The Fuglede-Kadison determinant, theme and variations

TL;DR

This paper reviews the Fuglede-Kadison determinant for finite von Neumann algebras, its generalizations, and discusses its relevance to $L^2$-torsion in topology, connecting algebraic and topological concepts.

Contribution

It provides a comprehensive review of the Fuglede-Kadison determinant and explores its applications to $L^2$-torsion, highlighting connections between operator algebras and topology.

Findings

Clarifies the definition of Fuglede-Kadison determinants

Discusses generalizations for invertible elements in Banach algebras

Suggests relevance to $L^2$-torsion in topology

Abstract

We review the definition of determinants for finite von Neumann algebras, due to Fuglede and Kadison (1952), and a generalisation for appropriate groups of invertible elements in Banach algebras, from a paper by Skandalis and the author (1984). After some reminder on K-theory and Whitehead torsion, we hint at the relevance of these determinants to the study of $L^2$-torsion in topology.