Entropy and determinants for unitary representations

TL;DR

This paper introduces new entropy-like invariants for unitary representations of groups, connecting ergodic theory, operator algebras, and non-commutative determinants, extending classical limit theorems.

Contribution

It develops novel quantities for unitary representations that generalize entropy concepts and relate them to Fuglede--Kadison determinants, bridging ergodic theory and operator algebra.

Findings

New invariants for unitary representations evaluated as Fuglede--Kadison determinants

Determinantal formulas generalize Szeg\

},

Abstract

Ergodic theory includes several notions of entropy for probability-preserving actions of countable groups. These include Kolmogorov--Sinai entropy based on F\o lner sequences for amenable groups, entropy defined using a random ordering of the group, and Bowen's sofic entropy for sofic groups. In this work we pursue these notions across an analogy between ergodic theory and representation theory. We arrive at new quantities associated to unitary representations of groups and representations of other C*-algebras. Our main results show that these new quantities can often be evaluated as Fuglede--Kadison determinants. The resulting determinantal formulas offer various non-commutative generalizations of Szeg\H{o}'s limit theorem for Toeplitz determinants. They make contact with Arveson's theory of subdiagonal subalgebras, and also with some entropy formulas in the ergodic theory of actions by automorphisms of compact Abelian groups.