A remark on cyclic projections in von Neumann algebras

Question

Let $M$ be a von Neumann algebra in $B(H)$. Let $x$ be a positive element in $M$ and $\zeta\in \ker x^{\perp}$.

Q. True or false: $\zeta\in \overline{Mx\zeta}$ ?!

Answer 1

If $x$ is invertible, then the statement is clearly true. If $x$ is left-pseudo-invertible (i.e. there is an $y \in M$ with $yx=r(x^*)$), the the statement is true as well. In general, however, operators are not even pseudo-invertible. We can, however, approximate the pseudo inverse of positive elements. Thus, we first split $x$ using polar decomposition in a positive operator and a partial isometry. Then we see that the approximate pseudo inverse does the trick.

1. By Polar decomposition $x = v \sqrt{x^*x}$ for some $v \in M$ with $vv^*=r(x)$ and $v^*v=r(x^*)$. See e.g. [2, p.15].

2. With the Spectral Theorem we can find a sequence $a_i$ of positive operators that commute with $x^*x$ such that $\mathrm{uwlim}_i\ x^*x a_i = r(x^*x)$. See e.g. [1, proof Prop. 6]. (Recall $y_i \to y$ ultraweakly if for all normal states $\varphi$ we have $\varphi(y_i) \rightarrow \varphi(y)$.)

3. Consequently $ x^* v \sqrt{a_i}^* \sqrt{a_i}v^* x = \sqrt{x^*x} v^*v a_i v^*v \sqrt{x^*x} = \sqrt{x^*x} a_i \sqrt{x^*x} = x^*x a_i$ which converges ultra weakly to $r(x^*x)=r(x^*)$.

4. Thus $\sqrt{a_i}v^* x$ converges ultrastrongly to $r(x^*)$. (Recall $y_i \rightarrow y$ ultrastrongly if for all normal states $\varphi$ we have $\sqrt{\varphi(y_i^*y_i)} \rightarrow \sqrt{\varphi(y^*y)}$.)

5. Hence $\sqrt{a_i}v^* x - r(x^*)$ converges ultrastrongly to 0. As $\langle (\ )\zeta, \zeta\rangle$ is a normal state, we find $\bigl< \bigl(\sqrt{a_i}v^* x - r(x^*)\bigr)^*\bigl(\sqrt{a_i}v^* x - r(x^*)\bigr) \zeta, \zeta \bigr>^{1/2} \rightarrow 0$, which says $\|(\sqrt{a_i}v^* x - r(x^*))\zeta\| \rightarrow 0$.

6. Thus $r(x^*) \zeta$ converges to $\sqrt{a_i}v^* x\zeta$ and so $r(x^*)\zeta \in \overline{Mx\zeta}$.
7. Finally, note $\zeta \in (\ker x)^\perp = \overline{\mathrm{ran}\ x^*}$ and so $\zeta = r(x^*)\zeta \in \overline{Mx\zeta}$, as desired.

[1] "A universal property for sequential measurement" Westerbaan & Westerbaan (2016, Journal of Mathematical Physics) http://scitation.aip.org/content/aip/journal/jmp/57/9/10.1063/1.4961526 [2] "Lectures on von Neumann algebras" Topping