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Deformations of Hopf \(C^*\)-algebras. (Déformations de \(C^*\)-algèbres de Hopf.) (French) Zbl 0851.46040
Summary: Given a compact space \(X\), we generalize the notion of multiplicative unitary introduced by Baaj and Skandalis to the framework of Hilbert \(C(X)\)-modules and we study its continuity properties. We then associate to several deformations of Hopf \(C^*\)-algebras constructed by Woronowicz continuous fields of multiplicative unitaries and we prove that those deformations correspond to topological deformations.

MSC:
46L05 General theory of \(C^*\)-algebras
16W30 Hopf algebras (associative rings and algebras) (MSC2000)
46M05 Tensor products in functional analysis
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[1] BAAJ (S.) . - Thèse , Université Pierre et Marie Curie, Paris, 1980 .
[2] BAAJ (S.) . - Représentation régulière du groupe quantique E\mu (2) de Woronowicz , C. R. Acad. Sci. Paris, t. 314, 1992 , p. 1021-1026. MR 93f:46106 | Zbl 0771.17011 · Zbl 0771.17011
[3] BAAJ (S.) . - Représentation régulière du groupe quantique des déplacements de Woronowicz , preprint. · Zbl 0840.46036
[4] BAAJ (S.) et SKANDALIS (G.) . - Unitaires multiplicatifs et dualité pour les produits croisés de C*-algèbres , Ann. Sci. École Norm. Sup. (4), t. 26, 1993 , p. 425-488. Numdam | MR 94e:46127 | Zbl 0804.46078 · Zbl 0804.46078
[5] BARTLE (R.G.) and GRAVES (L.M.) . - Mappings between function spaces , Trans. Amer. Math. Soc., t. 72, 1952 , p. 400-413. MR 13,951i | Zbl 0047.10901 · Zbl 0047.10901
[6] BAUVAL (A.) . - Déformation du groupe de Lie-Poisson SU(2) , prépublication n^\circ 13 de l’URA 1408, Toulouse, 1992 .
[7] BAUVAL (A.) . - RKK(X)-nucléarité (d’après G. Skandalis) , preprint. · Zbl 0909.19003
[8] BLACKADAR (B.) . - K-theory for Operator Algebras , Math. Sci. Res. Inst. Publ., t. 5, 1986 . MR 88g:46082 | Zbl 0597.46072 · Zbl 0597.46072
[9] BLANCHARD (E.) . - Représentations de champs de C*-algèbres , C.R. Acad. Sci. Paris, t. 314, 1992 , p. 911-914. MR 93f:46087 | Zbl 0787.46042 · Zbl 0787.46042
[10] BOURBAKI (N.) . - Espace vectoriels topologiques . - Hermann et Cie, Paris, 1955 . · Zbl 0067.08302
[11] CHOI (M.D.) and EFFROS (E.G.) . - Nuclear C*-algebras and injectivity : the general case , Indiana Univ. Math. J., t. 26, 1977 , p. 443-446. MR 55 #3799 | Zbl 0378.46052 · Zbl 0378.46052
[12] COHEN (P.J.) . - Factorization in group algebras , Duke Math. J., t. 26, 1959 , p. 199-205. Article | MR 21 #3729 | Zbl 0085.10201 · Zbl 0085.10201
[13] CONNES (A.) . - Classification of injective factors , Ann. of Math., t. 104, 1976 , p. 73-115. MR 56 #12908 | Zbl 0343.46042 · Zbl 0343.46042
[14] DIXMIER (J.) . - Les C*-algèbres et leurs représentations . - Gauthiers-Villars, Paris, 1969 . MR 39 #7442 | Zbl 0174.18601 · Zbl 0174.18601
[15] DRINFEL’D (V.G.) . - Quantum groups . - Proceedings of the I.C.M., Berkeley, California, 1986 .
