The algebras of bounded and essentially bounded Lebesgue measurable functions

acceptéInternational audienceLet 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, 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) 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, T dis), where T dis is the discrete topology. Moreover, ∆ is a dense subset of the spectrum M of L b (X). Finally, the hull h(I), (which is homeomorphic to M (L ∞ (X))), of the ideal of all functions in L b (X) vanishing almost everywhere on X is a nowhere dense and extremely disconnected subset of the Corona M \ ∆ of L b (X)