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The question actually is limited to a very specific case.

The following takes place in a fixed Hilbert space.

Let $(p_i)_i, (q_i)_i, p, q$ projections (resp. nets of projections) so that $\underbrace{p_i\searrow p}_{(p_i)_i \textrm{mon. decr.}}$ and $\underbrace{q_i \longrightarrow q}_{_{SOT}}$.

Question. If $(\forall{i})\ p_i \wedge q_i=0$, does it also hold, that $p\wedge q=0$?

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    Hello? Would be nice to address the question… – 2012-12-13

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No. The example is in $H=\mathbb C^2$, and the nets are sequences indexed by the positive integers. Let $p$ be the projection onto the span of $(1,0)$, and let $p_i=p$ for all $i$. Let $q_i$ be the projection onto the span of $(1+1/i,1/i)$. Then $q_i\to q=p=p\land q\neq 0$, while $p_i\land q_i =0$ for all $i$.