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Non-separability of the Gelfand space of measure algebras

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Arkiv för Matematik

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.

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Author information

Authors and Affiliations

  1. Institute of Mathematics, Polish Academy of Sciences, PL-00-956, Warszawa, Poland

    Przemysław Ohrysko & Michał Wojciechowski

  2. Department of Mathematics, University of British Columbia, P.O. Box 2031, Haines Junction, YT Y0B 1L0, Canada

    Colin C. Graham

Authors
  1. Przemysław Ohrysko
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  2. Michał Wojciechowski
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  3. Colin C. Graham
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Corresponding author

Correspondence to Przemysław Ohrysko.

Additional information

The research of P. Ohrysko has been supported by National Science Centre, Poland grant no. 2014/15/N/ST1/02124.

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Ohrysko, P., Wojciechowski, M. & Graham, C.C. Non-separability of the Gelfand space of measure algebras. Ark Mat 54, 525–535 (2016). https://doi.org/10.1007/s11512-016-0240-8

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  • DOI: https://doi.org/10.1007/s11512-016-0240-8

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