A note on the topological group c₀

A well-known result of Ferri and Galindo asserts that the topological group c₀ is not reflexively representable and the algebra WAP(c₀) of weakly almost periodic functions does not separate points and closed subsets. However, it is unknown if the same remains true for a larger important algebra Tame(c₀) of tame functions. Respectively, it is an open question if c₀ is representable on a Rosenthal Banach space. In the present work we show that Tame(c₀) is small in a sense that the unit sphere S and 2S cannot be separated by a tame function f ∈ Tame(c₀). As an application we show that the Gromov's compactification of c₀ is not a semigroup compactification. We discuss some questions.