Renault's equivalence theorem for reduced groupoid \(C^\ast\)-algebras
The purpose of the paper is threefold: firstly to give a precise statement and proof of the equivalence result for reduced groupoid \(C^\ast\)-algebras; secondly to illustrate that the equivalence result for reduced algebras is compatible with the result for the full algebras and Rieffel induction; and thirdly, and possibly most importantly, to highlight the role of the linking groupoid, which is the main tool in proofs.NEWLINENEWLINE The main results of the paper imply that if \(G\) and \(H\) are equivalent groups, then their reduced groupoid \(C^\ast\)-algebras \(C^\ast_r(G)\) and \(C^\ast_r(H)\) are Morita equivalent via a quotient \(X_r\) of \(X\) (\(X\) is Renault's imprimitivity bimodule). This equivalence is compatible in a natural way with the Equivalence Theorem for full groupoid \(C^\ast\)-algebras.
scientific article; zbMATH DE number 6090242