Lemma 37.7.10.label Let $H$ be a complex Hilbert space, $A \subset B(H)$ be a von Neumann algebra, and $\seqi{P}, \seqi{Q}\subset \text{Proj}(A)$ such that:
- (a)
$\seqi{P}$ is mutually orthogonal.
- (b)
$\seqi{Q}$ is mutually orthogonal.
- (c)
For each $i \in I$, $P_{i} \sim Q_{i}$.
then $\sum_{i \in I}P_{i} \sim \sum_{i \in I}Q_{i}$.
Proof. For each $i \in I$, let $V_{i} \in A$ such that $P_{i} = V_{i}^{*}V_{i}$ and $Q_{i} = V_{i}V_{i}^{*}$, then $V_{i}$ is a partial isometry with initial space $P_{i}(H)$ and final space $Q_{i}(H)$. As $\seqi{P}$ is mutually orthogonal and $\seqi{Q}$ is mutually orthogonal, the sum $\sum_{i \in I}V_{i}$ converges in strong operator topology to an operator $V$, where
\[V^{*}V = \sum_{i, j \in I}V_{i}^{*}V_{j} = \sum_{i \in I}V_{i}^{*}V_{i} = \sum_{i \in I}P_{i}\]
and
\[VV^{*} = \sum_{i, j \in I}V_{i}V_{j}^{*} = \sum_{i \in I}V_{i}V_{i}^{*} = \sum_{i \in I}Q_{i}\]
$\square$
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