Vol. 5, No. 3, 2020

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

Koen van den Dungen and Bram Mesland

Vol. 5 (2020), No. 3, 501–537

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 (A,B) of σ-unital C-algebras, we can then associate a semigroup 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 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.

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Kasparov theory
Mathematical Subject Classification 2010
Primary: 19K35
Received: 20 September 2019
Revised: 7 November 2019
Accepted: 24 November 2019
Published: 28 July 2020
Koen van den Dungen
Mathematisches Institut
Universität Bonn
Bram Mesland
Mathematisch Instituut
Universiteit Leiden