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

Abstract

Let $G$ be a group of subexponential growth and $\mathcal{C}\stackrel{q}{\to }G$ a Fell bundle. We show that any Banach ${}^{\ast }$-algebra that sits between the associated ${\ell }^1$-algebra ${\ell }^1(G|\mathcal{C})$ and its $C^{\ast }$-envelope has the same topological stable rank and real rank as ${\ell }^1(G|\mathcal{C})$. 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.

14 pages. The last section was restructured, the rest of the paper only had minimal changes. To appear in Bull. Lond. Math. Soc

References in corpus (4)

• Topologically free actions and ideals in twisted Banach algebra crossed products
• Multiplier algebras of $L^p$-operator algebras
• Positive definiteness and Fell bundles over discrete groups
• Polynomial growth and functional calculus in algebras of integrable cross-sections

Cited by in corpus (1)

• Twisted crossed products of Banach algebras