Controlled KK-theory, and a Milnor exact sequence

TL;DR

This paper introduces controlled KK-theory groups for pairs of C*-algebras, relating them to KK-groups via a Milnor exact sequence, and identifies Rordam's KL-group with an inverse limit of these controlled groups.

Contribution

It defines controlled KK-theory groups and establishes their connection to KK-groups through a Milnor exact sequence, extending previous results without assuming the UCT.

Findings

Controlled KK-theory groups are introduced for pairs of C*-algebras.

A Milnor exact sequence relates controlled KK-groups to KK-groups.

Rordam's KL-group is identified with an inverse limit of controlled KK-groups.

Abstract

We introduce controlled $KK$-theory groups associated to a pair $(A,B)$ of separable $C^*$-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 R\o{}rdam's $KL$-group is identified with an inverse limit of our controlled $KK$-groups. In the case that the $C^*$-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.