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C-Tensor Categories and Subfactors for Totally Disconnected Groups

Conference paper
Part of the Abel Symposia book series (ABEL, volume 12)

Abstract

We associate a rigid C-tensor category \(\mathcal{C}\) to a totally disconnected locally compact group G and a compact open subgroup K < G. We characterize when \(\mathcal{C}\) has the Haagerup property or property (T), and when \(\mathcal{C}\) is weakly amenable. When G is compactly generated, we prove that \(\mathcal{C}\) is essentially equivalent to the planar algebra associated by Jones and Burstein to a group acting on a locally finite bipartite graph. We then concretely realize \(\mathcal{C}\) as the category of bimodules generated by a hyperfinite subfactor.

Notes

Acknowledgements

Y.A. was supported by the Research Fellow of the Japan Society for the Promotion of Science and the Program for Leading Graduate Schools, MEXT, Japan.

S.V. was supported in part by European Research Council Consolidator Grant 614195, and by long term structural funding—Methusalem grant of the Flemish Government.

References

  1. 1.
    Y. Arano, Unitary spherical representations of Drinfeld doubles. J. Reine Angew. Math., to appear. arXiv:1410.6238. doi:10.1515/crelle-2015-0079 Google Scholar
  2. 2.
    B. Bekka, P. de la Harpe and A. Valette, Kazhdan’s property (T). Cambridge University Press, Cambridge, 2008.CrossRefzbMATHGoogle Scholar
  3. 3.
    A. Brothier, Weak amenability for subfactors. Internat. J. Math. 26 (2015), art. id. 1550048.Google Scholar
  4. 4.
    N.P. Brown and N. Ozawa, C-algebras and finite-dimensional approximations. Graduate Studies in Mathematics 88. American Mathematical Society, Providence, 2008.Google Scholar
  5. 5.
    R.D. Burstein, Automorphisms of the bipartite graph planar algebra. J. Funct. Anal. 259 (2010), 2384–2403.Google Scholar
  6. 6.
    K. De Commer, A. Freslon and M. Yamashita, CCAP for universal discrete quantum groups. Comm. Math. Phys. 331 (2014), 677–701.Google Scholar
  7. 7.
    S. Deprez and S. Vaes, A classification of all finite index subfactors for a class of group-measure space II1 factors. J. Noncommut. Geom. 5 (2011), 523–545.Google Scholar
  8. 8.
    Y. de Cornulier, Finitely presentable, non-Hopfian groups with Kazhdan’s property (T) and infinite outer automorphism group. Proc. Amer. Math. Soc. 135 (2007), 951–959.Google Scholar
  9. 9.
    S. Ghosh and C. Jones, Annular representation theory for rigid C-tensor categories. J. Funct. Anal. 270 (2016), 1537–1584.Google Scholar
  10. 10.
    P. Jolissaint, A characterization of completely bounded multipliers of Fourier algebras. Colloq. Math. 63 (1992), 311–313.Google Scholar
  11. 11.
    V.F.R Jones, The planar algebra of a bipartite graph. In Knots in Hellas ’98, World Scientific, 1999, pp. 94–117.Google Scholar
  12. 12.
    V.F.R. Jones, Planar algebras, I. arXiv:math.QA/9909027Google Scholar
  13. 13.
    R. Longo and J.E. Roberts, A theory of dimension. K-Theory 11 (1997), 103–159.Google Scholar
  14. 14.
    S. Morrison, E. Peters and N. Snyder, Skein theory for the D2n planar algebras. J. Pure Appl. Algebra 214 (2010), 117–139.Google Scholar
  15. 15.
    M. Müger, From subfactors to categories and topology, I. Frobenius algebras in and Morita equivalence of tensor categories. J. Pure Appl. Algebra 180 (2003), 81–157.Google Scholar
  16. 16.
    S. Neshveyev and L. Tuset, Compact quantum groups and their representation categories. Cours Spécialisés 20. Société Mathématique de France, Paris, 2013.Google Scholar
  17. 17.
    S. Neshveyev and M. Yamashita, Drinfeld center and representation theory for monoidal categories. Commun. Math. Phys., to appear. arXiv:1501.07390. doi:10.1007/s00220-016-2642-7 Google Scholar
  18. 18.
    S. Neshveyev and M. Yamashita, A few remarks on the tube algebra of a monoidal category. Preprint. arXiv:1511.06332Google Scholar
  19. 19.
    A. Ocneanu, Chirality for operator algebras. In Subfactors (Kyuzeso, 1993), World Sci. Publ., River Edge, 1994, pp. 39–63.Google Scholar
  20. 20.
    N. Ozawa, Examples of groups which are not weakly amenable. Kyoto J. Math. 52 (2012), 333–344.Google Scholar
  21. 21.
    S. Popa, Symmetric enveloping algebras, amenability and AFD properties for subfactors. Math. Res. Lett. 1 (1994), 409–425.Google Scholar
  22. 22.
    S. Popa, An axiomatization of the lattice of higher relative commutants of a subfactor. Invent. Math. 120 (1995), 427–445.Google Scholar
  23. 23.
    S. Popa, Some properties of the symmetric enveloping algebra of a subfactor, with applications to amenability and property T. Doc. Math. 4 (1999), 665–744.Google Scholar
  24. 24.
    S. Popa, On a class of type II1 factors with Betti numbers invariants. Ann. of Math. 163 (2006), 809–899.Google Scholar
  25. 25.
    S. Popa, D. Shlyakhtenko and S. Vaes, Cohomology and L 2-Betti numbers for subfactors and quasi-regular inclusions. Preprint. arXiv:1511.07329Google Scholar
  26. 26.
    S. Popa and S. Vaes, Representation theory for subfactors, λ-lattices and C-tensor categories, Commun. Math. Phys. 340 (2015), 1239–1280.Google Scholar
  27. 27.
    S. Vaes, Strictly outer actions of groups and quantum groups. J. Reine Angew. Math. 578 (2005), 147–184.Google Scholar
  28. 28.
    A. Wassermann, Ergodic actions of compact groups on operator algebras, I. Ann. of Math. 130 (1989), 273–319.Google Scholar
  29. 29.
    Y. Zhu, Hecke algebras and representation ring of Hopf algebras. In First International Congress of Chinese Mathematicians (Beijing, 1998), AMS/IP Stud. Adv. Math. 20, Amer. Math. Soc., Providence, 2001, pp. 219–227.Google Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Graduate School of MathematicsUniversity of TokyoTokyoJapan
  2. 2.Department of MathematicsKU LeuvenLeuvenBelgium

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