Weak sequential properties of the multiplication operators on Banach algebras

Abstract

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