Unitaires multiplicatifs et dualité pour les produits croisés de C * -algèbres
Baaj, Saad ; Skandalis, Georges
Annales scientifiques de l'École Normale Supérieure, Tome 26 (1993), p. 425-488 / Harvested from Numdam
Access to full text
Full (PDF)
Full (DJVU) 
@article{ASENS_1993_4_26_4_425_0,
     author = {Baaj, Saad and Skandalis, Georges},
     title = {Unitaires multiplicatifs et dualit\'e pour les produits crois\'es de $\mathrm {C}^\ast $-alg\`ebres},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     volume = {26},
     year = {1993},
     pages = {425-488},
     doi = {10.24033/asens.1677},
     mrnumber = {94e:46127},
     zbl = {0804.46078},
     language = {fr},
     url = {http://dml.mathdoc.fr/item/ASENS_1993_4_26_4_425_0}
}

Baaj, Saad; Skandalis, Georges. Unitaires multiplicatifs et dualité pour les produits croisés de $\mathrm {C}^\ast $-algèbres. Annales scientifiques de l'École Normale Supérieure, Tome 26 (1993) pp. 425-488. doi : 10.24033/asens.1677. http://gdmltest.u-ga.fr/item/ASENS_1993_4_26_4_425_0/

[1] E. Abe, Hopf algebras, Cambridge University Press, London, New York, 1977.

[2] S. Baaj et G. Skandalis, C*-algèbres de Hopf et théorie de Kasparov équivariante. K-theory, vol. 2, 1989, p. 683-721. | MR 90j:46061 | Zbl 0683.46048

[3] N. Bourbaki, Intégration, Chap. 7 à 8, Hermann, 1963. | Zbl 0156.03204

[4] V. G. Drinfeld, Quantum Groups, Proc. ICM Berkeley, 1986, p. 798-820. | MR 89f:17017 | Zbl 0667.16003

[5] M. Enock, Produit croisé d'une algèbre de von Neumann par une algèbre de Kac. I. J.F.A., vol. 26, 1977, p. 16-46. | MR 57 #13513 | Zbl 0366.46053

[6] M. Enock et J. M. Schwartz, Une dualité dans les algèbres de von Neumann, Bull. S.M.F. Suppl. mémoire, vol. 44, 1975, p. 1-144. | Numdam | MR 56 #1091 | Zbl 0343.46044

[7] M. Enock et J. M. Schwartz, Produit croisé d'une algèbre de von Neumann par une algèbre de Kac. II Publ. RIMS Kyoto Univ., vol. 16, n° 1, 1980, p. 189-232. | MR 81m:46084 | Zbl 0441.46056

[8] M. Enock et J. M. Schwartz, Extension de la catégorie des algèbres de Kac, Ann. de l'Inst. Fourier, vol. 36, fasc. 1, 1986, p. 105-131. | Numdam | MR 88a:46076 | Zbl 0586.43003

[9] M. Enock et J. M. Schwartz, Algèbres de Kac moyennables. Pacific J. of Math., vol. 125, n° 2, 1986, p. 363-379. | MR 88f:46126 | Zbl 0597.43002

[10] J. Ernest, Hopf von Neumann algebras. Proc. Conf. Funct. Anal. (Irvine, Calif.) Academic Press, 1967, p. 195-215. | MR 36 #6956 | Zbl 0219.43004

[11] L. Van Heeswijck, Duality in the Theory of Crossed Products, Math. Scand, vol. 44, 1979, p. 313-329. | MR 83d:46082 | Zbl 0419.46042

[12] G. I. Kac, Ring Groups and the Duality Principle, Trans. Moscow Math. Soc., 1963, p. 291-339. Translated from Trudy Moskov. Mat. Ob., vol. 12, 1963, p. 259-301. | MR 28 #164 | Zbl 0144.37902

[13] G. I. Kac, Ring Groups and the Duality Principle II, Trans. Moscow Math. Soc., 1965, p. 94-126. Translated from Trudy Moskov. Math. Ob., vol. 13, 1965, p. 84-113. | Zbl 0162.45101

[14] G. I. Kac, Certain arithmetic properties of ring groups, Funk. Anal. i. Prilozen, vol. 6, 1972, p. 88-90. | MR 46 #3687 | Zbl 0258.16007

[15] G. I. Kac et V. G. Paljutkin, Finite Group Rings, Trans. Moscow Math. Soc., 1966, p. 251-294. Translated from Trudy Moskov. Mat. Obsc., vol. 15, 1966, p. 224-261. | Zbl 0218.43005

