The Toeplitz noncommutative solenoid and its KMS states

TL;DR

This paper constructs Toeplitz extensions of noncommutative solenoid C*-algebras, analyzes their KMS states, and reveals a homeomorphism between the extreme points of the KMS simplex and a solenoid group, highlighting the structure of equilibrium states.

Contribution

It introduces Toeplitz extensions of noncommutative solenoids and characterizes their KMS states, linking the extreme boundary to a solenoid group action, which is a novel structural insight.

Findings

Extreme points of KMS states form a solenoid

Unique zero-temperature KMS state factors through the noncommutative solenoid

Homeomorphism between KMS extreme points and the solenoid group

Abstract

We use Katsura's topological graphs to define Toeplitz extensions of Latr\'emoli\`ere and Packer's noncommutative-solenoid C*-algebras. We identify a natural dynamics on each Toeplitz noncommutative solenoid and study the associated KMS states. Our main result shows that the space of extreme points of the KMS simplex of the Toeplitz noncommutative torus at a strictly positive inverse temperature is homeomorphic to a solenoid; indeed, there is an action of the solenoid group on the Toeplitz noncommutative solenoid that induces a free and transitive action on the extreme boundary of the KMS simplex. With the exception of the degenerate case of trivial rotations, at inverse temperature zero there is a unique KMS state, and only this one factors through Latr\'emoli\`ere and Packer's noncommutative solenoid.