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Generalization of Longo-Rehren construction to subfactors of infinite depth and amenability of fusion algebras. (English) Zbl 0954.46038

The aim of the paper was two-fold. First, the author wanted to extend the Longo-Rehren construction of subfactors [R. Longo and K.-H. Rehren, Rev. Math. Phys. 7, No. 4, 567-597 (1995; Zbl 0836.46055)] to the case of infinite depth, and secondly, he wanted to give a more direct approach to the amenability of subfactors based on Work by F. Hiai and M. Izumi [Int. J. Math. 9, No. 6, 669-722 (1998)] on amenability of fusion algebras. Regarding the first, he also proves that Popa’s symmetric enveloping inclusion and the subfactor constructed by Longo and Rehren are isomorphic.

MSC:

46L37 Subfactors and their classification
46L80 \(K\)-theory and operator algebras (including cyclic theory)

Citations:

Zbl 0836.46055
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References:

[1] Choda, M.; Kosaki, H., Strongly outer actions for an inclusion of factors, J. Funct. Anal., 122, 315-332 (1994) · Zbl 0802.46072
[2] Connes, A., Classification of injective factors, Ann. Math., 104, 73-115 (1976) · Zbl 0343.46042
[3] A. Connes, On the classification of von Neumann algebras and their automorphisms, Sympos. Math.20, 435-478.; A. Connes, On the classification of von Neumann algebras and their automorphisms, Sympos. Math.20, 435-478.
[4] Connes, A., Noncommutative Geometry (1994), Academic Press: Academic Press San Diego · Zbl 1106.58004
[5] Evans, D. E.; Kawahigashi, Y., Quantum Symmetries on Operator Algebras (1998), Oxford Univ. Press: Oxford Univ. Press London · Zbl 0924.46054
[6] Goto, S., Orbifold construction for non-AFD subfactors, Internat. J. Math., 5, 725-746 (1994) · Zbl 0818.46069
[7] Greenleaf, F. P., Invariant Means on Topological Groups and Their Applications (1969), Van Nostrand: Van Nostrand New York · Zbl 0174.19001
[8] F. Fidaleo, and, T. Isola, The canonical endomorphism for infinite index inclusions, preprint, 1996.; F. Fidaleo, and, T. Isola, The canonical endomorphism for infinite index inclusions, preprint, 1996. · Zbl 0933.46059
[9] Hiai, F.; Izumi, M., Amenability and strong amenability for fusion algebras with applications to subfactor theory, Internat. J. Math., 9, 669-722 (1998) · Zbl 0978.46043
[10] M. Izumi, private communication.; M. Izumi, private communication.
[11] Jones, V. F.R., Index for subfactors, Invent. Math., 72, 1-25 (1983) · Zbl 0508.46040
[12] Longo, R., A duality for Hopf algebras and for subfactors, Comm. Math. Phys., 159, 133-150 (1994) · Zbl 0802.46075
[13] Longo, R.; Rehren, K.-H., Nets of subfactors, Rev. Math., 7, 567-597 (1995) · Zbl 0836.46055
[14] Masuda, T., An analogue of Longo’s canonical endomorphism for bimodule theory and its application to asymptotic inclusions, Internat. J. Math., 8, 249-265 (1997) · Zbl 0909.46044
[15] T. Masuda, Extension of automorphisms of a subfactor to the symmetric enveloping algebra, preprint, 1999.; T. Masuda, Extension of automorphisms of a subfactor to the symmetric enveloping algebra, preprint, 1999. · Zbl 0957.46508
[16] Ocneanu, A., Quantized group string algebras and Galois theory for algebras, Operator Algebras and Applications, Vol. 2 (Warwick, 1987). Operator Algebras and Applications, Vol. 2 (Warwick, 1987), London Mathematical Society Lecture Note Series, 136 (1988), Cambridge Univ. Press: Cambridge Univ. Press Cambridge, p. 119-172
[17] Ocneanu, A., Quantum Symmetry, Differential Geometry of Finite Graphs and Classification of Subfactors. Quantum Symmetry, Differential Geometry of Finite Graphs and Classification of Subfactors, University of Tokyo Seminary Notes (1991)
[18] A. Ocneanu, An invariant coupling between 3-manifolds and subfactors, with connections to topological and conformal quantum field theory, preprint, 1991.; A. Ocneanu, An invariant coupling between 3-manifolds and subfactors, with connections to topological and conformal quantum field theory, preprint, 1991.
[19] A. Ocneanu, Seminar talk at University of California, Berkeley, June 1993.; A. Ocneanu, Seminar talk at University of California, Berkeley, June 1993.
[20] Pimsner, M.; Popa, S., Entropy and index for subfactors, Ann. Sci. Ecole Norm. Sup., 19, 57-106 (1986) · Zbl 0646.46057
[21] Pimsner, M.; Popa, S., Iterating the basic constructions, Trans. Amer. Math. Soc., 310, 127-134 (1988) · Zbl 0706.46047
[22] S. Popa, Correspondences, preprint, 1986.; S. Popa, Correspondences, preprint, 1986.
[23] Popa, S., Classification of amenable subfactors of type II, Acta. Math., 172, 163-255 (1994) · Zbl 0853.46059
[24] Popa, S., Symmetric enveloping algebras, amenability and AFD properties for subfactors, Math. Res. Lett., 1, 409-425 (1994) · Zbl 0902.46042
[25] S. Popa, Some properties of the symmetric enveloping algebra of a subfactor with applications to amenability and property \(T\); S. Popa, Some properties of the symmetric enveloping algebra of a subfactor with applications to amenability and property \(T\)
[26] Yamagami, S., A note on Ocneanu’s approach to Jones index theory, Internat. J. Math., 4, 859-871 (1993) · Zbl 0793.46040
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