Nombres de Betti $L^2$ et facteurs de type ${\rm II}

Nombres de Betti

L^{2}

et facteurs de type

{II}_{1}

[

L^{2}

Betti numbers and applications of the

p

-adic cohomology type II

_{1}

factors ]

Connes, Alain

Séminaire Bourbaki : volume 2002/2003, exposés 909-923, Astérisque no. 294 (2004), Talk no. 920, p. 321-333

Abstract
Résumé

Damien Gaboriau showed recently that the $L^{2}$ Betti numbers of measured foliations with contractile leaves are invariants of the associated equivalence relation. Sorin Popa used this result, together with rigidity properties of type II $_{1}$ factors whose fundamental group is trivial.

Damien Gaboriau a montré récemment que les nombres de Betti $L^{2}$ des feuilletages mesurés à feuilles contractiles sont des invariants de la relation d’équivalence associée. Sorin Popa a utilisé ce résultat joint à des propriétés de rigidité des facteurs de type II $_{1}$ pour en déduire l’existence de facteurs de type II $_{1}$ dont le groupe fondamental est trivial.

Zbl 1133.46032 | 1 citation in Numdam

Classification: 46L35, 57R30
Keywords:

L^{2}

Betti numbers, foliation, type II

_{1}

factor, fundamental group of a type II

_{1}

factor

@incollection{SB_2002-2003__45__321_0,
     author = {Connes, Alain},
     title = {Nombres de Betti $L^2$ et facteurs de type ${\rm II}\_1$},
     booktitle = {S\'eminaire Bourbaki : volume 2002/2003, expos\'es 909-923},
     author = {Collectif},
     series = {Ast\'erisque},
     publisher = {Association des amis de Nicolas Bourbaki, Soci\'et\'e math\'ematique de France},
     address = {Paris},
     number = {294},
     year = {2004},
     note = {talk:920},
     pages = {321-333},
     zbl = {1133.46032},
     language = {fr},
     url = {http://www.numdam.org/item/SB_2002-2003__45__321_0}
}

Connes, Alain. Nombres de Betti $L^2$ et facteurs de type ${\rm II}_1$, in Séminaire Bourbaki : volume 2002/2003, exposés 909-923, Astérisque, no. 294 (2004), Talk no. 920, pp. 321-333. http://www.numdam.org/item/SB_2002-2003__45__321_0/

References

[1] M. Atiyah - “Elliptic operators, discrete groups and von Neumann algebras”, in Colloque “Analyse et Topologie” en l'honneur d'Henri Cartan, Astérisque, vol. 32, Société Mathématique de France, 1976, p. 43-72. | Numdam | MR 420729 | Zbl 0323.58015

[2] F. Boca - “On the method for constructing irreducible finite index subfactors of Popa”, Pacific J. Math. 161 (1993), p. 201-231. | MR 1242197 | Zbl 0795.46044

[3] A. Borel - “The $L^{2}$ -cohomology of negatively curved Riemannian symmetric spaces”, Ann. Acad. Sci. Fenn. Ser. A I Math. 10 (1985), p. 95-105. | MR 802471 | Zbl 0586.57022

[4] J. De Cannière & U. Haagerup - “Multipliers of the Fourier algebra of some simple Lie groups and their discrete subgroups”, Amer. J. Math. 107 (1984), p. 455-500. | MR 784292 | Zbl 0577.43002

[5] J. Cheeger & M. Gromov - “ $L^{2}$ -cohomology and group cohomology”, Topology 25 (1986), p. 189-215. | MR 837621 | Zbl 0597.57020

[6] P.-A. Cherix, M. Cowling, P. Jolissaint, P. Julg & A. Valette - Groups with the Haagerup property (Gromov's a-T-menability), Progress in Math., Birkhäuser, 2001. | MR 1852148 | Zbl 1030.43002

[7] M. Choda - “Group factors of the Haagerup type”, Proc. Japan Acad. Ser. A Math. Sci. 59 (1983), p. 174-177. | MR 718798 | Zbl 0523.46038

