Arkiv för Matematik

  • Ark. Mat.
  • Volume 54, Number 2 (2016), 525-535.

Non-separability of the Gelfand space of measure algebras

Przemysław Ohrysko, Michał Wojciechowski, and Colin C. Graham

Full-text: Open access


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 M0(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.


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

Article information

Ark. Mat. Volume 54, Number 2 (2016), 525-535.

Received: 27 February 2016
Revised: 17 June 2016
First available in Project Euclid: 30 January 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

2016 © Institut Mittag-Leffler


Ohrysko, Przemysław; Wojciechowski, Michał; Graham, Colin C. Non-separability of the Gelfand space of measure algebras. Ark. Mat. 54 (2016), no. 2, 525--535. doi:10.1007/s11512-016-0240-8.

Export citation


  • Bailey, W. J., Brown, G. and Moran, W., Spectra of independent power measures, Math. Proc. Cambridge Philos. Soc. 72 (1972), 27–35.
  • Brown, G. and Moran, W., On orthogonality of Riesz products, Math. Proc. Cambridge Philos. Soc. 76 (1974), 173–181.
  • Graham, C. C., A Riesz product proof of the Wiener-Pitt theorem, Proc. Amer. Math. Soc. 44 (1974), 312–314.
  • Graham, C. C. and McGehee, O. C., Essays in Commutative Harmonic Analysis, Springer, New York, 1979.
  • Kaniuth, E., A Course in Commutative Banach Algebras, Springer, Berlin, 2008.
  • Rudin, W., Fourier Analysis on Groups, Wiley Classics Library, 1990.
  • Schreider, Y. A., The structure of maximal ideals in rings measure with convolution, Mat. Sb. 27 (1950), 297–318, Amer. Math. Soc. Trans. no., 81.
  • Wiener, N. and Pitt, H. R., Absolutely convergent Fourier-Stieltjes transforms, Duke Math. J. 4 (1938), 420–436.
  • Williamson, J. H., A theorem on algebras of measures on topological groups, Proc. Edinb. Math. Soc. 11 (1959), 195–206.
  • Zafran, M., On spectra of multipliers, Pac. J. Appl. Math. 47 (1973), 609–626.
  • Żelazko, W., Banach Algebras, Elsevier, Amsterdam, 1973.