[16] DUPRÉ (M.J.) . - The classification and structure of C*-algebra bundles , Mem. Amer. Math. Soc., t. 222, 1979 . MR 83c:46069 | Zbl 0416.46053 · Zbl 0416.46053
[17] EFFROS (E.G.) and LANCE (E.C.) . - Tensor products of operator algebras , Adv. in Math., t. 25, 1977 , p. 1-34. MR 56 #6402 | Zbl 0372.46064 · Zbl 0372.46064
[18] ELLIOTT (G.) , NATSUME (T.) and NEST (R.) . - The Heisenberg Group and K-Theorie , K-Theory, t. 7, 1993 , p. 409-428. MR 94m:46109 | Zbl 0803.46076 · Zbl 0803.46076
[19] ENOCK (M.) et SCHWARTZ (J.M.) . - Une dualité dans les algèbres de von Neumann , Mémoire Soc. Math. France, suppl. Bull., n^\circ 44, 1975 , p. 1-144. Numdam | MR 56 #1091 | Zbl 0343.46044 · Zbl 0343.46044
[20] ENOCK (M.) et SCHWARTZ (J.M.) . - Algèbres de Kac moyennables , Pacific J. Math., t. 125, 2, 1986 , p. 363-379. Article | MR 88f:46126 | Zbl 0597.43002 · Zbl 0597.43002
[21] FELL (J.M.G.) . - The structure of algebras of operator fields , Acta Math., t. 106, 1962 , p. 237-268. Zbl 0101.09301 · Zbl 0101.09301
[22] GELFAND (I.M.) and NAIRMARK (M.A.) . - On the embedding of normed rings into the ring of operators in Hilbert space , Mat. Sb., t. 12, 1943 , p. 197-213. Zbl 0060.27006 · Zbl 0060.27006
[23] GREEN (P.) . - The local structure of twisted covariance algebras , Acta Math., t. 140, 1978 , p. 191-250. MR 58 #12376 | Zbl 0407.46053 · Zbl 0407.46053
[24] DE LA HARPE (P.) et VALETTE (A.) . - La propriété (T) de Kazhdan pour les groupes localement compacts , Astérisque n^\circ 175, 1989 . · Zbl 0759.22001
[25] KAC (G.I.) . - Ring groups and the duality principle , Trans. Moscow Math. Soc., 1963 , p. 291-339 (traduit de Trudy Moskov. Math. Obshch., t. 12, 1963 , p. 259-301). MR 28 #164 | Zbl 0144.37902 · Zbl 0144.37902
[26] KASPAROV (G.G.) . - Hilbert C*-modules : theorems of Stinespring aand Voiculescu , J. Operator Theory, t. 4, 1980 , p. 133-150. MR 82b:46074 | Zbl 0456.46059 · Zbl 0456.46059
[27] KASPAROV (G.G.) . - Equivariant KK-theory and the Novikov conjecture , Invent. Math., t. 91, 1988 , p. 147-201. MR 88j:58123 | Zbl 0647.46053 · Zbl 0647.46053
[28] KATAYAMA (Y.) . - Takesaki’s duality for a non degenerate coaction , Math. Scand., t. 55, 1985 , p. 141-151. MR 86b:46112 | Zbl 0598.46042 · Zbl 0598.46042
[29] KIRCHBERG (E.) . - On non-semisplit extension, tensor products and exactness of group C*-algebras , Invent. Math., t. 112, 1993 , p. 449-489. MR 94d:46058 | Zbl 0803.46071 · Zbl 0803.46071
[30] KIRCHBERG (E.) and WASSERMANN (S.) . - Operations on continuous bundles of C*-algebras , preprint. · Zbl 0835.46057
[31] LANCE (C.) . - Tensor product and nuclear C*-algebras , Operator Algebras and Applications, Proc. Sympos. Pure Math., t. 38, I, 1982 , p. 379-399. MR 84b:46069 | Zbl 0498.46044 · Zbl 0498.46044
[32] LEE (R.-Y.) . - On the C*-algebras of operator fields , Indiana Univ. Math. J., t. 25, 1976 , p. 303-314. MR 53 #14150 | Zbl 0306.46073 · Zbl 0306.46073
[33] MASUDA (T.) , MIMACHI (K.) , NAKAGAMI (Y.) , NOUMI (M.) and UENO (K.) . - Representations of quantum groups and a q-analogue of orthogonal polynomials , C. R. Acad. Sci. Paris, t. 307, 1988 , p. 559-564. MR 90a:17013 | Zbl 0658.22010 · Zbl 0658.22010
[34] MICHAEL (E.) . - Continuous selection I , Ann. Math., t. 63, 1956 , p. 361-382. MR 17,990e | Zbl 0071.15902 · Zbl 0071.15902
[35] NAGY (G.) . - A framework for deformation quantization , thesis, Berkeley, 1992 .