[16] G. I. Kac et V. G. Paljutkin, Example of Ring Groups Generated by Lie Groups (en russe) Ukr. Mat. J., vol. 16, 1, 1964, p. 99-105. | MR 31 #4857

[17] G. I. Kac et L. I. Vainerman, Nonunimodular Ring-Groups and Hopf-von Neumann Algebras, Math. USSSR Sb., vol. 23, 1974, p. 185-214. Translated from Matem. Sb., vol. 94, (136), 1974, vol. 2, p. 194-225. | MR 50 #536 | Zbl 0309.46052

[18] Y. Katayama, Takesaki's Duality for a Non Degenerate Coaction, Math. Scand., vol. 55, 1985, p. 141-151. | MR 86b:46112 | Zbl 0598.46042

[19] E. Kirchberg, Representation of Coinvolutive Hopf-W*-Algebras and Non Abelian Duality. Bull. Acad. Pol. Sc., vol. 25, 1977, p. 117-122. | MR 56 #6415 | Zbl 0417.46062

[20] M. G. Krein, Hermitian-Positive Kernels in Homogeneous Spaces, Amer. Math. Soc. Transl., (2), vol. 34, 1963, p. 109-164. Translated from Ukr. Mat. Z., vol. 2, n° 1, 1950, p. 10-59. | MR 12,719b | Zbl 0131.12101

[21] M. B. Landstad, Duality Theory for Covariant Systems, Trans. A.M.S., vol. 248, 1979, p. 223-267. | MR 80j:46107 | Zbl 0397.46059

[22] M. B. Landstad, Duality for Dual Covariance Algebras, Comm. Math. Phys., vol. 52, 1977, p. 191-202. | MR 56 #8750 | Zbl 0362.46046

[23] M. B. Landstad, J. Phillips, I. Raeburn, C. E. Sutherland, Representations of Crossed Products by Coactions and Principal Bundles. Trans. AMS, vol. 299, n° 2, 1987, p. 747-784. | MR 88f:46127 | Zbl 0722.46031

[24] G. W. Mackey, Borel Structures in Groups and their Duals, Trans. A.M.S., vol. 85, 1957, p. 134-165. | MR 19,752b | Zbl 0082.11201

[25] S. Mac Lane, Categories for the working mathematicians, GTM 5. | Zbl 0561.01017

[26] S. Mac Lane, Natural Associativity and Commutativity, Rice Univ. Studies, vol. 49, 1963, p. 4-28. | MR 30 #1160 | Zbl 0244.18008

[27] S. H. Majid, Non-Commutative Geometric Groups by a Bicrossproduct Construction : Hopf Algebras at the Planck Scale, Thesis, Harvard Univ, 1988.

[28] S. H. Majid, Hopf von Neumann algebra Bicrossproducts, Kac Algebras Bicrossproducts, and the Classical Yang-Baxter Equation, J.F.A., vol. 95, n° 2, 1991, p. 291-319. | MR 92b:46088 | Zbl 0741.46033

[29] S. H. Majid, Physics for Algebraists, Non-Commutative and Non-Cocommutative Hopf Algebras by a Bicrossproduct Construction, J. of Algebra, vol. 130, n° 1, 1990, p. 17-64. | MR 91j:16050 | Zbl 0694.16008

[30] G. Moore and N. Seiberg, Classical and Quantum Conformal Field Theory, Comm. Math. Phys., vol. 123, 1989, p. 177-254. | MR 90e:81216 | Zbl 0694.53074

[31] Y. Nakagami, Dual Action on a von Neumann Algebra and Takesaki's Duality for a Locally Compact Group, Publ. R.I.M.S. Kyoto Univ., vol. 12, 1977, p. 727-775. | MR 56 #16393 | Zbl 0363.46062

[32] Y. Nakagami and M. Takesaki, Duality for Crossed Products of von Neumann Algebras. Lect. Notes in Math., vol. 731, 1979. | MR 81e:46053 | Zbl 0423.46051

[33] G. K. Pedersen, C*-Algebras and their Automorphism Groups. Academic press, 1979. | MR 81e:46037 | Zbl 0416.46043

[34] P. Podles et S. L. Woronowicz, Quantum Deformation of Lorentz Group, Comm. Math. Phys., vol. 130, 1990, p. 381-431. | MR 91f:46100 | Zbl 0703.22018

[35] M. A. Rieffel, Some Solvable Quantum Groups, Proc. Conf. Craiova Romania Sept. 1989 (à paraître).

