Groupoid models for relative Cuntz-Pimsner algebras of groupoid correspondences
Ralf Meyer
TL;DR
The paper builds a universal groupoid model for the relative Cuntz–Pimsner algebras associated to groupoid correspondences by introducing a universal action on a space Ω(R) and forming the transformation groupoid Ω(R) ⋊ I(G, X). It shows this groupoid C*-algebra C*(Ω(R) ⋊ I(G, X)) is canonically isomorphic to the relative Cuntz–Pimsner algebra of the C*(X) − C*(G) correspondence relative to C*(G_R), with the model arising from a universal inverse semigroup action. The construction unifies existing constructions for Cuntz algebras, graph algebras, topological graphs, and self-similar groupoid actions, and provides a universal property framework for analyzing morphisms and simplicity criteria. The results yield a flexible tool for understanding groupoid-related C*-algebras and their interrelations, including Morita equivalences and universal representations, across a broad class of examples.
Abstract
A groupoid correspondence on an etale, locally compact groupoid induces a C*-correspondence on its groupoid C*-algebra. We show that the Cuntz-Pimsner algebra for this C*-correspondence relative to an ideal associated to an open invariant subset of the groupoid is again a groupoid C*-algebra for a certain groupoid. We describe this groupoid explicitly and characterise it by a universal property that specifies its actions on topological spaces. Our construction unifies the construction of groupoids underlying the C*-algebras of topological graphs and self-similar groups.