# Characterizing the separability of the Gelfand space of a semisimple commutative Banach algebra

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".