Banach algebras associated to twisted étale groupoids: simplicity and pure infiniteness
Krzysztof Bardadyn, Bartosz Kwaśniewski, Andrew McKee
TL;DR
This work generalizes simplicity and pure infiniteness from twisted étale groupoid C*-algebras to the broader setting of Banach algebras associated with twisted groupoids. It develops a unified framework of reduced, essential, and exotic groupoid Banach algebras $F_{ ext{R}}( ext{G}, ext{L})$, graded by inverse semigroups of bisections, and uses generalized conditional expectations to relate these algebras to $C_0(X)$ via $j$-maps. The authors establish that topological freeness and minimality yield simplicity for essential algebras, and provide $n$-filling and locally contracting criteria for pure infiniteness, extending results to $L^P$-operator algebras and non-Hausdorff settings. Applications span twisted partial actions, Roe-type algebras, tight inverse semigroup algebras, graph algebras, and self-similar group actions on graphs, thereby solving open questions for $L^p$-algebras and unifying several strands of groupoid and Banach-algebra theory. The framework significantly broadens the applicability of groupoid methods beyond the C*-algebra paradigm, enabling new constructions and classifications in Banach-algebra contexts.
Abstract
We define reduced and essential Banach algebras associated to a twisted étale (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.