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Bicommutant categories from fusion categories

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Abstract

Bicommutant categories are higher categorical analogs of von Neumann algebras that were recently introduced by the first author. In this article, we prove that every unitary fusion category gives an example of a bicommutant category. This theorem categorifies the well-known result according to which a finite dimensional \(*\)-algebra that can be faithfully represented on a Hilbert space is in fact a von Neumann algebra.

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Notes

  1. This condition is present in the original definition [10] of Ghez, Lima, and Roberts, but is omitted from many other references (e.g., from [6, 15, 35]). It is automatic for categories that admit direct sums, but it can otherwise fail.

  2. The formula for the inner product makes most sense if one rewrites formally \([D\phi :D\psi ]_t\) as \(\phi ^{it}\psi ^{-it}\) and \(\phi (a)\) as \(\mathrm {Tr}(\phi a)\). It then simplifies to \(\mathrm {Tr}(\phi ^{1+it}\psi ^{-it})|_{t=i/2}=\mathrm {Tr}(\phi ^{1/2}\psi ^{1/2})\). Similarly, for next formula, one may replace formally \(\sigma _t^\psi (b)\) by \(\psi ^{it}b\psi ^{-it}\). Note that these formal symbols are genuinely meaningful and can be implemented as (unbounded) operators on some Hilbert space, see, e.g., [34].

  3. Later on, we will restrict attention to separable von Neumann algebras (i.e., ones which admit faithful actions on separable Hilbert spaces), in which case we will take \({{\mathrm{Bim}}}(R)\) to be the category of R–R-bimodules whose underlying Hilbert space is separable. The reason for that restriction will become evident in Sect. 5.

  4. Unlike the modular flow, which depends on a choice of state, the crossed product \(R\rtimes _\sigma \mathbb {R}\) does not depend on any choices, up to canonical isomorphism.

  5. The result in [5] is only stated for type \(\mathrm{III}\) factors, but the proof never uses the type \(\mathrm{III}\) assumption.

References

  1. Bartels, A., Douglas, C., Henriques, A.: Conformal nets II: conformal blocks. (2014). arxiv:1409.8672

  2. Bartels, A., Douglas, C.L., Henriques, A.: Dualizability and index of subfactors. Quantum Topol. 5(3), 289–345 (2014). doi:10.4171/QT/53. arXiv:1110.5671

    MathSciNet  MATH  Google Scholar 

  3. Connes, A.: Classification of injective factors. Cases \(II_{1},\) \(II_{\infty },\) \(III_{\lambda },\) \(\lambda \ne 1\). Ann. Math. 104, 73–115 (1976)

    Article  MathSciNet  Google Scholar 

  4. Connes, A.: Noncommutative Geometry. Academic Press Inc., San Diego, CA (1994)

    MATH  Google Scholar 

  5. Connes, A., Takesaki, M.: The flow of weights on factors of type \({\rm III}\). Tôhoku Math. J. 29(4), 473–575 (1977). doi:10.2748/tmj/1178240493

  6. Doplicher, S., Roberts, J.E.: A new duality theory for compact groups. Invent. Math. 98(1), 157–218 (1989). doi:10.1007/BF01388849

    Article  MathSciNet  MATH  Google Scholar 

  7. Evans, D.E., Kawahigashi, Y.: Quantum Symmetries on Operator Algebras. Oxford Mathematical Monographs. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York (1998). xvi+829 pp. ISBN: 0-19-851175-2

  8. Falguières, S., Raum, S.: Tensor \({\rm C}^{*}\)-categories arising as bimodule categories of \({\rm II}_{1}\) factors. Adv. Math. 237, 331–359 (2013). doi:10.1016/j.aim.2012.12.020. arxiv:1112.4088v2

  9. Galindo, C., Hong, S.-M., Rowell, E.: Generalized and quasi-localizations of braid group representations. Int. Math. Res. Not. 3, 693–731 (2013). arxiv:1105.5048

  10. Ghez, P., Lima, R., Roberts, J.E.: \(W^\ast \)-categories. Pac. J. Math. 120(1), 79–109 (1985)

    Article  MATH  Google Scholar 

  11. Haagerup, U.: The standard form of von Neumann algebras. Math. Scand. 37(2), 271–283 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  12. Haagerup, U.: \(L^{p}\)-spaces associated with an arbitrary von Neumann algebra. Algèbres d’opérateurs et leurs applications en physique mathématique (Proc. Colloq., Marseille, 1997) Colloq. Internat. CNRS, vol. 274. CNRS, Paris, 1979, pp. 175–184 (1977)

  13. Haagerup, U.: Connes’ bicentralizer problem and uniqueness of the injective factor of type \({\rm III}_1\). Acta Math. 158(1–2), 95–148 (1987). doi:10.1007/BF02392257

    Article  MathSciNet  MATH  Google Scholar 

  14. Henriques, A.: What Chern–Simons theory assigns to a point. (2015). arXiv:1503.06254

  15. Hayashi, T., Yamagami, S.: Amenable tensor categories and their realizations as AFD bimodules. J. Funct. Anal. 172(1), 19–75 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  16. Izumi, M.: The structure of sectors associated with Longo–Rehren inclusions. II. Examples. Rev. Math. Phys. 13(5), 603–674 (2001). doi:10.1142/S0129055X01000818

