C*-algebras from actions of congruence monoids

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dc.contributor.authorBruce, Chris
dc.contributor.supervisorLaca, Marcelo
dc.date.accessioned2020-04-21T05:35:02Z
dc.date.available2020-04-21T05:35:02Z
dc.date.copyright2020en_US
dc.date.issued2020-04-20
dc.degree.departmentDepartment of Mathematics and Statistics
dc.degree.levelDoctor of Philosophy Ph.D.en_US
dc.description.abstractWe initiate the study of a new class of semigroup C*-algebras arising from number-theoretic considerations; namely, we generalize the construction of Cuntz, Deninger, and Laca by considering the left regular C*-algebras of ax+b-semigroups from actions of congruence monoids on rings of algebraic integers in number fields. Our motivation for considering actions of congruence monoids comes from class field theory and work on Bost–Connes type systems. We give two presentations and a groupoid model for these algebras, and establish a faithfulness criterion for their representations. We then explicitly compute the primitive ideal space, give a semigroup crossed product description of the boundary quotient, and prove that the construction is functorial in the appropriate sense. These C*-algebras carry canonical time evolutions, so that our construction also produces a new class of C*-dynamical systems. We classify the KMS (equilibrium) states for this canonical time evolution, and show that there are several phase transitions whose complexity depends on properties of a generalized ideal class group. We compute the type of all high temperature KMS states, and consider several related C*-dynamical systems.en_US
dc.description.scholarlevelGraduateen_US
dc.identifier.bibliographicCitationC. Bruce, C*-algebras from actions of congruence monoids on rings of algebraic integers, Trans. Amer. Math. Soc. 373 (2020), no. 1, 699–726. DOI: 10.1090/tran/7966.en_US
dc.identifier.bibliographicCitationC. Bruce, Phase transitions on C*-algebras from actions of congruence monoids on rings of algebraic integers, to appear in Int. Math. Res. Not. IMRN. Preprint: arXiv:1901.04075.en_US
dc.identifier.urihttp://hdl.handle.net/1828/11689
dc.languageEnglisheng
dc.language.isoenen_US
dc.rightsAvailable to the World Wide Weben_US
dc.subjectC*-algebrasen_US
dc.subjectSemigroup C*-algebrasen_US
dc.subjectOperator algebrasen_US
dc.subjectPrimitive idealsen_US
dc.subjectKMS statesen_US
dc.subjectC*-dynamical systemen_US
dc.subjectNumber fieldsen_US
dc.subjectRings of algebraic integersen_US
dc.subjectCongruence monoidsen_US
dc.subjectvon Neumann algebrasen_US
dc.subjectNoncommutative geometryen_US
dc.subjectFaithful representationsen_US
dc.subjectGroupoid C*-algebrasen_US
dc.subjectType III_1 factorsen_US
dc.titleC*-algebras from actions of congruence monoidsen_US
dc.typeThesisen_US

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