A more symmetric picture for Kasparov's KK-bifunctor

TL;DR

This paper introduces a generalized, more symmetric notion of quasihomomorphisms for C*-algebras, enhancing the construction and understanding of Kasparov's KK-bifunctor by allowing pairs of maps from A that are not necessarily *-homomorphisms.

Contribution

It extends the concept of quasihomomorphisms to a broader class, making the KK-bifunctor more symmetric and flexible for constructing KK-elements.

Findings

Generalized quasihomomorphisms coincide with KK-groups under certain conditions.

Provides a more symmetric and flexible framework for KK-theory.

Enables construction of KK-elements via pairs of maps from A.

Abstract

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