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 $\textrm {AP}_\Lambda =\{f\in \textrm {AP}: \sigma (f)\subseteq \Lambda \}$ 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 $\textrm {AP}_\mathbb {R}=\textrm {AP}$. Also considered are general subalgebras of AP.