Defining states on von Neumann algebras from filters on the projection lattices

Question

Let $M$ be a von Neumann algebra, $P(M)$ be its projection lattice, and $\mathcal{F}$ a proper filter on $P(M)$. Does there exist a state $\varphi$ (not necessarily normal) s.t. $\varphi(p) = 1$ for all $p \in \mathcal{F}$? Does this hold even more generally, say for the projection lattices of $AW^*$-algebras?

Answer 1

Filters are directed downward. Given a filter $F$, for every $p\in F$ let $\phi_p$ be a state that takes the value $1$ on $p$, then find a cluster point of the net $(\phi_p)_{p\in F}$. This will be a state that takes the value $1$ on everything in $F$. I think this works fine for $AW{}^*$-algebras.