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Consider a type III von Neumann algebra and an isometry $W$. How does one show that there exists a sequence of unitaries $U_n$ that converge strongly to $W$?

Connes-Stormer in their "Homogeneity of the state space of Factor of type III$_1$" last page say: "Since $M$ is of type of III standard arguments show that we can find a sequence of unitaries in $M$ converging strongly to $W$."

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Let $\{P_n\}$ be an increasing sequence of projections with $P_n\nearrow 1$ and $P_n\ne1$ for all $n$. Since $M$ is a type III factor, all nonzero projections are equivalent, so let $V_n$ be a partial isometry with $$V_n^*V_n=1-P_n,\ \ \ \ V_nV_n^*=1-WP_nW^*.$$ Now let $$ U_n=WP_n+V_n. $$ Note that $W^*V_n=0$ (proof below), so $U_n$ is a unitary. As $P_n\to1$, $U_n\to W$.


If $R,S$ are partial isometries and $R^*RS^*S=0$, then $RS^*=0$. So see this, multiply by $R$ on the left and by $R^*$ on the right, to get $RS^*SR^*=0$, which is $(RS^*)(RS^*)^*=0$.

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