Polynomial growth and functional calculus in algebras of integrable cross-sections

Let $\mathsf{G}$ be a locally compact group with polynomial growth of order $d$, a polynomial weight $\nu$ on $\mathsf{G}$ and a Fell bundle $\mathcal{C}\mathsf{G}$. We study the Banach ${}^{\ast }$-algebras $L^1(\mathsf{G}\mathrm{\mid }\mathcal{C})$ and $L^{1,\nu }(\mathsf{G}\mathrm{\mid }\mathcal{C})$, consisting of integrable cross-sections with respect to $\mathrm{d}x$ and $\nu (x)\mathrm{d}x$, respectively. By exploring new relations between the $L^p$-norms and the norm of the Hilbert $C^{\ast }$-module $L_{\mathrm{e}}^2(\mathsf{G}\mathrm{\mid }\mathcal{C})$, we are able to show that the growth of the self-adjoint, compactly supported, continuous cross-sections is polynomial. More precisely, they satisfy $$\mathrm{\parallel }{e}^{it\mathrm{\Phi }}\mathrm{\parallel }=O(\mathrm{\mid }t{\mathrm{\mid }}^n),\text{ as }\mathrm{\mid }t\mathrm{\mid }\to \mathrm{\infty },$$ for values of $n$ that only depend on $d$ and the weight $\nu$. We use this fact to develop a smooth functional calculus for such elements. We also give some sufficient conditions for these algebras to be symmetric. As consequences, we show that these algebras are locally regular, ${}^{\ast }$-regular and have the Wiener property (when symmetric), among other results. Our results are already new for convolution algebras associated with $C^{\ast }$-dynamical systems.