On the topological ranks of Banach ^*-algebras associated with groups of subexponential growth

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