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Title page for ETD etd-03252016-125400


Type of Document Dissertation
Author Jones, Corey Michael
URN etd-03252016-125400
Title Annular representation theory with applications to approximation and rigidity properties for rigid C*-tensor categories
Degree PhD
Department Mathematics
Advisory Committee
Advisor Name Title
Vaughan Jones Committee Chair
Dietmar Bisch Committee Member
Jesse Peterson Committee Member
John Ratcliffe Committee Member
Thomas Kephart Committee Member
Keywords
  • Representation theory
  • Tensor categories
  • Operator algebras
Date of Defense 2016-03-24
Availability unrestricted
Abstract
We study annular algebras associated to a rigid C*-tensor category, a generalization of both Ocneanu's tube algebra and Jones' affine annular category. We show that all ``sufficiently large' annular algebras are strongly morita equivalent, hence have equivalent representation theories. We then demonstrate the existence of a universal C*-algebra for the tube algebra, analogous to the universal C*-algebra of a discrete group. Using this perspective, we show that a ``piece' of the representation theory of the tube algebra is precisely the admissible representation theory of the fusion algebra introduced by Popa and Vaes, allowing for properties such as amenability, the Haagerup property, and property (T) for the category to be interpreted in terms on annular representations. We treat several examples, and partially characterize the C*-algebras associated to the Temperley-Lieb-Jones categories with negative q parameter. As an application of our annular perspective, we show that quantum G_2 categories have property (T) for positive q not equal to 1.
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