An analytical criterion for the local finiteness of a countable semigroup

We prove that a countable semigroup $S$ is locally finite if and only if the Arens–Michael envelope of the semigroup algebra of $S$ is a $(DF)$-space. This is a counterpart to the author's recent result asserting that $S$ is finitely generated if and only if the Arens–Michael envelope is a Fréchet space.