Non-separability of the Gelfand space of measure algebras

TL;DR

This paper demonstrates that the Gelfand space of measure algebras on non-discrete abelian groups is non-separable by showing it contains uncountably many disjoint open sets, impacting spectral recovery methods.

Contribution

It establishes the non-separability of the Gelfand space for measure algebras on locally compact non-discrete abelian groups, including the ideal of measures with vanishing Fourier-Stieltjes transforms.

Findings

Gelfand space contains uncountably many disjoint open subsets.

The space's non-separability prevents spectral recovery from countable subsets.

Results apply to the ideal of measures with Fourier-Stieltjes transforms vanishing at infinity.

Abstract

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.