$C^*$-isomorphisms, Jordan isomorphisms, and numerical range preserving maps

Let $\mathbf V = B(H)$ or $S(H)$, where $B(H)$ is the algebra of a bounded linear operator acting on the Hilbert space $H$, and $S(H)$ is the set of self-adjoint operators in $B(H)$. Denote the numerical range of $A \in B(H)$ by $W(A) = \{ (Ax,x): x \in H, (x,x) = 1\}.$ It is shown that a surjective map $\phi: \mathbf V \rightarrow \mathbf V$ satisfies

$W(AB+BA) = W(\phi(A)\phi(B)+\phi(B)\phi(A)) \qquad \hbox{ for all } A, B \in \bV$

if and only if there is a unitary operator $U \in B(H)$ such that $\phi$ has the form

$X \mapsto \pm U^*XU \quad \hbox{ or } \quad X \mapsto \pm U^*X^tU,$

where $X^t$ is the transpose of $X$ with respect to a fixed orthonormal basis. In other words, the map $\phi$ or $-\phi$ is a $C^*$-isomorphism on $B(H)$ and a Jordan isomorphism on $S(H)$. Moreover, if $H$ has finite dimension, then the surjective assumption on $\phi$ can be removed.