Abstract

A groupoid correspondence on an étale, locally compact groupoid induces a $C^{\ast }$-correspondence on its groupoid $C^{\ast }$-algebra. We show that the Cuntz–Pimsner algebra for this $C^{\ast }$-correspondence relative to an ideal associated to an open invariant subset of the groupoid is again a groupoid $C^{\ast }$-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 $C^{\ast }$-algebras of topological graphs and self-similar groups.

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