×

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
PDFBibTeX XMLCite
Full Text: DOI Numdam EuDML

References:

[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 · doi:10.2307/1990709
[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 · doi:10.1512/iumj.1977.26.26034
[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 · doi:10.1215/S0012-7094-59-02620-1
[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 · doi:10.2307/1971057
[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 · doi:10.1016/0001-8708(77)90085-8
[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 · doi:10.1007/BF00961535
[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 · doi:10.2140/pjm.1986.125.363
[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 · doi:10.1007/BF02545788
[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 · doi:10.1007/BF02392308
[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 · doi:10.1007/BF01404917
[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 · doi:10.1007/BF01232444
[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 · doi:10.1512/iumj.1976.25.25026
[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 · doi:10.2307/1969615
[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 · doi:10.1007/BF02096641
[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 · doi:10.2307/1996542
[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 · doi:10.1007/BF02473358
[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 · doi:10.1007/BF01442857
[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 · doi:10.1007/BF01256492
[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 · doi:10.1007/BF02098041
[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 · doi:10.2748/tmj/1178243737
[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 · doi:10.2748/tmj/1178244174
[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 · doi:10.1112/plms/s3-50.1.131
[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 · doi:10.2307/2944351
[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 · doi:10.1007/BF01219077
[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 · doi:10.2977/prims/1195176848
[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 · doi:10.1007/BF01393687
[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 · doi:10.1007/BF02100032
[54] WORONOWICZ (S.L.) . - Compact quantum groups , preprint. · Zbl 0997.46045
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.