Controlled $KK$-theory and a Milnor exact sequence

Abstract

We introduce controlled $KK$-theory groups associated to a pair $(A,B)$ of separable $C^{\ast }$-algebras. Roughly, these consist of elements of the usual $K$-theory group $K_0(B)$ that approximately commute with elements of $A$. Our main results show that these groups are related to Kasparov’s $KK$-groups by a Milnor exact sequence, in such a way that Rørdam’s $KL$-group is identified with an inverse limit of our controlled $KK$-groups.
In the case that the $C^{\ast }$-algebras involved satisfy the UCT, our Milnor exact sequence agrees with the Milnor sequence associated to a $KK$-filtration in the sense of Schochet, although our results are independent of the UCT. Applications to the UCT will be pursued in subsequent work.