C*-algebras constructed from factor groupoids and their analysis through relative K-theory and excision

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dc.contributor.authorHaslehurst, Mitch
dc.contributor.supervisorPutnam, Ian F.
dc.date.accessioned2022-08-30T19:20:19Z
dc.date.available2022-08-30T19:20:19Z
dc.date.copyright2022en_US
dc.date.issued2022-08-30
dc.degree.departmentDepartment of Mathematics and Statisticsen_US
dc.degree.levelDoctor of Philosophy Ph.D.en_US
dc.description.abstractWe address the problem of finding groupoid models for C*-algebras given some prescribed K-theory data. This is a reasonable question because a groupoid model for a C*-algebra reveals much about the structure of the algebra. A great deal of progress towards solving this problem has been made using constructions with inductive limits, subgroupoids, and dynamical systems. This dissertation approaches the question with a more specific methodology in mind, with factor groupoids. In the first part, we develop a portrait of relative K-theory for C*-algebras using the general framework of Banach categories and Banach functors due to Max Karoubi. The purpose of developing such a portrait is to provide a means of analyzing the K-theory of an inclusion of C*-algebras, or more generally of a *-homomorphism between two C*-algebras. Another portrait may be obtained using a mapping cone construction and standard techniques (it is shown that the two presentations are naturally and functorially isomorphic), but for many examples, including the ones considered in the second part, the portrait obtained by Karoubi's construction is more convenient. In the second part, we construct examples of factor groupoids and analyze their C*-algebras. A factor groupoid setup (two groupoids with a surjective groupoid homomorphism between them) induces an inclusion of two C*-algebras, and therefore the portrait of relative K-theory developed in the first part, together with an excision theorem, can be used to elucidate the structure. The factor groupoids are obtained as quotients of AF-groupoids and certain extensions of Cantor minimal systems using iterated function systems. We describe the K-theory in both cases, and in the first case we show that the K-theory of the resulting C*-algebras can be prescribed through the factor groupoids.en_US
dc.description.scholarlevelGraduateen_US
dc.identifier.bibliographicCitationM. Haslehurst, "Relative K-theory for C*-algebras", Rocky Mountain J. Math., to appear.en_US
dc.identifier.bibliographicCitationM. Haslehurst, "Some examples of factor groupoids", preprint arXiv:2204.13888.en_US
dc.identifier.urihttp://hdl.handle.net/1828/14156
dc.languageEnglisheng
dc.language.isoenen_US
dc.rightsAvailable to the World Wide Weben_US
dc.subjectmathematicsen_US
dc.subjectoperator theoryen_US
dc.subjectc*-algebraen_US
dc.subjectk-theoryen_US
dc.subjectdynamical systemsen_US
dc.subjectgroupoidsen_US
dc.titleC*-algebras constructed from factor groupoids and their analysis through relative K-theory and excisionen_US
dc.typeThesisen_US

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