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Entropy and determinants for unitary representations

Tim Austin

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.

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.

Paper Structure

This paper contains 36 sections, 52 theorems, 216 equations.

Table of Contents

  1. Introduction
  2. A Szegő limit theorem over amenable groups
  3. Analogies with entropy and ergodic theory
  4. Random orders and Schur complements
  5. Almost periodic entropy
  6. Operator algebras, C*-algebras and representations
  7. Linear algebra
  8. C*-algebras, von Neumann algebras, and representations
  9. Types and completely positive maps
  10. Tracial functionals and determinants
  11. Group algebras and positive definite functions
  12. Algebras of matrices and the pairing isomorphism
  13. Comparing and decomposing completely positive maps
  14. Determinants of completely positive maps
  15. A Szegő-like theorem over amenable groups
  16. ...and 21 more sections

Key Result

Theorem A

Any right Fø lner sequence $(F_n)_{n\ge 1}$ satisfies

Theorems & Definitions (111)

  • Theorem A
  • Theorem B
  • Theorem C
  • Lemma 2.1
  • Proposition 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • Lemma 2.5
  • proof
  • ...and 101 more