The Toeplitz noncommutative solenoid and its Kubo–Martin–Schwinger states. (3rd April 2017)

• Record Type: Journal Article
• Authors: BROWNLOWE, NATHAN; HAWKINS, MITCHELL; SIMS, AIDAN
• Is Part Of: Ergodic theory and dynamical systems. Volume 39:Number 1(2019)
• Page Start: 105
• Page End: 131
• Publication Date: 2017-04-03
• Subjects: Ergodic theory -- Periodicals; Differentiable dynamical systems -- Periodicals; 515.42
• DOI: 10.1017/etds.2017.7
• Languages: English
• ISSNs: 0143-3857

Abstract

Abstract : We use Katsura's topological graphs to define Toeplitz extensions of Latrémolière and Packer's noncommutative-solenoid $C^{\ast }$ -algebras. We identify a natural dynamics on each Toeplitz noncommutative solenoid and study the associated Kubo–Martin–Schwinger (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 Latrémolière and Packer's noncommutative solenoid.