[8] A. Connes - “Classification of injective factors”, Ann. of Math. 104 (1976), p. 73-115. | MR 454659 | Zbl 0343.46042

[9] -, “Sur la théorie non-commutative de l'intégration”, in Algèbres d'opérateurs, Séminaire Les Plans-sur-Bex, 1978, Lect. Notes in Math., vol. 725, Springer, Berlin, 1979, p. 19-143. | Zbl 0412.46053

[10] -, “A type II $_{1}$ factor with countable fundamental group”, J. Operator Theory 4 (1980), p. 151-153. | MR 587372 | Zbl 0455.46056

[11] -, “Correspondences”, Notes manuscrites, 1980.

[12] -, “Classification des facteurs”, Proc. Symp. Pure Math., vol. 38, American Mathematical Society, 1982, p. 43-109. | MR 679497

[13] -, Noncommutative geometry, Academic Press, 1994. | MR 1303779

[14] A. Connes, J. Feldman & B. Weiss - “An amenable equivalence relation is generated by a single transformation”, Ergodic Theory Dynam. Systems 4 (1982), p. 431-450. | MR 662736 | Zbl 0491.28018

[15] A. Connes & V. F. R. Jones - “A II $_{1}$ factor with two non-conjugate Cartan subalgebras”, Bull. Amer. Math. Soc. (N.S.) 6 (1982), p. 211-212. | MR 640947 | Zbl 0501.46056

[16] -, “Property T for von Neumann algebras”, Bull. London Math. Soc. 17 (1985), p. 57-62. | MR 766450 | Zbl 0564.05003

[17] M. Cowling & U. Haagerup - “Completely bounded multipliers and the Fourier algebra of a simple Lie group of real rank one”, Invent. Math. 96 (1989), p. 507-549. | MR 996553 | Zbl 0681.43012

[18] C. Delaroche & A. Kirillov - “Sur les relations entre l'espace dual d'un groupe et la structure de ses sous-groupes fermés”, in Sém. Bourbaki (1967/1968), collection hors série de la S.M.F., vol. 10, Société Mathématique de France, 1995, exp. no 343, p. 507-528. | Numdam | MR 1610473 | Zbl 0214.04602

[19] J. Dixmier - “Sous-anneaux abéliens maximaux dans les facteurs de type fini”, Ann. of Math. 59 (1954), p. 279-286. | MR 59486 | Zbl 0055.10702

[20] H. Dye - “On groups of measure preserving transformations I, II”, Amer. J. Math. 81 (1959), p. 119-159, & 85 (1963), p. 551-576. | MR 131516 | Zbl 0087.11501

[21] J. Feldman & C. C. Moore - “Ergodic equivalence relations, cohomology, and von Neumann algebras I, II”, Trans. Amer. Math. Soc. 234 (1977), p. 289-324, 325-359. | MR 578656 | Zbl 0369.22010

[22] D. Gaboriau - “Coût des relations d'équivalence et des groupes”, Invent. Math. 139 (2000), p. 41-98. | MR 1728876 | Zbl 0939.28012

[23] -, “Invariants $ℓ^{2}$ de relations d’équivalence et de groupes”, Publ. Math. Inst. Hautes Études Sci. (2002). | Numdam | Zbl 1022.37002

[24] L. Ge - “Prime factors”, Proc. Nat. Acad. Sci. U.S.A. 93 (1996), p. 12762-12763. | MR 1417467 | Zbl 0863.46040

[25] V. Y. Golodets & N. I. Nesonov - “T-property and nonisomorphic factors of type II and III”, J. Funct. Anal. 70 (1987), p. 80-89. | MR 870754 | Zbl 0614.46053

[26] U. Haagerup - “An example of non-nuclear C $^{*}$ -algebra which has the metric approximation property”, Invent. Math. 50 (1979), p. 279-293. | MR 520930 | Zbl 0408.46046

[27] P. De La Harpe & A. Valette - La propriété T de Kazhdan pour les groupes localement compacts, Astérisque, vol. 175, Société Mathématique de France, 1989. | Numdam | Zbl 0759.22001

