Let $H$ be a [Hilbert space](/page/Hilbert%20Space) and let $M\subseteq \mathcal{L}(H)$ be a von Neumann algebra. On the set $\mathcal{P}(M)$ of projections in $M$, Murray-von Neumann equivalence $\sim$ is an [equivalence relation](/page/Equivalence%20Relation). Murray-von Neumann subequivalence $\precsim$ is reflexive and transitive. Moreover, for all $p,q\in\mathcal{P}(M)$, if $p\sim q$, then $p\precsim q$ and $q\precsim p$.