Banach algebras associated to twisted étale groupoids: inverse semigroup disintegration and representations on $L^p$-spaces

Abstract

We introduce Banach algebras associated to twisted étale groupoids $(\mathcal{G},\mathcal{L})$ and to twisted inverse semigroup actions. This provides a unifying framework for numerous recent papers on $L^p$-operator algebras and the theory of groupoid $C^*$-algebras. We prove disintegrations theorems that allow to study Banach algebras associated to $(\mathcal{G},\mathcal{L})$ as universal Banach algebras generated by $C_0(X)$ and a twisted inverse semigroup $S$ of partial isometries subject to some relations. They work best when the target of a representation is a dual Banach algebra. For representations on dual Banach spaces, they allow to extend representations to twisted Borel convolution algebras, which is crucial when the groupoid is non-Hausdorff. We establish fundamental norm estimates and hierarchy for full and reduced $L^p$-operator algebras for $(\mathcal{G},\mathcal{L})$ and $p \in [1,\infty]$, whose special cases have been studied recently by Gardella-Lupini, Choi-Gardella-Thiel and Hetland-Ortega. We show that in the constructions of $L^p$-analogues of Cuntz or graph algebras, by Phillips and Cortiñas-Rodr\'ıguez, the use of spatial partial isometries is not an assumption, in fact it is forced by the relations. We also introduce tight inverse semigroup Banach algebras that cover ample groupoid Banach algebras, and discuss Banach algebras associated to directed graphs. Our results cover non-Hausdorff étale groupoids and both real and complex algebras. Some of the results are new already for complex $C^*$-algebras.