Classification of amenable subfactors of type II. (English) Zbl 0853.46059

Popa, Sorin

Acta Math. 172, No. 2, 163-255 (1994).

A central problem arising in the theory of subfactors is the classification of subfactors \(N\subset M\) of finite index of the hyperfinite (or approximately finite dimensional) factors \(M\). The physically relevant invariant for such a subfactor is the lattice of its higher relative commutants \(\{M'_i\cap M_j\}_{i,j}\) in the Jones tower \(N\subset M\subset M_1\subset\dots\); this is called the standard invariant of \(N\subset M\), denoted \(\mathcal G_{N,M}\). The inclusions between these higher relative commutants are described by Jones’ principal graph \(\Gamma_{N,M}\). An abstract characterization of the invariant \(\mathcal G_{N,M}\) is given in [A. Ocneanu, Quantized groups, string algebras, and Galois theory for von Neumann algebras, London Math. Soc. Lecture Notes 136, 119-172 (1988; Zbl 0696.46048] as the paragroup of all irreducible correspondences (bimodules) generated under Connes’ composition (or fusion) rule by \(N\subset M\), and \(\Gamma_{N,M}\) as its “fusion rule matrix” (a Cayley-type graph).
The main result of the paper under review is that for a large class of subfactors, called strongly amenable, the standard invariant \(\mathcal G_{N,M}\) is the complete invariant. It should be noted that all the examples of subfactors coming out this far for quantum field theories and polynomial invariants for knots are strongly amenable. Also, the classical problems of classifying actions of amenable discrete groups and compact Lie groups on hyperfinite factors were given equivalent formulations in terms of classification of certain subfactors that are strongly amenable, in an earlier work by A. Wasserman and the author [Actions of compact Lie groups on von Neumann algebras, C. R. Acad. Sci. Paris Sér. I Math. 315, No. 4, 421-426 (1992; Zbl 0794.46050)]. The result is in some sense the best that can be obtained, as it is shown that strongly amenable subfactors give the largest class of subfactors which can be reconstructed in a direct way from their standard invariants.
The definition of strong amenability goes as follows. First a representation of an inclusion of type II\(_1\) subfactors \(N\subset M\) (of finite index) is defined as an embedding of \(N\subset M\) into an inclusion of von Neumann algebras \(\mathcal N\subset^{\mathcal E}\mathcal M\), with \(N\subset\mathcal N\), \(M\subset\mathcal M\) and with \(\mathcal E:\mathcal M\to\mathcal N\) a conditional expectation of \(\mathcal M\) onto \(\mathcal N\) that restricted to \(\mathcal M\) agrees with the trace preserving expectation of \(M\) onto \(N\). The inclusion \(N\subset M\) is then called amenable if whenever represented smoothly in some \(\mathcal N\subset^{\mathcal E}\mathcal M\) there exists an \(M\)-hypertrace \(\phi\) on \(\mathcal M\) satisfying \(\phi=\phi\circ\mathcal E\); this is equivalent to the existence of a norm one projection of \(\mathcal N\subset\mathcal M\) onto \(N\subset M\). Finally \(N\subset M\) is strongly amenable if in addition the graph \(\Gamma_{N,M}\) is ergodic. Equivalently, \(N\subset M\) is strongly amenable if and only if it is isomorphic to the canonical model \(N^{\text{st}} \subset M^{\text{st}}\) coming from the standard invariant \(\mathcal G_{N,M}\), i.e. from the higher relative commutants. For inclusions of type II\(_\infty\) factors \(N^\infty\subset M^\infty\) of finite index, (strong) amenability is defined as the (strong) amenability of the type II\(_1\) inclusion \(pN^\infty p\subset p M^\infty p\) obtained by reducing with finite projections in \(N^\infty\).

