Equilibrium states on higher-rank Toeplitz noncommutative solenoids

TL;DR

This paper studies equilibrium states on higher-dimensional noncommutative tori and their Toeplitz extensions, revealing a rich structure of states parametrized by measures on solenoids.

Contribution

It introduces Toeplitz noncommutative solenoids as direct limits of Toeplitz extensions and characterizes their equilibrium states under natural dynamics.

Findings

Existence of a large simplex of equilibrium states at each positive inverse temperature.

Parametrization of equilibrium states by probability measures on solenoids.

Explicit computation of equilibrium states for the considered noncommutative algebras.

Abstract

We consider a family of higher-dimensional noncommutative tori, which are twisted analogues of the algebras of continuous functions on ordinary tori, and their Toeplitz extensions. Just as solenoids are inverse limits of tori, our Toeplitz noncommutative solenoids are direct limits of the Toeplitz extensions of noncommutative tori. We consider natural dynamics on these Toeplitz algebras, and compute the equilibrium states for these dynamics. We find a large simplex of equilibrium states at each positive inverse temperature, parametrised by the probability measures on an (ordinary) solenoid.