无界KK-理论中的同伦等价
Homotopy equivalence in unbounded KK-theory
摘要：
本文提出了无界$K\!K$ -圈的新概念，它是无界Kasparov模的一个温和的推广，并且它的直和是有定义的.对于$σ$ -酉$C^{*}$ -代数对$(A,B)$，我们可以结合一个由无界圈的同伦等价类构成的半群$\overline{U\!K\!K}(A,B)$，并证明了这个半群实际上是一个阿贝尔群.在$A$可分的情况下，我们的群$\overline{U\!K\!K}(A,B)$通过有界变换同构于Kasparov的$K\!K$ -理论群$K\!K(A,B)$.我们还讨论了退化圈的各种概念，并证明了无界圈上的同伦关系与算子同伦和退化圈的加法所产生的关系是一致的。
英文摘要：
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 $σ$-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.
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