Lemma 44.3.5.label Let $H$ be a complex Hilbert space, $(M, \tau)$ be a tracial von Neumann algebra on $H$, and $V \subset H$ be a vector subspace such that for every $\eps > 0$, there exists $p \in \text{Proj}(M)$ with $pH \subset V$ and $\tau(p) > 1- \eps$, then $V$ is dense in $H$.
Proof. For any $p, q \in \text{Proj}(M)$ with $pH, qH \subset V$, $(p \vee q)(H) = \ol{pH + qH}\subset \ol V$ and $\tau(p \vee q) \ge \tau(p) \vee \tau(q)$. As such, there exists an increasing sequence $\seq{p_n}\subset \text{Proj}(M)$ such that $p_{n}H \subset \ol V$ for all $n \in \natp$ and $\tau(p_{n}) \upto 1$ as $n \to \infty$.
Since $\tau$ is faithful and normal, $\tau(p_{n}) \upto 1$ implies that $\bigvee_{n \in \natp}p_{n} = 1_{M}$. By Theorem 43.3.1, $1_{M} = \sotlim_{n \to \infty}p_{n}$. As $p_{n}H \subset \ol V$ for all $n \in \natp$, $H \subset \ol V$.$\square$
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