Controlled KK-theory, and a Milnor exact sequence

2020/11/22 by Willett, Rufus, Yu, Guoliang
#19K35 #46L80 #46L85 #FOS: Mathematics #K-Theory and Homology (math.KT) #Operator Algebras (math.OA)

paper · doi:10.48550/arxiv.2011.10906

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 K₀(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^*-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.

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