Homotopy equivalence in unbounded KK-theory

Koen van den Dungen, B. Mesland
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引用次数: 7

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.
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无界kk理论中的同伦等价
我们提出了无界$K\!K$ -循环的新概念,温和地推广了无界卡斯帕罗夫模,其直接和是定义良好的。对于一对$(A,B)$$\sigma$ -单$C^{*}$ -代数,我们可以关联一个无界环的同伦等价类的半群$\overline{U\!K\!K}(A,B)$,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卡斯帕罗夫的$K\!K$ -理论群$K\!K(A,B)$。讨论了简并环的各种概念,并证明了无界环上的同伦关系与简并环的算子同伦和加法所产生的关系是一致的。
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