QUASI-SYMMETRIC POLYNOMIALLY BOUNDED FReCHET ALGEBRAS
Keywords:
Fourier Transform, Reductive Groups, Harish-Chandra's Schwartz algebras, Symmetric Polynomial Algebras, FrEchet Algebras.
Abstract
This paper concerns the notion of a symmetric algebra and its generalization to a quasi-symmetric algebra. We study the structure of these algebras in respect to their hull-kernel regularity and existence of some ideals, especially the hull-minimal ideals. An immediate application of our results is to the Harish-Chandra Schwartz Frechet algebras, Cp(G), of a (connected) semi-simple G.
References
[1]: Arthur, J. G., Some tempered distributions on semi-simple groups of real rank one. Ann. of Math. 100 (1974): 553-584.
[2]: Barker, W. H., The spherical Bochner theorem on semi-simple Lie groups. J. Funct. Anal. 20 (1975): 179-207.
[3]: Barker, W. H., Positive de nite distributions on unimodular Lie groups. Duke Math. J. 43. 1 (1976): 71-79.
[4]: Barker, W. H., Tempered, invariant, positive-definite distributions on SU(1; 1)/{+-}. Illinois J. Math. 28. 1 (1984): 83-102.
[5]: Bassey, U. N. and Oyadare, O. O., On Hull-minimal ideals in the Schwartz algebras of spherical functions on a semi-simple Lie group. International Journal of Functional Analysis, Operator Theory and Applications. 3 (1) (2011) : 37 - 52.
[6]: Bonsall, F. F. and Duncan, J., Complete normed algebras. Berlin: Springer-Verlag, 1973.
[7]: Dixmier, J., Operateurs de rang fini dans les representations unitaires. Inst. Hautes Etudes. Sci. Publ. Math. 6 (1960): 13-25.
[8]: Dixmier, J., Les C*-algebres et leurs representations. Paris: Gauthier-Villars, 1964.
[9]: Harish-Chandra, Spherical functions on a semi-simple Lie group, I. Amer. J. Math. 80. 2 (1958): 241-310.
[10]: Hille, E. and Phillips, R. S., Functional analysis and semi-groups. 31. Rhode Island: Amer. Math. Soc., Colloq. Publ., Providence, 1957.
[11]: Kadison, R. V. and Ringrose, J. R., Fundamentals of the theory of operators algebras, vol: 1, Elementary theory. London: Academic Press, 1983.
[12]: Ludwig, J., Hull-minimal Ideals in the Schwartz algebra of the Heisenberg group, Studia Mathematica, 130, 1 (1998), pp. 77-98.
[13]: Oyadare, O. O., On harmonic analysis of spherical convolutions on semi-simple Lie groups, Theoretical Mathematics and Applications, 5 (3) (2015), p. 19- 36.
[14]: Oyadare, O. O., On ideal theory for the Schwartz algebras of spherical functions on semi-simple Lie groups. Ph.D Thesis, University of Ibadan (2016). x + 383pp.
[15]: Oyadare, O. O., Functional analysis of canonical wave-packets on real reductive groups, arXiv:1912:07542v1: [math.FA], 13 Dec. 2019.
[16]: Oyadare, O. O., Non-spherical Harish-Chandra Fourier transforms on real reductive groups, J. Fourier Anal. Appl. 28, 15 (2022). http://doi.org/10.1007/s00041-09906-w
[17]: Oyadare, O. O., The full Bochner theorem on real reductive groups, Algebras, Groups and Geometries 39, (2023); p. 207- 220.
[18]: Oyadare, O. O., Series analysis and Schwartz algebras of spherical convolutions on semisimple Lie groups Algebras, Groups and Geometries 40(1), (2024a); p. 41 - 59: See also arXiv.1706:09045 [math.RT].
[19]: Oyadare, O. O., On the operator-valued Fourier transform of the Harish-Chandra Schwartz Algebra, arXiv:2407:20755v1: [math.RT], 2024b.
