I'm struggling with an seemingly obvious fact, at least I've never found a proof for the statement.
If I have a von Neumann algebra and an increasing net of projections therein, then these projections should converge strongly to the projection onto the closure of the union of the ranges, right? But how can one prove this? I always fail because at some point I have to interchange two limits somehow. The closure part seems to be what throws me off here... Can anyone push me in the right direction here?
Thank you!
To fix notation, we have $M\subset B(H)$, $\{p_j\}\subset M$ the net of projections, and $p$ the projection onto $\overline{\bigcup_j p_jH}$.
We clearly have $p_j\leq p$ for all $j$. It is not hard to establish that a bounded increasing net of selfadjoint operators has a least upper bound. So $p_j\nearrow q$, and $q$ is a projection, and $q\leq p$.
If $q\ne p$, then $qH$ is a proper closed subspace of $pH$. The projection $p-q$ is orthogonal to all $p_j$; then, since $(p-q)p_j=0$ for all $j$, $(p-q)\bigcup_j p_jH=0$, and by continuity (as $p-q$ is bounded), we get $(p-q)\overline{\bigcup_jP_jH}=0$. So $(p-q)p=0$, but this is $$0=(p-q)p=p-pq=p-q.$$ Thus $q=p$.