A Class of Abstract Linear Representations for Convolution Function Algebras overHomogeneous Spaces of Compact Groups

Farashahi, Arash Ghaani

Canadian journal of mathematics, Tome 70 (2018) no. 1, pp. 97-116

Résumé

This paper introduces a class of abstract linear representations on Banach convolution function algebras over homogeneous spaces of compact groups. Let $G$ be a compact group and $H$ a closed subgroup of $G$ . Let $\mu $ be the normalized $G$ -invariant measure over the compact homogeneous space $G/H$ associated with Weil's formula and $1\,\le \,p\,<\,\infty $ . We then present a structured class of abstract linear representations of the Banach convolution function algebras ${{L}^{p}}\left( G/H,\,\mu\right)$ .

Détail

DOI : 10.4153/CJM-2016-043-9

Mots-clés : 43A85, 47A67, 20G05, homogeneous space, linear representation, continuous unitary representation, convolution function algebra, compact group, convolution, involution

Bibliographie

[1] [1] Derighetti, A., Á propos des convoluteurs d'un groupe quotient. Bull. Sci. Math. (2) 107(1983), no. 1, 3–23. Google Scholar

[2] [2] Derighetti, A., Convolution operators on groups. Lecture Notes of the Unione Matematica Italiana, 11.Springer, Heidelberg, 2011. Google Scholar | DOI

[3] [3] Dixmier, J., C* -algebras. North-Holland Mathematical Library, 15. North-Holland, Amsterdam, 1977. Google Scholar

[4] [4] Feichtinger, H. G., On a class of convolution algebras of functions. Ann. Inst. Fourier (Grenoble) 27(1977), no. 3, 135–162. Google Scholar

[5] [5] Feichtinger, H. G., Banach convolution algebras of functions. II. Monatsh. Math. 87(1979), no. 3,181–207. http://dx.doi.Org/10.1007/BF01303075 Google Scholar

[6] [6] Feichtinger, H. G., On a new Segal algebra. Monatsh. Math. 92(1981), no. 4, 269–289. http://dx.doi.Org/10.1007/BF01320058 Google Scholar

[7] [7] Folland, G. B., A course in abstract harmonic analysis. Studies in Advanced Mathematics. CRC Press, Boca Raton, FL, 1995. Google Scholar

[8] [8] Ghaani Farashahi, A., Abstract non-commutative harmonic analysis of coherent state transforms. Ph.D. thesis, Ferdowsi University of Mashhad, 2012. Google Scholar

[9] [9] Ghaani Farashahi, A., Convolution and involution on function spaces of homogeneous spaces. Bull. Malays. Math. Sci. Soc. (2) 36(2013) no. 4,1109–1122. Google Scholar

[10] [10] Ghaani Farashahi, A., Abstract harmonic analysis of relative convolutions over canonical homogeneous spaces of semidirect product groups. J. Aust. Math. Soc. 101(2016), no. 2,171–187, http://dx.doi.Org/10.1017/S1446788715000798 Google Scholar

[11] [11] Ghaani Farashahi, A., Abstract harmonic analysis of wave-packet transforms over locally compact abelian groups. Banach J. Math. Anal. 11(2017), no. 1, 50–71. Google Scholar | DOI

[12] [12] Ghaani Farashahi, A., Abstract operator-valued Fourier transforms over homogeneous spaces of compact groups. Groups, Geometry, Dynamics, to appear. Google Scholar

[13] [13] Ghaani Farashahi, A., Abstract convolution function algebras over homogeneous spaces of compact groups. Illinois J. Math., to appear. Google Scholar

[14] [14] Ghaani Farashahi, A., Abstract Plancherel (trace) formulas over homogeneous spaces of compact groups. Can. Math. Bull, to appear. http://dx.doi.Org/10.4153/CMB-2O16-037-6 Google Scholar

[15] [15] Hewitt, E. and Ross, K. A., Abstract harmonic analysis. Vol. 1-2,1963,1970. Google Scholar

[16] [16] Kisil, V., Relative convolutions. I. Properties and applications. Adv. Math. 147(1999), no. 1, 35–73. http://dx.doi.Org/10.1006/aima.1 999.1 833 Google Scholar

[17] [17] Kisil, V., Erlangen program at large: an overview. Trends Math., Birkhäuser/Springer, Basel, 2012, pp. 1–94. http://dx.doi.Org/10.1007/978-3-0348-0417-2_1 Google Scholar

[18] [18] Kisil, V., Geometry of Mobius transformations. Elliptic, parabolic and hyperbolic actions of SL(). Imperial College Press, London, 2012. Google Scholar | DOI

[19] [19] Kisil, V., Operator covariant transform and local principle. J. Phys. A 45(2012), no. 24, pp. 244022, 10. Google Scholar | DOI

[20] [20] Kisil, V., Calculus of operators: covariant transform and relative convolutions. Banach J. Math. Anal. 8(2014), no. 2, 156–184. Google Scholar | DOI

[21] [21] Murphy, G. J., C* -algebras and operator theory. Academic Press, Boston, MA, 1990. Google Scholar

[22] [22] Reiter, H. and Stegeman, J. D., Classical harmonic analysis and locally compact groups. Second edition. London Mathematical Society Monographs, 22. Oxford University Press, New York, 2000. Google Scholar