Homogeneous Banach spaces as Banach convolution modules over $M(G)$

Abstract

We develop an elementary approach to convolution and Fourier analysis on a locally compact Abelian group (G), based on bounded measures and bounded uniform partitions of unity. In earlier work, the author introduced convolution and the Fourier--Stieltjes transform on the Banach space (M(G)) of bounded measures, viewed as linear functionals, in a direct Euclidean setting. The present paper constructs arbitrarily fine bounded uniform partitions of unity on general locally compact Abelian groups. The construction is designed to avoid structure theory and does not presuppose Haar measure or Lebesgue integration. It is then used to establish a natural convolution-module structure of (M(G)) on a broad class of homogeneous Banach spaces on (G). This class includes (L^p(G)), for (1\leq p<\infty), the Fourier--Stieltjes algebra, and, in particular, Segal algebras. After introducing Haar measure, we identify (L^1(G)) with the closure in (M(G)) of the measure-embedded space (C_c(G)). We prove that the homogeneous Banach spaces under consideration are essential (L^1(G))-modules. Consequently, standard approximate identities act in the expected manner and converge strongly to the identity operator. The method follows the spirit of Hans Reiter: it avoids the customary reliance on LCA-group structure theory and on vector-valued integration arguments based on duality. It is intended as a foundation for a subsequent elementary treatment of the extended Fourier transform in the Banach Gelfand triple generated by a Segal algebra.