Banach Algebra of Bounded Complex Radon Measures on Homogeneous Space
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RESEARCH PAPER
Banach Algebra of Bounded Complex Radon Measures on Homogeneous Space T. Derikvand1 • R. A. Kamyabi-Gol2 • M. Janfada3 Received: 28 January 2020 / Accepted: 11 July 2020 Ó Shiraz University 2020
Abstract Let H be a compact subgroup of a locally compact group G. In this paper we define a convolution on M(G/H) , the space of all bounded complex Radon measures on the homogeneous space G/H. Then we prove that the measure space M(G/H) with the newly well-defined convolution is a non-unital Banach algebra that possesses an approximate identity. Finally, it is shown that this Banach algebra is not involutive and also L1 ðG=HÞ with the new convolution is a two-sided ideal of it. Keywords Complex Radon measure Homogeneous spaces Convolution Banach algebra Mathematics Subject Classification Primary 43A15 Secondary 43A85
1 Introduction and Preliminaries Let G be a locally compact group, and let M(G) be the space of all bounded complex Radon measures on it. The convolution of any two measures l1 and l2 in M(G) is defined by Z Z l1 l2 ðf Þ ¼ f ðxyÞdl1 ðxÞdl2 ðyÞ; ðf 2 Cc ðGÞÞ: G
G
ð1:1Þ It is well-known that ðMðGÞ; Þ is a unital Banach algebra, it is called the measure algebra and plays a key role in harmonic analysis, (See, e.g., Deitmar and Echterhoof 2009 and Fell and Doran 1988). Now let H be a compact & T. Derikvand [email protected] R. A. Kamyabi-Gol [email protected] M. Janfada [email protected] 1
Department of Mathematics, Marvdasht Branch, Islamic Azad University, Marvdasht, Iran
2
Department of Pure Mathematics, Centre of Excellence in Analysis on Algebraic Structures (CEAAS), Ferdowsi University of Mashhad, P.O. Box 1159, Mashhad 91775, Iran
3
Department of Pure Mathematics, Ferdowsi University of Mashhad, P.O. Box 1159, Mashhad 91775, Iran
subgroup of locally compact group G, and the homogeneous space G/H is a Hausdorff space on which G acts transitively by left. We should clear that H is not normal subgroup necessarily, so G/H does not possess a group structure but it will be a locally compact Hausdorff space. Let M(G/ H) denote the space of all bounded complex Radon measures on G/H. Compared with the measure algebra M(G) , it is worthwhile to investigate the existence of a convolution on M(G/H) which makes it into a Banach algebra. Farashahi (2018) studied this problem in the case that H is a closed subgroup of a compact group G; However, the theory of homogeneous spaces in which H is a compact subgroup of a locally compact group G has many applications in physics and engineering. For example, if the Euclidian group E(2) acts transitively on R2 , then the isotropy subgroup of origin is the orthogonal group O(2). In that sequel, the homogeneous space E(2)/O(2) provides definition of X-ray transform that is used in many areas such as radio astronomy, positron emission tomography, crystallography, etc (See, e.g., Deans 1983, Ch. 1 and Helgason 2011). Now, we review some preliminaries and results in homogeneous spaces theory. Let dy be the
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