Proposition 45.2.2.label Let $H$ be a complex Hilbert space, $M \subset B(H)$ be a von Neumann algebra, and $p \in \text{Proj}(M)$, then:
- (1)
For any central projection $e \in Z(pMp)$, $e = Z(e)p$.
- (2)
$Z(pMp) = Z(M)p$.
Proof, [Proposition 4.2.1, AP]. (1): Since $(p - e)Me = (p - e)pMpe = 0$, $(p - e)yey^{*} = 0$ for all $y \in U(M)$, so $(p - e)z(e) = 0$ by Proposition 43.8.4, and $e = Z(e)p$.
(2): By Theorem 43.7.2.$\square$
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