The algebras of bounded and essentially bounded Lebesgue measurable functions

The algebras of bounded and essentially bounded Lebesgue measurable functions AbstractLet 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). http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Demonstratio Mathematica de Gruyter

The algebras of bounded and essentially bounded Lebesgue measurable functions

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Publisher
De Gruyter Open
Copyright
© 2017 Raymond Mortini and Rudolf Rupp
ISSN
0420-1213
eISSN
2391-4661
D.O.I.
10.1515/dema-2017-0010
Publisher site
See Article on Publisher Site

Abstract

AbstractLet 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).

Journal

Demonstratio Mathematicade Gruyter

Published: Apr 25, 2017

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