When does a partial isometry have unitary extension?

Question

$u$ extends $v$ means that $u^*v=v^*v$.

If $v^2=0$ or $v^*v=vv^*$ it certainly does. If $v^*v<vv^*=1$ it certainly does not.

Does finite Von Neumann algebra sound like to have the property that every partial isometry has a unitary extension?

Answer 1

Yes. In a finite von Neumann algebra, if $v^*v=p\sim q$ then $1-p\sim 1-q$. If $w$ is a partial isometry for the second equivalence, then $v+w$ is a unitary.