A classical theorem of Jacobs, de Leeuw and Glicksberg shows that a representation of a group on a reflexive Banach space may be decomposed into a returning subspace and a weakly mixing subspace. This may be realized as arising from the idempotent in the weakly almost periodic compactification of the group associated with the minimal idempotent, or, equivalently with the largest precompact topology on the group. I wish to exhibit generalizations of Jacobs-de Leeuw-Glicksberg decompositions and the associated projections. I wish to specialize these to Eberlein-compactications, i.e. those which may be realized as weak operator closed semi-groups of contractions on Hilbert spaces, and indicate some applications to the structure theory of the Fourier-Stieltjes algebras of locally compact groups.
