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 WT 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 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.
Published Version
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