The equivalence relation groupoid and the cyclic homology (Q2883275)

Given a smooth compact manifold \(M\) an prescribed open cover \(\{U_i\}^n_{i=1}\) of \(M\) gives rise to a groupoid \({\mathcal R}\) which is a non-commutative version of \(M\), see [\textit{A. Connes}, Noncommutative geometry. Transl. from the French by Sterling Berberian. San Diego, CA: Academic Press (1994; Zbl 0818.46076)]. Moreover, a \(C^*\)-algebra \(C^*({\mathcal R})\) is associated to this groupoid by using the algebraic operations on the vector space \(C_0({\mathcal R})\). In this geometrical context, the authors establish a Morita equivalence between the \(C^*\)-algebra \(C^*({\mathcal R})\) and the \(C^*\)-algebra \(C(M)\); their \(K_0\)-groups isomorphism is obtained by using a result of \textit{R. Exel} [K-Theory 7, No. 3, 285--308 (1993; Zbl 0792.46051)]. Finally, following \textit{J.-L. Brylinski} and \textit{V. Nistor}'s paper [K-Theory 8, No. 4, 341--365 (1994; Zbl 0812.19003)], the localized cyclic homology of \(C^*({\mathcal R})\) is described.