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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Automorphisms of subfactors from commuting squares

Author: Anne Louise Svendsen
Translated by:
Journal: Trans. Amer. Math. Soc. 356 (2004), 2515-2543
MSC (2000): Primary 46L37, 46L40
Published electronically: January 21, 2004
MathSciNet review: 2048528
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Abstract: We study an infinite series of irreducible, hyperfinite subfactors, which are obtained from an initial commuting square by iterating Jones' basic construction. They were constructed by Haagerup and Schou and have $A_{\infty}$as principal graphs, which means that their standard invariant is ``trivial''. We use certain symmetries of the initial commuting squares to construct explicitly non-trivial outer automorphisms of these subfactors. These automorphisms capture information about the subfactors which is not contained in the standard invariant.

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  • 1. Asaeda, M. and Haagerup, U. (1999). Exotic subfactors of finite depth with Jones indices ${(5+\sqrt{13})}/{2}$ and ${(5+\sqrt{17})}/{2}$. Communications in Mathematical Physics, 202, 1-63. MR 2000c:46120
  • 2. Bisch, D. (1998). Principal graphs of subfactors with small Jones index. Mathematische Annalen, 311, 223-231. MR 2000k:46087
  • 3. Bratteli, O. (1972). Inductive limits of finite dimensional $C^*$-algebras. Transactions of the American Mathematical Society, 171, 195-234. MR 47:844
  • 4. Evans, D. E. and Kawahigashi, Y. (1998). Quantum symmetries on operator algebras. Oxford University Press. MR 99m:46148
  • 5. Goodman, F., de la Harpe, P. and Jones, V. F. R. (1989). Coxeter graphs and towers of algebras. MSRI Publications (Springer), 14. MR 91c:46082
  • 6. Goto, S. (1995). Symmetric flat connections, triviality of Loi's invariant, and orbifold subfactors. Publications of the RIMS, Kyoto University, 31, 609-624. MR 97e:46081
  • 7. Haagerup, U. (1994). Principal graphs of subfactors in the index range $4< 3+\sqrt2$. in Subfactors -- Proceedings of the Taniguchi Symposium, Katata --, (ed. H. Araki, et al.), World Scientific, 1-38. MR 96d:46081
  • 8. Jones, V. F. R. (1983). Index for subfactors. Inventiones Mathematicae, 72, 1-25. MR 84d:46097
  • 9. Jones, V. F. R. and Sunder, V. S. (1997). Introduction to subfactors. London Math. Soc. Lecture Notes Series 234, Cambridge University Press. MR 98h:46067
  • 10. Kawahigashi, Y. (1995). On flatness of Ocneanu's connections on the Dynkin diagrams and classification of subfactors. Journal of Functional Analysis, 127, 63-107. MR 95j:46075
  • 11. Kawahigashi, Y. (1993). Centrally trivial automorphisms and an analogue of Connes's $\chi(M)$ for subfactors. Duke Mathematical Journal, 71, 93-118. MR 94k:46131
  • 12. Loi, P. H. (1996). On automorphisms of subfactors. Journal of Functional Analysis, 141, 275-293. MR 98b:46082
  • 13. Ocneanu, A. (1988). Quantized group, string algebras and Galois theory for algebras. Operator algebras and applications, Vol. 2 (Warwick, 1987), (ed. D. E. Evans and M. Takesaki), London Mathematical Society Lecture Note Series Vol. 136, Cambridge University Press, 119-172. MR 91k:46068
  • 14. Ocneanu, A. (1991). Quantum symmetry, differential geometry of finite graphs and classification of subfactors, University of Tokyo Seminary Notes 45 (Notes recorded by Kawahigashi, Y.).
  • 15. Popa, S. (1983). Orthogonal pairs of $*$-subalgebras in finite von Neumann algebras. Journal of Operator Theory, 9, 253-268. MR 84h:46077
  • 16. Popa, S. (1994). Classification of amenable subfactors of type II. Acta Mathematica, 172, 163-255. MR 95f:46105
  • 17. Sato, N. (1997). Two subfactors arising from a non-degenerate commuting square --An answer to a question raised by V. F. R. Jones--. Pacific Journal of Mathematics, 180, 369-376. MR 99c:46073
  • 18. Schou, J. (1990). Commuting squares and index for subfactors. Ph.D. thesis at Odense University.
  • 19. Svendsen, A.L. (2002). Commuting squares and automorphisms of subfactors. Ph.D. thesis at University of California at Santa Barbara.
  • 20. Wenzl, H. (1988). Hecke algebras of type $A_n$ and subfactors. Inventiones Mathematicae, 92, 345-383. MR 90b:46118

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Additional Information

Anne Louise Svendsen
Affiliation: Department of Mathematics, University of Oslo, P.O. Box 1053, Blindern, N - 0316 Oslo, Norway
Address at time of publication: Department of Mathematics, University of Copenhagen, Universitetsparken 5, DK-2100 Copenhagen 0, Denmark

Received by editor(s): December 9, 2002
Received by editor(s) in revised form: June 2, 2003
Published electronically: January 21, 2004
Article copyright: © Copyright 2004 American Mathematical Society

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