The Bass and topological stable ranks for algebras of almost periodic functions on the real line. II. (English) Zbl 1430.46036

Summary: 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 \(\operatorname{AP}_\Lambda=\{f\in\operatorname{AP}:\sigma(f)\subseteq\Lambda\}\) 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 [R. Mortini and R. Rupp, Trans. Am. Math. Soc. 368, No. 5, 3059–3073 (2016; Zbl 1347.46037)], 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.


46J10 Banach algebras of continuous functions, function algebras
42A75 Classical almost periodic functions, mean periodic functions
30H05 Spaces of bounded analytic functions of one complex variable


Zbl 1347.46037
Full Text: DOI Euclid


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