$K$ -理论与变换群上的2-上圈同伦
$K$-theory and homotopies of 2-cocycles on transformation groups
摘要：
本文是作者研究局部紧Hausdorff广群$\mathcal{G}$上的2-上圈$ω= \{ω_t\}_{t \in [0,1]}$的同伦何时诱导约化扭广群$C^*$ -代数的$K$ -理论群的同构问题的第一步：$K_*(C^*_r(\mathcal{G}, ω_0)) \cong K_*(C^*_r(\mathcal{G}, ω_1)).$概括了Echterhoff、Lück、Phillips和Walters自2010年以来的工作，证明了如果$\mathcal{G} = G \ltimes X$是第二可数局部紧变换群，则当$G$满足Baum-Connes猜想时，$G \ltimes X$上的2-上圈同伦$ω= \{ω_t\}_{t \in [0,1]}$产生同构$K_*(C^*_r(G \ltimes X, ω_0)) \cong K_*(C^*_r(G \ltimes X, ω_1)).$
英文摘要：
This paper constitutes a first step in the author's program to investigate the question of when a homotopy of 2-cocycles $ω= \{ω_t\}_{t \in [0,1]}$ on a locally compact Hausdorff groupoid $\mathcal{G}$ induces an isomorphism of the $K$-theory groups of the reduced twisted groupoid $C^*$-algebras: $K_*(C^*_r(\mathcal{G}, ω_0)) \cong K_*(C^*_r(\mathcal{G}, ω_1)).$ Generalizing work of Echterhoff, Lück, Phillips, and Walters from 2010, we show that if $\mathcal{G} = G \ltimes X$ is a second countable locally compact transformation group, then whenever $G$ satisfies the Baum-Connes conjecture with coefficients, a homotopy $ω= \{ω_t\}_{t \in [0,1]}$ of 2-cocycles on $G \ltimes X$ gives rise to an isomorphism $K_*(C^*_r(G \ltimes X, ω_0)) \cong K_*(C^*_r(G \ltimes X, ω_1)).$
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