Reviewer: M.Engliš (Praha)

MSC:

46L37

Subfactors and their classification

Keywords:

operator algebras; factors; Jones tower; amenable subfactors

Citations:

Zbl 0696.46048; Zbl 0794.46050

References:

[1]	[A]Avez, A., Entropie des groupes de type fini.C. R. Acad. Sci. Paris Sér. I Math., 275 (1972), 1363–1366. · Zbl 0252.94013
[2]	[AD]Anantharam-Delaroche, C., Correspondences and related properties of subfactors. Preprint, 1992.
[3]	[BDH]Baillet, M., Denizeau, Y. &Havet, J., Indice d’une esperance conditionelle.Compositio Math., 66 (1988), 199–236. · Zbl 0657.46041
[4]	[Bi1]Bisch, D., Entropy of groups and subfactors.J. Funct. Anal., 103 (1992), 190–208. · Zbl 0829.46050
[5]	[Bi2] –, On the existence of central sequences in subfactors.Trans. Amer. Math. Soc., 321 (1990), 117–128. · Zbl 0711.46048
[6]	[Bi3]Bisch, D., A note on intermediate subfactors. To appear inPacific J. Math. · Zbl 0814.46053
[7]	[BN]Bion-Nadal, J., Subfactors associated toE 6.J. Operator Theory, 28 (1992), 27–50. · Zbl 0815.46052
[8]	[C1]Connes, A., Une classification des facteurs de type III.Ann. Sci. École Norm. Sup. (4), 6 (1973), 133–252. · Zbl 0274.46050
[9]	[C2] –, Outer conjugacy classes of automorphisms of factors.Ann. Sci. École Norm. Sup. (4), 8 (1975), 383–419. · Zbl 0342.46052
[10]	[C3] –, Classification of injective factors.Ann. of Math., 104 (1976), 73–115. · Zbl 0343.46042
[11]	[C4] –, Classification des facteurs.Proc. Sympos. Pure Math., 38:2 (1982), 43–109.
[12]	[C5] –, Periodic automorphisms of the hyperfinite type II1 factors.Acta. Sci. Math. (Szeged), 39 (1977), 39–66. · Zbl 0382.46027
[13]	[C6] –, Compact metric spaces, Fredholm modules and hyperfiniteness.Pacific J. Math., 137 (1989), 120–135. · Zbl 0718.46051
[14]	[Cho]Choda, M., The conjugacy classes of subfactors and the outer conjugacy classes of the automorphism group.Math. Japon., 32 (1987), 379–388. · Zbl 0656.46046
[15]	[Chr]Christensen, E., Subalgebras of finite algebras.Math. Ann., 243 (1979), 17–29. · Zbl 0406.46052
[16]	[CS]Connes, A. &Størmer, E., Entropy for automorphisms of II1 von Neumann algebras.Acta Math., 134 (1975), 288–306. · Zbl 0326.46032
[17]	[CT]Connes, A. &Takesaki, M., The flow of weights on a factor of type III.Tôhoku Math. J., 29 (1977), 473–575. · Zbl 0408.46047
[18]	[D]Dixmier, J.,Les algèbres d’opérateurs sur l’espace hilbertien (algèbres de von Neumann). Gauthier-Villars, Paris, 1957, 1969. · Zbl 0088.32304
[19]	[DHR]Doplicher, S., Haag, R. &Roberts, J., Local observables and particle statistics.Comm. Math. Phys., 23 (1971), 199–230.
[20]	[E]Effros, E., Amenability and virtual diagonals for von Neumann algebras.J. Funct. Anal., 78 (1988), 137–154. · Zbl 0655.46053
[21]	[EvKa]Evans, D. & Kawahigashi, Y., Orbifold subfactors from Hecke algebras. Preprint, 1992.
[22]	[FeM]Feldman, J. &Moore, C. C., Ergodic equivalence relations, cohomology and von Neumann algebras I and II.Trans. Amer. Math. Soc., 236 (1977), 289–361. · Zbl 0369.22009
[23]	[Fr]Frölich, J., Statistics of fields, the Yang-Baxter equation and the theory of knots and links, inNon-Perturbative Quantum Field Theory. 1987 Cargese Lectures. Plenum Press, New York, 1988.
[24]	[FRS]Fredenhagen, K., Rehren, K. &Schroer B., Superselection sectors, with braid group statistics and exchange algebras I: General theory.Comm. Math. Phys., 125 (1989), 201–230. · Zbl 0682.46051
[25]	[GHJ]Goodman, F., De la Harpe, P. & Jones, V. F. R.,Coxeter Graphs and Towers of Algebras. Math. Sci. Res. Inst. Publ. 14. Springer-Verlag, 1989. · Zbl 0698.46050
