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

Abstract

Let $\Lambda$ be a sub-semigroup of the reals. We show that 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 $\Lambda$ are infinite whenever the algebraic dimension of the $\mathbb{Q}$-vector space generated by $\Lambda$ is infinite. This extends Suárez's result for ${\mathrm{AP}}_{\mathbb{R}}=\mathrm{AP}$. Also considered are general subalgebras of AP.

arXiv comment15 pages