The Bass and topological stable ranks for algebras of almost periodic functions on the real line, II

Abstract

Let $\Lambda$ be either a subgroup of the integers $\mathbb{Z}$, a semigroup in $\mathbb{N}$, or $\Lambda =\mathbb{Q}$ (resp., ${\mathbb{Q}}^{+}$). We determine the Bass and topological stable ranks of the algebras ${\mathrm{AP}}_{\Lambda }=\{f\in \mathrm{AP}:\sigma (f)\subseteq \Lambda \}$ of almost periodic functions on the real line and with Bohr spectrum in $\Lambda$. This answers a question in the first part of this series of articles under the same heading, where it was shown that, in contrast to the present situation, these ranks were infinite for each semigroup $\Lambda$ of real numbers for which the $\mathbb{Q}$-vector space generated by $\Lambda$ had infinite dimension.