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


Authors: Raymond Mortini and Rudolf Rupp
Journal: Trans. Amer. Math. Soc. 368 (2016), 3059-3073
MSC (2010): Primary 46J10; Secondary 42A75, 30H05
DOI: https://doi.org/10.1090/tran/6398
Published electronically: July 22, 2015
MathSciNet review: 3451869
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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 Suárez's result for $ {\rm AP}_\mathbb{R}={\rm AP}$. Also considered are general subalgebras of AP.


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Raymond Mortini
Affiliation: Département de Mathématiques et Institut Élie Cartan de Lorraine, UMR 7502, Université de Lorraine, Ile du Saulcy, F-57045 Metz, France
Email: Raymond.Mortini@univ-lorraine.fr

Rudolf Rupp
Affiliation: Fakultät für Angewandte Mathematik, Physik und Allgemeinwissenschaften, TH-Nürnberg, Kesslerplatz 12, D-90489 Nürnberg, Germany
Email: Rudolf.Rupp@th-nuernberg.de

DOI: https://doi.org/10.1090/tran/6398
Keywords: Almost periodic functions, Bass stable rank, topological stable rank, bounded analytic functions, polydisk-algebra, reducibility of function pairs
Received by editor(s): November 14, 2013
Received by editor(s) in revised form: January 27, 2014
Published electronically: July 22, 2015
Article copyright: © Copyright 2015 American Mathematical Society