Composition of subfactors : new examples of infinite depth subfactors

Dietmar Bisch; Uffe Haagerup

Annales scientifiques de l'École Normale Supérieure (1996)

  • Volume: 29, Issue: 3, page 329-383
  • ISSN: 0012-9593

How to cite


Bisch, Dietmar, and Haagerup, Uffe. "Composition of subfactors : new examples of infinite depth subfactors." Annales scientifiques de l'École Normale Supérieure 29.3 (1996): 329-383. <>.

author = {Bisch, Dietmar, Haagerup, Uffe},
journal = {Annales scientifiques de l'École Normale Supérieure},
keywords = {inclusions of factors; finite Jones index; fusion of - and - bimodules; irreducibility; finite depth; strong amenability; amenability; irreducible, amenable subfactors; hyperfinite factor},
language = {eng},
number = {3},
pages = {329-383},
publisher = {Elsevier},
title = {Composition of subfactors : new examples of infinite depth subfactors},
url = {},
volume = {29},
year = {1996},

AU - Bisch, Dietmar
AU - Haagerup, Uffe
TI - Composition of subfactors : new examples of infinite depth subfactors
JO - Annales scientifiques de l'École Normale Supérieure
PY - 1996
PB - Elsevier
VL - 29
IS - 3
SP - 329
EP - 383
LA - eng
KW - inclusions of factors; finite Jones index; fusion of - and - bimodules; irreducibility; finite depth; strong amenability; amenability; irreducible, amenable subfactors; hyperfinite factor
UR -
ER -


