Explore advanced concepts in category theory, subfactor theory, and quantum groups in this 51-minute lecture by Sergey Neshveyev. Delve into the connections between Drinfeld center, Drinfeld double, Ocneanu's tube algebra, and annular representation theory. Learn about recent progress in defining C*-completion of fusion algebras for tensor categories and its implications for approximation properties. Discover how this unifies concepts like central property (T) for discrete quantum groups and Popa's property (T) for subfactors. Examine topics such as fusion algebra, maximal completion, Temperley-Lieb categories, regular half-braidings, Morita equivalence, and central vectors. Gain insights from this joint work with Makoto Yamashita, presented as part of the Hausdorff Trimester Program "Von Neumann Algebras."