Homotopy equivalence in unbounded $\operatorname{\mathit{KK}}$-theory

Abstract

We propose a new notion of unbounded KK-cycle, mildly generalizing unbounded Kasparov modules, for which the direct sum is well-defined. To a pair $\left(A,B\right)$ of $\sigma$-unital $C^{\ast }$-algebras, we can then associate a semigroup $\overline{UKK}\left(A,B\right)$ 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 $\overline{UKK}\left(A,B\right)$ is isomorphic to Kasparov’s KK-theory group KK$\left(A,B\right)$ 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.