  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 $X$.
