SEMISIMPLE BANACH ALGEBRAS GENERATED BY STRONGLY CONTINUOUS REPRESENTATIONS OF MOORE GROUPS

We prove that if τ is a strongly continuous bounded representation of a Moore group G on a Banach space X, and if the Arveson spectrum of τ is scattered, then the closure with respect to the weak operator topology in L(X) of the algebra generated by the transforms ∫ _Gf(t) τ (t)d t with f ∈ L¹(G) is a semisimple Banach algebra.

We prove that if τ is a strongly continuous bounded representation of a Moore group G on a Banach space X, and if the Arveson spectrum of τ is scattered, then the closure with respect to the weak operator topology in L(X) of the algebra generated by the transforms ∫ _Gf(t) τ (t)d t with f ∈ L¹(G) is a semisimple Banach algebra.