Semisimple Banach algebras generated by strongly continuous representations of locally compact abelian groups (Q1340843)

Let \(T\) be a strongly continuous representation of a locally compact Abelian group \(G\) in a Banach space. It is shown that the weak operator closure of the algebra generated by \(\widehat f(T)= \int_G f(g) T(g) dg\), \(f\in L^1(G)\), is a semisimple Banach algebra if the spectrum of \(T\) is scattered (each nontrivial subset contains an isolated point). The spectrum of the representation \(T\) is defined as \(\{\chi\in G^*: \widehat f(\chi)= 0\) whenever \(\widehat f(T)= 0\}\).