代数最大数值范围及其在$C^*$-代数上三重积的保持者
Algebraic Maximal Numerical Range and its preservers of Triple Products on $C^*$-Algebras
AI总结：
研究$C^*$-代数中三重积保持映射的性质，证明其为乘法双射并确定特定条件下保持者为$*$-同构乘以三次单位中心元素。
AI中文摘要：
设$\mathcal{A}$和$\mathcal{B}$为单位$C^*$-代数，令$V_0(a)=\{f(a): f\in\mathcal S(\mathcal A), f(a^*a)=\\|a\\|^2\}$为$a\in\mathcal{A}$的代数最大数值范围，其中$\mathcal S(\mathcal A)$为$\mathcal A$的所有态的集合。我们研究$V_0(a)$的性质，并刻画保持三重积$V_0$的满射映射。我们证明，若$\Phi:\mathcal{A}→\mathcal{B}$满足$V_0(\Phi(a)\Phi(b)\Phi(c))=V_0(abc)$对所有$a,b,c∈\mathcal{A}$，则映射$a→\Phi(1_{\mathcal{A}})^{-1}\Phi(a)$是乘法双射。此外，对于没有类型$I_1$中心分解的冯·诺依曼代数或实秩为零的素$C^*$-代数，此类保持者恰好是乘以中心元素$u∈Z(\mathcal{B})$的$*$-同构，其中$u^3=1$。
英文摘要：
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 $Φ\colon\mathcal{A}\to\mathcal{B}$ satisfies \(V_0(Φ(a)Φ(b)Φ(c))=V_0(abc) \text{~for all~} a,b,c\in\mathcal{A},\) then the map $a\mapsto Φ(1_{\mathcal{A}})^{-1}Φ(a)$ 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 $u^3=1$.
