Let $H$ be a Hilbert space and $\mathcal{X}\subseteq \mathcal{B}(H)$ be an operator space. We denote $d(T,\mathcal{X})=\inf_{X\in\mathcal{X}}||T-X||,\,\,T\in\mathcal{B}(H)$$d(T,\mathcal{X})=\inf_{X\in\mathcal{X}}\|T-X\|,\,\,T\in\mathcal{B}(H)$ and $r_{\mathcal{X}}(T)=\sup\left\{|\langle{T \xi,\eta\rangle}|: ||\xi||=||\eta||=1,\,\,\langle{X \xi,\eta\rangle}=0,\,\forall\,X\in\mathcal{X}\right\}.$$r_{\mathcal{X}}(T)=\sup\left\{|\langle{T \xi,\eta\rangle}|: \|\xi\|=\|\eta\|=1,\,\,\langle{X \xi,\eta\rangle}=0,\,\forall\,X\in\mathcal{X}\right\}.$
We say that $\mathcal{X}$ is hyperreflexive if there exists $k>0$ such that $d(T,\mathcal{X})\leq k \,r_{\mathcal{X}}(T)$ for every $T\in\mathcal{B}(H).$
Fact: Let $\mathcal{A}$ and $\mathcal{B}$ be von Neumann algebras and $\pi\colon \mathcal{A}\to \mathcal{B}$ be a $w^*$-continuous $*$-homomorphism and suppose that $\mathcal{A}$ is hyperreflexive. The kernel $Ker(\pi)$$\ker(\pi)$ of the map $\pi$ is an ideal of $\mathcal{A}$ and thus it is of the form $P \mathcal{A} P$ for some central projection $P\in\mathcal{A}.$ By known results it follows thar $Ker(\pi)$$\ker(\pi)$ is hyperreflexive.
Search: I have the expectation that an analogus result may hold for kernels of specific derivations, i.e. let $\mathcal{A}$ be a hyperreflexive von Neumann algebra acting on the Hilbert space $H$ and $P\in\mathcal{B}(H)$ be a projection. Consider the inner derivation $\delta(X)=XP-PX,\,X\in\mathcal{A}$ of $\mathcal{A}$ into $\mathcal{B}(H).$ Can we prove that $Ker(\delta)=\mathcal{A}\cap \left\{P\right\}^{\prime}$$\ker(\delta)=\mathcal{A}\cap \left\{P\right\}^{\prime}$ is also hyperreflexive?
Maybe it is a hard question but interesting too.
