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 $f\in L^\infty(\mu)$, then $\{\gamma f:\gamma\in\widehat G\}$ is not relatively weakly compact.

ii) If $G$ is a discrete abelian group and $f\in \ell^\infty(G)\backslash C_o(G)$, then $\{\gamma f:\gamma\in E\}$ is not relatively weakly compact if $E\subset \widehat G$ 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 $\widehat\Gamma$ the interpolation (except at a finite subset depending on $U$) can be done using at most 4 point masses.

iv) A new proof that $B(G)\subset WAP(G)$ for abelian groups is given that identifies the weak limits of translates of Fourier-Stieltjes transforms.

v) Analogous results for $C_o(G)$, $A_p(G)$, and $M_p(G)$ are given.

vi) Semigroup compactifications of groups are studied, both abelian and non-abelian: the weak* closure of $\widehat G$ in $L^\infty(\mu)$, 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 $\mathcal{M}$, the weak* closure of $\rho(\Gamma)$ is studied.