Unitary dual functors for unitary multitensor categories

Abstract

We classify which dual functors on a unitary multitensor category are compatible with the dagger structure in terms of groupoid homomorphisms from the universal grading groupoid to ${\mathbb{R}}_{>0}$ where the latter is considered as a groupoid with one object. We then prove that all unitary dual functors induce unitarily equivalent bi-involutive structures. As an application, we provide the unitary version of the folklore correspondence between shaded planar ${\mathrm{C}}^{\ast }$ algebras with finite dimensional box spaces and unitary multitensor categories with a chosen unitary dual functor and chosen generator. We make connection with the recent work of Giorgetti-Longo to determine when the loop parameters in these planar algebras are scalars. Finally, we show that we can correct for many non-spherical choices of dual functor by adding the data of a spherical state on ${\mathrm{End}}_{\mathcal{C}}(1_{\mathcal{C}})$, similar to the spherical state for a graph planar algebra.

32 pages, many figures. Comments welcome!