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Fourier algebra of a hypergroup. I

Published online by Cambridge University Press:  09 April 2009

Varadharajan Muruganandam
Affiliation:
Department of Mathematics Pondicherry UniversityPondicherry 605 014India e-mail: [email protected]
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Abstract

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In this article we study the Fourier space of a general hypergroup and its multipliers. The main result of this paper characterizes commutative hypergroups whose Fourier space forms a Banach algebra under pointwise product with an equivalent norm. Among those hypergroups whose Fourier space forms a Banach algebra, we identify a subclass for which the Gelfand spectrum of the Fourier algebra is equal to the underlying hypergroup. This subclass includes for instance, Jacobi hypergroups, Bessel-Kingman hypergroups.

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 2007

References

[1]Arveson, W. B., ‘On groups of automorphisms of operator algebras’, J. Funct. Analysis 15 (1974), 217243.CrossRefGoogle Scholar
[2]Bloom, W. R. and Heyer, H., Harmonic analysis of probability measures on hypergroups, de Gruyter Studies in Math. 20 (Walter de Gruyter, Berlin, 1995).Google Scholar
[3]Bloom, W. R. and Xu, Z., ‘The Hardy-Littlewood maximal function for Chébli-Trimèche hypergroups’, Contemp. Math. 183 (1995), 4570.CrossRefGoogle Scholar
[4]Bloom, W. R. and Xu, Z., ‘Fourier transforms of Schwartz functions on Chébli-Trimèche hypergroups’, Monatsh. Math. 125 (1998), 89109.CrossRefGoogle Scholar
[5]Blower, G., ‘Stationary processes for translation operators’, Proc. London. Math. Soc. 72 (1996), 697720.Google Scholar
[6]Chihara, T. S., An introduction to orthogonal polynomials (Gordon and Breach, New York, 1978).Google Scholar
[7]DeCannière, J. and Haagerup, U., ‘Multipliers of the Fourier algebra of some simple Lie groups and their discrete subgroups’, Amer. J. Math. 107 (1985), 455500.Google Scholar
[8]Dixmier, J., C*-algebras, volume 15 (North-Holland, Amsterdam, 1977).Google Scholar
[9]Dixmier, J., Von Neumann algebras, volume 27 (North-Holland, Amsterdam, 1981).Google Scholar
[10]Eymard, P., ‘L'algèbre de Fourier d'un groupe localement compact’, Bull. Soc. Math. France 92 (1964), 181236.Google Scholar
[11]Flensted-Jensen, M. and Koornwinder, T. H., ‘Jacobi functions: the addition formula and the positivity of the dual convolution structure’, Ark. Math. 17 (1979), 139151.Google Scholar
[12]Gasper, G., ‘Banach algebras for Jacobi series and positivity of a kernel’, Ann. of Math. (1) 95 (1972), 261280.Google Scholar
[13]Godement, R., ‘Les fonctions de type positif et la théorie des groupes’, Trans. Amer. Math. Soc. 63 (1948), 184.Google Scholar
[14]Grothendieck, A., Topological vector spaces, Notes on Mathematics and its Applications (Gordon and Breach, New York, 1973).Google Scholar
[15]Hartmann, K., Henrichs, R. W. and Lasser, R., ‘Duals of orbit spaces in groups with relatively compact inner automorphism groups are hypergroups’, Monatsh. Math. 88 (1979), 229238.Google Scholar
[16]Herz, C. S., ‘Harmonic synthesis for subgroups’, Ann. Inst. Fourier (Grenoble) 23 (1973), 91123.Google Scholar
[17]Jewett, R. I., ‘Spaces with an abstract convolution of measures’, Adv. Math. 18 (1975), 1101.Google Scholar
[18]Koornwinder, T. H., ‘Jacobi functions and analysis of noncompact semisimple Lie groups’, in:Special functions: Group theoretical aspects and applications (ed. Askey, R. A. et al. ) (D. Reidel, Dordrecht, 1984) pp. 185.Google Scholar
[19]Laine, T. P., ‘The product formula and convolution structure for the generalized Chebyshev polynomials’, SIAM. J. Math. Anal. 11 (1980), 133146.CrossRefGoogle Scholar
[20]Lasser, R., ‘Almost periodic functions on hypergroups’, Math. Ann. 252 (1980), 183196.Google Scholar
[21]Lasser, R., ‘Orthogonal polynomials and hypergroups’, Rend. Math. Appl. 3 (1983), 185209.Google Scholar
[22]Muruganandam, V., ‘Fourier algebra of a hypergroup. II. Spherical hypergroups’, Math. Nach., to appear.Google Scholar
[23]Pederson, G. K., C*-algebras and their automorphism groups, London Math. Soc. Monographs 14 (Academic Press, London, 1979).Google Scholar
[24]Rickart, C. E., General theory of Banach algebras, The University Series in Higher Math. (D. Van Nostrand, Princeton, N.J., 1960).Google Scholar
[25]Ross, K. A., ‘Centers of hypergroups’, Trans. Amer. Math. Soc. 213 (1978), 251269.Google Scholar
[26]Takesaki, M., Theory of operator algebras I (Springer, New York, 1979).Google Scholar
[27]Trimèche, K., Generalized wavelets and hypergroups (Gordon and Breach, Amsterdam, 1997).Google Scholar
[28]Voit, M., ‘On the Fourier transformation of positive, positive definite measures on commutative hypergroups, and dual convolution structures’, Manuscripta Math. 72 (1991), 141153.Google Scholar
[29]Voit, M., ‘A product formula for orthogonal polynomials associated with infinite distance-transitive graphs’, J. Approx. Theory 120 (2003), 337354.Google Scholar
[30]Vrem, R. C., ‘Harmonic analysis on compact hypergroups’, Pacific. J. Math. 85 (1979), 239251.Google Scholar
[31]Zeuner, H., ‘Properties of the cosh hypergroup’, in: Probability Measures on Groups IX, Proc. Conf. Oberwolfach, 1988, Springer Lecture Notes in Math. 1379 (Springer, Berlin, 1989) pp. 425434.Google Scholar
[32]Zeuner, H., ‘Duality of commutative hypergroups’, in: Probability Measures on Groups X, Proc. Conf. Oberwolfach, 1988 (Plenum Press, New York, 1990) pp. 467488.Google Scholar