# Isometries in type III von Neumann algebra

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 $$W$$."

## 1 Answer

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 $$(RS^*)(RS^*)^*=0$$.