Entropy and determinants for unitary representations

Abstract

Ergodic theory includes several notions of entropy for probability-preserving actions of countable groups. These include Kolmogorov--Sinai entropy based on Følner 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 Szegő'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.

80p. [v2] Part I of v1. has been rewritten and expanded; some new main theorems added, material re-organized, some notation changed. Part II of v1. now a separate listing at arXiv:2507.08909. [v3] Title changed; keywords and MSC2020 added; some other minor corrections. [v4] Moderate re-writing throughout for brevity; some new organization and proofs in Section 4