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Article

Penneys, David
Unitary dual functors for unitary multitensor categories. (English). Higher Structures, vol. 4 (2020), issue 2, pp. 22-56
MSC: 18D05, 18D10, 46L37 | MR 4133163 | Zbl 1457.18019 | DOI: 10.21136/HS.2020.09
Keywords:
tensor category; planar algebra; pivotal structure; unitary dual functor
Summary:
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 $\Bbb R_{>0}$ 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 C$^*$ 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 End$_C(1c)$, similar to the spherical state for a graph planar algebra. This is the published version of http://arxiv.org/pdf/1808.00323.
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