Locally compact quantum groups

Kustermans, Johan; Vaes, Stefaan

doi:10.1016/s0012-9593(00)01055-7

Kustermans, Johan ; Vaes, Stefaan

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 33 (2000) no. 6, pp. 837-934.

MR Zbl | 4 citations dans Numdam

DOI : 10.1016/s0012-9593(00)01055-7

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     author = {Kustermans, Johan and Vaes, Stefaan},
     title = {Locally compact quantum groups},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     pages = {837--934},
     publisher = {Elsevier},
     volume = {Ser. 4, 33},
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Kustermans, Johan; Vaes, Stefaan. Locally compact quantum groups. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 33 (2000) no. 6, pp. 837-934. doi : 10.1016/s0012-9593(00)01055-7. http://www.numdam.org/articles/10.1016/s0012-9593(00)01055-7/

Bibliographie
Cité par

[1] Abe E., Hopf Algebras, Cambridge Tracts in Mathematics, Vol. 74, Cambridge University Press, Cambridge, 1980. | MR | Zbl

[2] Baaj S., Représentation régulière du groupe quantique des déplacements de Woronowicz, Astérisque 232 (1995) 11-48. | Numdam | MR | Zbl

[3] Baaj S., Représentation régulière du groupe quantique Eµ(2), C. R. Acad. Sci. Paris, Série I 314 (1992) 1021-1026. | MR | Zbl

[4] Baaj S., Multiplicateurs non bornés, Thèse 3ème cycle, Université Paris-6, 1980.

[5] Baaj S., Prolongement d'un poids, C. R. Acad. Sci., Paris, Série A 288 (1979) 1013-1015. | MR | Zbl

[6] Baaj S., Skandalis G., Unitaires multiplicatifs et dualité pour les produits croisés de C*-algèbres, Ann. Scient. Éc. Norm. Sup., 4e série 26 (1993) 425-488. | EuDML | Numdam | MR | Zbl

[7] Banica T., Le groupe quantique compact libre U(n), Commun. Math. Phys. 190 (1997) 143-172. | MR | Zbl

[8] Busby R.C., Double centralizers and extensions of C*-algebras, Trans. Amer. Math. Soc. 132 (1968) 79-99. | MR | Zbl

[9] Ciorănescu I., Zsidó L., Analytic generators for one-parameter groups, Tôhoku Math. J. 28 (1976) 327-362. | MR | Zbl

[10] Combes F., Poids associé à une algèbre hilbertienne à gauche, Compos. Math. 23 (1971) 49-77. | Numdam | MR | Zbl

[11] Combes F., Poids sur une C*-algèbre, J. Math. Pures et Appl. 47 (1968) 57-100. | MR | Zbl

[12] D'Antoni C., Zsidó L., Groups of linear isometries on multiplier C*-algebras, Pac. J. Math. 193 (2) (2000) 279-306. | MR | Zbl

[13] Drinfel'D V.G., Quantum groups, Proceedings ICM Berkeley (1986) 798-820. | MR | Zbl

[14] Effros E., Ruan Z.-J., Discrete quantum groups I : The Haar measure, Int. J. Math. 5 (5) (1994) 681-723. | MR | Zbl

[15] Enock M., Schwartz J.-M., Kac Algebras and Duality of Locally Compact Groups, Springer-Verlag, Berlin, 1992. | MR | Zbl

[16] Enock M., Vallin J.-M., C*-algèbres de Kac et algèbres de Kac, Proc. London Math. Soc. 66 (3) (1993) 619-650. | MR | Zbl

[17] Haagerup U., Operator-valued weights in von Neumann algebras, I, J. Funct. Anal. 32 (1979) 175-206. | MR | Zbl

[18] Haagerup U., Normal weights on W*-algebras, J. Funct. Anal. 19 (1975) 302-317. | MR | Zbl

[19] Kirchberg E., Lecture on the conference 'Invariants in operator algebras', Copenhagen, August 1992.

[20] Koelink H.T., Addition formula for a 2-parameter family of Askey-Wilson polynomials, Preprint, KU Leuven, 1994.

[21] Koelink H.T., On quantum groups and q-special functions, PhD-thesis, Rijksuniversiteit Leiden, 1991.

