KK-theory of reduced free-product \(C^*\)-algebras (Q1919642)

Let \(A_r\) denote the reduced free-product \(C^*\)-algebra of a set of \(K\)-nuclear \(C^*\)-algebras endowed with states, and let \(A\) be the full free-product \(C^*\)-algebra. One of the purposes of this paper is to get a candidate for the inverse in \(KK(A_r,A)\) of the canonical morphism from \(A\) to \(A_r\) and to show that this morphism always realizes a so named \(K\)-theoretical equivalence. The tools developed here allow to give a unified treatment and to extend to a larger set of \(C^*\)-algebras the computation of the \(KK\)-groups of full free-product \(C^*\)-algebras. To this purpose the notion of \(K\)-pointed \(C^*\)-algebra is introduced and the main properties of such \(C^*\)-algebras are considered.