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The Fourier Algebra for Locally Compact Groupoids

Published online by Cambridge University Press:  20 November 2018

Alan L. T. Paterson
Alan L. T. Paterson*
Affiliation:
Department of Mathematics, University of Mississippi, University, MS 38677, U.S.A. e-mail: [email protected]
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Abstract

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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.

Keywords

43A32Fourier algebralocally compact groupoidsHilbert modulespositive definite functionscompletely bounded maps
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 56 , Issue 6 , 01 December 2004 , pp. 1259 - 1289
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), 181–236.Google Scholar
[6] Hahn, P., Haar measure for measure groupoids. Trans. Amer. Math. Soc. 242(1978), 1–33.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), 69–82.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), 47–72.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), 3–22.Google Scholar
[14] Oty, K., Fourier-Stieltjes algebras of r-discrete groupoids. J. Operator Theory 41(1999), 175–197.Google Scholar
[15] Paschke, W.L., Inner product modules over B*-algebras. Trans. Amer.Math. Soc. 182(1973), 443–468.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), 89–104.Google Scholar
[20] Peetre, J., Réctification à l’article “Une caractérisation abstraite des opérateurs différentiels”. Math. Scand. 8(1960), 116–120.Google Scholar
[21] Ramsay, A., Topologies for measured groupoids. J. Funct. Anal. 47(1982), 314–343.Google Scholar
[22] Ramsay, A. and Walter, M. E., Fourier-Stieltjes algebras of locally compact groupoids. J. Funct, Anal. 148(1997), 314–367.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), 67–97.Google Scholar
[25] Renault, J.N., The Fourier algebra of a measured groupoid and its multipliers. J. Funct. Anal. 145(1997), 455–490.Google Scholar
[26] Vallin, J., Bimodules de Hopf et poids operatoriels de Haar. J. Operator Theory 35(1996), 39–65.Google Scholar
[27] Vallin, J., Unitaire pseudo-multiplicatif associé à un groupoϊde. J. Operator Theory 44(2000), 249–300.Google Scholar
[28] Walter, M., W*-algebras and nonabelian harmonic analysis. J. Funct. Anal. 11(1972), 17–38.Google Scholar
[29] Walter, M., Dual algebras. Math. Scand. 58(1986), 77–104.Google Scholar