An analytic 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 its semigroup algebra is a $(DF)$-space. This is a counterpart to a recent result of the author, which asserts that $S$ is finitely generated if and only if the Arens-Michael envelope is a Fr\'echet space.