Is there a way of endowing the unit ball $B_X$ of a Banach space $X$ (we may assume that $X$ is an AL space, if that helps) with a topology $\tau$, so that $\tau=\sigma(Y^*,Y)$ (the weak* topology) if $X=Y^*$, for some Banach space $Y$? In other words, is it possible to equip the unit ball of a Banach space $X$ with a topology that corresponds with the weak* topology if $X$ is a dual space, but is well-defined if $X$ is not a dual space?

As Nik Weaver observes in this post, "... on any dual Banach space there is no locally convex vector space topology strictly stronger than the weak* topology that makes the unit ball compact." So, given an arbitrary Banach space (or AL space) $X$, could one endow it with something like the "strongest locally convex vector space topology making the unit ball compact"?