Question: Let $N\subseteq B(H)$ be a finite von Neumann algebra with a faithful normal trace $\tau$ and let $n\in N$. Let $B$ be a von Neumann subalgebra of $N$. Let $\mathbb{E}_B: N\rightarrow B$ be the conditional expectation of N onto B with respect to $\tau$. Let $e_{(0,\infty)}(\mathbb{E}_B(nn^*))$ be the spectral projection of $\mathbb{E}_B(nn^*)$ on $(0,\infty)$. Is it true that the operator $e_{(0,\infty)}(\mathbb{E}_B(nn^*))(\mathbb{E}_B(nn^*))^{-1/2}n$ is densly defined and closed on $H$?
Note that a linear operator $T:D(T)\subseteq H\rightarrow H$ is densely defined if the domain of $T$, i.e. $D(T)$ is dense in $H$. The linear operator $T$ is called closed if it is densely defined and $\{\xi_n\}_{n\in\mathbb{N}}\subset D(T),\,(\xi_n,T\xi_n)\rightarrow (x_0,y_0)\implies x_0\in D(T),\,Tx_0=y_0.$
Thanks in advance for any help or suggestion.
P.S. I asked the above question in here, but did not receive any positive response.
