Annealed almost periodic entropy

TL;DR

This paper introduces new notions of entropy for representations of C*-algebras, especially for free group C*-algebras with random matrix representations, linking operator algebra entropy with random matrix large deviations.

Contribution

It defines annealed and zeroth-order AP entropy notions and derives explicit formulas and large deviations principles in the context of free group C*-algebras with random unitary matrices.

Findings

Explicit formulas for entropy in free group C*-algebras with random matrices.

New large deviations principles in random matrix theory.

Analogies with ergodic theory and Szeg ext{"o} limit theorems.

Abstract

This work studies certain notions of entropy that can be associated to (i) a representation of a separable, unital C*-algebra $\mathfrak{A}$ and (ii) an auxiliary random sequence $(\pi_n)_{n\ge 1}$ of finite-dimensional representations of $\mathfrak{A}$. This continues a previous research program into the properties of these entropy notions when each $\pi_n$ is deterministic, which uncovered a range of analogies with entropy in ergodic theory and also with non-commutative generalizations of Szeg\H{o}'s limit theorems. We associate two new notions of entropy to data as in (i) and (ii) above: `annealed' AP entropy, which is roughly a kind of first-moment average of deterministic AP entropies; and `zeroth-order' AP entropy, which controls the large deviations probabilities that certain positive definite functions appear in the representations $\pi_n$ at all. After developing some of this general theory, we then focus on the special case in which $\mathfrak{A}$ is the group C*-algebra of a finitely-generated free group and each $\pi_n$ is generated by choosing a tuple of $n$-by-$n$ unitary matrices independently at random from Haar measure. In that case, explicit formulas can be derived for some of our notions of entropy, and new large deviations principles in random matrix theory are obtained as a consequence.