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The algebras of bounded and essentially bounded Lebesgue measurable functions

Mortini R., Rupp R.

Demonstratio Mathematica

2017

Vol. 50, nr 1

94--99

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).

Some algebraic properties of the Wiener-Laplace algebra

Mortini R., Sasane A.

Journal of Applied Analysis

2010

Vol. 16, nr 1

79-94

We denote by W+(C+) the set of all complex-valued functions defined in the losed right half plane C+ := {s ∈ C | Re(s) ≥ 0} that differ from the Laplace transform of functions from L1 (O, ∞) by a constant. Equipped with pointwise operations, W+(C+) forms a ring. It is known that W + (C+) is a pre-Bézout ring. The following properties are shown for W+(C+): W+(C+) is not a GCD domain, that is, there exist functions F1, F2 in W+(C+) that do not possess a greatest common divisor in W + (C+). W+(C+) is not coherent, and in fact, we give an example of two principal ideals whose intersection is not finitely generated. We will also observe that W+(C+) is a Hermite ring, by showing that the maximal ideal space of W+(C+), equipped with the Gelfand topology, is contractible.