标题： 局部化，Whitehead群，以及Atiyah猜想 显示英文标题

标题： Localization, Whitehead groups, and the Atiyah Conjecture

摘要： 设 Wh^w(G) 为整数群环 ZG 上的方阵的 K_1 群，这些矩阵不一定可逆，但在通过希尔伯特空间完成之后诱导弱同构。 设 D(G) 为 ZG 在与群冯诺依曼代数相关的算子代数 U(G) 中的除法闭包。 设 C 是包含所有自由群并且在定向并和具有初等可解商的扩张下封闭的群的最小类。 设 G 是属于 C 的无扭群。那么我们证明 Wh^w(G) 同构于 K_1(D(G))。 此外我们证明 D(G) 是一个斜域，因此 K_1(\D (G)) 是 D(G) 中乘法单位群的交换化。 显示英文摘要

摘要： Let Wh^w(G) be the K_1-group of square matrices over the integral group ring ZG which are not necessarily invertible but induce weak isomorphisms after passing to Hilbert space completions. Let D(G) be the division closure of ZG in the algebra U(G) of operators affiliated to the group von Neumann algebra. Let C be the smallest class of groups which contains all free groups and is closed under directed unions and extensions with elementary amenable quotients. Let G be a torsionfree group which belongs to C. Then we prove that Wh^w(G) is isomorphic to K_1(D(G)). Furthermore we show that D(G) is a skew field and henc K_1(\D(G)) is the abelianization of the multiplicative group of units in D(G).