Semisimplicity of Some Class of Operator Algebras on Banach Space

Abstract

Let G be a locally compact abelian group and let T = {T(9)}(g is an element of G) 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 {f (T) : f is an element of L-1 (G)}, where f (T) f(G) f (g)T (g) dg is the Fourier transform of f G L' (G) with respect to the group T. Then WT 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 WT is semisimple. Some related problems are also discussed.