Let $V$ and $V'$ be Abelian Von Neumann algebras of projections on some Hilbert space $H$, and let $V_{1}$ and $V_{2}$ be minimal sub-algebras of $V$ and $V'$ generated by projections $P_{1} \in V$ and $P_{2} \in V'$, respectively. Suppose that $P_{1}$ and $P_{2}$ do not commute. Is it always possible to find some non-zero $P_{3}$ such that $P_{3}$ is orthogonal to both $P_{1}$ and $P_{2}$?
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$\begingroup$ For a counter-example take $H = \mathbb C^2$. $\endgroup$– Jesse PetersonCommented Apr 25, 2014 at 14:36
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$\begingroup$ Sorry, I forgot to specify that I'm assuming that $H$ has dimension greater than 2. That does that change things? $\endgroup$– King KongCommented May 13, 2014 at 8:34
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