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- W3124802064 abstract "We propose a new notion of unbounded $K!K$-cycle, mildly generalising unbounded Kasparov modules, for which the direct sum is well-defined. To a pair $(A,B)$ of $sigma$-unital $C^{*}$-algebras, we can then associate a semigroup $overline{U!K!K}(A,B)$ 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 $overline{U!K!K}(A,B)$ is isomorphic to Kasparov's $K!K$-theory group $K!K(A,B)$ 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." @default.
- W3124802064 created "2021-02-01" @default.
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- W3124802064 date "2020-07-28" @default.
- W3124802064 modified "2023-09-27" @default.
- W3124802064 title "Homotopy equivalence in unbounded KK-theory" @default.
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- W3124802064 doi "https://doi.org/10.2140/akt.2020.5.501" @default.
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