On the continuity of intertwining operators over generalized convolution algebras

Let $\mathsf{G}$ be a locally compact group, $\mathcal{C}\mathsf{G}$ a Fell bundle and $\mathfrak{B}=L^1(\mathsf{G}\mathrm{\mid }\mathcal{C})$ the algebra of integrable cross-sections associated to the bundle. We give conditions that guarantee the automatic continuity of an intertwining operator $\theta :{\mathcal{X}}_1\to {\mathcal{X}}_2$, where ${\mathcal{X}}_1$ is a Banach $\mathfrak{B}$-bimodule and ${\mathcal{X}}_2$ is a weak Banach $\mathfrak{B}$-bimodule, in terms of the continuity ideal of $\theta$. 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 $\mathsf{G}$ is infinite, finitely-generated, has polynomial growth and $\alpha$ is a free (partial) action of $\mathsf{G}$ on the compact space $X$, then every homomorphism of ${\mathrm{\ell }}_{\alpha }^1(\mathsf{G},C(X))$ into a Banach algebra is automatically continuous.