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We study analogues of weak almost periodicity in Banach spaces on locally compact groups. i) If μ, is a continous measure on the locally compact abelian group G and ∫∈L ∞ ∈ L∞(μ), then {γ∫: γ ∈Ğ} is not relatively weakly compact. ii) If G is a discrete abelian group and ∫∈ l ∞ (G)\C o (G), then {γ∫: γ ∈E) is not relatively weakly compact if E C G has non-empty interior. That result will follow from an existence theorem for Io-sets, as follows. iii) Every infinite subset of a discrete abelian group F contains an infinite Io-set such that for every neighbourhood U of the identity of Γ 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) C WAP(G) for abelian groups is given that identifies the weak limits of translates of Fourier-Stieltjes transforms. v) Analogous results for C o (G), Ap(G), and Mp(G) are given. vi) Semigroup compactifications of groups are studied, both abelian and non-abelian: the weak* closure of G in L ∞ (p), for abelian G; and when p is a continuous homomorphism of the locally compact group Γ into the unitary elements of a von Neumann algebra M, the weak* closure of ρ(Γ) is studied.

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