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](https://mathoverflow.net/questions/427447/compactness-of-the-unit-ball-of-a-banach-space-for-topologies-finer-than-the-wea), "... 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"*?



An abstract characterisation of weak* topologies
This is not possible in general. The obstruction does not come from spaces that are not dual spaces, but from the spaces that appear in several different ways as dual spaces. Indeed, the restriction of the $\sigma(Y^\*,Y)$-topology to the unit ball of $Y^\*$ determines $Y$ uniquely : by the [Krein-Smulian theorem](https://en.wikipedia.org/wiki/Krein%E2%80%93Smulian_theorem), $Y$ coincides with the subspace of elements of $(Y^\*)^\*$ whose restriction to the unit ball is $\sigma(Y^\*,Y)$-continuous. In particular, if $X$ admits several preduals that are not isometrically isomorphic, then there are several non-comparable *maximal locally convex vector space topologies making the unit ball compact*, and there is no strongest such topology.


Having a unique predual is a somewhat exceptional situation (this is the case for von Neumann algebras). The standard example of Banach space with many preduals is $\ell\_1$. It has lots of very wild preduals, including the not-so-wild spaces $C(K)$ for $K$ countable and compact. Very concretely, two non-(isometrically isomorphic) preduals of $\ell\_1$ are given by the space $c$ of converging sequences of complex numbers, and its subspace $c\_0$ of sequences converging to $0$. See [this question](https://math.stackexchange.com/questions/80727/are-these-two-banach-spaces-isometrically-isomorphic/80811#80811). See also the introduction to [this article](https://arxiv.org/abs/1101.5696) for more examples and references.


