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2020 Homotopy equivalence in unbounded $\operatorname{\mathit{KK}}$-theory
Koen van den Dungen, Bram Mesland
Ann. K-Theory 5(3): 501-537 (2020). DOI: 10.2140/akt.2020.5.501

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 (A,B) of σ-unital C-algebras, we can then associate a semigroup U K K ¯ ( 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 U K K ¯ ( 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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Koen van den Dungen. Bram Mesland. "Homotopy equivalence in unbounded $\operatorname{\mathit{KK}}$-theory." Ann. K-Theory 5 (3) 501 - 537, 2020. https://doi.org/10.2140/akt.2020.5.501

Information

Received: 20 September 2019; Revised: 7 November 2019; Accepted: 24 November 2019; Published: 2020
First available in Project Euclid: 11 August 2020

zbMATH: 07237240
MathSciNet: MR4132745
Digital Object Identifier: 10.2140/akt.2020.5.501

Subjects:
Primary: 19K35

Keywords: Kasparov theory

Rights: Copyright © 2020 Mathematical Sciences Publishers

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Vol.5 • No. 3 • 2020
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