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

Published online by Cambridge University Press:  03 April 2017

NATHAN BROWNLOWE
Affiliation:
School of Mathematics and Statistics, University of Sydney, NSW 2006, Australia email nathan.brownlowe@sydney.edu.au
MITCHELL HAWKINS
Affiliation:
School of Mathematics and Applied Statistics, University of Wollongong, NSW 2522, Australia email mrhawkins1989@gmail.com, asims@uow.edu.au
AIDAN SIMS
Affiliation:
School of Mathematics and Applied Statistics, University of Wollongong, NSW 2522, Australia email mrhawkins1989@gmail.com, asims@uow.edu.au

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.

Type
Original Article
Copyright
© Cambridge University Press, 2017 

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