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Demonstratio Mathematica

Editor-in-Chief: Vetro, Calogero


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CiteScore 2018: 0.47
SCImago Journal Rank (SJR) 2018: 0.265
Source Normalized Impact per Paper (SNIP) 2018: 0.714

Mathematical Citation Quotient (MCQ) 2018: 0.17

ICV 2018: 121.16

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ISSN
2391-4661
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Volume 50, Issue 1

Issues

The algebras of bounded and essentially bounded Lebesgue measurable functions

Raymond Mortini
  • Corresponding author
  • Université de Lorraine, Département de Mathématiques et Institut Élie Cartan de Lorraine, UMR 7502, Ile du Saulcy, F-57045 Metz, France
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  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Rudolf Rupp
  • Fakultät für Angew. Mathematik, Physik und Allgemeinwissenschaften, TH-Nürnberg, Kesslerplatz 12, D-90489 Nürnberg, Germany
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  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
Published Online: 2017-05-04 | DOI: https://doi.org/10.1515/dema-2017-0010

Abstract

Let X be a set in ℝn with positive Lebesgue measure. It is well known that the spectrum of the algebra L(X) of (equivalence classes) of essentially bounded, complex-valued, measurable functions on X is an extremely disconnected compact Hausdorff space.We show, by elementary methods, that the spectrum M of the algebra ℒb(X, ℂ) of all bounded measurable functions on X is not extremely disconnected, though totally disconnected. Let ∆ = { δx : x ∈ X} be the set of point evaluations and let g be the Gelfand topology on M. Then (∆, g) is homeomorphic to (X, Τdis),where Tdis is the discrete topology. Moreover, ∆ is a dense subset of the spectrum M of ℒb(X, ℂ). Finally, the hull h(I), (which is homeomorphic to M(L(X))), of the ideal of all functions in ℒb(X, ℂ) vanishing almost everywhere on X is a nowhere dense and extremely disconnected subset of the Corona M \ ∆ of ℒb(X, ℂ).

Keywords: bounded Lebesgue measurable functions; essentially bounded functions; spectra and maximal ideal spaces; extremely disconnected space; totally disconnected space

MSC 2010: Primary 46J10; Secondary 54G05; 54C20

References

  • [1] Dales H. G., Banach algebras and automatic continuity, Oxford Sci. Pub., Clarendon Press, Oxford, 2000Google Scholar

  • [2] Gamelin T. W., Uniform algebras, Chelsea, New York, 1984Google Scholar

  • [3] Garnett J. B., Bounded analytic functions, Academic Press, New York, 1981Google Scholar

  • [4] Gillman L., Jerison M., Rings of continuous functions, Springer, New York, 1976Google Scholar

  • [5] Gonshor H., Remarks on the algebra of bounded functions, Math. Z., 1969, 108, 325-328Google Scholar

  • [6] Mortini R., Wick B., Spectral characteristics and stable ranks for the Sarason algebra H1 + C, Michigan Math. J., 2010, 59, 395-409Google Scholar

  • [7] Palmer T. W., Banach algebras and the general theory of *-algebras, Vol 1+2, Cambridge Univ. Press, London, 1994Google Scholar

  • [8] Pears A. R., Dimension theory of general spaces, Cambridge Univ. Press London, 1975Google Scholar

  • [9] Rudin W., Real and complex analysis, third edition, McGraw-Hill, New York, 1986Google Scholar

  • [10] Takesaki M., Theory of operator algebra I, Springer, New York, 2002Google Scholar

  • [11] Yood B., Banach algebras of bounded functions, Duke Math. J., 1949, 16, 151-163Google Scholar

About the article

Received: 2015-10-19

Accepted: 2016-02-10

Published Online: 2017-05-04

Published in Print: 2017-04-25


Citation Information: Demonstratio Mathematica, Volume 50, Issue 1, Pages 94–99, ISSN (Online) 2391-4661, DOI: https://doi.org/10.1515/dema-2017-0010.

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© 2017 Raymond Mortini and Rudolf Rupp. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License. BY-NC-ND 4.0

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