Semisimple Banach Algebras Generated by Strongly Continuous Representations of Locally Compact Abelian Groups

We prove that if T is a strongly based continuous bounded representation of a locally compact abelian group G on a Banach Space X, and if the spectrum of T is countable, then the Banach algebra generated by f̂ (T) = ∫Gf(g) T(g) dg, f ∈ L1(G), is semisimple.