Definition 37.8.3 (Minimal Projection).label Let $H$ be a complex Hilbert space, $A \subset B(H)$ be a von Neumann algebra, and $P \in \text{Proj}(A)$, then the following are equivalent:
- (1)
$PAP = \complex P$.
- (2)
There exists no $Q \in \text{Proj}(A)$ with $0 < Q < P$.
If the above holds, then $P$ is minimal.
Proof. (1) $\Rightarrow$ (2): Let $Q \in \text{Proj}(A)$ with $Q \le P$, then $Q = PQP$. As $PAP = \complex P$, either $PQP = 0$ or $PQP = P$.
(2) $\Rightarrow$ (1): Given that there exists no projections strictly between $0$ and $P$, the only non-zero projection in $PAP$ is $P$ itself. By Theorem 37.6.2, the linear span of projections in $PAP$ is norm-dense in $PAP$. Therefore $PAP = \complex P$.$\square$
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