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

Abstract

We study analogues of weak almost periodicity in Banach spaces on locally compact groups. i) If μ \mu is a continous measure on the locally compact abelian group G G and f ∈ L ∞ ( μ ) f\in L^\infty (\mu ) , then { γ f : γ ∈ G ^ } \{\gamma f:\gamma \in \widehat G\} is not relatively weakly compact. ii) If G G is a discrete abelian group and f ∈ ℓ ∞ ( G ) ∖ C o ( G ) f\in \ell ^\infty (G)\backslash C_o(G) , then { γ f : γ ∈ E } \{\gamma f:\gamma \in E\} is not relatively weakly compact if E ⊂ G ^ E\subset \widehat G has non-empty interior. That result will follow from an existence theorem for I o I_o -sets, as follows. iii) Every infinite subset of a discrete abelian group Γ \Gamma contains an infinite I o I_o -set such that for every neighbourhood U U of the identity of Γ ^ \widehat \Gamma the interpolation (except at a finite subset depending on U U ) can be done using at most 4 point masses. iv) A new proof that B ( G ) ⊂ W A P ( G ) 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 ) C_o(G) , A p ( G ) A_p(G) , and M p ( G ) M_p(G) are given. vi) Semigroup compactifications of groups are studied, both abelian and non-abelian: the weak* closure of G ^ \widehat G in L ∞ ( μ ) L^\infty (\mu ) , for abelian G G ; and when ρ \rho is a continuous homomorphism of the locally compact group Γ \Gamma into the unitary elements of a von Neumann algebra M \mathcal {M} , the weak* closure of ρ ( Γ ) \rho (\Gamma ) is studied.