Lemma 44.4.2.label Let $(M, \tau)$ be a tracial von Neumann algebra in standard form and $x \in \td M_{+}$, then for any non-decreasing sequence $\seq{p_n}\subset \text{Proj}(M)$ with $p_{n}H \subset D(x)$ and $\tau(p_{n}) \upto 1$ as $n \to \infty$,
\[\tau(x) = \limv{n}\dpn{x p_n\wh 1_M, p_n\wh 1_M}{L^2(M, \tau)}= \limv{n}\normn{x^{1/2}p_n \wh 1_M}_{L^2(M, \tau)}^{2}\]
Proof. Since $\tau$ is faithful and normal, $\tau(p_{n}) \upto 1$ and Theorem 43.3.1 imply that $p_{n} \upto 1_{M}$ in the strong operator topology.
($\le$): Let $E: \cb_{[0, \infty)}\to M$ be the resolution of the identity for $x$. For each $\alpha > 0$,
\begin{align*}\liminf_{n \to \infty}\normn{x^{1/2}p_n \wh 1_M}_{L^2(M, \tau)}^{2}&= \liminf_{n \to \infty}\int_{[0, \infty)}\lambda E_{p_n \wh 1_M, p_n \wh 1_M}(d\lambda) \\&\ge \liminf_{n \to \infty}\dpn{(x \wedge \alpha)p_n \wh 1_M, p_n \wh 1_M}{L^2(M, \tau)}\\&= \dpn{(x \wedge \alpha)\wh 1_M, \wh 1_M}{L^2(M, \tau)}\\&= \int_{[0, \infty)}(\lambda \wedge \alpha) E_{\wh 1_M, \wh 1_M}(d\lambda)\end{align*}
By the Monotone Convergence Theorem,
\begin{align*}\liminf_{n \to \infty}\normn{x^{1/2}p_n \wh 1_M}_{L^2(M, \tau)}^{2}&\ge \sup_{\alpha > 0}\int_{[0, \infty)}(\lambda \wedge \alpha) E_{\wh 1_M, \wh 1_M}(d\lambda) \\&\ge \int_{[0, \infty)}\lambda E_{\wh 1_M, \wh 1_M}(d\lambda) = \tau(x)\end{align*}
($\ge$): Assume without loss of generality that $\tau(x) < \infty$. In which case, $\int_{[0, \infty)}|\lambda^{1/2}|^{2} E_{\wh 1_M, \wh 1_M}(d\lambda) < \infty$, and $\wh 1_{M} \in D(x^{1/2})$. Let $J: L^{2}(M, \tau) \to L^{2}(M, \tau)$ be the canonical conjugation operator, then for each $p \in \text{Proj}(M)$ with $pH \subset D(x)$, $JpJ\hat 1_{M} = Jp \hat 1_{M} = p \hat 1_{M}$. By Theorem 44.2.8, $JpJ \in M'$, so
\begin{align*}\normn{x^{1/2}p \wh 1_M}_{L^2(M, \tau)}^{2}&= \dpn{x^{1/2}JpJ \wh 1_M, x^{1/2} JpJ \wh 1_M}{L^2(M, \tau)}\\&= \dpn{JpJx^{1/2} \wh 1_M, JpJx^{1/2} \wh 1_M}{L^2(M, \tau)}\\&= \normn{JpJx^{1/2} \wh 1_M}_{L^2(M, \tau)}^{2} \le \normn{x^{1/2} \wh 1_M}_{L^2(M, \tau)}^{2} = \tau(x)\end{align*}
and $\limsup_{n \to \infty}\normn{x^{1/2}p_n \wh 1_M}_{L^2(M, \tau)}^{2} \le \tau(x)$.$\square$
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