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



Publication Details

Brownlowe, N., Hawkins, M. & Sims, A. (2019). The Toeplitz noncommutative solenoid and its Kubo-Martin-Schwinger states. Ergodic Theory and Dynamical Systems, 39 (1), 105-131.


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.

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