Lemma 37.8.13.label Let $H$ be a complex Hilbert space, $A \subset B(H)$ be a factor, $T \in A$, $\seqi{P}\subset \text{Proj}(A)$ be non-zero minimal projections such that $I = \sum_{i \in I}P_{i}$, and $\bracsn{V_{i, j}}_{i, j \in I}\subset A$ such that $P_{i} = V_{i, j}^{*}V_{i, j}$ and $P_{j} = V_{i, j}V_{i, j}^{*}$ for all $i, j \in I$, then there exists $\bracsn{\mu_{i, j}}_{i, j \in I}\subset \complex$ such that $T = \sum_{i, j \in I}\mu_{i, j}V_{i, j}$.
Proof. Let $i, j \in I$, then since $P_{i}$ and $P_{j}$ are minimal,
If $P_{i}TP_{j} \ne 0$, then $P_{j}T^{*}P_{i}TP_{j}$ and $P_{i}TP_{j}T^{*}P_{i}$ are positive, and there exists $\lambda > 0$ such that $\lambda P_{i}TP_{j}$ is a partial isometry with initial space $P_{j}(H)$ and final space $P_{i}(H)$. By Lemma 37.8.4, there exists $\mu_{i, j}\in \complex$ such that $P_{i}TP_{j} = \mu_{i, j}V_{j, i}$. Therefore
$\square$
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