Translation in Measure Algebras and the Correspondence to Fourier Transforms Vanishing at Infinity

Let G denote a locally compact (not necessarily abelian) group and M(G) the collection of finite regular Borel measures on G. The set M(G) is a semisimple Banach algebra with identity under convolution *. It can be identified with the dml space of CO(G), the space of continuous complex-valued functions on G that vanish at infinity, with the sup-norm. The group G has a left-invariant regular Borel measure din(x) that is unique up to a constant and is called the left Haar measure of G. Let C ‘(G) denote the space of bounded continuous functions on G. For each x e G, we define on C ‘(G) the left-translation operator by the relation