Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-26T08:31:30.432Z Has data issue: false hasContentIssue false
  • English
  • Français

C*-pseudo-multiplicative unitaries, Hopf C*-bimodules and their Fourier algebras

Part of: Selfadjoint operator algebras

Published online by Cambridge University Press:  04 February 2011

Thomas Timmermann
Thomas Timmermann
Affiliation:
Universität Münster, Einsteinstrasse 62, 48149 Münster, Germany ([email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

We introduce C*-pseudo-multiplicative unitaries and concrete Hopf C*-bimodules for the study of quantum groupoids in the setting of C*-algebras. These unitaries and Hopf C*-bimodules generalize multiplicative unitaries and Hopf C*-algebras and are analogues of the pseudo-multiplicative unitaries and Hopf–von Neumann-bimodules studied by Enock, Lesieur and Vallin. To each C*-pseudo-multiplicative unitary, we associate two Fourier algebras with a duality pairing and in the regular case two Hopf C*-bimodules. The theory is illustrated by examples related to locally compact Hausdorff groupoids. In particular, we obtain a continuous Fourier algebra for a locally compact Hausdorff groupoid.

Keywords

quantum groupoidgroupoidduality

MSC classification

Secondary: 46L55: Noncommutative dynamical systems
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 11 , Issue 1 , January 2012 , pp. 189 - 220
Copyright
Copyright © Cambridge University Press 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Baaj, S., Représentation régulière du groupe quantique E μ(2) de Woronowicz, C. R. Acad. Sci. Paris Sér. I Math. 314(13) (1992), 1021–1026.Google Scholar
2.Baaj, S., Représentation régulière du groupe quantique des déplacements de Woronowicz, Astérisque 232(1995), 11–48.Google Scholar
3.Baaj, S. and Skandalis, G., Unitaires multiplicatifs et dualité pour les produits croisés de C*-algèbres, Annales Sci. Éc. Norm. Sup. (4) 26(4) (1993), 425–488.CrossRefGoogle Scholar
4.Blanchard, É., Déformations de C*-algèbres de Hopf, Bull. Soc. Math. France 124(1) (1996), 141–215.CrossRefGoogle Scholar
5.Böhm, G., Nill, F. and Szlachá;nyi, K., Weak Hopf algebras, I, Integral theory and C*-structure, J. Alg. 221(2) (1999), 385–438.CrossRefGoogle Scholar
6.Böhm, G. and Szlachá;nyi, K., Weak Hopf algebras, II, Representation theory, dimensions, and the Markov trace, J. Alg. 233(1) (2000), 156–212.CrossRefGoogle Scholar
7.Böhm, G. and Szlachá;nyi, K., Weak C*-Hopf algebras and multiplicative isometries, J. Operat. Theory 45(2) (2001), 357–376.Google Scholar
8.Echterhoff, S., Kaliszewski, S., Quigg, J. and Raeburn, I., A categorical approach to imprimitivity theorems for C*-dynamical systems, Memoirs of the American Mathematical Society, Volume 180 (American Mathematical Society, Providence, RI, 2006).Google Scholar
9.Enock, M., Quantum groupoids of compact type, J. Inst. Math. Jussieu 4(1) (2005), 29–133.CrossRefGoogle Scholar
10.Enock, M. and Vallin, J.-M., Inclusions of von Neumann algebras, and quantum groupoids, J. Funct. Analysis 172(2) (2000), 249–300.CrossRefGoogle Scholar
11.Enock, M. and Vallin, J.-M., Inclusions of von Neumann algebras and quantum groupoids, II, J. Funct. Analysis 178(1) (2000), 156–225.CrossRefGoogle Scholar
12.Hahn, P., Haar measure for measure groupoids, Trans. Am. Math. Soc. 242(1978), 1–33.CrossRefGoogle Scholar
