Let A be a Banach algebra
with a bounded left approximate identity. We denote, respectively, by wap(A) and
AA∗ the subspaces of A∗ consisting of the weakly almost periodic functionals and the
functional of the form af, where af(x) = f(ax). The main results can be summarized
as follows:
wap(A) ⊆ AA∗ and the equality wap(A) = AA∗ holds if A is a right ideal
in its second dual.
If A is Arens regular and a right ideal in its second dual then A∗ has the
RNP (Radon-Nikodym Property).
If A is a right ideal in its second dual then A is Arens regular and has the
Dunford-Pettis property iff A∗ has the RNP and the Schur property.
As applications we give very short (and probably new) proofs of several well-known
results about topological groups, group algebras and their weakly almost periodic
functionals. Our applications also include some new proofs and results about the
projective tensor products of Banach spaces and algebras.
