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

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. © 2006 Elsevier Inc. All rights reserved