Skip to main content
SpringerLink
Log in

Minimal Index and Dimension for 2-C*-Categories with Finite-Dimensional Centers

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

In the first part of this paper, we give a newlook at inclusions of von Neumann algebras with finite-dimensional centers and finite Jones’ index. The minimal conditional expectation is characterized by means of a canonical state on the relative commutant, that we call the spherical state; the minimal index is neither additive nor multiplicative (it is submultiplicative), contrary to the subfactor case. So we introduce amatrix dimension with the good functorial properties: it is always additive and multiplicative. Theminimal index turns out to be the square of the norm of the matrix dimension, as was known in the multi-matrix inclusion case. In the second part, we show how our results are valid in a purely 2-C*-categorical context, in particular they can be formulated in the framework of Connes’ bimodules over von Neumann algebras.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Bertozzini, P., Conti, R., Lewkeeratiyutkul, W., Suthichitranont, N.: On Strict Higher C *-Categories. arXiv:1709.09339 (2017)

  2. Baillet M., Denizeau Y., Havet J.-F.: Indice d’une espérance conditionnelle. Compos. Math. 66, 199–236 (1988)

    MATH  Google Scholar 

  3. Bartels A., Douglas C.L., Henriques A.: Dualizability and index of subfactors. Quantum Topol. 5, 289–345 (2014)

    MathSciNet  MATH  Google Scholar 

  4. Bischoff, M., Kawahigashi, Y., Longo, R., Rehren, K.-H.: Tensor Categories and Endomorphisms of Von Neumann Algebras. With Applications to Quantum Field Theory, vol. 3 of Springer Briefs in Mathematical Physics. Springer, Cham (2015)

  5. Bratteli O.: Inductive limits of finite dimensional C *-algebras. Trans. Am. Math. Soc. 171, 195–234 (1972)

    MathSciNet  MATH  Google Scholar 

  6. Burns, M.: Subfactors, planar algebras and rotations. Ph.D. thesis, University of California at Berkeley. arXiv:1111.1362v1 (2003)

  7. Combes F., Delaroche C.: Groupe modulaire d’une espérance conditionnele dans une algèbre de von Neumann. Bull. Soc. Math. France 103, 385–426 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  8. Connes A.: Noncommutative Geometry. Academic Press, New York (2013)

    MATH  Google Scholar 

  9. Etingof P., Gelaki S., Nikshych D., Ostrik V.: Tensor Categories, vol. 205 of Mathematical Surveys and Monographs. American Mathematical Society, Providence (2015)

    MATH  Google Scholar 

  10. Fidaleo F., Isola T.: Minimal expectations for inclusions with atomic centres. Int. J. Math. 7, 307–327 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  11. Goodman F.M., de la Harpe P., Jones V.F.R.: Coxeter Graphs and Towers of Algebras, Vol. 14 of Mathematical Sciences Research Institute Publications. Springer, New York (1989)

    Book  Google Scholar 

  12. Ghez P., Lima R., Roberts J.E.: W *-categories. Pac. J. Math. 120, 79–109 (1985)

    Article  MATH  Google Scholar 

  13. Giorgetti, L.,Yuan, W.: Realization of rigid C *-tensor categories viaTomita bimodules. arXiv:1712.09311 (2017)

  14. Haagerup U.: Operator-valued weights in von Neumann algebras. II. J. Funct. Anal. 33, 339–361 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  15. Havet J.-F.: Espérance conditionnelle minimale. J. Oper. Theory 24, 33–55 (1990)

    MATH  Google Scholar 

  16. Hiai F.: Minimizing indices of conditional expectations onto a subfactor. Publ. Res. Inst. Math. Sci. 24, 673–678 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  17. Horn R.A., Johnson C.R.: Matrix Analysis. 2nd edn. Cambridge University Press, Cambridge (2013)

    Google Scholar 

  18. Henriques A., Penneys D.: Bicommutant categories from fusion categories. Sel.Math. (N.S.) 23, 1669–1708 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  19. Isola T.: Modular structure of the crossed product by a compact group dual. J. Oper. Theory 33, 3–31 (1995)

    ADS  MathSciNet  MATH  Google Scholar 

  20. Johnson C.R., Bru R.: The spectral radius of a product of nonnegative matrices. Linear Algebra Appl. 141, 227–240 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  21. Jolissaint P.: Index for pairs of finite von Neumann algebras. Pac. J. Math. 146, 43–70 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  22. Jolissaint P.: Indice d’espérances conditionnelles et algèbres de von Neumann finies. Math. Scand. 68, 221–246 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  23. Jones V.F.R.: Index for subfactors. Invent. Math. 72, 1–25 (1983)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  24. Jones V.F.R.: Hecke algebra representations of braid groups and link polynomials. Ann. Math. (2) 2(126), 335–388 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  25. Kosaki H., Longo R.: A remark on the minimal index of subfactors. J. Funct. Anal. 107, 458–470 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  26. Kosaki H.: Extension of Jones’ theory on index to arbitrary factors. J. Funct. Anal. 66, 123–140 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  27. Kosaki, H.: Type III Factors and Index Theory, vol. 43 of Lecture Notes Series. Seoul National University Research Institute of Mathematics Global Analysis Research Center, Seoul (1998)

