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 $M$?
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 $M$?
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 $B(H)$.