Equilibrium on Toeplitz extensions of higher dimensional noncommutative tori

TL;DR

This paper studies equilibrium states of a class of C*-algebras related to higher-dimensional noncommutative tori, revealing how these states are parametrized by tracial states and classical measures, generalizing known phase transition results.

Contribution

It characterizes equilibrium states of Toeplitz extensions of higher-dimensional noncommutative tori, linking them to tracial states and classical probability measures, extending previous phase transition analyses.

Findings

Equilibrium states are parametrized by tracial states of the noncommutative torus.

These states correspond to probability measures on classical tori.

Results generalize phase transition phenomena in Toeplitz noncommutative tori.

Abstract

The C*-algebra generated by the left-regular representation of $\mathbb{N}^n$ twisted by a $2$-cocycle is a Toeplitz extension of an $n$-dimensional noncommutative torus, on which each vector $r \in [0,\infty)^n$ determines a one-parameter subgroup of the gauge action. We show that the equilibrium states of the resulting C*-dynamical system are parametrised by tracial states of the noncommutative torus corresponding to the restriction of the cocycle to the vanishing coordinates of $r$. These in turn correspond to probability measures on a classical torus whose dimension depends on a certain degeneracy index of the restricted cocycle. Our results generalise the phase transition on the Toeplitz noncommutative tori used as building blocks in recent work of Brownlowe, Hawkins and Sims, and of Afsar, an Huef, Raeburn and Sims.