Let $\mathcal{R}_1 \subseteq \mathcal{R}_2 \subseteq \cdots $ be an infinite strictly increasing sequence of von Neumann factors (we can assume type III factors if that's easier) acting on the same Hilbert space.
Let $\mathcal{R}$ be the limit von Neumann algebra $( \bigcup_n \mathcal{R}_n)''$.
Is every projection in $\mathcal{R}$ necessarily the strong-operator limit of an increasing sequence of projections in $\bigcup_n \mathcal{R}_n$?
I believe that $\mathcal{R}$ is just the strong-operator closure of $\bigcup_n \mathcal{R}_n$ [although we can't say, as I originally did, that the latter is a C*-algebra -- see comment below].
Assuming that, I believe the existence of polar decompositions in $\bigcup_n \mathcal{R}_n$ entails that every self-adjoint operator in $\mathcal{R}$ is a strong-operator limit of a sequence $T_1, T_2, \ldots$ of self-adjoint operators in $\bigcup_n \mathcal{R}_n$. [Edit: I now think this holds only if the algebras are acting on a separable $H$. Go ahead and assume that.]
But this would not by itself give a positive answer to the question since in general there are sequences of self-adjoint operators that are not projections, whose limits are projections.
Any suggestions appreciated -- I am not seeing how to proceed. It might help to note that if the answer were positive, then every projection in $\mathcal{R}$ would also be the limit of a decreasing sequence of projections in $\bigcup_n \mathcal{R}_n$, since if the increasing $T_1, T_2, \ldots$ converge to $(1-T)$, then the decreasing $(1-T_1), (1-T_2), \ldots$ converge to $T$.