Let $H$ be a [Hilbert space](/page/Hilbert%20Space), let $M \subseteq \mathcal{L}(H)$ be a von Neumann algebra with identity operator $1$, and let $z \in Z(M)$ be a projection. Define $zM=\{zx:x\in M\}$ and $(1-z)M=\{(1-z)x:x\in M\}$, regarded as von Neumann algebras with units $z$ and $1-z$, respectively, and with the weak operator topology inherited from $\mathcal{L}(H)$. Equip $zM\oplus(1-z)M$ with its coordinatewise von Neumann algebra structure and coordinatewise ultraweak topology. Then the map

\begin{align*} \Phi:M \to zM \oplus (1-z)M \end{align*}

defined by

\begin{align*} \Phi(x)=(zx,(1-z)x) \end{align*}

is a normal $*$-isomorphism of von Neumann algebras.