Hostname: page-component-586b7cd67f-2brh9 Total loading time: 0 Render date: 2024-11-22T09:48:01.994Z Has data issue: false hasContentIssue false

The Fourier Algebra for Locally Compact Groupoids

Published online by Cambridge University Press:  20 November 2018

Alan L. T. Paterson*
Affiliation:
Department of Mathematics, University of Mississippi, University, MS 38677, U.S.A. e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We introduce and investigate using Hilbert modules the properties of the Fourier algebra$A(G)$ for a locally compact groupoid $G$. We establish a duality theorem for such groupoids in terms of multiplicative module maps. This includes as a special case the classical duality theorem for locally compact groups proved by P. Eymard.

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 2004

References

[1] Connes, A., Noncommutative geometry. Academic Press, New York, 1994.Google Scholar
[2] Davidson, K., C*-algebras by example. Fields Institute Monographs, American Mathematical Society, Providence, RI, 1996.Google Scholar
[3] Dixmier, J., C*-algebras. North-Holland,, Amsterdam, 1977.Google Scholar
[4] Effros, E.G. and Ruan, Z-J., Operator spaces. London Mathematical Society Monographs New Series 23, Clarendon Press, Oxford, 2000.Google Scholar
[5] Eymard, P., L'algèbre de Fourier d’un groupe localement compact. Bull. Soc. Math. France 92(1964), 181236.Google Scholar
[6] Hahn, P., Haar measure for measure groupoids. Trans. Amer. Math. Soc. 242(1978), 133.Google Scholar
[7] Helgason, S., Differential geometry, Lie groups and symmetric spaces. Academic Press, New York, 1978.Google Scholar
[8] Herz, C., The Theory of p-spaces with an application to convolution operators. Trans. Amer. Math. Soc. 154(1971), 6982.Google Scholar
[9] Hewitt, E. and Ross, K.A., Abstract harmonic analysis I, Springer-Verlag, Gottingen, 1970.Google Scholar
[10] Khoshkam, M. and Skandalis, G., Regular representation of groupoid C*-algebras and applications to inverse semigroups. J. Reine Angew.Math. 546(2002), 4772.Google Scholar
[11] Lance, E C., Hilbert C*-modules. London Mathematical Society Lecture Note Series 210, Cambridge University Press, Cambridge, 1995.Google Scholar
[12] Muhly, P.S., Coordinates in operator algebra. to appear, CBMS Regional Conference Series in Mathematics, American Mathematical Society, Providence, 180pp.Google Scholar
[13] Muhly, P.S., Renault, J.N. and Williams, D. P., Equivalence and isomorphism for groupoid C*-algebras. J. Operator Theory 17(1987), 322.Google Scholar
[14] Oty, K., Fourier-Stieltjes algebras of r-discrete groupoids. J. Operator Theory 41(1999), 175197.Google Scholar
[15] Paschke, W.L., Inner product modules over B*-algebras. Trans. Amer.Math. Soc. 182(1973), 443468.Google Scholar
[16] Paulsen, V.I., Completely bounded maps and dilations. Pitman Research Notes in Mathematics, 146, John Wiley, New York, 1986.Google Scholar
[17] Paterson, A. L.T., Amenability. Mathematical Surveys and Monographs, 29, American Mathematical Society, Providence, RI, 1988.Google Scholar
[18] Paterson, A. L.T., Groupoids, inverse semigroups and their operator algebras. Progress in Mathematics, 170, Birkhäuser, Boston, 1999.Google Scholar
[19] Paterson, A. L.T., Continuous family groupoids. Homology, Homotopy and Applications, 2(2000), 89104.Google Scholar
[20] Peetre, J., Réctification à l’article “Une caractérisation abstraite des opérateurs différentiels”. Math. Scand. 8(1960), 116120.Google Scholar
[21] Ramsay, A., Topologies for measured groupoids. J. Funct. Anal. 47(1982), 314343.Google Scholar
[22] Ramsay, A. and Walter, M. E., Fourier-Stieltjes algebras of locally compact groupoids. J. Funct, Anal. 148(1997), 314367.Google Scholar
[23] Renault, J.N., A groupoid approach to C*-algebras. Lecture Notes in Mathematics, 793, Springer-Verlag, Berlin, 1980.Google Scholar
[24] Renault, J.N., Répresentation de produits croisés d’algèbres de groupoϊdes, J. Operator Theory, 18(1987), 6797.Google Scholar
[25] Renault, J.N., The Fourier algebra of a measured groupoid and its multipliers. J. Funct. Anal. 145(1997), 455490.Google Scholar
[26] Vallin, J., Bimodules de Hopf et poids operatoriels de Haar. J. Operator Theory 35(1996), 3965.Google Scholar
[27] Vallin, J., Unitaire pseudo-multiplicatif associé à un groupoϊde. J. Operator Theory 44(2000), 249300.Google Scholar
[28] Walter, M., W*-algebras and nonabelian harmonic analysis. J. Funct. Anal. 11(1972), 1738.Google Scholar
[29] Walter, M., Dual algebras. Math. Scand. 58(1986), 77104.Google Scholar