The algebras of bounded and essentially bounded Lebesgue measurable functions

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).

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Opracowanie ze środków MNiSW w ramach umowy 812/P-DUN/2016 na działalność upowszechniającą naukę (zadania 2017).