## Abstract

Let X be a set in ℝ^{n} 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 ℒ_{b}(X, ℂ) of all bounded measurable functions on X is not extremely disconnected, though totally disconnected. Let ∆ = { δ_{x} : x ∈ X} be the set of point evaluations and let g be the Gelfand topology on M. Then (∆, g) is homeomorphic to (X, Τ_{dis}),where T_{dis} is the discrete topology. Moreover, ∆ is a dense subset of the spectrum M of ℒ_{b}(X, ℂ). Finally, the hull h(I), (which is homeomorphic to M(L^{∞}(X))), of the ideal of all functions in ℒ_{b}(X, ℂ) vanishing almost everywhere on X is a nowhere dense and extremely disconnected subset of the Corona M \ ∆ of ℒ_{b}(X, ℂ).

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