Section de Mathématiques, Université de Genève, CH 1211 Genève 4, Switzerland.
Proc Natl Acad Sci U S A. 2013 Oct 1;110(40):15864-77. doi: 10.1073/pnas.1202059110. Epub 2013 Sep 30.
We review the definition of determinants for finite von Neumann algebras, due to Fuglede and Kadison [Fuglede B, Kadison R (1952) Ann Math 55:520-530], and a generalization for appropriate groups of invertible elements in Banach algebras, from a paper by Skandalis and the author (1984). After some discussion of K-theory and Whitehead torsion, we indicate the relevance of these determinants to the study of -torsion in topology. Contents are as follows:1. The classical setting. 2. On von Neumann algebras and traces. 3. Fuglede-Kadison determinant for finite von Neumann algebras. 4. Motivating question. 5. Brief reminder of K0, K1, K1(top), and Bott periodicity. 6. Revisiting the Fuglede-Kadison and other determinants. 7. On Whitehead torsion. 8. A few lines on L2-torsion.
我们回顾了由 Fuglede 和 Kadison [Fuglede B，Kadison R（1952）Ann Math 55：520-530] 提出的有限 von Neumann 代数的行列式定义，以及 Skandalis 和作者（1984）的一篇论文中对适当的巴拿赫代数可逆元群的推广。在讨论了 K-理论和 Whitehead 挠率之后，我们指出了这些行列式与拓扑中 -挠率研究的相关性。内容如下：1. 经典设置。2. 关于 von Neumann 代数和迹。3. 有限 von Neumann 代数的 Fuglede-Kadison 行列式。4. 激发问题。5. K0、K1、K1(top)和 Bott 周期性的简要回顾。6. 重新审视 Fuglede-Kadison 和其他行列式。7. 关于 Whitehead 挠率。8. L2-挠率的几行。