    Article  MathSciNet  MATH  Google Scholar 

  17. Joyal, A., Street, R.: The geometry of tensor calculus. I. Adv. Math. 88(1), 55–112 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  18. Kosaki, H.: Canonical \({L}^p\)-spaces associated with an arbitrary abstract von Neumann algebra. ProQuest LLC, Ann Arbor, MI, 1980, Thesis (Ph.D.), University of California, Los Angeles

  19. Krieger, W.: On ergodic flows and the isomorphism of factors. Math. Ann. 223(1), 19–70 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  20. Majid, S.: Representations, duals and quantum doubles of monoidal categories. In: Proceedings of the Winter School on Geometry and Physics (Srní, 1990), vol. 26, pp. 197–206 (1991)

  21. Müger, M.: From subfactors to categories and topology. I. Frobenius algebras in and Morita equivalence of tensor categories. J. Pure Appl. Algebra 180(1–2), 81–157 (2003). doi:10.1016/S0022-4049(02)00247-5. arXiv:math.CT/0111204

  22. Müger, M.: From subfactors to categories and topology. II. The quantum double of tensor categories and subfactors. J. Pure Appl. Algebra 180(1–2), 159–219 (2003). doi:10.1016/S0022-4049(02)00248-7. arXiv:math.CT/0111205

  23. Murray, F.J., von Neumann, J.: On rings of operators. IV. Ann. Math. 44, 716–808 (1943)

    Article  MathSciNet  MATH  Google Scholar 

  24. Popa, S.: Correspondences. INCREST Preprint, 1986. http://www.math.ucla.edu/~popa/popa-correspondences.pdf

  25. Popa, S.: Classification of subfactors: the reduction to commuting squares. Invent. Math. 101(1), 19–43 (1990). doi:10.1007/BF01231494

    Article  MathSciNet  MATH  Google Scholar 

  26. Popa, S.: Classification of amenable subfactors of type II. Acta Math. 172(2), 163–255 (1994). doi:10.1007/BF02392646

    Article  MathSciNet  MATH  Google Scholar 

  27. Popa, S.: An axiomatization of the lattice of higher relative commutants of a subfactor. Invent. Math. 120(3), 427–445 (1995). doi:10.1007/BF01241137

    Article  MathSciNet  MATH  Google Scholar 

  28. Popa, S.: Classification of subfactors and their endomorphisms. CBMS Regional Conference Series in Mathematics, vol. 86, Published for the Conference Board of the Mathematical Sciences, Washington, DC, (1995)

  29. Ponto, Kate, Shulman, Michael: Shadows and traces in bicategories. J. Homot. Relat. Struct. 8(2), 151–200 (2013). doi:10.1007/s40062-012-0017-0. arxiv:0910.1306

    Article  MathSciNet  MATH  Google Scholar 

  30. Sauvageot, Jean-Luc: Sur le produit tensoriel relatif d’espaces de Hilbert. J. Oper. Theory 9(2), 237–252 (1983)

    MathSciNet  MATH  Google Scholar 

  31. Selinger, P.: A survey of graphical languages for monoidal categories. New structures for physics. Lecture Notes in Physics, vol. 813, pp. 289–355. Springer, Heidelberg (2011). doi:10.1007/978-3-642-12821-9_4

  32. Takesaki, M.: Theory of Operator Algebras. III, Encyclopaedia of Mathematical Sciences, vol. 127, Springer, Berlin. Operator Algebras and Non-commutative. Geometry 8, (2003). doi:10.1007/978-3-662-10453-8

  33. Tambara, D.: A duality for modules over monoidal categories of representations of semisimple Hopf algebras. J. Algebra 241(2), 515–547 (2001). doi:10.1006/jabr.2001.8771

    Article  MathSciNet  MATH  Google Scholar 

  34. Yamagami, S.: Algebraic aspects in modular theory. Publ. Res. Inst. Math. Sci. 28(6), 1075–1106 (1992). doi:10.2977/prims/1195167738

    Article  MathSciNet  MATH  Google Scholar 

  35. Yamagami, S.: Frobenius duality in \(C^*\)-tensor categories. J. Oper. Theory 52(1), 3–20 (2004)

    MathSciNet  MATH  Google Scholar 

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Acknowledgments

This project began at the 2015 Mathematisches Forschungsinstitut Oberwolfach workshop on Subfactors and conformal field theory. The authors would like to thank the organizers and MFO for their hospitality. André Henriques was supported by the Leverhulme trust and the EPSRC grant “Quantum Mathematics and Computation” during his stay in Oxford. David Penneys was partially supported by an AMS-Simons travel Grant and NSF DMS Grant 1500387.

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  1. Utrecht University, Hans Freudenthal building, Budapestlaan 6, 3584, CD, Utrecht, The Netherlands

    André Henriques

  2. University of Oxford, Andrew Wiles Building, Woodstock Rd, Oxford, OX2 6GG, UK

    André Henriques

  3. The Oho State University, Columbus, Ohio, USA

    David Penneys

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Henriques, A., Penneys, D. Bicommutant categories from fusion categories. Sel. Math. New Ser. 23, 1669–1708 (2017). https://doi.org/10.1007/s00029-016-0251-0

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