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The embedding theorem for finite depth subfactor planar algebras. (English) Zbl 1230.46055
Summary: We define a canonical planar \(\ast\)-algebra from a strongly Markov inclusion of finite von Neumann algebras. In the case of a connected unital inclusion of finite dimensional \(C^\ast\)-algebras with the Markov trace, we show that this planar algebra is isomorphic to the bipartite graph planar algebra of the Bratteli diagram of the inclusion. Finally, we show that a finite depth subfactor planar algebra is a planar subalgebra of the bipartite graph planar algebra of its principal graph.

MSC:
46L37 Subfactors and their classification
18D10 Monoidal, symmetric monoidal and braided categories (MSC2010)
57M20 Two-dimensional complexes (manifolds) (MSC2010)
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[1] M. Baillet,Y. Denizeau, and J.-F. Havet, Indice d’une espérance conditionelle. Compositio Math. 66 (1988), 199-236. · Zbl 0657.46041 · numdam:CM_1988__66_2_199_0 · eudml:89903
[2] S. Bigelow, S. Morrison, E. Peters, and N. Snyder, Constructing the extended Haagerup planar algebra. Preprit 2009. Acta Math. · Zbl 1270.46058 · doi:10.1007/s11511-012-0081-7 · arxiv:0909.4099
[3] D. Bisch, Bimodules, higher relative commutants and the fusion algebra associated to a subfactor. In Fillmore, Peter A. (ed.) et al., Operator algebras and their appli- cations ( Waterloo, ON, 1994/1995 ), 13-63. Amer. Math. Soc., Providence, RI, 1997. · Zbl 0894.46046 · books.google.com
[4] M. Burns, Subfactors, planar algebras, and rotations. University of California Berkeley, 2003. Ph.D. Thesis. · Zbl 1283.46040 · arxiv:1111.1362
[5] D. E. Evans and Y. Kawahigashi, Quantum Symmetries on Operator Algebras. Clarendon Press, Oxford, 1998. · Zbl 0924.46054
[6] F. M. Goodman, P. de la Harpe, and V. F. R. Jones, Coxeter graphs and towers of algebras. Springer Verlag, New York, 1989. · Zbl 0698.46050
[7] P. Jolissaint, Index for pairs of finite von Neumann algebras. Pacific J. Math. 146 (1990), 43-70. · Zbl 0663.46052 · doi:10.2140/pjm.1990.146.43 · projecteuclid.org
[8] V. F. R. Jones, Index for subfactors. Invent. Math. , 72 (1993), 1-25. · Zbl 0508.46040 · doi:10.1007/BF01389127 · eudml:143011
[9] V. F. R. Jones, Planar algebras I. Preprint 1999. · Zbl 1328.46049 · arxiv:math/9909027
[10] V. F. R. Jones, The planar algebra of a bipartite graph. In C. McA. Gordon (ed.) et al., Knots in Hellas ’98 (Delphi), 94-117. World Sci. Publ., River Edge, NJ, 2000. · Zbl 1021.46047 · math.berkeley.edu
[11] V. F. R. Jones, Quadratic tangles in planar algebras. Prerpint 2010. · Zbl 1257.46033 · doi:10.1215/00127094-1723608 · euclid:dmj/1346936107
[12] V. F.R. Jones and V. S. Sunder, Introduction to subfactors. Cambridge University Press, Cambridge, 1997. · Zbl 0903.46062
[13] V. Kodiyalam and V. S. Sunder, On Jones’ planar algebras. J. Knot Theory Ramifications 13 (2004), 219-247. · Zbl 1054.46045 · doi:10.1142/S021821650400310X
[14] S. Morrison, E. Peters, and N. Snyder, Skein theory for the D2n planar algebras. J. Pure Appl. Alg. 214 (2010), 117-139. · Zbl 1191.46051 · doi:10.1016/j.jpaa.2009.04.010 · arxiv:0808.0764
[15] S. Morrison and K. Walker, The graph planar algebra embedding theorem, Preprint 2010. · tqft.net
[16] D. Penneys, A cyclic approach to the annular Temperley-Lieb category, Preprint 2009. · Zbl 1256.46042 · doi:10.1142/S0218216511010012 · arxiv:0912.1320
[17] E. Peters, A planar algebra construction of the Haagerup subfactor. Internat. J. Math., 21 (2010), 987-1045. · Zbl 1203.46039 · doi:10.1142/S0129167X10006380 · arxiv:0902.1294
[18] M. Pimsner and S. Popa, Entropy and index for subfactors. Ann. Sci. École Norm. Sup. (4) 19 (1986), 57-106. · Zbl 0646.46057 · numdam:ASENS_1986_4_19_1_57_0 · eudml:82174
[19] M. Pimsner and S. Popa, Iterating the basic construction. Trans. Amer. Math. Soc. 310 (1988), 127-133. · Zbl 0706.46047 · doi:10.2307/2001113
[20] S. Popa, Classification of subfactors: the reduction to commuting squares. Invent. Math. 101 (1990), 19-43. · Zbl 0757.46054 · doi:10.1007/BF01231494 · eudml:143795
[21] S. Popa, Classification of amenable subfactors of type II. Acta Math. 172 (1994), 163-255. · Zbl 0853.46059 · doi:10.1007/BF02392646
[22] M. Takesaki, Theory of operator algebras I. Volume 124 of the Encyclopaedia of Math- ematical Sciences. Springer-Verlag, Berlin, 2002. · Zbl 0436.46043
[23] Y. Watatani, Index for C -subalgebras. Amer. Math. Soc., Providence, RI, 1990. · Zbl 0697.46024 · books.google.com
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