Let $H$ be a [Hilbert space](/page/Hilbert%20Space), and let $M\subseteq \mathcal{L}(H)$ be a unital von Neumann algebra, so $M$ is a unital self-adjoint algebra and $M=M''$, where
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\begin{align*}
M':=\{T\in\mathcal{L}(H):Ta=aT\text{ for every }a\in M\}
\end{align*}
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and $M'':=(M')'$. Let $\mathcal{P}(M)$ denote the set of projections in $M$, meaning the operators $p\in M$ with $p=p^*=p^2$, ordered by $p\le q$ iff $pH\subseteq qH$.
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For every family $(p_i)_{i\in I}$ in $\mathcal{P}(M)$, the meet $\bigwedge_{i\in I}p_i$ and the join $\bigvee_{i\in I}p_i$ exist in $\mathcal{P}(M)$. Moreover,