Homotopy equivalence in unbounded KK-theory

Abstract

We propose a new notion of unbounded $KK$-cycle, mildly generalising unbounded Kasparov modules, for which the direct sum is well-defined. To a pair $(A,B)$ of $\sigma$-unital $C^{\ast }$-algebras, we can then associate a semigroup $\overline{UKK}(A,B)$ of homotopy equivalence classes of unbounded cycles, and we prove that this semigroup is in fact an abelian group. In case $A$ is separable, our group $\overline{UKK}(A,B)$ is isomorphic to Kasparov's $KK$-theory group $KK(A,B)$ via the bounded transform. We also discuss various notions of degenerate cycles, and we prove that the homotopy relation on unbounded cycles coincides with the relation generated by operator-homotopies and addition of degenerate cycles.

33 pages