Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



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
Published electronically: July 22, 2015
MathSciNet review: 3451869
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

  • [1] A. S. Besicovitch, Almost periodic functions, Dover Publications, Inc., New York, 1955. MR 0068029 (16,817a)
  • [2] Harald Bohr, Zur Theorie der fast periodischen Funktionen, I. Eine Verallgemeinerung der Theorie der Fourierreihen. Acta Math. 45 (1925), no. 1, 29-127 (German). MR 1555192,
  • [3] Harald 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), no. 1-2, 101-214 (German). MR 1555201,
  • [4] A. Böttcher, On the corona theorem for almost periodic functions, Integral Equations Operator Theory 33 (1999), no. 3, 253-272. MR 1671480 (2000b:46086),
  • [5] Albrecht Böttcher, Yuri I. Karlovich, and Ilya M. Spitkovsky, Convolution operators and factorization of almost periodic matrix functions, Operator Theory: Advances and Applications, vol. 131, Birkhäuser Verlag, Basel, 2002. MR 1898405 (2003c:47047)
  • [6] Gustavo Corach and Fernando Daniel Suárez, Extension problems and stable rank in commutative Banach algebras, Topology Appl. 21 (1985), no. 1, 1-8. MR 808718 (87a:46086),
  • [7] Gustavo Corach and Fernando Daniel Suárez, Dense morphisms in commutative Banach algebras, Trans. Amer. Math. Soc. 304 (1987), no. 2, 537-547. MR 911084 (88k:46064),
  • [8] C. Corduneanu, Almost periodic functions, Interscience Publishers [John Wiley & Sons], New York-London-Sydney, 1968. With the collaboration of N. Gheorghiu and V. Barbu; Translated from the Romanian by Gitta Bernstein and Eugene Tomer; Interscience Tracts in Pure and Applied Mathematics, No. 22. MR 0481915 (58 #2006)
  • [9] Ryszard Engelking and Karol Sieklucki, Topology: a geometric approach, Sigma Series in Pure Mathematics, vol. 4, Heldermann Verlag, Berlin, 1992. Translated from the Polish original by Adam Ostaszewski. MR 1199810 (94d:54001)
  • [10] Ka-Lam Kueh, A note on Kronecker's approximation theorem, Amer. Math. Monthly 93 (1986), no. 7, 555-556. MR 856296 (87i:11085),
  • [11] Kalle Mikkola and Amol 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. MR 2652531 (2012b:43003),
  • [12] Raymond Mortini, An example of a subalgebra of $ H^\infty $ on the unit disk whose stable rank is not finite, Studia Math. 103 (1992), no. 3, 275-281. MR 1202011 (95c:46077)
  • [13] Raymond Mortini and Rudolf Rupp, A note on some uniform algebra generated by smooth functions in the plane, J. Funct. Spaces Appl., 2012, Art. ID 905650, 15 pp.. MR 2881767,
  • [14] Raymond Mortini and Rudolf Rupp, Real-symmetric extensions of invertible tuples of multivariable continuous functions, Complex Anal. Oper. Theory 7 (2013), no. 1, 135-162. MR 3010793,
  • [15] Marc A. Rieffel, Dimension and stable rank in the $ K$-theory of $ C^{\ast }$-algebras, Proc. London Math. Soc. (3) 46 (1983), no. 2, 301-333. MR 693043 (84g:46085),
  • [16] Fernando Daniel Suárez, The algebra of almost periodic functions has infinite topological stable rank, Proc. Amer. Math. Soc. 124 (1996), no. 3, 873-876. MR 1307566 (96f:46098),
  • [17] L. Vasershtein, Stable rank of rings and dimensionality of topological spaces, Funct. Anal. Appl. 5 (1971), 102-110; translation from Funkts. Anal. Prilozh. 5 (1971), no. 2, 17-27. MR 0284476 (44 #1701)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 46J10, 42A75, 30H05

Retrieve articles in all journals with MSC (2010): 46J10, 42A75, 30H05

Additional Information

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

Rudolf Rupp
Affiliation: Fakultät für Angewandte Mathematik, Physik und Allgemeinwissenschaften, TH-Nürnberg, Kesslerplatz 12, D-90489 Nürnberg, Germany

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

American Mathematical Society