[22] Koelink H.T., Verding J., Spectral analysis and the Haar functional on the quantum SU(2) group, Commun. Math. Phys. 177 (1996) 399-415. | MR | Zbl

[23] Kustermans J., Locally compact quantum groups in the universal setting, Int. J. Math., to appear. #math/9902015.

[24] Kustermans J., KMS-weights on C*-algebras, Preprint, Odense Universitet, 1997. #funct-an/9704008.

[25] Kustermans J., One-parameter representations on C*-algebras, Preprint, Odense Universitet, 1997. #funct-an/9707010.

[26] Kustermans J., The functional calculus of regular operators on Hilbert C*-modules revisited, Preprint, Odense Universitet, 1997. #funct-an/9706007.

[27] Kustermans J., Vaes S., A simple definition for locally compact quantum groups, C. R. Acad. Sci. Paris, Série I 328 (10) (1999) 871-876. | MR | Zbl

[28] Kustermans J., Vaes S., The operator algebra approach to quantum groups, Proc. Natl. Acad. Sc. 97 (2) (2000) 547-552. | MR | Zbl

[29] Kustermans J., Vaes S., Locally compact quantum groups in the von Neumann algebraic setting, Preprint, KU, Leuven, 2000. #math.OA/0005219.

[30] Kustermans J., Vaes S., Weight theory for C*-algebraic quantum groups, Preprint, KU Leuven & University College Cork, 1999. #math/9901063.

[31] Kustermans J., Van Daele A., C*-algebraic quantum groups arising from algebraic quantum groups, Int. J. Math. 8 (8) (1997) 1067-1139. #q-alg/9611023. | MR | Zbl

[32] Lance C., Hilbert C*-Modules. A Toolkit for Operator Algebraists, London Math. Soc. Lect. Note Series, Vol. 210, Cambridge University Press, Cambridge, 1995. | Zbl

[33] Masuda T., Nakagami Y., A von Neumann algebra framework for the duality of the quantum groups, Publ. RIMS, Kyoto University 30 (1994) 799-850. | MR | Zbl

[34] Masuda T., Nakagami Y., Woronowicz S.L., Lectures at the Fields Institute and at the University of Warsaw, 1995.

[35] Pedersen G.K., C*-Algebras and their Automorphism Groups, Academic Press, London, 1979. | MR | Zbl

[36] Pedersen G.K., Takesaki M., The Radon-Nikodym theorem for von Neumann algebras, Acta Math. 130 (1973) 53-87. | MR | Zbl

[37] Podleś P., Quantum spheres, Lett. Math. Phys. 14 (1987) 193-202. | MR | Zbl

[38] Podleś P., Woronowicz S.L., Quantum deformation of Lorentz group, Commun. Math. Phys. 130 (1990) 381-431. | MR | Zbl

[39] Quaegebeur J., Verding J., A construction for weights on C*-algebras. Dual weights for C*-crossed products, Int. J. Math. 10 (1) (1999) 129-157. | MR | Zbl

[40] Quaegebeur J., Verding J., Left invariant weights and the left regular corepresentation for locally compact quantum semi-groups, Preprint, KU Leuven, 1994.

[41] Rieffel M.A., Compact quantum groups associated with toral subgroups, Contemp. Math. 145 (1993) 465-491. | MR | Zbl

[42] Rieffel M.A., Deformation quantization of Heisenberg manifolds, Commun. Math. Phys. 122 (1989) 531-562. | MR | Zbl

[43] Rieffel M.A., Integrable and proper actions on C*-algebras, and square-integrable representations of groups, 1998. #math/9809098.

[44] Rieffel M.A., Some solvable quantum groups. Operator algebras and topology, in : Arveson W.B. et al. (Eds.), Proceedings of the OATE 2 Conference Bucharest, Romania, 1989, Pitman Research Notes in Mathematics, Vol. 270, 1992, pp. 146-159. | Zbl

[45] Sakai S., C*-Algebras and W*-Algebras, Springer-Verlag, Berlin, 1971. | MR | Zbl

[46] Stratila S., Modular Theory in Operator Algebras, Abacus Press, Tunbridge Wells, England, 1981. | MR | Zbl

[47] Stratila S., Zsidó L., Lectures on von Neumann Algebras, Abacus Press, Tunbridge Wells, England, 1979. | MR | Zbl

[48] Takesaki M., Theory of Operator Algebras I, Springer-Verlag, New York, 1979. | MR | Zbl

