equivalent projections in finite factors are unitarily equivalent

Question

Why are two Murray von Neuman equivalent projections $p$ and $q$ in a finite factor unitarily equivalent?

Answer 1

In a finite factor, you have that $p,q$ are equivalent if and only if $\tau(p)=\tau(q)$. So $p\sim q$ if and only $1-p\sim 1-q$. If $v$ is partial isometry with $v^*v=p$, $vv^*=q$, and $w$ is a partial isometry with $w^*w=1-p$, $ww^*=1-q$, then $u=v+w$ is a unitary with $upu^*=q$.