Semisimplicity of Some Class of Operator Algebras on Banach Space

Let G be a locally compact abelian group and let be a representation of G by means of isometries on a Banach space. We define W_T as the closure with respect to the weak operator topology of the set where is the Fourier transform of f ∈L¹(G) with respect to the group T. Then W_T is a commutative Banach algebra. In this paper we study semisimlicity problem for such algebras. The main result is that if the Arveson spectrum sp(T) of T is scattered (i.e. it does not contain a nonempty perfect subset) then the algebra W_T is semisimple. Some related problems are also discussed.