Semisimplicity of Some Class of Operator Algebras on Banach Space

Abstract.

Let G be a locally compact abelian group and let ${\bf T}{\text{ = }}{\left\{ {T{\left( g \right)}} \right\}}_{{g \in G}}$ 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 ${\left\{ {\ifmmode\expandafter\hat\else\expandafter\^\fi{f}{\left( {\text{T}} \right)}:f \in L^{1} {\left( G \right)}} \right\}},$ where $\ifmmode\expandafter\hat\else\expandafter\^\fi{f}{\left( {\text{T}} \right)} = {\int\limits_G {f{\left( g \right)}T{\left( g \right)}dg} }$ 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.