Closed densely defined operators affiliated with a von Neumann algebra $M$

Question

Let $H$ be a Hilbert space and $(M,\tau)$ be a tracial von Neumann algebra acting on $H$. Let $T$ be a closed densely defined operators affiliated with $M$. Let $T=u|T|$ be the polar decomposition of $T$. Now consider the spectral projection of $T$ relative to $[0,n]$, and denote it by $p_n$. I want to show that $p_nH \subset \text{domain}(T)$ and $\lim_n \tau(p_n)=1.$
I know that, since $T$ is a closed densely defined operators affiliated with $M$, so $u$ and the spectral projections of $|T|$ are in $M$. So $p_n \in M$ for all $n \ge 1.$ I am not able to proceed from here. Please help me to solve this. Thank you.

Answer 1

From the spectral theorem you have $$ |T|p_n=\int_{[0,n]}\lambda\,dE_{|T|}(\lambda)\leq n\,E_{|T|}([0,n])=n\,p_n. $$ Then $|T|p_n$ is bounded.

Fix $x$. By the density, there exists a sequence $\{x_k\}\subset\operatorname{domain} T$ with $x_k\to p_nx$. We have \begin{align} \|Tp_nx_k-Tp_nx_j\|&=\|Tp_n(x_k-x_j)\|=\|u|T|p_n(x_k-x_j)\|\leq\|\,|T|p_n(x_k-x_j)\|\\[0.3cm] &\leq n\|p_n(x_k-x_j)\|\leq n\,\|x_k-x_j\|. \end{align} So $\{Tp_nx_j\}_j$ is Cauchy, since $\{x_k\}_k$ is. From $T$ being closed we get that $p_nx\in\operatorname{domain} T$

The trace thing is just the fact that $p_n\nearrow1$, again from the Spectral Theorem.