Algebraic K-theory of Fredholm modules and KK-theory
  Kasparov $KK$-groups $KK(A,B)$ are represented as homotopy groups of the Pedersen-Weibel nonconnective algebraic $K$-theory spectrum of the additive category of Fredholm $(A,B)$-bimodules for $A$ and $B$, respectively, a separable and $\sigma$-unital trivially graded real or complex $C^*$-algebra acted upon by a fixed compact metrizable group. 
Tamaz Kandelaki