## Automorphisms of subfactors from commuting squares

HTML articles powered by AMS MathViewer

- by Anne Louise Svendsen
- Trans. Amer. Math. Soc.
**356**(2004), 2515-2543 - DOI: https://doi.org/10.1090/S0002-9947-04-03447-6
- Published electronically: January 21, 2004
- PDF | Request permission

## 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.## References

- M. Asaeda and U. Haagerup,
*Exotic subfactors of finite depth with Jones indices $(5+\sqrt {13})/2$ and $(5+\sqrt {17})/2$*, Comm. Math. Phys.**202**(1999), no. 1, 1–63. MR**1686551**, DOI 10.1007/s002200050574 - Dietmar Bisch,
*Principal graphs of subfactors with small Jones index*, Math. Ann.**311**(1998), no. 2, 223–231. MR**1625762**, DOI 10.1007/s002080050185 - Ola Bratteli,
*Inductive limits of finite dimensional $C^{\ast }$-algebras*, Trans. Amer. Math. Soc.**171**(1972), 195–234. MR**312282**, DOI 10.1090/S0002-9947-1972-0312282-2 - David E. Evans and Yasuyuki Kawahigashi,
*Quantum symmetries on operator algebras*, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1998. Oxford Science Publications. MR**1642584** - Frederick M. Goodman, Pierre de la Harpe, and Vaughan F. R. Jones,
*Coxeter graphs and towers of algebras*, Mathematical Sciences Research Institute Publications, vol. 14, Springer-Verlag, New York, 1989. MR**999799**, DOI 10.1007/978-1-4613-9641-3 - Satoshi Goto,
*Symmetric flat connections, triviality of Loi’s invariant and orbifold subfactors*, Publ. Res. Inst. Math. Sci.**31**(1995), no. 4, 609–624. MR**1371787**, DOI 10.2977/prims/1195163917 - Uffe Haagerup,
*Principal graphs of subfactors in the index range $4<[M:N]<3+\sqrt 2$*, Subfactors (Kyuzeso, 1993) World Sci. Publ., River Edge, NJ, 1994, pp. 1–38. MR**1317352** - V. F. R. Jones,
*Index for subfactors*, Invent. Math.**72**(1983), no. 1, 1–25. MR**696688**, DOI 10.1007/BF01389127 - V. Jones and V. S. Sunder,
*Introduction to subfactors*, London Mathematical Society Lecture Note Series, vol. 234, Cambridge University Press, Cambridge, 1997. MR**1473221**, DOI 10.1017/CBO9780511566219 - Yasuyuki Kawahigashi,
*On flatness of Ocneanu’s connections on the Dynkin diagrams and classification of subfactors*, J. Funct. Anal.**127**(1995), no. 1, 63–107. MR**1308617**, DOI 10.1006/jfan.1995.1003 - Yasuyuki Kawahigashi,
*Centrally trivial automorphisms and an analogue of Connes’s $\chi (M)$ for subfactors*, Duke Math. J.**71**(1993), no. 1, 93–118. MR**1230287**, DOI 10.1215/S0012-7094-93-07105-0 - Phan H. Loi,
*On automorphisms of subfactors*, J. Funct. Anal.**141**(1996), no. 2, 275–293. MR**1418506**, DOI 10.1006/jfan.1996.0128 - Adrian Ocneanu,
*Quantized groups, string algebras and Galois theory for algebras*, Operator algebras and applications, Vol. 2, London Math. Soc. Lecture Note Ser., vol. 136, Cambridge Univ. Press, Cambridge, 1988, pp. 119–172. MR**996454** - 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.). - Sorin Popa,
*Orthogonal pairs of $\ast$-subalgebras in finite von Neumann algebras*, J. Operator Theory**9**(1983), no. 2, 253–268. MR**703810** - Sorin Popa,
*Classification of amenable subfactors of type II*, Acta Math.**172**(1994), no. 2, 163–255. MR**1278111**, DOI 10.1007/BF02392646 - Nobuya Sato,
*Two subfactors arising from a non-degenerate commuting square. An answer to a question raised by V. F. R. Jones*, Pacific J. Math.**180**(1997), no. 2, 369–376. MR**1487569**, DOI 10.2140/pjm.1997.180.369 - Schou, J. (1990). Commuting squares and index for subfactors. Ph.D. thesis at Odense University.
- Svendsen, A.L. (2002). Commuting squares and automorphisms of subfactors. Ph.D. thesis at University of California at Santa Barbara.
- Hans Wenzl,
*Hecke algebras of type $A_n$ and subfactors*, Invent. Math.**92**(1988), no. 2, 349–383. MR**936086**, DOI 10.1007/BF01404457

## Bibliographic 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
- Email: annelsv@math.uio.no, svendsen@math.ku.dk
- Received by editor(s): December 9, 2002
- Received by editor(s) in revised form: June 2, 2003
- Published electronically: January 21, 2004
- © Copyright 2004 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**356**(2004), 2515-2543 - MSC (2000): Primary 46L37, 46L40
- DOI: https://doi.org/10.1090/S0002-9947-04-03447-6
- MathSciNet review: 2048528