$KK$-theory as the $K$-theory of $C^ *$-categories

Abstract

Let complex $C^*$ algebras be endowed with a norm-continuous action of a fixed compact second countable group. From a separable $C^*$-algebra $A$ and a $\sigma $-unital $C^{*}$-algebra $B$, we construct a $C^{*}$-category Rep ($A,B$) and an isomorphism \[\kappa :K^{i+1}(\text{Rep} (A,B))\rightarrow KK^i(A,B),\;\;\;i\in \mathbb{Z}_2,\] where on the left-hand side are Karoubi's topological $K$-groups, and on the right-hand side are Kasparov's equivariant bivariant $K$-groups.