Annular representation theory with applications to approximation and rigidity properties for rigid C*-tensor categories

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.