[49] Taylor D.C., The strict topology for double centralizer algebras, Trans. Amer. Math. Soc. 150 (1970) 633-643. | MR | Zbl

[50] Vaes S., The unitary implementation of a locally compact quantum group action, J. Funct. Anal., to appear. #math.OA/0005262. | Zbl

[51] Vaes S., Examples of locally compact quantum groups through the bicrossed product construction, in : Proceedings of the XIIIth International Conference on Mathematical Physics, London, 2000, to appear. | Zbl

[52] Vaes S., A Radon-Nikodym theorem for von Neumann algebras, J. Oper. Th., to appear. #math.OA/9811122. | Zbl

[53] Vaes S., Hopf C*-algebras, Masters Thesis, KU Leuven, 1998.

[54] Vaes S., Van Daele A., Hopf C*-algebras, Proc. London Math. Soc., to appear. #math.OA/9907030. | Zbl

[55] Vainerman L.I., Kac G.I., Nonunimodular ring-groups and Hopf-von Neumann algebras, Math. USSR, Sbornik 23 (1974) 185-214. | MR | Zbl

[56] Vainerman G.I., Kac G.I., Nonunimodular ring-groups and Hopf-von Neumann algebras, Soviet Math. Dokl. 14 (1973) 1144-1148. | Zbl

[57] Vaksman L.L., Soibel'Man Y.S., Algebra of functions on the quantum group SU(2), Funct. Anal. Appl. 22 (1988) 170-181. | MR | Zbl

[58] Vallin J.-M., C*-algèbres de Hopf et C*-algèbres de Kac, Proc. London Math. Soc. 50 (3) (1985) 131-174. | MR | Zbl

[59] Van Daele A., An algebraic framework for group duality, Adv. in Math. 140 (1998) 323-366. | MR | Zbl

[60] Van Daele A., Discrete quantum groups, J. Algebra 180 (1996) 431-444. | MR | Zbl

[61] Van Daele A., The Haar measure on a compact quantum group, Proc. Amer. Math. Soc. 123 (1995) 3125-3128. | MR | Zbl

[62] Van Daele A., Multiplier Hopf Algebras, Trans. Amer. Math. Soc. 342 (1994) 917-932. | MR | Zbl

[63] Van Daele A., Wang S.Z., Universal quantum groups, Int. J. Math. 7 (2) (1996) 255-263. | MR | Zbl

[64] Van Daele A., Woronowicz S.L., Duality for the quantum E(2) group, Pacific J. Math. 173 (2) (1996) 375-385. | MR | Zbl

[65] Verding J., Weights on C*-algebras, PhD Thesis, KU Leuven, 1995.

[66] Wang S.Z., Free Products of compact quantum groups, Commun. Math. Phys. 167 (1995) 671-692. | MR | Zbl

[67] Woronowicz S.L., Compact quantum groups, in : Symétries Quantiques (Les Houches, 1995), North-Holland, Amsterdam, 1998, pp. 845-884. | MR | Zbl

[68] Woronowicz S.L., From multiplicative unitaries to quantum groups, Int. J. Math. 7 (1) (1996) 127-149. | MR | Zbl

[69] Woronowicz S.L., Quantum E(2) group and its Pontryagin dual, Lett. Math. Phys. 23 (1991) 251-263. | MR | Zbl

[70] Woronowicz S.L., Unbounded elements affiliated with C*-algebras and non-compact quantum groups, Commun. Math. Phys. 136 (1991) 399-432. | MR | Zbl

[71] Woronowicz S.L., Tannaka-Krein duality for compact matrix pseudogroups. Twisted SU(N) groups, Invent. Math. 93 (1988) 35-76. | MR | Zbl

[72] Woronowicz S.L., Compact matrix pseudogroups, Commun. Math. Phys. 111 (1987) 613-665. | MR | Zbl

[73] Woronowicz S.L., Twisted SU(2) group. An example of a non-commutative differential calculus, Publ. RIMS, Kyoto University 23 (1987) 117-181. | MR | Zbl

[74] Woronowicz S.L., Pseudospaces, pseudogroups and Pontrjagin duality, in : Proceedings of the International Conference on Mathematical Physics (Lausanne, 1979), Lecture Notes in Physics, Vol. 116, 1980, pp. 407-412. | Zbl

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