Consider a type III von Neumann algebra $\mathcal{M}$ and an isometry $w$. How does one show that there exists a sequence of unitaries $u_n\in\mathcal{M}$ that converge strongly to $w$?
For instance, if I take a particular projection operator in the hyperfinite III$_1$ realized in the sense of Araki-Woods as an infinite tensor product of type I algebras. Can I explicitly find the sequence of unitaries that approximate it?
