The uniqueness theorem for Kasparov theory

Abstract

Answering a question of Carri\'on et al in their recent landmark paper on C*-algebra classification, we prove a general uniqueness theorem for $KK$-theory. Given arbitrary separable C*-algebras $A$ and $B$ and a Cuntz pair consisting of two absorbing representations $\varphi,\psi: A\to\mathcal{M}(B\otimes\mathcal{K})$, the induced element of $KK(A,B)$ vanishes if and only if $\varphi$ and $\psi$ are strongly asymptotically unitarily equivalent. This improves upon the Lin-Dadarlat-Eilers stable uniqueness theorem. The conclusion is deduced by first showing the $K_1$-injectivity of an auxiliary C*-algebra associated to the C*-pair $(A,B)$, which is sometimes called the Paschke dual algebra in the literature. Most of the article is concerned with the treatment of an umbrella theorem, which yields such a uniqueness theorem for other variants of $KK$-theory. This encompasses nuclear $KK$-theory, ideal-related $KK$-theory, equivariant $KK$-theory, or any combinations thereof.