Planar algebras and the Ocneanu-Szymanski theorem
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- by Paramita Das and Vijay Kodiyalam
- Proc. Amer. Math. Soc. 133 (2005), 2751-2759
- DOI: https://doi.org/10.1090/S0002-9939-05-07789-0
- Published electronically: April 19, 2005
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Abstract:
We give a very simple ‘planar algebra’ proof of the part of the Ocneanu-Szymański theorem which asserts that for a finite index, depth two, irreducible $II_1$-subfactor $N \subset M$, the relative commutants $N^\prime \cap M_1$ and $M^\prime \cap M_2$ admit mutually dual Kac algebra structures. In the hyperfinite case, the same techniques also prove the other part, which asserts that $N^\prime \cap M_1$ acts on $M$ with invariants $N$.References
- Gabriella Böhm, Florian Nill, and Kornél Szlachányi, Weak Hopf algebras. I. Integral theory and $C^*$-structure, J. Algebra 221 (1999), no. 2, 385–438. MR 1726707, DOI 10.1006/jabr.1999.7984
- Paramita Das, Weak Hopf $C^*$-algebras and depth two subfactors, J. Funct. Anal. 214 (2004), no. 1, 74–105. MR 2079886, DOI 10.1016/j.jfa.2004.02.010
- V. F. R. Jones, Planar Algebras I, New Zealand Journal of Mathematics, To appear.
- Vijay Kodiyalam, Zeph Landau, and V. S. Sunder, The planar algebra associated to a Kac algebra, Proc. Indian Acad. Sci. Math. Sci. 113 (2003), no. 1, 15–51. Functional analysis (Kolkata, 2001). MR 1971553, DOI 10.1007/BF02829677
- Dmitri Nikshych and Leonid Vainerman, A characterization of depth 2 subfactors of $\textrm {II}_1$ factors, J. Funct. Anal. 171 (2000), no. 2, 278–307. MR 1745634, DOI 10.1006/jfan.1999.3522
- 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
- Sorin Popa, Classification of amenable subfactors of type II, Acta Math. 172 (1994), no. 2, 163–255. MR 1278111, DOI 10.1007/BF02392646
- Wojciech Szymański, Finite index subfactors and Hopf algebra crossed products, Proc. Amer. Math. Soc. 120 (1994), no. 2, 519–528. MR 1186139, DOI 10.1090/S0002-9939-1994-1186139-1
Bibliographic Information
- Paramita Das
- Affiliation: The Institute of Mathematical Sciences, Taramani, Chennai, India 600113
- Address at time of publication: Department of Mathematics and Statistics, University of New Hampshire, Durham, New Hampshire 03824
- Email: pdas@imsc.res.in, pnt2@unh.edu
- Vijay Kodiyalam
- Affiliation: The Institute of Mathematical Sciences, Taramani, Chennai, India 600113
- MR Author ID: 321352
- Email: vijay@imsc.res.in
- Received by editor(s): December 2, 2002
- Received by editor(s) in revised form: June 25, 2003
- Published electronically: April 19, 2005
- Communicated by: David R. Larson
- © Copyright 2005
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 133 (2005), 2751-2759
- MSC (1991): Primary 54C40, 14E20; Secondary 46E25, 20C20
- DOI: https://doi.org/10.1090/S0002-9939-05-07789-0
- MathSciNet review: 2146224