×

Multiplicative unitaries and duality for crossed products of \(C^*\)-algebras. (Unitaires multiplicatifs et dualité pour les produits croisés de \(C^*\)-algèbres.) (French) Zbl 0804.46078

Summary: Let \(H\) be a Hilbert space. A unitary operator \(V\in {\mathcal L}(H\otimes H)\) is said to be multiplicative if it satisfies the pentagone equation \(V_{12} V_{13} V_{23}= V_{23} V_{12}\). In many papers concerned on operator algebras with duality, a multiplicative unitary plays a fundamental role. In this paper we look for additional conditions that a multiplicative unitary should satisfy in order to correspond to a “locally compact quantum group”. We introduce two conditions: “regularity” and “irreducibility”. To any multiplicative unitary satisfying these conditions we associate two pairwise dual Hopf \(C^*\)- algebras. Moreover, we establish Takesaki-Takai duality results, using an adaptation of the method of [M. Enock, J. Funct. Anal. 26, 16-47 (1977; Zbl 0366.46053)].
If the Hilbert space is finite-dimensional or if the unitary V satisfies a commutativity condition, regularity and irreducibility are automatic. If the unitary V is of compact or discrete type, its regularity implies its irreducibility.

MSC:

46L55 Noncommutative dynamical systems
47L50 Dual spaces of operator algebras
46L05 General theory of \(C^*\)-algebras

Citations:

Zbl 0366.46053
PDF BibTeX XML Cite
Full Text: DOI Numdam EuDML

References:

[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 · Zbl 0683.46048
[3] N. BOURBAKI , Intégration , Chap. 7 à 8, Hermann, 1963 . Zbl 0156.03204 · Zbl 0156.03204
[4] V. G. DRINFELD , Quantum Groups , Proc. ICM Berkeley, 1986 , p. 798-820. MR 89f:17017 | Zbl 0667.16003 · 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 · 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 · 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^\circ 1, 1980 , p. 189-232. Article | MR 81m:46084 | Zbl 0441.46056 · 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 · Zbl 0586.43003
[9] M. ENOCK et J. M. SCHWARTZ , Algèbres de Kac moyennables . Pacific J. of Math., vol. 125, n^\circ 2, 1986 , p. 363-379. Article | MR 88f:46126 | Zbl 0597.43002 · 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 · 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 · 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 · 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 · 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 · 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 · 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 · 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 · 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 · 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^\circ 1, 1950 , p. 10-59. MR 12,719b | Zbl 0131.12101 · 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 · Zbl 0397.46059
[22] M. B. LANDSTAD , Duality for Dual Covariance Algebras , Comm. Math. Phys., vol. 52, 1977 , p. 191-202. Article | MR 56 #8750 | Zbl 0362.46046 · 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^\circ 2, 1987 , p. 747-784. MR 88f:46127 | Zbl 0722.46031 · 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 · Zbl 0082.11201
[25] S. MAC LANE , Categories for the working mathematicians , GTM 5. Zbl 0561.01017 · 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 · 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^\circ 2, 1991 , p. 291-319. MR 92b:46088 | Zbl 0741.46033 · 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^\circ 1, 1990 , p. 17-64. MR 91j:16050 | Zbl 0694.16008 · Zbl 0694.16008
[30] G. MOORE and N. SEIBERG , Classical and Quantum Conformal Field Theory , Comm. Math. Phys., vol. 123, 1989 , p. 177-254. Article | MR 90e:81216 | Zbl 0694.53074 · 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. Article | MR 56 #16393 | Zbl 0363.46062 · 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 · Zbl 0423.46051
[33] G. K. PEDERSEN , C*-Algebras and their Automorphism Groups . Academic press, 1979 . MR 81e:46037 | Zbl 0416.46043 · Zbl 0416.46043
[34] P. PODLES et S. L. WORONOWICZ , Quantum Deformation of Lorentz Group , Comm. Math. Phys., vol. 130, 1990 , p. 381-431. Article | MR 91f:46100 | Zbl 0703.22018 · Zbl 0703.22018
[35] M. A. RIEFFEL , Some Solvable Quantum Groups , Proc. Conf. Craiova Romania Sept. 1989 (à paraître). · Zbl 0804.17010
[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 · Zbl 0617.16005
[37] M. ROSSO , Algèbres enveloppantes quantifiées, groupes quantiques compacts de matrices et calcul différentiel non commutatif . Prépublication. · Zbl 0721.17013
[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 · 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 · 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 · 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 · 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 · 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 · 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 · 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 · Zbl 0268.46058
[46] M. TAKEUCHI , Matched Pairs of Groups and Bismash Products of Hopf Algebras , Comm. Algebra, vol. 9, n^\circ 8, 1981 , p. 841-882. MR 83f:16013 | Zbl 0456.16011 · Zbl 0456.16011
[47] T. TANNAKA , Über den Dualität der nicht-kommutativen topologischen Gruppen , Tôhoku Math. J., vol. 45, 1938 , p. 1-12. Zbl 0020.00904 | 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. Article | MR 36 #313 | Zbl 0184.17402 · 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^\circ 1, p. 75-76. MR 49 #463 | Zbl 0312.22007 · 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 · 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^\circ 274, 1977 , p. 451-457. MR 83c:46065 | Zbl 0503.46049 · Zbl 0503.46049
[52] A. WEIL , L’intégration dans les groupes topologiques et ses applications . Act. Sc. Ind., n^\circ 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 · Zbl 0676.46050
[54] S. L. WORONOWICZ , Compact Matrix Pseudogroups , Comm. Math. Phys., vol. 111, 1987 , p. 613-665. Article | MR 88m:46079 | Zbl 0627.58034 · 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 · Zbl 0664.58044
[56] S. L. WORONOWICZ , Differential Calculus on Compact Matrix Pseudogroups (Quantum Groups), Comm. Math. Phys., vol. 122, 1989 , p. 125-170. Article | MR 90g:58010 | Zbl 0751.58042 · Zbl 0751.58042
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.