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Unitary anchored planar algebras

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arxiv 2301.11114 v2 pith:4FWRJSZP submitted 2023-01-26 math.QA math.CT

André Henriques , David Penneys , James Tener This is my paper · ORCID
classification math.QAmath.CT
keywords unitaryalgebrasanchoredcategoriesplanararticleequivalencehilbert
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abstract

In our previous article [arXiv:1607.06041], we established an equivalence between pointed pivotal module tensor categories and anchored planar algebras. This article introduces the notion of unitarity for both module tensor categories and anchored planar algebras, and establishes the unitary analog of the above equivalence. Our constructions use Baez's 2-Hilbert spaces (i.e., semisimple $C^*$-categories equipped with unitary traces), the unitary Yoneda embedding, and the notion of unitary adjunction for dagger functors between 2-Hilbert spaces.

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  1. Orthonormal bases for higher Hilbert spaces

    math.QA 2026-08 conditional novelty 6.0 of 10

    For finite-dimensional 3-Hilbert spaces, orthonormal bases exist uniquely up to contractible choice, and the Yoneda embedding into the presheaf 3-Hilbert space is an isometric equivalence.

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