Let $\mathcal{M}$ and $\mathcal{N}$ be two given von Neumann algebras with faithful states $\tau$ and $\eta$, respectively. Let $\varphi:(\mathcal{N},\eta)\rightarrow(\mathcal{M},\tau)$ be a $^{*}$-embedding (i.e. an injective $C^{*}$-homomorphism) which preserves the states, i.e. $\tau(\varphi(x))=\eta(x)$ for all $x\in\mathcal{N}$. Define an $L^2$-inner product on $\mathcal{M}$ by $\langle a,b\rangle=\tau (ab^*)$ for $a,b\in\mathcal{M}$. Let $L^2(\mathcal{M},\tau)$ be the completion of $\mathcal{M}$ with respect to the corresponding $L^2$-norm. Similarly consider $L^2 (\mathcal{N},\eta)$. Then for each $x,y\in\mathcal{N}$, $\langle\varphi(x),\varphi(y)\rangle=\tau(\varphi(x)\varphi(y)^{*})=\tau(\varphi(xy^{*}))=\eta(xy^{*})=\langle x,y\rangle$ and thus $\varphi$ extends to an isometry $\varphi:L^{2}(\mathcal{N},\eta)\rightarrow L^{2}(M,\tau)$. $\mathcal{M}$ and $\mathcal{N}$ are Let $P$ be the orthogonal projection of $L^2(\mathcal{M},\tau)$ onto $\varphi (L^2(\mathcal{N},\eta))$.
Question: Is it true that $P(\varphi (x)\cdot a)=\varphi (x)\cdot P(a)$ for each $a\in\mathcal{M}$ and $x\in\mathcal{N}$?