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 \[ \kappa :K^{i+1}(\Rep (A,B))\rightarrow KK^i(A,B),\;\;\;i\in \Ztwo, \] 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.
Homology, Homotopy and Applications, Vol. 2, 2000, No. 10, pp. 127-145