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

Abstract

Let $G$ be a locally compact group with polynomial growth of order d, a polynomial weight ν on $G$ and a Fell bundle $C\stackrel{q}{\to }G$. We study the Banach ^⁎-algebras $L^1\left(G\right|C)$ and $L^{1,\nu }\left(G\right|C)$, consisting of integrable cross-sections with respect to $dx$ and $\nu \left(x\right)dx$, respectively. By exploring new relations between the $L^p$-norms and the norm of the Hilbert $C^{⁎}$-module $L_e^2\left(G\right|C)$, we are able to show that the growth of the self-adjoint, compactly supported, continuous cross-sections is polynomial. More precisely, they satisfy$\Vert e^{it\Phi }\Vert =O\left(\right|t{|}^n),\text{as}|t|\to \infty ,$ for values of n that only depend on d and the weight ν. 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.