Banach algebras on locally compact groups

In this chapter, we shall consider the classical Banach algebras of harmonic analysis that are associated with a locally compact group G. The class of these algebras includes the group algebra \(L^1(G)\) , the measure algebra M(G), the related algebras \(L^p(G)\) (which are Banach algebras when G is compact and \(1\le p\le \infty \) ), and Beurling algebras on semigroups and locally compact groups. The product in each of these algebras is given by convolution, denoted by \(\,\star \,\) . For example, \((M(G),\,\star \,)\) is an isometric dual Banach algebra, with Banach-algebra predual \(C_{\,0}(G)\) , for each locally compact group G. The definitions and some basic properties of these algebras will be recalled in \(\S 4.1\) .