Groupoid models for relative Cuntz-Pimsner algebras of groupoid correspondences

TL;DR

This paper establishes a connection between groupoid correspondences and Cuntz-Pimsner algebras, showing they can be represented as groupoid C*-algebras, unifying various constructions in the field.

Contribution

It introduces a method to model relative Cuntz-Pimsner algebras of groupoid correspondences as explicit groupoid C*-algebras, with a universal property characterization.

Findings

The construction yields a new groupoid model for certain C*-algebras.

It unifies existing models for topological graph and self-similar group C*-algebras.

Provides an explicit description and universal property for the resulting groupoid.

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.