Twisted Steinberg algebras, regular inclusions and induction

Abstract

Given a field $\mathbb{K}$ and an ample (not necessarily Hausdorff) groupoid $G$, we define the concept of a line bundle over $G$ inspired by the well-known concept from the theory of C*-algebras. If $E$ is such a line bundle, we construct the associated twisted Steinberg algebra in terms of sections of $E$, extending the original construction introduced independently by Steinberg in 2010 and by Clark, Farthing, Sims and Tomforde in a 2014 paper (originally announced in 2011). We also generalize (strictly, in the non-Hausdorff case) the 2023 construction of (cocycle) twisted Steinberg algebras of Armstrong, Clark, Courtney, Lin, McCormick and Ramagge. We then extend Steinberg’s theory of induction of modules, not only to the twisted case but to the much more general case of regular inclusions of algebras. Among our main results, we show that, under appropriate conditions, every irreducible module is induced by an irreducible module over a certain abstractly defined isotropy algebra. We also describe a process of disintegration of modules and use it to prove a version of the Effros–Hahn conjecture, showing that every primitive ideal coincides with the annihilator of a module induced from isotropy.