2
$\begingroup$

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 $p\wedge q=0$?

  • 0
    What does $\underbrace{p_i\searrow p}_{(p_i)_i \textrm{mon. decr.}}$ precisely mean?2012-12-13
  • 0
    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).2012-12-13
  • 0
    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?2012-12-13
  • 1
    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.2012-12-13
  • 0
    Hello? Would be nice to address the question…2012-12-13

1 Answers 1