Let $M$ be a von Neumann algebra in $B(H)$. Let $p$ and $q$ 
be projections in $M$. Assume that they are equivalent in $B(H)$, i.e there is a partial isometry $u$ in $B(H)$ with $p=uu^\*$ and $q=u^\*u$. 


Question: Are $p$ and $q$ equivalent in $M$?



Equivalent projections in von Neumann algebras
The answer to your question is negative.  

Take two inequivalent projections in a type $II\_1$ factor $R\subset B(H)$.  

These projections have infinite dimensional range in $H$.
They are therefore equivalent in $B(H)$.


