## Arkiv för Matematik

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

### Non-separability of the Gelfand space of measure algebras

#### 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.

#### Note

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

#### Article information

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

Dates
Revised: 17 June 2016
First available in Project Euclid: 30 January 2017

https://projecteuclid.org/euclid.afm/1485802756

Digital Object Identifier
doi:10.1007/s11512-016-0240-8

Mathematical Reviews number (MathSciNet)
MR3546365

Zentralblatt MATH identifier
1368.43004

Rights

#### Citation

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. https://projecteuclid.org/euclid.afm/1485802756

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