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

中文翻译：

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度量代数的Gelfand空间的不可分性

我们证明在任意局部紧凑的非离散阿贝尔群上，度量代数的Gelfand空间存在着无数的成对不相交的开放子集，这表明该空间是不可分离的（实际上，我们证明了对理想\（M_ {0}（G）\）由带有Fourier-Stieltjes变换的量度组成，并在无穷远处消失，这是一个更强的说法。作为推论，我们获得了度量代数中元素的光谱无法从Gelfand变换的Gelfand空间的一个可数子集的图像中恢复的情况，该图像对于代数中的所有元素都是通用的。