Fuglede-Kadison determinant: theme and variations.

Abstract

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.