The algebras of bounded and essentially bounded Lebesgue measurable functions

• Autorzy: Mortini R. , Rupp R.

Identyfikatory

DOI

10.1515/dema-2017-0010

• 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

EN

bounded Lebesgue measurable functions; essentially bounded functions; spectra and maximal ideal spaces; extremally disconnected space; totally disconnected space

PL

funkcja mierzalna Lebesque; funkcje ograniczone; przestrzeń ekstremalnie niespójna; przestrzeń całkowicie niespójna; przestrzeń topologiczna

• 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

autor

Rupp R.

• Fakultät für Angew. Mathematik, Physik und Allgemeinwissenschaften, TH-Nürnberg, Kesslerplatz 12, D-90489 Nürnberg, Germany

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).