[26]	[H]Haagerup, U., In preparation.
[27]	[HaG]Handelman D. & Gould, J., Positive multiplications in difference equations. Preprint, 1991.
[28]	[Hi]Hiai, F., Minimizing indices of conditional expectations onto a subfactor.Publ. Res. Inst. Math. Sci., 24 (1988), 673–678. · Zbl 0679.46050
[29]	[HSc]Haagerup, U. & Schou, J., Some new subfactors of the hyperfinite type II1 factor. Preprint.
[30]	[Iz]Izumi, M., Application of fusion rules to classification of subfactors.Publ. Res. Inst. Math. Sci., 27 (1991), 953–994. · Zbl 0765.46048
[31]	[IzKa]Izumi, M. &Kawahigashi, Y., Classification of subfactors with principal graphD n (1) .J. Funct. Anal., 112 (1993), 257–286. · Zbl 0791.46039
[32]	[J1]Jones, V. F. R.,Actions of Finite Groups on the Hyperfinite Type II 1 Factor. Mem. Amer. Math. Soc., 237. Providence, RI, 1980.
[33]	[J2] – Index for subfactors.Invent. Math., 72 (1983), 1–25. · Zbl 0508.46040
[34]	[J3]Jones, V. F. R., Subfactors of II1 factors and related topics, inProceedings of the International Congress of Mathematicians, Berkeley, 1986, pp. 939–947.
[35]	[J4] – Hecke algebra representation of braid groups and link polynomials.Ann. of Math., 126 (1987), 335–388. · Zbl 0631.57005
[36]	[K]Kadison, R. V., Normality in operator algebras.Duke Math. J., 29 (1962), 459–464. · Zbl 0177.17802
[37]	[Ka]Kawahigashi, Y., On flatness of Ocneanu’s connection on the Dynkin diagrams and classification of subfactors. Preprint, 1990.
[38]	[KaiVe]Kaimanovitch, V. &Vershik, A., Random walks on discrete groups: boundary and entropy.Ann. Probab., 11 (1983), 457–490. · Zbl 0641.60009
[39]	[Ke]Kesten, H., Symmetric random walks on groups.Trans. Amer. Math. Soc., 92 (1959), 336–354. · Zbl 0092.33503
[40]	[Ko]Kosaki, H., Extension of Jones theory on index to arbitrary factors.J. Funct. Anal., 66 (1986), 123–140. · Zbl 0607.46034
[41]	[KoL]Kosaki, H. & Longo, R., A remark on the minimal index for subfactors. Preprint, 1991.
[42]	[L1]Longo, R., Index of subfactors and statistics of quantum fields I.Comm. Math. Phys., 126 (1989), 217–247. · Zbl 0682.46045
[43]	[L2] –, Index of subfactors and statistics of quantum fields II.Comm. Math. Phys., 130 (1990), 285–309. · Zbl 0705.46038
[44]	[L3] –, Minimal index and braided subfactors.J. Funct. Anal., 109 (1992), 98–112. · Zbl 0798.46047
[45]	[Lo]Loi, P., On the theory of index and type III factors. Thesis, 1988
[46]	[MvN]Murray, F. &von Neumann, J., Rings of operators IV.Ann. of Math., 44 (1943), 716–808. · Zbl 0060.26903
[47]	[Oc1]Ocneanu, A.,Actions of Discrete Amenable Groups on von Neumann Algebras. Lecture Notes in Math., 1138. Springer-Verlag, Berlin-Heidelberg-New York, 1985. · Zbl 0608.46035
[48]	[Oc2]Ocneanu, A., Quantized groups, string algebras, and Galois theory for von Neumann algebras, inOperator Algebras and Applications. London Math. Soc. Lecture Note Ser., 136, pp. 119–172. Cambridge Univ. Press, 1988. · Zbl 0696.46048
[49]	[Ocetal]Ocneanu, A., Freyd, P., Yetter, D., Hoste, J., Likorish, W. &Millet, K., A new polynomial invariant of knots and links.Bull. Amer. Math. Soc., 12 (1985), 183–189. · Zbl 0575.42003
[50]	[Ok]Okamoto, S., Invariants for subfactors arizing from Coxeter graphs. Thesis.
[51]	[Or]Orey, S.,Limit Theorems for Markov Chain Transition Probabilities. Van Nostrand Reinhold Math. Studies, 34. London, 1971. · Zbl 0295.60054
[52]	[PiPo1]Pimsner, M. &Popa, S., Entropy and index for subfactors.Ann. Sci. École Norm. Sup. (4), 19 (1986), 57–106. · Zbl 0646.46057
