Equivariant \(KK\)-theory and \(C^*\)-extensions
The author gives a description of equivariant \(KK\)-theory in terms of extensions of \(C^\ast\)-algebras generalizing that of non-equivariant case given by \textit{G. G. Kasparov} [Math. USSR, Izv. 16, 513-572 (1981); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 44, 571-636 (1980; Zbl 0448.46051)]. The main difference from the non-equivariant case is the way of stabilizing. One should use the \(C^*\)-algebra \(K_G\) of compact operators on the Hilbert space \(H=\bigoplus_{i=1}^\infty L^2(G)\) with the action of the group \(G\) given by the direct sum of its regular representations. The main result states that for \(G{-}C^*\)-algebras \(A\) and \(B\) the group \(KK^1_G(A,B)\) is isomorphic to the group of equivalence classes of invertible \(G\)-extensions of \(A\otimes K_G\) by \(B\otimes K_G\), where the equivalence relation can be taken to be (like in the non-equivariant case) either homotopy or unitary equivalence up to addition by degenerate extensions. It is also shown that a \(G\)-extension is invertible iff it admits a completely positive contractive equivariant section after tensoring by \(K_G\).
scientific article; zbMATH DE number 1463123