An angle between intermediate subfactors and its rigidity

Bakshi, Keshab; Das, Sayan; Liu, Zhengwei; Ren, Yunxiang

doi:10.1090/tran/7738

An angle between intermediate subfactors and its rigidity
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by Keshab Chandra Bakshi, Sayan Das, Zhengwei Liu and Yunxiang Ren PDF

Trans. Amer. Math. Soc. 371 (2019), 5973-5991 Request permission

Abstract:

We introduce a new notion of an angle between intermediate subfactors and prove various interesting properties of the angle and relate it to the Jones index. We prove a uniform $60$ to $90$ degree bound for the angle between minimal intermediate subfactors of a finite index irreducible subfactor. From this rigidity we can bound the number of minimal (or maximal) intermediate subfactors by the kissing number in geometry. As a consequence, the number of intermediate subfactors of an irreducible subfactor has at most exponential growth with respect to the Jones index. This answers a question of Longo’s published in 2003.

References

Dietmar Bisch, A note on intermediate subfactors, Pacific J. Math. 163 (1994), no. 2, 201–216. MR 1262294
Dietmar Bisch, Bimodules, higher relative commutants and the fusion algebra associated to a subfactor, Operator algebras and their applications (Waterloo, ON, 1994/1995) Fields Inst. Commun., vol. 13, Amer. Math. Soc., Providence, RI, 1997, pp. 13–63. MR 1424954, DOI 10.1007/s002220050137
Dietmar Bisch and Vaughan Jones, Algebras associated to intermediate subfactors, Invent. Math. 128 (1997), no. 1, 89–157. MR 1437496, DOI 10.1007/s002220050137
Dietmar Bisch and Vaughan Jones, Singly generated planar algebras of small dimension, Duke Math. J. 101 (2000), no. 1, 41–75. MR 1733737, DOI 10.1215/S0012-7094-00-10112-3
Bill Casselman, The difficulties of kissing in three dimensions, Notices Amer. Math. Soc. 51 (2004), no. 8, 884–885. MR 2145822
Erik Christensen, Subalgebras of a finite algebra, Math. Ann. 243 (1979), no. 1, 17–29. MR 543091, DOI 10.1007/BF01420203
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
Pinhas Grossman and Masaki Izumi, Classification of noncommuting quadrilaterals of factors, Internat. J. Math. 19 (2008), no. 5, 557–643. MR 2418197, DOI 10.1142/S0129167X08004807
Pinhas Grossman and Vaughan F. R. Jones, Intermediate subfactors with no extra structure, J. Amer. Math. Soc. 20 (2007), no. 1, 219–265. MR 2257402, DOI 10.1090/S0894-0347-06-00531-5
Robert Guralnick and Feng Xu, On a subfactor generalization of Wall’s conjecture, J. Algebra 332 (2011), 457–468. MR 2774698, DOI 10.1016/j.jalgebra.2011.01.003
P. R. Halmos, Two subspaces, Trans. Amer. Math. Soc. 144 (1969), 381–389. MR 251519, DOI 10.1090/S0002-9947-1969-0251519-5
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
Zeph A. Landau, Exchange relation planar algebras, Proceedings of the Conference on Geometric and Combinatorial Group Theory, Part II (Haifa, 2000), 2002, pp. 183–214. MR 1950890, DOI 10.1023/A:1021296230310
Roberto Longo, Conformal subnets and intermediate subfactors, Comm. Math. Phys. 237 (2003), no. 1-2, 7–30. Dedicated to Rudolf Haag. MR 2007172, DOI 10.1007/s00220-003-0814-8
F. J. Murray and J. Von Neumann, On rings of operators, Ann. of Math. (2) 37 (1936), no. 1, 116–229. MR 1503275, DOI 10.2307/1968693
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, Correspondences, preprint, Ser. in Mathematics 13 (1986), no. 56.
Sorin Popa, Relative dimension, towers of projections and commuting squares of subfactors, Pacific J. Math. 137 (1989), no. 1, 181–207. MR 983336
Sorin Popa, Classification of amenable subfactors of type II, Acta Math. 172 (1994), no. 2, 163–255. MR 1278111, DOI 10.1007/BF02392646
Sorin Popa, An axiomatization of the lattice of higher relative commutants of a subfactor, Invent. Math. 120 (1995), no. 3, 427–445. MR 1334479, DOI 10.1007/BF01241137
Mihai Pimsner and Sorin Popa, Entropy and index for subfactors, Ann. Sci. École Norm. Sup. (4) 19 (1986), no. 1, 57–106. MR 860811
Takashi Sano and Yasuo Watatani, Angles between two subfactors, J. Operator Theory 32 (1994), no. 2, 209–241. MR 1338739
Tamotsu Teruya and Yasuo Watatani, Lattices of intermediate subfactors for type III factors, Arch. Math. (Basel) 68 (1997), no. 6, 454–463. MR 1444657, DOI 10.1007/s000130050078
John von Neumann, Continuous geometry, Princeton Mathematical Series, No. 25, Princeton University Press, Princeton, N.J., 1960. Foreword by Israel Halperin. MR 0120174
G. E. Wall, Some applications of the Eulerian functions of a finite group, J. Austral. Math. Soc. 2 (1961/1962), 35–59. MR 0125156
Yasuo Watatani, Lattices of intermediate subfactors, J. Funct. Anal. 140 (1996), no. 2, 312–334. MR 1409040, DOI 10.1006/jfan.1996.0110
Feng Xu, On a problem about tensor products of subfactors, Adv. Math. 246 (2013), 128–143. MR 3091803, DOI 10.1016/j.aim.2013.06.026
Feng Xu, On maximal subfactors from quantum groups, J. Funct. Anal. 268 (2015), no. 9, 2735–2753. MR 3325536, DOI 10.1016/j.jfa.2015.02.007
Feng Xu, Symmetries of subfactors motivated by Aschbacher-Guralnick conjecture, Adv. Math. 289 (2016), 345–361. MR 3439688, DOI 10.1016/j.aim.2015.10.029