# zbMATH — the first resource for mathematics

An axiomatization of the lattice of higher relative commutants of a subfactor. (English) Zbl 0831.46069
Summary: We consider certain conditions for abstract lattices of commuting squares, that we prove are necessary and sufficient for them to arise as lattices of higher relative commutants of a subfactor. We call such lattices standard and use this axiomatization to prove that their sublattices are standard too. We consider a method for producing sublattices and deduce from this and [the author, Ergodic Theory Dyn. Syst. 15, No. 5, 993-1003 (1995)] some criteria for bipartite graphs to be graphs of subfactors.

##### MSC:
 46L37 Subfactors and their classification
Full Text:
##### References:
 [1] D. Bisch: On the structure of finite depth subfactors In: Algebraic Methods in Operator Theory, Birkhäuser, Basel-Boston-Stuttgart, 1994, pp. 175-194 · Zbl 0809.46069 [2] F. Boca: On the method of constructing irreducible finite index subfactors of Popa. Pac. J. Math161 (1993) 201-231 · Zbl 0795.46044 [3] F. Goodman, P. de la Harpe, V.F.R. Jones: Coxeter graphs and towers of algebras. MSRI Publ. 14, Springer, Berlin, 1989 · Zbl 0698.46050 [4] U. Haagerup: Principal graphs of subfactors in the index range 4 < [M :N] < $$3 + \sqrt 2$$ In: subfactors, World Scientific, Singapore-New Jersey-London Hong Kong, 1994, pp. 1-39 · Zbl 0933.46058 [5] U. Haagerup (In preparation) [6] M. Izumi: Applications of fusion rules to classification of subfactors. Publ. RIMS Kyoto Univ.27 (1991) 953-994 · Zbl 0765.46048 · doi:10.2977/prims/1195169007 [7] M. Izumi, Y. Kawahigashi: Classification of subfactors with principal graphD n (1) . J. Funct. Anal.112 (1993) 257-286 · Zbl 0791.46039 · doi:10.1006/jfan.1993.1033 [8] V.F.R. Jones: Index for subfactors. Invent. Math.72 (1983) 1-25 · Zbl 0508.46040 · doi:10.1007/BF01389127 [9] V.F.R. Jones (In preparation) [10] V.F.R. Jones: An affine Hecke algebra quotient in the Brauer algebra, preprint (1994) · Zbl 0852.20035 [11] R. Longo: Index of subfactors and statistics of quantum fields I. Commn. Math. Phys.130 (1989) 217-247 · Zbl 0682.46045 · doi:10.1007/BF02125124 [12] A. Ocneanu: Quantized groups, string algebras and Galois theory for von Neumann algebran, In: Operator Algebras and Applications. London Math. Soc. Lect. Notes Series 136, London, 1988, pp. 119-172 [13] M. Pimsner, S. Popa: Entropy and index for subfactors. Ann. Sci. Ex. Norm. Sup.19 (1986) 57-106 · Zbl 0646.46057 [14] M. Pimsner, S. Popa: Iterating the basic construction. Trans. Am. Math. Soc.310 (1988) 127-133 · Zbl 0706.46047 · doi:10.1090/S0002-9947-1988-0965748-8 [15] S. Popa: Classification of subfactors of type II. Acta Math.172 No. 2 (1994) 163-255 · Zbl 0853.46059 · doi:10.1007/BF02392646 [16] S. Popa: Classification of subfactors and of their endomorphisms. CBMS Lecture Notes Series, 1994 [17] S. Popa: Markov traces on universal Jones algebras and subfactors of finite index. Invent. Math.111 (1993) 375-405 · Zbl 0787.46047 · doi:10.1007/BF01231293 [18] S. Popa: Free independent sequences in typeII 1, factors and related problems (to appear in Asterisque) [19] S. Popa: Some ergodic properties for infinite graphs associated to subfactors to appear in Ergod. Th. & Dynam. Sys. [20] S. Popa: Symmetric enveloping algebras amenability and AFD properties for subfactors. Math, Research Letters1 (1994) 409-425 · Zbl 0902.46042 [21] F. Radulescu: Random matrices, amalgamated free products and subfactors of the von Neumann algebra of the free group. Invent. Math.115 (1994) 347-389 · Zbl 0861.46038 · doi:10.1007/BF01231764 [22] J. Schou: Commuting squares and index for subfactors. thesis, University of Odense (1990) [23] S.Sunder:II 1 factors, their bimodules and hypergroups. Trans. Am. Math. Soc. (1993) · Zbl 0757.46053
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.