A localization algebra approach to the Baum-Connes conjecture for extensions

Abstract

For an extension $1\to N\to \hat{I}“\stackrel{q}{\to }\hat{I}“/N\to 1$ of discrete countable groups, it is known that the Baum-Connes conjecture with coefficients holds for $\hat{I}“$ if it holds for $\hat{I}“/N$ and $q^{-1}(F)$ for any finite subgroup $F$ of $\hat{I}“/N$. In this paper, we employ a localization algebra approach to prove this result.

We changed the paper's title, since we just used a localization algebra approach to prove a known result for the Baum-Connes conjecturte in this version. In the first version of the paper, there exists an essential mistake in Theorem 3.13 (thanks to Prof. Ralf Meyer and Prof. Shintaro Nishikawa), thus we strengthened the conditions in that theorem