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dc.contributor.advisoran Huef, Astrid
dc.contributor.advisorRaeburn, Iain
dc.contributor.advisorOrloff Clark, Lisa
dc.contributor.authorAfsar, Zahra
dc.date.available2016-05-09T22:04:21Z
dc.date.copyright2016
dc.identifier.citationAfsar, Z. (2016). Equilibrium States on Toeplitz Algebras (Thesis, Doctor of Philosophy). University of Otago. Retrieved from http://hdl.handle.net/10523/6444en
dc.identifier.urihttp://hdl.handle.net/10523/6444
dc.description.abstractThis thesis describes the equilibrium states (the KMS states) of dynamical systems arising from local homeomorphisms. It has two main components. First, we consider a local homeomorphism on a compact space and the associated Hilbert bimodule. This Hilbert bimodule has both a Toeplitz algebra and a Cuntz-Pimsner algebra, which is a quotient of the Toeplitz algebra. Both algebras carry natural gauge actions of the circle, and hence one can obtain natural dynamics by lifting these actions to actions of the real numbers. We study KMS states of these dynamics at, above, and below a certain critical value. For inverse temperature larger than the critical value, we find a large simplex of KMS states on the Toeplitz algebra. For the Cuntz-Pimsner algebra, the KMS states all have inverse temperatures below the critical value. Our results for the Cuntz-Pimsner algebra overlap with recent work of Thomsen, but our proofs are quite different. At the critical value, we build a KMS state of the Toeplitz algebra which factors through the Cuntz-Pimsner algebra. To understand KMS states below the critical value, we study the backward shift on the infinite path space of an ordinary directed graph. Merging our results for the Cuntz-Pimsner algebra of shifts with the recent work about KMS states of the graph algebras, we show that Thomsen's bounds on of the possible inverse temperature of KMS states are sharp. In the second component, we consider a family of *-commuting local homeomorphisms on a compact space and build a compactly aligned product system of Hilbert bimodules (in the sense of Fowler). This product system also has two interesting algebras, the Nica-Toeplitz algebra and the Cuntz-Pimsner algebra. For these algebras, the gauge action is an action of a higher-dimensional torus, and there are many possible dynamics obtained by composing with different embeddings of the real line in the torus. We use the techniques from the first component of the thesis to study the KMS states for these dynamics. For large inverse temperature, we describe the simplex of the KMS states on the Nica-Toeplitz algebra. To study KMS states for smaller inverse temperature, we consider a preferred dynamics for which there is a single critical inverse temperature, which we can normalise to be 1. We then find a KMS1 state for the Nica-Toeplitz algebra which factors through the Cuntz-Pimsner algebra. We then illustrate our results by considering different backward shifts on the infinite path space of some higher-rank graphs.
dc.format.mimetypeapplication/pdf
dc.language.isoen
dc.publisherUniversity of Otago
dc.rightsAll items in OUR Archive are provided for private study and research purposes and are protected by copyright with all rights reserved unless otherwise indicated.
dc.subjectToeplitz algebra
dc.subjectCuntz–Pimsner algebra
dc.subjectgauge action
dc.subjectKMS state.
dc.subjectKMS state
dc.subjectNica Toeplitz Algebra
dc.subject*-commuting maps
dc.subjectProduct sytems.
dc.titleEquilibrium States on Toeplitz Algebras
dc.typeThesis
dc.date.updated2016-05-09T04:01:56Z
dc.language.rfc3066en
thesis.degree.disciplineDepartment of Mathematics and Statistics
thesis.degree.nameDoctor of Philosophy
thesis.degree.grantorUniversity of Otago
thesis.degree.levelDoctoral
otago.openaccessOpen
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