Theorem 37.7.14 (The Comparability Theorem).label Let $H$ be a complex Hilbert space, $A \subset B(H)$ be a von Neumann algebra, and $P, Q \in \text{Proj}(A)$, then there exists a central projection $R$ such that $RP \preceq RQ$ and $(I - R)Q \preceq (I - R)P$.
Proof, [Theorem 25.4, Zhu93]. By Zorn’s lemma, there exists maximal families $\seqi{P}, \seqi{Q}\subset \text{Proj}(A)$ such that:
- (i)
$\seqi{P}$ is mutually orthogonal.
- (ii)
$\seqi{Q}$ is mutually orthogonal.
- (iii)
For each $i \in I$, $P_{i} \sim Q_{i}$.
- (iv)
For each $i \in I$, $P_{i} \le P$ and $Q_{i} \le Q$.
Let $P_{0} = \sum_{i \in I}P_{i}$ and $Q_{0} = \sum_{i \in I}Q_{i}$, then $P_{0} \sim Q_{0}$ by Lemma 37.7.10. By maximality, there exists no non-zero $P', Q' \in \text{Proj}(A)$ such that $P' \le P - P_{0}$, $Q' \le Q - Q_{0}$, and $P' \sim Q'$. By Proposition 37.7.5, $Z(P - P_{0}) Z(Q - Q_{0}) = 0$.
Let $R = Z(Q - Q_{0})$, then $Q - Q_{0} \le R$ and $P - P_{0} \le (I - R)$, so $(P - P_{0})R = 0$ and $(Q - Q_{0})R = Q - Q_{0}$. By Lemma 37.7.6,
and
$\square$
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