Relative weak compactness of orbits in Banach spaces associated with locally compact groups

We study analogues of weak almost periodicity in Banach spaces on locally compact groups. i) If mu is a continous measure on the locally compact abelian group G and f is an element of L-infinity (mu), then {gamma f : gamma is an element of (G) over cap} is not relatively weakly compact. ii) If G is a discrete abelian group and f is an element of l(infinity) (G) \ C-o (G), then {gamma f : gamma is an element of E} is not relatively weakly compact if E subset of (G) over cap has non-empty interior. That result will follow from an existence theorem for I-o-sets, as follows. iii) Every infinite subset of a discrete abelian group Gamma contains an infinite I-o-set such that for every neighbourhood U of the identity of (Gamma) over cap the interpolation (except at a finite subset depending on U) can be done using at most 4 point masses. iv) A new proof that B (G) subset of WAP (G) for abelian groups is given that identies the weak limits of translates of Fourier-Stieltjes transforms. v) Analogous results for C-o (G), A(p) (G), and M-p (G) are given. vi) Semigroup compactifications of groups are studied, both abelian and non-abelian: the weak* closure of (G) over cap in L-infinity(mu), for abelian G; and when rho is a continuous homomorphism of the locally compact group Gamma into the unitary elements of a von Neumann algebra M, the weak* closure of rho(Gamma) is studied.