Weak* continuity of positive parts

Given a finite set $\cal F$ of functions in $\ell_1$, choose a function $z_{\cal F}$ in $\ell_\infty$ s.t. $\langle z_{\cal F}, x \rangle =0$ for all $x$ in $\cal F$ s.t. $z_{\cal F}$ has at least one positive coordinate, and normalized s.t. $\langle z^+_{\cal F}, u \rangle = 1$, where $u := \sum_{n=1}^\infty 2^{-n} e_n$ and $e_n$ is the $n$th unit vector in $\ell_1$. The net $(z_{\cal F})$ converges weak$^*$ to zero when the finite subsets of $\ell_1$ are directed by inclusion.