orthogonal projections and equivalent projections in a von Neumann algebra

Question

Let $M$ be a von Neumanna algebra.

Does there exist relationship between equivalent projections in $M$ and orthogonal projections in $M$?

If $p$ and $q$ are equivalent non-trivial projections, is it possible that $pq=0$?

Answer 1

There aren't really any relationships in a general von Neumann algebra. Here are some type I examples: take $M = M_2$, the algebra of 2x2 matrices. The projections $$ 1 \oplus 0 = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix} \text{ and } \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix}$$ are equivalent (two projections in $M_2$ are equivalent if and only if they have the same rank) and orthogonal. But $$ 1 \oplus 0 \text{ and } \frac{1}{2}\begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix} $$ are equivalent and not orthogonal.

If we go to $M_3$, the 3x3 matrices, its easy to find two projections which are orthogonal and not equivalent (just take the projection onto a plane and the projection onto the normal).