13.Kustermans, J. and Vaes, S., Locally compact quantum groups, Annales Sci. École Norm. Sup. (4) 33(6) (2000), 837–934.CrossRefGoogle Scholar
14.Kustermans, J. and Vaes, S., Locally compact quantum groups in the von Neumann algebraic setting, Math. Scand. 92(1) (2003), 68–92.CrossRefGoogle Scholar
15.Lance, E. C., Hilbert C*-modules: a toolkit for operator algebraists, London Mathematical Society Lecture Note Series, Volume 210 (Cambridge University Press, 1995).Google Scholar
16.Lesieur, F., Measured quantum groupoids, Mémoires de la Société Mathématique de France, Volume 109 (Société Mathématique de France, Paris, 2008).Google Scholar
17.Masuda, T., Nakagami, Y. and Woronowicz, S. L., A C*-algebraic framework for quantum groups, Int. J. Math. 14(9) (2003), 903–1001.CrossRefGoogle Scholar
18.Nikshych, D., A duality theorem for quantum groupoids, in New Trends in Hopf Algebra Theory (La Falda, 1999), Contemporary Mathematics, Volume 267, pp. 237–243 (American Mathematical Society, Providence, RI, 2000).Google Scholar
19.O'uchi, M., Pseudo-multiplicative unitaries on Hilbert C*-modules, Far East J. Math. Sci. Special Volume, Part II (2001), 229–249.Google Scholar
20.Paterson, A. L. T., Groupoids, inverse semigroups, and their operator algebras, Progress in Mathematics, Volume 170 (Birkhäuser, Boston, MA, 1999).Google Scholar
21.Paterson, A. L. T., The Fourier algebra for locally compact groupoids, Can. J. Math. 56(6) (2004), 1259–1289.CrossRefGoogle Scholar
22.Renault, J., A groupoid approach to C*-algebras, Lecture Notes in Mathematics, Volume 793 (Springer, 1980).Google Scholar
23.Renault, J., The Fourier algebra of a measured groupoid and its multipliers, J. Funct. Analysis 145(2) (1997), 455–490.CrossRefGoogle Scholar
24.Sauvageot, J.-L., Produits tensoriels de Z-modules et applications, in Operator Algebras and Their Connections with Topology and Ergodic Theory (Buşteni, 1983), Lecture Notes in Mathematics, Volume 1132, pp. 468–485 (Springer, 1985).Google Scholar
25.Timmermann, T., The relative tensor product and a minimal fiber product in the setting of C*-algebras, J. Operat. Theory, in press.Google Scholar
26.Timmermann, T., From Hopf C*-families to concrete Hopf C*-bimodules, Technical report, SFB 478, WW-University Muenster (December 2007).Google Scholar
27.Timmermann, T., Pseudo-multiplicative unitaries on C*-modules and Hopf C*-families, J. Noncomm. Geom. 1(2007), 497–542.CrossRefGoogle Scholar
28.Timmermann, T., An invitation to quantum groups and duality, EMS Textbooks in Mathematics (European Mathematical Society, Zürich, 2008).Google Scholar
29.Timmermann, T., C*-pseudo-multiplicative unitaries and Hopf C*-bimodules, preprint (arXiv:0908.1850, 2009).Google Scholar
30.Timmermann, T., Coactions of Hopf C*-bimodules, J. Operat. Theory, in press.Google Scholar
31.Timmermann, T., A definition of compact C*-quantum groupoids, in Operator Structures and Dynamical Systems (Leiden, 2008), Contemporary Mathematics (American Mathematical Society, Providence, RI, 2009).Google Scholar
32.Vallin, J.-M., Bimodules de Hopf et poids opératoriels de Haar, J. Operat. Theory 35(1) (1996), 39–65.Google Scholar
33.Vallin, J.-M., Unitaire pseudo-multiplicatif associé à un groupoïde: applications à la moyennabilité, J. Operat. Theory 44(2) (2000), 347–368.Google Scholar