  1. [ABH] J. ANDERSON, B. BLACKADAR and U. HAAGERUP, Minimal projections in the reduced groups C*-algebra of ℤn * ℤM, (J. of Operator Theory, Vol. 26, 1991, pp. 3-23). Zbl0785.46050MR94c:46110
  2. [Av] AVEZ, Entropy des groupes de type fini (C. R. Acad. Sci. Paris, Vol. 275A, 1972, pp. 1363-1366). Zbl0252.94013MR48 #3090
  3. [BC] Ch. BERG and J. P. R. CHRISTENSEN, Sur la norme des opérateurs de convolution (Invent. math., Vol. 23, 1974, pp. 173-178). MR49 #3449
  4. [Bi1] D. BISCH, Entropy of groups and subfactors (Journal of Funct. Analysis, Vol. 103, 1992, pp. 190-208). Zbl0829.46050MR93e:46076
  5. [Bi2] D. BISCH, A note on intermediate subfactors (Pacific Journal of Math., Vol. 163, 1994, pp. 201-216). Zbl0814.46053MR95c:46105
  6. [Bi3] D. BISCH, On the structure of finite depth subfactors (in Algebraic Methods in Operator Theory, Birkhäuser, 1994, pp. 175-194). Zbl0809.46069MR95j:46073
  7. [Bu] W. BURNSIDE, Theory of groups of finite order, Cambridge University Press, 1911. JFM42.0151.02
  8. [CD] G. CHOQUET, J. DENY, Sur l'équation de convolution µ = µ * σ (C. R. Acad. Sci. Paris, Vol. 250A, 1960, pp. 799-801). Zbl0093.12802MR22 #9808
  9. [Co] A. CONNES, Notes on correspondences, unpublished manuscript. 
  10. [CM] H. S. M. COXETER and W. O. J. MOSER, Generators and relations for discrete groups (Springer-Verlag, New York, Berlin, Heidelberg, 2nd edition, 1972). Zbl0239.20040MR50 #2313
  11. [DG] Y. DERRIENNIC and Y. GUIVARC'H, Théorème de renouvellement pour les groups non-moyennables (C. R. Acad. Sci. Paris, Série A, Vol. 277, 1973, pp. 613-615). Zbl0272.60005MR48 #7332
  12. [Fu] H. FURSTENBERG, Random walks and discrete subgroups of Lie groups (in Adv. Prob. Related Topics, Dekker, New York, Vol. 1, 1971, pp. 1-63). Zbl0221.22008MR44 #1794
  13. [GHJ] F. GOODMAN, P. de la HARPE, and V. JONES, Coxeter graphs and towers of algebras, Springer-Verlag, MSRI publications, 1989. Zbl0698.46050MR91c:46082
  14. [G] F. P. GREENLEAF, Invariant means on topological groups and their applications, Van Nostrand, New York, 1969. Zbl0174.19001MR40 #4776
  15. [Ha] U. HAAGERUP, Principal graphs of subfactors in the index range 4 &lt; [M : N] &lt; 3 + √2 (in ”Subfactors”, Proceedings of the Taniguchi Symposium on Operator Algebras, 1994, pp. 1-38). Zbl0933.46058MR96d:46081
  16. [Hi] F. HIAI, Entropy and growth for derived towers of subfactors (in “Subfactors”, Proceedings of the Taniguchi Symposium on Operator Algebras, 1994, pp. 206-232). Zbl0932.46049MR96c:46060
  17. [HPS] G. HOLE, S. C. PORT and C. J. STONE, Introduction to stochastical processes, Houghton Mifflin Co., 1972. Zbl0258.60003
  18. [Jo] V. F. R. JONES, Index for subfactors (Invent. Math., Vol. 72, 1983, pp. 1-25). Zbl0508.46040MR84d:46097
  19. [KV] V. A. KAIMANOVICH and A. M. VERSHIK, Random walks on discrete groups : Boundary and entropy (Ann. Probab., Vol. 11, No. 3, 1983, pp. 457-490). Zbl0641.60009MR85d:60024
  20. [Ka1] V. A. KAIMANOVICH, Poisson boundaries of random walks on discrete solvable groups (Proc. 10th Oberwolfach Conf. on Prob. Measures on groups, ed. H. Heyer, Plenum, New York, 1991, pp. 205-238). Zbl0823.60006MR94m:60014
  21. [Ka2] V. A. KAIMANOVICH, The Poisson boundary of covering Markov operators, to appear in Israel Journal of Math. Zbl0843.43001
  22. [Ka3] V. A. KAIMANOVICH, Bi-harmonic functions on groups (C. R. Acad. Sci. Paris, Vol. 314, 1992, pp. 259-262). Zbl0754.60015MR93e:60160
  23. [Ka4] V. A. KAIMANOVICH, Examples of non-commutative groups with non-trivial exit boundary (J. Soviet Math., Vol. 28, 1985, pp. 579-591). Zbl0558.60010
  24. [Ke] H. KESTEN, Symmetric random walks on groups (Trans. Amer. Math. Soc., Vol. 92, 1959, pp. 336-354). Zbl0092.33503MR22 #253
  25. [K] A. W. KNAPP, Doubly generated Fuchsian groups (Michigan Math. Journal, Vol. 15, 1968, pp. 289-304). Zbl0167.07002MR40 #1483
  26. [M] J. P. MATELSKI, The classification of discrete 2-generator subgroups of PSL(2, ℝ) (Israel J. Math., Vol. 42, 1982, pp. 309-317). Zbl0497.20036MR84d:10029
  27. [Me] J. L. MENNICKE, Finite factor groups of the unimodular group (Ann. of Math., Vol. 81, 1965, pp. 31-37). Zbl0135.06504MR30 #2083
  28. [Ne] M. NEWMAN, The structure of some subgroups of the modular group (Ill. Jour. of Math., Vol. 6, 1962, pp. 480-487). Zbl0104.25301MR25 #4001
  29. [Oc1] A. OCNEANU, Quantized group string algebras and Galois theory for operator algebras, in Operator Algebras and Applications 2 (London Math. Soc. Lect., Notes Series, Vol. 136, 1988, pp. 119-172). Zbl0696.46048MR91k:46068
  30. [Oc2] A. OCNEANU, Quantum symmetry, differential geometry of finite graphs and classification of subfactors University of Tokyo Seminary Notes (notes recorded by Y. Kawahigashi), Vol. 45, 1991. 
  31. [Po1] S. POPA, Correspondences, Increst preprint, 1986. 
  32. [Po2] S. POPA, Classification of subfactors : reduction to commuting squares (Invent. Math., Vol. 101, 1990, pp. 19-43). Zbl0757.46054MR91h:46109
  33. [Po3] S. POPA, Sous-facteurs, actions des groupes et cohomologie (C. R. Acad. Sci. Paris, Vol. 309, 1989, pp. 771-776). Zbl0684.46051MR91i:46069
  34. [Po4] S. POPA, Classification of amenable subfactors of type II (Acta Math., Vol. 172, 1994, pp. 352-445). Zbl0853.46059MR95f:46105
  35. [Po5] S. POPA, Symmetric enveloping algebras, amenability and AFD properties for subfactors (Mathematical Research Letters, Vol. 1, 1994, pp. 409-425). Zbl0902.46042MR95i:46095
  36. [Po6] S. POPA, Approximate innerness and central freeness for subfactors : A classification result, preprint 1993. 
  37. [R] G. ROSENBERGER, All generating pairs of all two-generator Fuchsian groups (Arch. Math., Vol 46, 1986, pp. 198-204). Zbl0563.20043MR87g:20080
  38. [Se] E. SENETA, Non-negative matrices and Markov chains, Springer Verlag, 1981. Zbl0471.60001MR85i:60058
  39. [Sp] F. SPITZER, Principles of Random walks, 2nd ed. Springer Verlag, New York, Heidelberg, Berlin, 1976. Zbl0359.60003MR52 #9383
  40. [Su] V. S. SUNDER, II1 factors, their bimodules and hypergroups (Trans. Amer. Math. Soc., Vol. 330, 1992, pp. 227-256). Zbl0757.46053MR92f:46076
  41. [Sz] W. SZYMANSKI, Finite index subfactors and Hopf algebra crossed products (Proc. of the Amer. Math. Soc, Vol. 120, 1994, pp. 519-528). Zbl0802.46076MR94d:46061
  42. [Voi] D. VOICULESCU, Symmetries of some reduced free product C*-algebras (in Operator Algebras and Their Connections with Topology and Ergodic Theory, Lecture Notes in Mathematics, Springer Verlag, Vol. 1132, 1985, pp. 556-588). Zbl0618.46048MR87d:46075
  43. [Wi] J. WIERZBICKI, The estimation of the depth from an intermediate subfactor, preprint 1993. 
  44. [Wo] W. WOESS, Random walks on infinite graphs and groups - a survey on selected topics (Bull. London Math. Soc., Vol. 26, 1994, pp. 1-60). Zbl0830.60061MR94i:60081
  45. [Ya] S. YAMAGAMI, A note on Ocneanu's approach to Jones' index theory (Internat. J. of Math., Vol. 4, 1993, pp. 859-871). Zbl0793.46040MR95f:46114

NotesEmbed ?


You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.


Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.