intersection of two von Nuemann algebras

Question

Let $M_1$ and $M_2$ be two von Neumann algebras. $p=p^*=p^2$ and $p\in M_1\cap M_2$. Can we conclude that $pM_1p\cap pM_2p=p(M_1\cap M_2)p$?

It is trivial that $p(M_1\cap M_2)p\subset pM_1p\cap pM_2p$.

If $x\in pM_1p\cap pM_2p$, we have $x=pyp=pzp$, where $y\in M_1,z\in M_2$.

Can we prove that $y\in M_1$ or $z\in M_1$?

Answer 1

If $x\in pM_1p\cap pM_2p$, then $x\in M_1\cap M_2$, so $$ x = pxp \in p(M_1\cap M_2)p. $$