On the Jacobs-de Leeuw-Glicksberg decomposition

For any JdLG-admissible representation $\pi$ of a semigroup $S$ on a Banach space $E$, we show that the reversible part is weakly equivalent to a unitary representation on a Hilbert space that decomposes into a direct sum of finite dimensional representations, and we give an alternative characterization of the almost weakly stable part in terms of the unique invariant mean on the space of weakly almost periodic functions. In the case that $S$ is a bi-amenable measured semigroup, we characterize the almost weakly stable part using invariant means and averages along F\o lner sequences. Moreover, we give a description of the unique projection onto the reversible part whose kernel is the almost weakly stable part in terms of ultrafilters.