Weak sequential properties of the multiplication operators on Banach algebras

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