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 \(\mathscr {B}\).
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Communicated by Daniel Aron Alpay.
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Bourhim, A., Mabrouk, M. Maps Preserving the Numerical Radius Distance Between \(C^*\)-Algebras. Complex Anal. Oper. Theory 13, 2371–2380 (2019). https://doi.org/10.1007/s11785-019-00894-2
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DOI: https://doi.org/10.1007/s11785-019-00894-2