On the continuity of intertwining operators over generalized convolution algebras

Let G be a locally compact group, Cq→ G a Fell bundle and B=L¹( G\,\, C) the algebra of integrable cross-sections associated to the bundle. We give conditions that guarantee the automatic continuity of an intertwining operator θ: X₁→ X₂, where X₁ is a Banach B-bimodule and X₂ is a weak Banach B-bimodule, in terms of the continuity ideal of θ. We provide examples of algebras where this conditions are met, both in the case of derivations and algebra morphisms. In particular, we show that, if G is infinite, finitely-generated, has polynomial growth and α is a free (partial) action of G on the compact space X, then every homomorphism of ¹_α( G,C(X)) into a Banach algebra is automatically continuous.