×

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.

MSC:

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

Citations:

Zbl 1347.46037
PDF BibTeX XML Cite
Full Text: DOI Euclid

References:

[1] M. Artin, Algebra, Prentice Hall, Englewood Cliffs, NJ, 1991.
[2] C. Badea, The stable rank of topological algebras and a problem of R. G. Swan, J. Funct. Anal. 160 (1998), no. 1, 42-78. · Zbl 0921.46053
[3] A. S. Besicovitch, Almost Periodic Functions, Dover, New York, 1955. · Zbl 0065.07102
[4] H. Bohr, Zur theorie der fast periodischen funktionen, I: Eine verallgemeinerung der theorie der fourierreihen, Acta Math. 45 (1925), no. 1, 29-127. · JFM 50.0196.01
[5] H. Bohr, Zur Theorie der Fastperiodischen Funktionen, II: Zusammenhang der fastperiodischen Funktionen mit Funktionen von unendlich vielen Variabeln; gleichmässige Approximation durch trigonometrische Summen, Acta Math. 46 (1925), nos. 1-2, 101-214. · JFM 51.0212.02
[6] A. Böttcher, Y. I. Karlovich, and I. M. Spitkovsky, Convolution Operators and Factorization of Almost Periodic Matrix Functions, Oper. Theory Adv. Appl. 131, Birkhäuser, Basel, 2002.
[7] G. Corach and F. D. Suárez, Stable rank in holomorphic function algebras, Illinois J. Math. 29 (1985), no. 4, 627-639. · Zbl 0606.46034
[8] G. Corach and F. D. Suárez, On the stable range of uniform algebras and \(H^{\infty}\), Proc. Amer. Math. Soc. 98 (1986), no. 4, 607-610. · Zbl 0625.46060
[9] G. Corach and F. D. Suárez, Dense morphisms in commutative Banach algebras, Trans. Amer. Math. Soc. 304 (1987), no. 2, 537-547. · Zbl 0633.46054
[10] C. Corduneanu, Almost Periodic Functions, 2nd English ed., Chelsea, New York, 1989. · Zbl 0672.42008
[11] P. W. Jones, D. Marshall, and T. Wolff, Stable rank of the disc algebra, Proc. Amer. Math. Soc. 96 (1986), no. 4, 603-604. · Zbl 0626.46043
[12] K. Mikkola and A. Sasane, Bass and topological stable ranks of complex and real algebras of measures, functions and sequences, Complex Anal. Oper. Theory 4 (2010), no. 2, 401-448. · Zbl 1211.46056
[13] R. Mortini, The Faá di Bruno formula revisited, Elem. Math. 68 (2013), no. 1, 33-38. · Zbl 1272.05003
[14] R. Mortini and R. Rupp, The Bass and topological stable ranks for algebras of almost periodic functions on the real line, Trans. Amer. Math. Soc. 368 (2016), no. 5, 3059-3073. · Zbl 1347.46037
[15] F. D. Suárez, The algebra of almost periodic functions has infinite topological stable rank, Proc. Amer. Math. Soc. 124 (1996), no. 3, 873-876. · Zbl 0842.46034
[16] S. Treil, The stable rank of \(H^{\infty}\) equals \(1\), J. Funct. Anal. 109 (1992), no. 1, 130-154. · Zbl 0784.46037
[17] L. N. Vaserstein, The stable range of rings and the dimension of topological spaces (in Russian), Funktsional. Anal. i Prilozhen. 5 (1971), no. 2, 17-27; English translation in Funct. Anal. Appl. 5 (1971), 102-110.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.