Symmetrization of Cuntz' Picture for the Kasparov $KK$-Bifunctor

Given $C^*$-algebras $A$ and $B$, we generalize the notion of a quasi-homomorphism from $A$ to $B$ in the sense of Cuntz by considering quasi-homomorphisms from some $C^*$-algebra $C$ to $B$ such that $C$ surjects onto $A$ and the two maps forming the quasi-homomorphism agree on the kernel of this surjection. Under an additional assumption, the group of homotopy classes of such generalized quasi-homomorphisms coincides with $KK(A, B)$. This makes the definition of the Kasparov bifunctor slightly more symmetric and provides more flexibility in constructing elements of $KK$-groups. These generalized quasi-homomorphisms can be viewed as pairs of maps directly from $A$ (instead of various $C$'s), but these maps need not be $*$-homomorphisms.