Lemma 44.3.4.label Let $H$ be a complex Hilbert space, $(M, \tau)$ be a tracial von Neumann algebra with $M \subset B(H)$, $x \in \td M$, and $\eps > 0$, then there exists $p \in \text{Proj}(M)$ such that $pH \subset D(x)$, $xp \in M$, and $\tau(p) > 1 - \eps$.
Proof, [Lemma 7.2.5, AP]. Let $x = u|x|$ be the polar decomposition of $x$ and $E: \cb_{[0, \infty)}\to B(H)$ be the resolution of the identity for $|x|$, then $E(B)(H) \subset D(x)$ for all bounded $B \in \cb_{[0, \infty)}$. By Proposition 44.3.2, $E(B) \in\text{Proj}(M)$ for all $B \in \cb_{[0, \infty)}$. Since $\tau$ is normal, $\limv{n}\tau(E([0, n])) = \tau(1_{M}) = 1$, so there exists $n \in \natp$ with $\tau(E([0, n])) > 1 - \eps$. In which case, $E([0, n])H \subset D(x)$, and $\norm{xE([0, n])}_{M}\le n$.$\square$
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