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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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What does $\underbrace{p_i\searrow p}_{(p_i)_i \textrm{mon. decr.}}$ precisely mean? –  no identity Dec 13 '12 at 11:14
    
That $(p_i)_i$ is a monotone decreasing net (of projections) and her infimum is $p$. Convergence holds also in the sense of SOT (equiv. WOT). –  vNALG Dec 13 '12 at 11:20
    
If it is equivalent to SOT convergence why did you mention monotonity? I think $\underbrace{p_i\searrow p}_{(p_i)_i \textrm{mon. decr.}}$ is equivalent to the SOT convergence + monotonicity. I'm I right? –  no identity Dec 13 '12 at 11:25
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It is SOT and monotone convergence. Yes. Are the conditions of the problem clear now? Note, it could, for all I know, turn out to be sufficient, to only rely on the information $p_i \longrightarrow p$, $q_i \longrightarrow q$ and $p_i\wedge q_i=0$ to arrive at $p\wedge q=0$. I only mention monotonicity, because this extra nicety occurs in my situation, and it might help. –  vNALG Dec 13 '12 at 11:30
    
Hello? Would be nice to address the question… –  vNALG Dec 13 '12 at 12:37

1 Answer 1

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 $i$.

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