Weak Sequential Properties of the Multiplication Operators on Banach Algebras

Abstract

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