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We prove that there exists uncountably many pairwise disjoint open subsets ofthe Gelfand space of the measure algebra on any locally compact non-discreteabelian group which shows that this space is not separable (in fact, we provethis assertion for the ideal $M_{0}(G)$ consisting of measures withFourier-Stieltjes transforms vanishing at infinity which is a strongerstatement). As a corollary, we obtain that the spectras of elements in thealgebra of measures cannot be recovered from the image of one countable subsetof the Gelfand space under Gelfand transform, common for all elements in thealgebra.

Non-separability of the Gelfand space of measure algebras