not. Furthermore, we provide the first examples of minimal and effective but not topologically principal étale groupoids with totally disconnected unit spaces. Our examples come from self-similar group actions of uncountable groups.The main application of our work is to provide a description of the simple contracted inverse semigroup algebras, thereby answering a question of Munn from the seventies.Using Galois descent, we show that simplicity of étale groupoid and inverse semigroup algebras depends only on the characteristic of the field and can be lifted from positive characteristic to characteristic 0. We also provide examples of inverse semigroups and étale groupoids with simple algebras outside of a prescribed set of prime characteristics.

On the simplicity of Nekrashevych algebras of contracting self-similar groups

“…Given any inverse semigroup S, the tight ideal T K (S) of K 0 S is the ideal generated by all products f ∈F (e − f ), where e ∈ E(S), and F ⊆ E(S) is a finite cover of e. This ideal arises naturally as the kernel of the surjective homomorphism K 0 S → KG T (S), where KG T (S) is the Steinberg algebra [39] of the tight groupoid G T (S) of the inverse semigroup S as defined by Exel [15]; see [40,41] for details.…”

“…The singular ideal of K 0 S was introduced by the authors in [41]. An element a ∈ K 0 S is singular if, for all 0 = e ∈ E(S), there exists 0 = f ≤ e with af = 0.…”

“…The singular elements form a two-sided ideal I K (S) containing T K (S) called the singular ideal. One of the main results of [41] is then the following theorem.…”

“…The Steinberg algebra [39] of this groupoid is the algebraic counterpart of the Nekrashevych C * -algebra and is called the Nekrashevych algebra of the self-similar group (with coefficients in a field). Nekrashevych algebras over fields were first studied in [37]; further work was done in [11,28] and by the authors in [41]. When the group is trivial, the Nekrashevych algebra is the Leavitt algebra [32], a celebrated finitely presented simple algebra.…”

“…Associated to any self-similar group G is a congruence-free inverse semigroup S. The ample groupoid associated to the self similar group G is the tight groupoid of S (in the sense of Exel [15]). In [41], the authors show that the Steinberg algebra of the tight groupoid of a congruencefree inverse semigroup has a unique maximal ideal and give an explicit description of its preimage in the algebra of the inverse semigroup, called the singular ideal. We use this description of the singular ideal, specialized to the case of inverse semigroups associated to self-similar group actions, to attack the simplicity question for Nekrashevych algebras.…”