What question did this study set out to answer?

To investigate the topological stable rank and real rank of Banach algebras associated with groups of subexponential growth.

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February 5, 2026Open Access

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

Key Points

To investigate the topological stable rank and real rank of Banach algebras associated with groups of subexponential growth.
Analyzed the structure of Banach algebras between the associated algebra and its envelope.
Computed ranks for various classes of symmetrized twisted crossed products.
Applied results to demonstrate properties of twisted crossed products.
Identified that certain twisted crossed products have a topological stable rank of 1.
Established that the ranks of Banach algebras match those of the original groups in specific cases.

Abstract

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

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