Problem. Is the separability of the Gelfand space of a semi-simple commutative Banach algebra $A$ equivalent to the existence of a countable family $\{\varphi_n\}_{n\in\omega}$ of multiplicative linear functionals on $A$ such that for each $a\in A$ its spectrum coincides with the closure of the union $\bigcup_{n\in\omega}\varphi_n(a)$?

(The problem was posed 09.08.2015 by Michal Wojciechowski on page 14 of Volume 0 of the Lviv Scottish Book).

The prize for solution: A dinner in "Szkocka".


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