Weak sequential properties of the multiplication operators on Banach algebras

Abstract

Let A be a Banach algebra. For f ∈ A*, we inspect the weak sequential properties of the well-known map T_f : A → A*, T_f (a) = fa, where fa ∈ A* is defined by fa(x) = f (ax) for all x ∈ A. We provide equivalent conditions for when T_f is completely continuous for every f ∈ A*, and for when T_f maps weakly precompact sets onto L-sets for every f ∈ A*. Our results have applications to the algebra of compact operators K(X) on a Banach space X.