Can there exist an uncountable sequence $R_1 \subset R_2 ...$ of von Neumann algebras all acting on the same separable Hilbert space $H$, with a "limit" algebra $R$ such that $R_\alpha \subset R$ for all countable ordinals $\alpha$; and $R$ and the $R_\alpha$'s are all either type II or III? (Here $\subset$ means proper inclusion.)
I've tried to find a contradiction between the uncountability of the $(R_\alpha)$ sequence and the countable basis for $H$, but haven't been successful and now wonder whether there is any contradiction. In the other direction I've tried to construct a concrete example of a $(R_\alpha)$ sequence using a group-von-Neumann-algebra for $R$, but I have little experience manipulating those things and didn't get anywhere. Any help greatly appreciated!