Abdellatif Bourhim, Mohamed Mabrouk
Abstract: Let $\mathscr {A}$ and $\mathscr {B}$ be unital $C^*$-algebras, and let v(a) be the numerical radius of any element $a\in \mathscr {A}$. We show that if a map T from $\mathscr {A}$ onto $\mathscr {B}$ satisfies $v(T(a)-T(b))=v(a-b),$ $(a, b\in \mathscr {A})$, then $T(\mathbf{1 })-T(0)$ is a unitary central element in $\mathscr {B}$. This shows that the characterization of Bai, Hou and Xu for the numerical radius distance preservers on $C^*$-algebras can be obtained without the extra condition that $T(\mathbf{1 })-T(0)$ is in the center of $\mathscr {B}$.
Journal: Complex Analysis and Operator Theory
DOI: 10.1007/s11785-019-00894-2