Lemma 44.3.6.label Let $H$ be a complex Hilbert space, $(M, \tau)$ be a tracial von Neumann algebra on $H$, $x, y \in \td M$, and $p, q \in \text{Proj}(M)$ such that
- (a)
$pH \subset D(x)$ and $qH \subset D(y)$.
- (b)
$yq \in M$.
then there exists $s \in \text{Proj}(M)$ such that:
- (1)
$sH \subset D(xy)$ and $yrH \subset pH$.
- (2)
$\tau(s) \ge 1 - \tau(1_{M} - p) - \tau(1_{M} - q)$.
Proof, [Extracted from Theorem 7.2.8, AP]. Let $r \in M$ be the projection onto $\ker((1_{M} - p)yq)$ and $s = r \wedge q$.
(1): For any $\xi \in \ker((1_{M} - p)yq)$, $yq\xi \in \ker(1_{M} - p) = pH \subset D(x)$, so $rH \subset D(xyq)$, and $(q \wedge r)H \subset D(xy)$. By definition, $yrH \subset \ker(1_{M} - p) = pH$.
(2): Let $(1 - p)yq = v |(1_{M} - p)yq|$ be the polar decomposition of $(1_{M} - p)yq$, then $v \in M$ is a partial isometry with initial space $\ol{|(1_M - p)yq|H}$ and final space $\ol{(1_M - p)yqH}= \ker((1_{M} - p)yq)^{\perp}$. Thus $v^{*}v = (1 - r)$, $vv^{*} \le 1_{M} - p$, and $1_{M} - r \prec 1_{M} - p$. Therefore
$\square$
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