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

Abstract:

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 \[ W(AB+BA) = W(\phi (A)\phi (B)+\phi (B)\phi (A)) \qquad \text {for all $A$, $B \in \mathbf {V}$} \] if and only if there is a unitary operator $U \in B(H)$ such that $\phi$ has the form \[ X \mapsto \pm U^*XU \quad \mathrm {or} \quad X \mapsto \pm U^*X^tU, \] 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.