On the topological ranks of Banach ${}^{\ast }$-algebras associated with groups of subexponential growth

Let $G$ be a group of subexponential growth and $\mathcal{C}G$ a Fell bundle. We show that any Banach ${}^{\ast }$-algebra that sits between the associated ${\mathrm{\ell }}^1$-algebra ${\mathrm{\ell }}^1(G\mathrm{\mid }\mathcal{C})$ and its $C^{\ast }$-envelope has the same topological stable rank and real rank as ${\mathrm{\ell }}^1(G\mathrm{\mid }\mathcal{C})$. We apply this result to compute the topological stable rank and real rank of various classes of symmetrized twisted $L^p$-crossed products and show that some twisted $L^p$-crossed products have topological stable rank 1. Our results are new even in the case of (untwisted) group algebras.