Groupoid \(C^*\)-algebras and operator \(K\)-theory (Q2785087)

This paper is a concise interesting survey of the \(KK\)-theory equivariant with respect to a groupoid. The corresponding \(KK\)-functor, denoted by \(KK_{\mathcal G}(A,B),\) has three variables, namely the groupoid \({\mathcal G}\) and the \({\mathcal G}\)-algebras \(A,B.\) Besides its usual properties in \(A\) and \(B,\) in particular the existence of the Kasparov product, its functoriality in \({\mathcal G}\) is useful in computing \(K\)-groups and has applications to the Baum-Connes conjecture which states that for a groupoid \({\mathcal G}\) (under some mild assumptions) a certain map between a geometrically defined group \(K^{\text{top}}_*({\mathcal G})\) and \(KK_{\mathcal G}\) is an isomorphism.NEWLINENEWLINENEWLINEReviewer's remark: The paper of \textit{R. Popescu} [``E-théorie équivariante et groupoïdes'', C. R. Acad. Sci., Sér. I, Math. 331, No. 3, 223-228 (2000; Zbl 0970.19003)] is devoted to the similar problem.NEWLINENEWLINEFor the entire collection see [Zbl 0972.00029].