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The Toeplitz noncommutative solenoid and its Kubo-Martin-Schwinger states

Journal Article


Abstract


  • We use Katsura’s topological graphs to define Toeplitz extensions of Latrémolière and Packer’s noncommutative-solenoid -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.

UOW Authors


  •   Brownlowe, Nathan D. (external author)
  •   Hawkins, Mitchell (external author)
  •   Sims, Aidan

Publication Date


  • 2017

Citation


  • Brownlowe, N., Hawkins, M. & Sims, A. (2017). The Toeplitz noncommutative solenoid and its Kubo-Martin-Schwinger states. Ergodic Theory and Dynamical Systems, Online First 1-27.

Scopus Eid


  • 2-s2.0-85016776816

Number Of Pages


  • 26

Start Page


  • 1

End Page


  • 27

Volume


  • Online First

Place Of Publication


  • United Kingdom

Abstract


  • We use Katsura’s topological graphs to define Toeplitz extensions of Latrémolière and Packer’s noncommutative-solenoid -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.

UOW Authors


  •   Brownlowe, Nathan D. (external author)
  •   Hawkins, Mitchell (external author)
  •   Sims, Aidan

Publication Date


  • 2017

Citation


  • Brownlowe, N., Hawkins, M. & Sims, A. (2017). The Toeplitz noncommutative solenoid and its Kubo-Martin-Schwinger states. Ergodic Theory and Dynamical Systems, Online First 1-27.

Scopus Eid


  • 2-s2.0-85016776816

Number Of Pages


  • 26

Start Page


  • 1

End Page


  • 27

Volume


  • Online First

Place Of Publication


  • United Kingdom