Homomorphisms from L1(G) for G ∈ [FIA] − ∪ [Moore]

Let G be a locally compact group, let B be a Banach algebra, and let θ : L1(G) → B be a homomorphism. We investigate the continuity properties of θ when G ∈ [FIA] − ∪ [Moore]. If G ∈ [Moore], then the closure of J(θ), the continuity ideal of θ, has finite codimension in L1(G). Moreover, there is an ideal J of L1(G) such that J− has finite codimension and the restriction of θ to J is continuous. Using this result, we obtain a splitting of θ into a continuous part θcont and a singular part θsing, as for homomorphisms from C*-algebras. In case G ∈ [FIA ] − ∩ [Moore], we have J− = J(θ)−, and θsing is accessible to further structural analysis. For G ∈ [FIA] − ∪ [Moore], we show that if there is n ∈ N such that G has an infinite number of inequivalent, n-dimensional, irreducible unitary representations, then there is a discontinuous homomorphism from L1(G) into a Banach algebra. Finally, we prove that if G ∈ [FIA] − ∩ [Moore], then every homomorphism from L1(G) onto a dense subalgebra of a semisimple Banach algebra is continuous.