[53]	[PiPo2] –, Iterating the basic construction.Trans. Amer. Math. Soc., 310 (1988), 127–133. · Zbl 0706.46047
[54]	[PiPo3] –, Finite dimensional approximation for pairs of algebras and obstructions for the index.J. Funct. Anal., 98 (1991), 270–291. · Zbl 0746.46052
[55]	[Po1]Popa, S., On a problem of R. V. Kadison on maximal abelian *-subalgebras in factors.Invent. Math., 65 (1981), 269–281. · Zbl 0481.46028
[56]	[Po2] –, Singular maximal abelian *-subalgebras in continuous von Neumann algebras.J. Funct. Anal., 50 (1983), 151–166. · Zbl 0526.46059
[57]	[Po3] –, Maximal injective subalgebras in factors associated with free groups.Adv. In Math., 50 (1983), 27–48. · Zbl 0545.46041
[58]	[Po4]Popa, S., Correspondences. Preprint, 1986.
[59]	[Po5] –, Classification of subfactors: reduction to commuting squares.Invent. Math., 101 (1990), 19–43. · Zbl 0757.46054
[60]	[Po6] –, Sous-facteurs, actions des groupes et cohomologie.C. R. Acad. Sci. Paris Sér I Math., 309 (1989), 771–776. · Zbl 0684.46051
[61]	[Po7] –, Sur la classification des sous-facteurs d’indice fini du facteur hyperfini.C. R. Acad. Sci. Paris Sér I Math., 311 (1990), 95–100. · Zbl 0706.46048
[62]	[Po8] –, Relative dimension, towers of projections and commuting squares.Pacific J. Math., 137 (1989), 181–207. · Zbl 0699.46042
[63]	[Po9]Popa, S., Classification of actions of discrete amenable groups on subfactors. I.H.E.S. preprint, 1992. · Zbl 0766.43002
[64]	[Po10]Popa, S., On amenable factors of type II1, inOperator Algebras and Applications. London Math. Soc. Lecture Note Ser., 136, pp. 173–184. Cambridge Univ. Press, 1988.
[65]	[Po11]Popa, S., Hyperfinite subalgebras normalized by a given automorphism and related topics, inLecture Notes in Math., 1132, pp. 421–433. Springer-Verlag, 1984.
[66]	[Po12]Popa, S., Some equivalent characterizations of amenability for inclusions of factors. To appear.
[67]	[PoWa]Popa, S. &Wassermann, A., Actions of compact Lie groups on von Neumann algebras.C. R. Acad. Sci. Paris Sér I Math., 315 (1992), 421–426. · Zbl 0794.46050
[68]	[PS]Powers, R. &Størmer, E., Free states of the canonical anticommutation relations.Comm. Math. Phys., 16 (1970), 1–33. · Zbl 0186.28301
[69]	[Sk]Skau, C., Finite subalgebras of a von Neumann algebra.J. Funct. Anal., 25 (1977), 211–235. · Zbl 0347.46070
[70]	[StZ]Stratila, S. &Zsido, L.,Lectures on von Neumann Algebras, Abacus Press-Editura Academiei, London-Bucharest, 1979.
[71]	[SuVi]Sunder, S. & Vijayavagan, A., On the non-occurence of the Coxeter graphsE 7,D 2n+1 as principal graphs of an inclusion of II1 factor. Preprint, 1991.
[72]	[T1]Takesaki, M., Duality for crossed products and the structure of von Neumann algebras of type III.Acta Math., 131 (1973), 249–310. · Zbl 0268.46058
[73]	[T2] –, Conditional expectations in von Neumann algebras.J. Funct. Anal., 9 (1972), 306–321. · Zbl 0245.46089
[74]	[Wa1]Wassermann, A., Ergodic actions of compact groups on operator algebras III.Invent. Math., 9, (1988), 309–355. · Zbl 0692.46058
[75]	[Wa2]Wassermann, A., Coactions and Yang-Baxter equations for ergodic actions and subfactors, inOperator Algebras and Applications. London Math. Soc. Lecture Note Ser., 136, pp. 202–236. Cambridge Univ. Press, 1988. · Zbl 0809.46079
[76]	[We1]Wenzl, H., Hecke algebras of typeA n and subfactors.Invent. Math., 92 (1988), 349–383. · Zbl 0663.46055
[77]	[We2] –, Quantum groups and subfactors of typeB, C, andD.Comm. Math. Phys., 133 (1990), 383–432. · Zbl 0744.17021