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

Abstract: Let ${\sf G}$ be a locally compact group with polynomial growth of order $d$, a polynomial weight $\nu$ on ${\sf G}$ and a Fell bundle $\mathscr C\overset{q}{\to}{\sf G}$. We study the Banach $^*$-algebras $L^1({\sf G}\,\vert\,\mathscr C)$ and $L^{1,\nu}({\sf G}\,\vert\,\mathscr C)$, consisting of integrable cross-sections with respect to ${\rm d} x$ and $\nu(x){\rm d} x$, respectively. By exploring new relations between the $L^p$-norms and the norm of the Hilbert $C^*$-module $L^2_{\rm e}({\sf G}\,\vert\,\mathscr C)$, we are able to show that the growth of the self-adjoint, compactly supported, continuous cross-sections is polynomial. More precisely, they satisfy $$|{e^{it\Phi}}|=O(|t|^n),\quad\text{ as }|t|\to\infty,$$ 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, $^*$-regular and have the Wiener property (when symmetric), among other results. Our results are already new for convolution algebras associated with $C^*$-dynamical systems.