Document Type: Research Paper

Authors

1 Numerical Harmonic Analysis Group (NuHAG), Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, A-1090 Wien, Vienna, Austria.

2 Department of Pure Mathematics, Ferdowsi University of Mashhad, Center of Excellence in Analysis on Algebraic Structures (CEAAS), P. O. Box 1159-91775, Mashhad, Iran.

Abstract

This article presents a unified approach to the abstract notions of partial convolution and involution in $L^p$-function spaces over semi-direct product of locally compact groups. Let $H$ and $K$ be locally compact groups and $\tau:H\to Aut(K)$ be a continuous homomorphism.  Let $G_\tau=H\ltimes_\tau K$ be the semi-direct product of $H$ and $K$ with respect to $\tau$. We define left and right $\tau$-convolution on $L^1(G_\tau)$ and we show that, with respect to each of them, the function space $L^1(G_\tau)$ is a Banach algebra. We define $\tau$-convolution as a linear combination of the left and right $\tau$-convolution and we show that the $\tau$-convolution is commutative if and only if $K$ is abelian. We prove that there is a $\tau$-involution on $L^1(G_\tau)$ such that with respect to the $\tau$-involution and $\tau$-convolution, $L^1(G_\tau)$ is a non-associative Banach $*$-algebra. It is also shown that when $K$ is abelian, the $\tau$-involution and $\tau$-convolution make $L^1(G_\tau)$ into a Jordan Banach $*$-algebra. Finally, we also present the generalized notation of $\tau$-convolution for other $L^p$-spaces with $p>1$.

Keywords

Main Subjects

[1] A.A. Are jamaal, The continuous Zak transform and generalized Gabor frames, Mediterr. J. Math. 10 (2013), No. 1, 353-365.

[2] A.A. Are jamaal and A. Ghaani Farashahi, Zak transform for semidirect product of locally compact groups, Anal. Math. Phys. 3 (2013), No. 3, 263-276.

[3] A.A. Are jamaal and R.A. Kamyabi-Gol, On the square integrability of quasi regular representation on semidirect product groups, J. Geom. Anal. 19 (2009), No. 3, 541-552.

[4] A.A. Are jamaal and R.A. Kamyabi-Gol, On construction of coherent states associated with semidirect products, Int. J. Wavelets Multiresolut. Inf. Process. 6 (2008), No. 5, 749-759.

[5] W.R. Bloom and H. Heyer, Harmonic Analysis of Probability Measures on Hypergroups, de Gruyter Studies in Mathematics, 20, Walter de Gruyter (1995).

[6] G. Chirikjian and A. Kyatkin, Engineering Applications of Noncommutative Harmonic Analysis With Emphasis on Rotation and Motion Groups, Boca Raton, FL: CRC Press. xxii, 2001.

[7] A. Derighetti, Convolution operators on groups, Lecture Notes of the Unione Matematica Italiana, 11. Springer, Heidelberg; UMI, Bologna, 2011. xii+171 pp. ISBN: 978-3-642-20655-9.

[8] J. Dixmier, C-Algebras, North-Holland and Publishing company, 1977.

[9] J. Fell and R. Doran, Representations of -Algebras, Locally Compact Groups,mand Banach -Algebraic Bundles, Pure and Applied Mathematics, Vol. 1, Academic Press, 1998.

[10] J. Fell and R. Doran, Representations of -Algebras, Locally Compact Groups, and Banach -Algebraic Bundles, Pure and Applied Mathematics, Vol. 2, Academic Press, 1998.
[11] G.B. Folland, A course in Abstract Harmonic Analysis, CRC press, 1995.

[12] A. Ghaani Farashahi, Continuous partial Gabor transform for semi-direct productm of locally compact groups, Bull. Malays. Math. Sci. Soc. 38 (2015), No. 2, 779-803.

[13] A. Ghaani Farashahi, A uni ed group theoretical method for the partial Fourier analysis on semi-direct product of locally compact groups, Results Math. 67 (2015), No. 1-2, 235-251.

[14] A. Ghaani Farashahi, Cyclic wave packet transform on nite Abelian groups of prime order, Int. J. Wavelets Multiresolut. Inf. Process. 12 (2014), No. 6, 1450041, 14 pp.

[15] A. Ghaani Farashahi, Generalized Weyl-Heisenberg (GWH) groups, Anal. Math. Phys. 4 (2014), No. 3, 187-197.

[16] A. Ghaani Farashahi, Convolution and involution on function spaces of homogeneous spaces, Bull. Malays. Math. Sci. Soc., (2) 36 (2013), No. 4, 1109-1122.

[17] A. Ghaani Farashahi, Abstract Non-Commutative Harmonic Analysis of Coherent State Transforms, Ferdowsi University of Mashhad (FUM) (2012) PhD Thesis.

18. A. Ghaani Farashahi and M. Mohammad-Pour, A uni ed theoretical harmonic analysis approach to the cyclic wavelet transform (CWT) for periodic signals of prime dimensions, Sahand Commun. Math. Anal. Vol. 1, No. 2, 1-17 (2014).

[19] A. Ghaani Farashahi and R.A. Kamyabi-Gol, Frames and homogeneous spaces, J. Sci. Islam. Repub. Iran., 22 (2011), No. 4, 355-361, 372.

[20] S. Helgason, Di erential Geometry, Lie Groups, and Symmetric Spaces, Academic Press, New York, San Francisco, London, 1978.

[21] E. Hewitt and K.A. Ross, Abstract Harmonic Analysis, Vol 2, 1970.

[22] E. Hewitt and K.A. Ross, Absrtact Harmonic Analysis, Vol 1, 1963.

[23] G. Hochschild, The Structure of Lie Groups, Hpolden-day, San Francisco, 1965.

[24] R.I. Jewett, Spaces with an abstract convolution of measures, Advances in Math., 18 (1975), 1-101.

[25] R.A. Kamyabi-Gol and N. Tavallaei, Wavelet transforms via generalized quasiregular representations, Appl. Comput. Harmon. Anal., 26 (2009), No. 3, 291- 300.

[26] V. Kisil, Calculus of operators: covariant transform and relative convolutions, Banach J. Math. Anal. 8 (2014), No. 2, 156-184.

[27] V. Kisil, Geometry of Mobius transformations. Elliptic, parabolic and hyperbolic actions of SL2(R), Imperial College Press, London, 2012.

[28] V. Kisil, Relative convolutions. I. Properties and applications, Adv. Math. 147 (1999), No. 1, 35-73.

[29] V. Kisil, Connection between two-sided and one-sided convolution type operators on non-commutative groups, Integral Equations Operator Theory 22 (1995), No. 3, 317-332.

[30] H. Reiter and J.D. Stegeman, Classical Harmonic Analysis, 2nd Ed, Oxford University Press, New York, 2000.