On the continuity of intertwining operators over generalized convolution algebras

Let ${\sf G}$ be a locally compact group, $\mathscr C\overset{q}{\to}{\sf G}$ a Fell bundle and $\mathfrak B=L^1({\sf G}\,\vert\,\mathscr C)$ 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 ${\sf G}$ is infinite, finitely-generated, has polynomial growth and $\alpha$ is a free (partial) action of ${\sf G}$ on the compact space $X$, then every homomorphism of $\ell^1_\alpha({\sf G},C(X))$ into a Banach algebra is automatically continuous.