Compact Group Automorphisms, Addition Formulas and Fuglede-Kadison Determinants

TL;DR

This paper establishes a connection between entropy of group actions and Fuglede-Kadison determinants for amenable groups, providing new formulas and approximation methods for these determinants.

Contribution

It introduces an p-version of Bowen's entropy, addition formulas for group extensions, and an approximation formula for Fuglede-Kadison determinants.

Findings

Entropy equals the logarithm of the Fuglede-Kadison determinant for invertible elements.

Developed an p-version of Bowen's topological entropy.

Provided an approximation formula for Fuglede-Kadison determinants.

Abstract

For a countable amenable group \Gamma and an element f in the integral group ring Z\Gamma being invertible in the group von Neumann algebra of \Gamma, we show that the entropy of the shift action of \Gamma on the Pontryagin dual of the quotient of Z\Gamma by its left ideal generated by f is the logarithm of the Fuglede-Kadison determinant of f. For the proof, we establish an \ell^p-version of Rufus Bowen's definition of topological entropy, addition formulas for group extensions of countable amenable group actions, and an approximation formula for the Fuglede-Kadison determinant of f in terms of the determinants of perturbations of the compressions of f.