[20]: Rickart, C. E., General theory of Banach algebras. New York: Van Nostrand, 1974.
[2]: Barker, W. H., The spherical Bochner theorem on semi-simple Lie groups. J. Funct. Anal. 20 (1975): 179-207.
[3]: Barker, W. H., Positive de nite distributions on unimodular Lie groups. Duke Math. J. 43. 1 (1976): 71-79.
[4]: Barker, W. H., Tempered, invariant, positive-definite distributions on SU(1; 1)/{+-}. Illinois J. Math. 28. 1 (1984): 83-102.
[5]: Bassey, U. N. and Oyadare, O. O., On Hull-minimal ideals in the Schwartz algebras of spherical functions on a semi-simple Lie group. International Journal of Functional Analysis, Operator Theory and Applications. 3 (1) (2011) : 37 - 52.
[6]: Bonsall, F. F. and Duncan, J., Complete normed algebras. Berlin: Springer-Verlag, 1973.
[7]: Dixmier, J., Operateurs de rang fini dans les representations unitaires. Inst. Hautes Etudes. Sci. Publ. Math. 6 (1960): 13-25.
[8]: Dixmier, J., Les C*-algebres et leurs representations. Paris: Gauthier-Villars, 1964.
[9]: Harish-Chandra, Spherical functions on a semi-simple Lie group, I. Amer. J. Math. 80. 2 (1958): 241-310.
[10]: Hille, E. and Phillips, R. S., Functional analysis and semi-groups. 31. Rhode Island: Amer. Math. Soc., Colloq. Publ., Providence, 1957.
[11]: Kadison, R. V. and Ringrose, J. R., Fundamentals of the theory of operators algebras, vol: 1, Elementary theory. London: Academic Press, 1983.
[12]: Ludwig, J., Hull-minimal Ideals in the Schwartz algebra of the Heisenberg group, Studia Mathematica, 130, 1 (1998), pp. 77-98.
[13]: Oyadare, O. O., On harmonic analysis of spherical convolutions on semi-simple Lie groups, Theoretical Mathematics and Applications, 5 (3) (2015), p. 19- 36.
[14]: Oyadare, O. O., On ideal theory for the Schwartz algebras of spherical functions on semi-simple Lie groups. Ph.D Thesis, University of Ibadan (2016). x + 383pp.
[15]: Oyadare, O. O., Functional analysis of canonical wave-packets on real reductive groups, arXiv:1912:07542v1: [math.FA], 13 Dec. 2019.
[16]: Oyadare, O. O., Non-spherical Harish-Chandra Fourier transforms on real reductive groups, J. Fourier Anal. Appl. 28, 15 (2022). http://doi.org/10.1007/s00041-09906-w
[17]: Oyadare, O. O., The full Bochner theorem on real reductive groups, Algebras, Groups and Geometries 39, (2023); p. 207- 220.
[18]: Oyadare, O. O., Series analysis and Schwartz algebras of spherical convolutions on semisimple Lie groups Algebras, Groups and Geometries 40(1), (2024a); p. 41 - 59: See also arXiv.1706:09045 [math.RT].
[19]: Oyadare, O. O., On the operator-valued Fourier transform of the Harish-Chandra Schwartz Algebra, arXiv:2407:20755v1: [math.RT], 2024b.
[20]: Rickart, C. E., General theory of Banach algebras. New York: Van Nostrand, 1974.
Published
2024-12-10
How to Cite
Oyadare, O. O. (2024). QUASI-SYMMETRIC POLYNOMIALLY BOUNDED FReCHET ALGEBRAS. Unilag Journal of Mathematics and Applications, 4(1), 70-80. Retrieved from http://lagjma.unilag.edu.ng/article/view/2284
Section
Articles
Copyright (c) 2024 Olufemi O. Oyadare
This work is licensed under a Creative Commons Attribution 4.0 International License.
