positive matrix of elements of a von Neumann algebra

Question

Let $M$ be a von Neumann algebra. Suppose that $a,b\in M_+$, $x \in M$ and that $\begin{pmatrix} a & x \\ x^* & b \\ \end{pmatrix}$ is positive. Is it true that there exists $y \in M$ with $\|y\| \leq 1$ such that $$ x=a^{1/2}yb^{1/2} \ \ ? $$

I know that it is true for matrix algebras (Bhatia, Positive definite matrices, page 13 or Xingzhi Zhan, Matrix inequalities, page 15).

Answer 1

$\def\e{\varepsilon}$

Fix $\e>0$ and let
$$ x_\e=(a+\e I)^{-1/2} x(b+\e I)^{-1/2}. $$ Then \begin{align} \begin{bmatrix}I& x_\e\\ x_\e^*& I \end{bmatrix} &=\begin{bmatrix} (a+\e I)^{-1/2}&\!\!\!\!0\\ 0&\!\!\!\!(b+\e I)^{-1/2} \end{bmatrix} \!\! \begin{bmatrix} a+\e I&x\\ x^*& b+\e I \end{bmatrix} \!\!\begin{bmatrix} (a+\e I)^{-1/2}&0\\ \!\!\!\!0&\!\!\!\!(b+\e I)^{-1/2} \end{bmatrix}\\[0.3cm] &\geq0. \end{align} It is easy to check, from $\begin{bmatrix}I& x_\e\\ x_\e^*& I \end{bmatrix} \geq0$, that $\|x_\e\|\leq 1$. Using the ultraweak compactness of the unit ball in a von Neumann algebra, let $y$ be a cluster point of the net $\{x_\e\}$. Say $y=\lim_jx_{\e_j}$. Then $$ x=\lim_j x=\lim_j (a+\e I)^{1/2}x_{\e_j}(b+\e I)^{1/2}=a^{1/2}yb^{1/2}. $$