[28] G. Hjorth - “A lemma for cost attained”, UCLA preprint, 2002. | MR 2258624 | Zbl 1101.37004

[29] R. V. Kadison - “Problems on von Neumann algebras”, Baton Rouge Conference 1867, unpublished. | Zbl 1043.46045

[30] D. Kazhdan - “Connection of the dual space of a group with the structure of its closed subgroups”, Functional Anal. Appl. 1 (1967), p. 63-65. | MR 209390 | Zbl 0168.27602

[31] G. Levitt - “On the cost of generating an equivalence relation ”, Ergodic Theory Dynam. Systems 15 (1995), p. 1173-1181. | MR 1366313 | Zbl 0843.28010

[32] W. Luck - “Dimension theory of arbitrary modules over finite von Neumann algebras and $L^{2}$ -Betti numbers II. Applications to Grothendieck groups, $L^{2}$ -Euler characteristics and Burnside groups”, J. reine angew. Math. 496 (1998), p. 213-236. | MR 1605818 | Zbl 1001.55019

[33] G. Margulis - “Discrete groups of motion of manifolds of non-positive curvature”, Am. Math. Soc. Translations. 109 (1977), p. 33-45. | Zbl 0367.57012

[34] -, “Finitely-additive invariant measures on Euclidian spaces”, Ergodic Theory Dynam. Systems 2 (1982), p. 383-396. | MR 721730

[35] N. Monod & Y. Shalom - “Orbit equivalence rigidity and bounded cohomology”, preprint. | MR 2259246 | Zbl 1129.37003

[36] D. Ornstein & B. Weiss - “Ergodic theory of amenable group actions”, Bull. Amer. Math. Soc. (N.S.) 2 (1980), p. 161-164. | MR 551753 | Zbl 0427.28018

[37] N. Ozawa - “Solid von Neumann algebras”, math/0302082. | Zbl 1072.46040

[38] -, “There is no separable universal II $_{1}$ -factor”, math/0210411.

[39] N. Ozawa & S. Popa - “Some prime factorisation results for II $_{1}$ factors”, math/0302240. | Zbl 1060.46044

[40] S. Popa - “On a class of type II $_{1}$ factors with Betti numbers invariants”. | Zbl 1120.46045

[41] -, “Strong rigidity of II $_{1}$ factors coming from malleable actions of weakly rigid groups”, math/0305306. | Zbl 1120.46044

[42] -, “Correspondences”, INCREST preprint, unpublished, 1986.

[43] -, “Free independent sequences in type II $_{1}$ factors and related problems”, in Recent advances in operator algebras (Orléans, 1992), Astérisque, vol. 232, Société Mathématique de France, 1995, p. 187-202. | Numdam | MR 1372533 | Zbl 0840.46039

[44] S. Popa & D. Shlyakhtenko - “Cartan subalgebras and bimodule decomposition of II $_{1}$ factors”, Math. Scand. 92 (2003), p. 93-102. | MR 1951447 | Zbl 1067.46055

[45] S. Sakai - C $^{*}$ -algebras and W $^{*}$ -algebras, Springer-Verlag, Berlin-Heidelberg-New York, 1971. | MR 442701 | Zbl 1024.46001

[46] J.-L. Sauvageot - “Sur le produit tensoriel relatif d'espaces de Hilbert”, J. Operator Theory 9 (1983), p. 237-252. | MR 703809 | Zbl 0517.46050

[47] J.-P. Serre - Arbres, amalgames, SL(2), Astérisque, vol. 46, Société Mathématique de France, 1977. | Numdam | MR 476875 | Zbl 0369.20013

[48] D. Voiculescu - “The analogues of entropy and of Fisher's information theory in free probability II”, Invent. Math. 118 (1994), p. 411-440. | MR 1296352 | Zbl 0820.60001

[49] -, “The absence of Cartan subalgebras”, Geom. Funct. Anal. 6 (1996), p. 172-199. | MR 1371236

[50] R. Zimmer - Ergodic theory and semisimple groups, Birkhäuser-Verlag, Boston, 1984. | MR 776417 | Zbl 0571.58015