2020/04/17 by André Henriques, Henriques, André, David Penneys +1 · 1 citation
Mathematics · #18D10 (Primary) #46L10 (Secondary) #Advanced Operator Algebra Research #Algebraic structures and combinatorial models #Category Theory (math.CT) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Operator Algebras (math.OA) #Quantum Algebra (math.QA)
paper · pdf · doi:10.48550/arxiv.2004.08271
openalex publication_date 2020/04/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A bicommutant category is a higher categorical analog of a von Neumann algebra. We study the bicommutant categories which arise as the commutant C' of a fully faithful representation C\toBim(R) of a unitary fusion category C. Using results of Izumi, Popa, and Tomatsu about existence and uniqueness of representations of unitary (multi)fusion categories, we prove that if C and D are Morita equivalent unitary fusion categories, then their commutant categories C' and D' are equivalent as bicommutant categories. In particular, they are equivalent as tensor categories: ( C ≃Morita D ) \Longrightarrow ( C' ≃tensor D' ). This categorifies the well-known result according to which the commutants (in some representations) of Morita equivalent finite dimensional \rm C^*-algebras are isomorphic von Neumann algebras, provided the representations are `big enough'. We also introduce a notion of positivity for bi-involutive tensor categories. For dagger categories, positivity is a property (the property of being a \rm C^*-category). But for bi-involutive tensor categories, positivity is extra structure. We show that unitary fusion categories and Bim(R) admit distinguished positive structures, and that fully faithful representations C\toBim(R) automatically respect these positive structures.