Annular representation theory for rigid $C^{*}$-tensor categories

Abstract

We define annular algebras for rigid $C^{*}$-tensor categories, providing a unified framework for both Ocneanu's tube algebra and Jones' affine annular category of a planar algebra. We study the representation theory of annular algebras, and show that all sufficiently large (full) annular algebras for a category are isomorphic after tensoring with the algebra of matrix units with countable index set, hence have equivalent representation theories. Annular algebras admit a universal $C^{*}$-algebra closure analogous to the universal $C^{*}$-algebra for groups. These algebras have interesting corner algebras indexed by some set of isomorphism classes of objects, which we call centralizer algebras. The centralizer algebra corresponding to the identity object is canonically isomorphic to the fusion algebra of the category, and we show that the admissible representations of the fusion algebra of Popa and Vaes are precisely the restrictions of arbitrary (non-degenerate) $*$-representations of full annular algebras. This allows approximation and rigidity properties defined for categories by Popa and Vaes to be interpreted in the context of annular representation theory. This perspective also allows us to define "higher weight" approximation properties based on other centralizer algebras of an annular algebra. Using the analysis of annular representations due to Jones and Reznikoff, we identify all centralizer algebras for the $TLJ(\delta)$ categories for $\delta\ge 2$.