Algebraic Maximal Numerical Range and its preservers of Triple Products on $C^*$-Algebras

Let $\mathcal{A}$ and $\mathcal{B}$ be unital $C^*$-algebras, and let $V_0(a)=\{f(a): f\in\mathcal S(\mathcal A), f(a^*a)=\|a\|^2\}$ be the algebraic maximal numerical range of $a\in\mathcal{A}$, where $\mathcal S(\mathcal A)$ is the set of all states of $\mathcal A$.

We study the properties of $V_0(a)$ and characterize surjective maps preserving $V_0$ of triple products.

We show that if $\Phi\colon\mathcal{A}\to\mathcal{B}$ satisfies \(V_0(\Phi(a)\Phi(b)\Phi(c))=V_0(abc) \text{~for all~} a,b,c\in\mathcal{A},\) then the map $a\mapsto \Phi(1_{\mathcal{A}})^{-1}\Phi(a)$ is a multiplicative bijection.

Furthermore, for von Neumann algebras without central summands of type $I_1$ or prime $C^*$-algebras of real rank zero, such preservers are precisely $*$-isomorphisms multiplied by a central element $u\in Z(\mathcal{B})$ with $u^3=1$.