Algebraic Maximal Numerical Range and its preservers of Triple Products on $C^{\ast }$-Algebras

Abstract

Let $\mathcal{A}$ and $\mathcal{B}$ be unital $C^{\ast }$-algebras, and let $V_0(a)=\{f(a):f\in \mathcal{S}(\mathcal{A}),f(a^{\ast }a)=\Vert a{\Vert }^2\}$ 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 $\Phi :\mathcal{A}\to \mathcal{B}$ satisfies \(V_0(Φ(a)Φ(b)Φ(c))=V_0(abc) \text{~for all~} a,b,c\in\mathcal{A},\) 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 $C^{\ast }$-algebras of real rank zero, such preservers are precisely $\ast$-isomorphisms multiplied by a central element $u\in Z(\mathcal{B})$ with $u^3=1$.

18 pages