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http://yadda.icm.edu.pl:80/baztech/element/bwmeta1.element.baztech-828e389b-101a-440e-8fcc-87d5ce00d5d2

Czasopismo

Demonstratio Mathematica

Tytuł artykułu

The algebras of bounded and essentially bounded Lebesgue measurable functions

Autorzy Mortini, R.  Rupp, R. 
Treść / Zawartość http://www.degruyter.com/view/j/dema
Warianty tytułu
Języki publikacji EN
Abstrakty
EN 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 Hausdorspace. 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, Tdis), 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).
Słowa kluczowe
PL funkcja mierzalna Lebesque   funkcje ograniczone   przestrzeń ekstremalnie niespójna   przestrzeń całkowicie niespójna   przestrzeń topologiczna  
EN bounded Lebesgue measurable functions   essentially bounded functions   spectra and maximal ideal spaces   extremally disconnected space   totally disconnected space  
Wydawca De Gruyter
Czasopismo Demonstratio Mathematica
Rocznik 2017
Tom Vol. 50, nr 1
Strony 94--99
Opis fizyczny Bibliogr. 11 poz.
Twórcy
autor Mortini, R.
  • Université de Lorraine, Département de Mathématiques et Institut Élie Cartan de Lorraine, UMR 7502, Ile du Saulcy, F-57045 Metz, France, raymond.mortini@univ-lorraine.fr
autor Rupp, R.
  • Fakultät für Angew. Mathematik, Physik und Allgemeinwissenschaften, TH-Nürnberg, Kesslerplatz 12, D-90489 Nürnberg, Germany, Rudolf.Rupp@th-nuernberg.de
Bibliografia
[1] Dales H. G., Banach algebras and automatic continuity, Oxford Sci. Pub., Clarendon Press, Oxford, 2000
[2] Gamelin T. W., Uniform algebras, Chelsea, New York, 1984
[3] Garnett J. B., Bounded analytic functions, Academic Press, New York, 1981
[4] Gillman L., Jerison M., Rings of continuous functions, Springer, New York, 1976
[5] Gonshor H., Remarks on the algebra of bounded functions, Math. Z., 1969, 108, 325-328
[6] Mortini R., Wick B., Spectral characteristics and stable ranks for the Sarason algebra H∞ + C, Michigan Math. J., 2010, 59, 395-409
[7] Palmer T. W., Banach algebras and the general theory of *-algebras, Vol 1+2, Cambridge Univ. Press, London, 1994
[8] Pears A. R., Dimension theory of general spaces, Cambridge Univ. Press London, 1975
[9] Rudin W., Real and complex analysis, third edition, McGraw-Hill, New York, 1986
[10] Takesaki M., Theory of operator algebra I, Springer, New York, 2002
[11] Yood B., Banach algebras of bounded functions, Duke Math. J., 1949, 16, 151-163
Uwagi
PL Opracowanie ze środków MNiSW w ramach umowy 812/P-DUN/2016 na działalność upowszechniającą naukę (zadania 2017).
Kolekcja BazTech
Identyfikator YADDA bwmeta1.element.baztech-828e389b-101a-440e-8fcc-87d5ce00d5d2
Identyfikatory
DOI 10.1515/dema-2017-0010