Semisimple Banach algebras generated by compact groups of operators on a Banach space

Abstract

We prove that if τ is a strongly continuous representation of a compact group G on a Banach space X, then the weakly closed Banach algebra generated by the Fourier transforms G τ (t) dμ(t) with μ ∈ M(G) is a semisimple Banach algebra.