Definition 37.7.5 (Murray-von Neumann Equivalent).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 the following are equivalent:
- (1)
There exists $V \in A$ such that $P = V^{*}V$ and $Q = VV^{*}$.
- (2)
There exists a partial isometry $V \in A$ from $P(H)$ to $Q(H)$.
If the above holds, then $P$ and $Q$ are Murrey-von Neumann equivalent, denoted $P \sim Q$. The relation $\sim$ is an equivalence relation on $\text{Proj}(A)$.
Proof. (1) $\Rightarrow$ (2): Let $V \in A$ with $P = V^{*}V$ and $Q = VV^{*}$. By Proposition 38.2.3, $V$ is a partial isometry with initial space $\ker(P)^{\perp}$, and $V^{*}$ is a partial isometry with initial space $\ker(Q)^{\perp}$. Therefore $V$ is a partial isometry from $P(H)$ to $Q(H)$.
(2) $\Rightarrow$ (1): By Proposition 38.2.3, $P = V^{*}V$ is a projection onto $\ker(V)^{\perp}$, and $Q = VV^{*}$ is a projection onto $\ker(V^{*})^{\perp} = V(H)$.$\square$
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