Non-separability of the Gelfand space of measure algebras

Przemysław Ohrysko Institute of Mathematics, Polish Academy of Sciences Michał Wojciechowski Institute of Mathematics, Polish Academy of Sciences Colin C. Graham Department of Mathematics, University of British Columbia

Representation Theory Rings and Algebras mathscidoc:1701.30003

Arkiv for Matematik, 1-11, 2016.2
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.
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@inproceedings{przemysław2016non-separability,
  title={Non-separability of the Gelfand space of measure algebras},
  author={Przemysław Ohrysko, Michał Wojciechowski, and Colin C. Graham},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203408516126829},
  booktitle={Arkiv for Matematik},
  pages={1-11},
  year={2016},
}
Przemysław Ohrysko, Michał Wojciechowski, and Colin C. Graham. Non-separability of the Gelfand space of measure algebras. 2016. In Arkiv for Matematik. pp.1-11. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203408516126829.
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