[36] M. Rosso, Comparaison des groupes SU(2) quantiques de Drinfeld et Woronowicz, C. R. Acad. Sci., vol. 304, 1987, p. 323-326. | MR 88h:22033 | Zbl 0617.16005

[37] M. Rosso, Algèbres enveloppantes quantifiées, groupes quantiques compacts de matrices et calcul différentiel non commutatif. Prépublication.

[38] J. M. Schwartz, Sur la structure des algèbres de Kac I, J. Funct. Anal., vol. 34, 1979, p. 370-406. | MR 83a:46072a | Zbl 0431.46044

[39] J. M. Schwartz, Sur la structure des algèbres de Kac II, Proc. of the London Math. Soc., vol. 41, 1980, p. 465-480. | MR 83a:46072b | Zbl 0398.46050

[40] W. F. Stinespring, Integration theorems for gages and duality for unimodular groups, Trans. AMS, vol. 90, 1959, p. 15-56. | MR 21 #1547 | Zbl 0085.10202

[41] S. Stratila, D. Voiculescu et L. Zsido, On Crossed Products. I and II, Rev. Roumaine Math. P. et Appl., vol. 21, 1976, p. 1411-1449 et vol. 22, 1977, p. 83-117. | Zbl 0402.46038

[42] H. Takai, On a Duality for Crossed Products of C*-Algebras, J.F.A., vol. 19, 1975, p. 25-39. | MR 51 #1413 | Zbl 0295.46088

[43] M. Takesaki, A Characterization of Group Algebras as a Converse of Tannaka-Stinespring-Tatsuuma Duality Theorem, Amer. J. of Math., vol. 91, 1969, p. 529-564. | MR 39 #5752 | Zbl 0182.18103

[44] M. Takesaki, Duality and von Neumann Algebras, L.N.M., vol. 247, 1972, p. 665-779. | MR 53 #704 | Zbl 0238.46063

[45] M. Takesaki, Duality for Crossed Products and the Structure of von Neumann Algebras of type III, Acta Math., vol. 131, 1973, p. 249-310. | MR 55 #11068 | Zbl 0268.46058

[46] M. Takeuchi, Matched Pairs of Groups and Bismash Products of Hopf Algebras, Comm. Algebra, vol. 9, n° 8, 1981, p. 841-882. | MR 83f:16013 | Zbl 0456.16011

[47] T. Tannaka, Über den Dualität der nicht-kommutativen topologischen Gruppen, Tôhoku Math. J., vol. 45, 1938, p. 1-12. | JFM 64.0362.01 | Zbl 0020.00904

[48] N. Tatsuuma, A Duality Theorem for Locally Compact Groups, J. of Math. of Kyoto Univ., vol. 6, 1967, p. 187-293. | MR 36 #313 | Zbl 0184.17402

[49] L. I. Vainerman, Characterization of Dual Objects for Locally Compact Groups, Funct. Anal. Appl., vol. 8, 1974, p. 66-67. Translated from Funk. Anal. i. Prilozen, vol. 8, 1974, n° 1, p. 75-76. | MR 49 #463 | Zbl 0312.22007

[50] J. M. Vallin, C*-algèbres de Hopf et C*-algèbres de Kac. Proc. London Math. Soc., (3), vol. 50, 1985, p. 131-174. | MR 86f:46072 | Zbl 0577.46063

[51] D. Voiculescu, Amenability and Katz algebras. Algèbres d'opérateurs et leurs applications en physique mathématique. Colloques Internationaux, C.N.R.S., n° 274, 1977, p. 451-457. | MR 83c:46065 | Zbl 0503.46049

[52] A. Weil, L'intégration dans les groupes topologiques et ses applications. Act. Sc. Ind., n° 1145, Hermann, Paris, 1953.

[53] S. L. Woronowicz, Twisted SU (2) group. An example of a non commutative differential calculus. Publ. R.I.M.S., vol. 23, 1987, p. 117-181. | MR 88h:46130 | Zbl 0676.46050

[54] S. L. Woronowicz, Compact Matrix Pseudogroups, Comm. Math. Phys., vol. 111, 1987, p. 613-665. | MR 88m:46079 | Zbl 0627.58034

[55] S. L. Woronowicz, Tannaka-Krein Duality for Compact Matrix Pseudogroups. Twisted SU (N) Group, Inv. Math., vol. 93, 1988. | MR 90e:22033 | Zbl 0664.58044

[56] S. L. Woronowicz, Differential Calculus on Compact Matrix Pseudogroups (Quantum Groups), Comm. Math. Phys., vol. 122, 1989, p. 125-170. | MR 90g:58010 | Zbl 0751.58042