  28. Kawakami S., Watatani Y.: The multiplicativity of the minimal index of simple C *-algebras. Proc. Am. Math. Soc. 123, 2809–2813 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  29. Landsman, N.P.: Bicategories of operator algebras and Poisson manifolds. In: Mathematical Physics in Mathematics and Physics (Siena, 2000), vol. 30 of Fields Institute Communications, pp. 271–286. American Mathematical Society, Providence (2001)

  30. Leinster T.: Higher Operads, Higher Categories, vol. 298 of London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge (2004) arXiv:math/0305049

    Book  Google Scholar 

  31. Longo R.: Index of subfactors and statistics of quantum fields. I. Commun. Math. Phys. 126, 217–247 (1989)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  32. Longo R.: Index of subfactors and statistics of quantumfields. II. Correspondences, braid group statistics and Jones polynomial. Commun. Math. Phys. 130, 285–309 (1990)

    Article  ADS  MATH  Google Scholar 

  33. Longo R.: Minimal index and braided subfactors. J. Funct. Anal. 109, 98–112 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  34. Longo, R.: On Landauer’s principle and bound for infinite systems. Commun. Math. Phys. 363, 531–560 (2018)

  35. Longo R., Roberts J.E.: A theory of dimension. K-Theory 11, 103–159 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  36. Mac Lane S.: Categories for the Working Mathematician, vol. 5 of Graduate Texts in Mathematics, 2nd edn. Springer, New York (1998)

    Google Scholar 

  37. Popa S.: Classification of Subfactors and Their Endomorphisms, vol. 86 of CBMS Regional Conference Series in Mathematics. American Mathematical Society, Providence (1995)

    Book  Google Scholar 

  38. Pimsner M., Popa S.: Entropy and index for subfactors. Ann. Sci. Ecole Norm. Sup 19, 57–106 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  39. Pimsner M., Popa S.: Finite-dimensional approximation of pairs of algebras and obstructions for the index. J. Funct. Anal. 98, 270–291 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  40. Reed M., Simon B.: Methods of Modern Mathematical Physics. I. Functional Analysis 2nd edn. Academic Press, Inc., Harcourt Brace Jovanovich, Publishers. New York, London (1980)

    Google Scholar 

  41. Sawada, Y., Yamagami, S.: Notes on the bicategory of W*-bimodules. arXiv:1705.05600 (2017)

  42. Teruya T.: Index for von Neumann algebras with finite-dimensional centers. Publ. Res. Inst. Math. Sci. 28, 437–453 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  43. Zito P.A.: 2−C *-categories with non-simple units. Adv. Math. 210, 122–164 (2007)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

We thank the Isaac Newton Institute (INI) for Mathematical Sciences in Cambridge for hospitality during the program “Operator algebras: subfactors and their applications” (OAS), supported by EPSRC Grant Nos. EP/K032208/1 and EP/R014604/1. L.G. thanks also the Department of Mathematics of the Ohio University and of the Georg-August-Universität Göttingen for financial support, the latter during the workshop LQP41 “Foundations and constructive aspects of QFT” where part of our results were presented. We acknowledge the MIUR Excellence Department Project awarded to the Department of Mathematics, University of Rome Tor Vergata, CUP E83C18000100006.

Author information

Authors and Affiliations

  1. Dipartimento di Matematica, Università di Roma Tor Vergata, Via della Ricerca Scientifica, 1, 00133, Rome, Italy

    Luca Giorgetti & Roberto Longo

Authors
  1. Luca Giorgetti
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Roberto Longo
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Luca Giorgetti.

Additional information

Communicated by Y. Kawahigashi

Supported in part by the ERC Advanced Grant 669240 QUEST “Quantum Algebraic Structures and Models”, MIUR FARE R16X5RB55W QUEST-NET, OPAL-“Consolidate the Foundations”, GNAMPAINdAM.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Giorgetti, L., Longo, R. Minimal Index and Dimension for 2-C*-Categories with Finite-Dimensional Centers. Commun. Math. Phys. 370, 719–757 (2019). https://doi.org/10.1007/s00220-018-3266-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-018-3266-x

Search

Navigation