On the union of sets of semisimplicity

We introduce the notion of a set of semisimplicity, or $S_3$-set, as a set $\Lambda$ such that if $T$ is a representation of a LCA group $G$ with $Sp(T)\subset\Lambda$, then $T$ generates a semisimple Banach algebra. We prove that the union of $S_3$-sets is a $S_3$-set, provided their intersection is countable. In particular, the union of a countable set and a Helson $S$-set is a $S_3$-set.