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 ${\rm AP}_\Lambda=\{f\in {\rm AP}: \sigma(f)\subseteq \Lambda\}$ 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 Su\'arez's result for ${\rm AP}_\mathbb R={\rm AP}$. Also considered are general subalgebras of AP.