The stable uniqueness theorem for equivariant Kasparov theory

Abstract: This paper examines and strengthens the Cuntz--Thomsen picture of equivariant Kasparov theory for arbitrary second-countable locally compact groups, in which elements are given by certain pairs of cocycle representations between $\mathrm{C}^*$-dynamical systems. The main result is a stable uniqueness theorem that generalizes a fundamental characterization of ordinary $KK$-theory by Lin and Dadarlat--Eilers. Along the way, we prove an equivariant Cuntz--Thomsen picture analog of the fact that the equivalence relation of homotopy agrees with the (\emph{a priori} stronger) equivalence relation of stable operator homotopy. The results proved in this paper will be employed as the technical centerpiece in forthcoming work of the authors to classify certain amenable group actions on Kirchberg algebras by equivariant Kasparov theory.