Comparison of \(KE\)-theory and \(KK\)-theory (Q344491)

In his Ph.D. Thesis and in [J. Noncommut. Geom. 10, No. 3, 1083--1130 (2016; Zbl 1353.46054)], \textit{C. D. Dumitraşcu} considers an equivariant bivariant \(K\)-theory called \(KE\)-theory which is by definition a mix of Kasparov's \(KK\)-theory and A. Connes and N. Higson's picture of \(E\)-theory. Indeed there are natural concrete maps NEWLINE\[NEWLINEKK^G(A,B) \rightarrow KE^G(A,B) \rightarrow E^G(A,B),NEWLINE\]NEWLINE where \(G\) is a second countable locally compact group.NEWLINENEWLINEThe author of this article was suspicious that \(KE\)-theory could be either isomorphic to \(KK\)-theory or \(E\)-theory, and shows that the map \(KE^G(A,B) \rightarrow E^G(A,B)\) factors through a map \(KE^G(A,B) \rightarrow KK^G(A,B)\), the map NEWLINE\[NEWLINEKK^G(A,B) \rightarrow KE^G(A,B) \rightarrow KK^G(A,B)NEWLINE\]NEWLINE is the identity map, and hence, \(KK^G\)-theory is a direct summand of \(KE^G\)-theory.NEWLINENEWLINEThe proof uses the equivariant Cuntz picture of \(KK^G\)-theory and is otherwise the usual technical work with cycles and completely positive equivariant asymptotic morphisms.