Kasparov $KK$-groups $KK(A,B)$ are represented as homotopy groupsof the Pedersen-Weibel nonconnective algebraic $K$-theory spectrumof the additive category of Fredholm $(A,B)$-bimodules for $A$ and$B$, respectively, a separable and $\sigma$-unital triviallygraded real or complex $C^*$-algebra acted upon by a fixed compactmetrizable group.
Journal of Homotopy and Related Structures, Vol. 1(2006), No. 1, pp. 195-218 http://jhrs.rmi.acnet.ge/volumes/2006/n1a9/