Banach algebras associated to twisted \'{e}tale groupoids: simplicity and pure infiniteness

TL;DR

This paper extends fundamental results on simplicity and pure infiniteness from $C^*$-algebras to twisted Banach algebras associated with étale groupoids, including $L^p$-operator algebras, under various conditions.

Contribution

It introduces new definitions of reduced and essential Banach algebras for twisted étale groupoids and establishes criteria for simplicity and pure infiniteness, solving an open problem for $L^p$-operator algebras.

Findings

Essential Banach algebras have the ideal intersection property for topologically free groupoids.

Simplicity of the algebra is equivalent to the minimality of the groupoid.

Provides pure infiniteness criteria using $n$-filling or locally contracting conditions.

Abstract

We define reduced and essential Banach algebras associated to a twisted \'{e}tale (not necessarily Hausdorff) groupoid $(\mathcal{G},\mathcal{L})$ and extend some fundamental results from $C^*$-algebras to this context. We prove that for topologically free groupoids the associated essential Banach algebras have the ideal interesection property, and thus such an algebra is simple if and only if the groupoid is minimal. We give conditions under which reduced algebras are essential (for example Hausdorffness of $\mathcal{G}$ is sufficient). This in particular solves the simplicity problem posed recently by Gardella-Lupini for $L^p$-operator algebras associated to $\mathcal{G}$. In addition, using either the $n$-filling or locally contracting condition we give pure infiniteness criteria for essential simple Banach algebras associated to $(\mathcal{G},\mathcal{L})$. This extends the corresponding $C^*$-algebraic results that were previously known to hold in the untwisted Hausdorff case. The results work nicely, and allow for characterisation of the generalized intersection property, in the realm of $L^P$-operator algebras where $P \subseteq [1,\infty]$ is a non-empty set of parameters. Such algebras cover in particular $L^p$-operator algebras, for $p\in [1,\infty]$, and their Banach $*$-algebra versions. We apply our results to Banach algebra crossed products by twisted partial group actions, Roe-type Banach algebras with coefficients in finite-rank operators on a Banach space, twisted tight $L^P$-operator algebras of inverse semigroups, graph $L^P$-operator algebras, and algebras associated to self-similar group actions on graphs. We also interpret our results in terms of twisted inverse semigroup actions and their crossed products.