[36] NAGY (G.) . - On the Haar measure of the quantum SU(N) group , Comm. Math. Phys., t. 153, 1993 , p. 217-228. Article | MR 94e:46124 | Zbl 0779.17015 · Zbl 0779.17015
[37] PASCHKE (W.) . - Inner product modules over B*-algebras , Trans. Amer. Math. Soc., t. 182, 1973 , p. 443-468. MR 50 #8087 | Zbl 0239.46062 · Zbl 0239.46062
[38] PEDERSEN (G.K.) . - C*-algebras and their automorphism groups . - Academic press, 1979 . MR 81e:46037 | Zbl 0416.46043 · Zbl 0416.46043
[39] PODLES (P.) and WORONOWICZ (S.L.) . - Quantum deformation of Lorentz group , Comm. Math. Phys., t. 130, 1990 , p. 381-431. Article | MR 91f:46100 | Zbl 0703.22018 · Zbl 0703.22018
[40] RIEFFEL (M.A.) . - Continuous fields of C*-algebras coming from group cocycles and actions , Math. Ann., t. 283, 1989 , p. 631-643. MR 90b:46120 | Zbl 0646.46063 · Zbl 0646.46063
[41] RIEFFEL (M.A.) . - Deformation quantization of Heisenberg manifolds , Comm. Math. Phys., t. 122, 1989 , p. 531-562. Article | MR 90e:46060 | Zbl 0679.46055 · Zbl 0679.46055
[42] SHEU (J.L.) . - Quantization of the Poisson SU(2) and its Poisson homogeneous space, the 2-sphere , Comm. Math. Phys., t. 135, 1991 , p. 217-232. Article | MR 91m:58011 | Zbl 0719.58042 · Zbl 0719.58042
[43] SKANDALIS (G.) . - Operator algebras and duality , Proc. Intern. Congr. Math., t. 2, 1990 , p. 997-1009. MR 93d:46094 | Zbl 0819.46054 · Zbl 0819.46054
[44] STRATILA (S.) . - Modular theory in operator algebras . - Academiei and Abacus Press, 1981 . MR 85g:46072 | Zbl 0504.46043 · Zbl 0504.46043
[45] TAKESAKI (M.) . - On the cross norm of the direct product of C*-algebras , Tôhoku Math. J., t. 16, 1964 , p. 111-122. Article | MR 29 #2668 | Zbl 0127.07302 · Zbl 0127.07302
[46] TOMIYAMA (J.) . - Topological representation of C*-algebras , Tôhoku Math. J., t. 14, 1962 , p. 187-204. Article | MR 26 #619 | Zbl 0216.16201 · Zbl 0216.16201
[47] VALIN (J.M.) . - C*-algèbres de Hopf et C*-algèbres de Kac , Proc. London Math. Soc. (3), t. 50, 1985 , p. 131-174. MR 86f:46072 | Zbl 0577.46063 · Zbl 0577.46063
[48] VOICULESCU (D.) . - Amenability and Katz algebras, Algèbres d’opérateurs et leurs applications en physique mathématique , Colloq. Intern. CNRS, t. 274, 1977 , p. 451-457. MR 83c:46065 | Zbl 0503.46049 · Zbl 0503.46049
[49] WASSERMANN (S.) . - C*-algebras associated with groups with Kazhdan’s property T , Ann. Math., t. 134, 1991 , p. 423-431. MR 92g:46085 | Zbl 0754.46040 · Zbl 0754.46040
[50] WORONOWICZ (S.L.) . - Compact matrix pseudogroups , Comm. Math. Phys., t. 111, 1987 , p. 613-665. Article | MR 88m:46079 | Zbl 0627.58034 · Zbl 0627.58034
[51] WORONOWICZ (S.L.) . - Twisted SU(2) group. An example of a non commutative calculus , Publ. Res. Inst. Math. Sci., t. 23, 1987 , p. 117-181. MR 88h:46130 | Zbl 0676.46050 · Zbl 0676.46050
[52] WORONOWICZ (S.L.) . - Tannaka-Krein duality for compact matrix pseudogroups . Twisted SU(N) group, Invent. Math., t. 93, 1988 , p. 35-76. MR 90e:22033 | Zbl 0664.58044 · Zbl 0664.58044
[53] WORONOWICZ (S.L.) . - Unbounded Elements Affiliated with C*-Algebras and Non-Compact Quantum Groups , Comm. Math. Phys., t. 136, 1991 , p. 399-432. Article | MR 92b:46117 | Zbl 0743.46080 · Zbl 0743.46080
[54] WORONOWICZ (S.L.) . - Compact quantum groups , preprint. · Zbl 0997.46045
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