Maps preserving $$\varvec{A}^{\varvec{*}}\varvec{A+AA}^{\varvec{*}}$$ A ∗ A + A A ∗ on $$\varvec{C}^{\varvec{*}}$$ C ∗ -algebras

Abstract

Let \(\mathcal {A}\) be a \(C^*\)-algebra of real-rank zero and \(\mathcal {B}\) be a \(C^{*}\)-algebra with unit I. It is shown that the mapping \(\Phi : {{\mathcal {A}}}\longrightarrow {{\mathcal {B}}}\) which preserves arithmetic mean and satisfies $$\begin{aligned} \Phi (A^{*}A)=\frac{\Phi (A)^{*}\Phi (A)+\Phi (A)\Phi (A)^{*}}{2}, \end{aligned}$$ for all normal elements \(A\in \mathcal {A}\), is an \({\mathbb {R}}\)-linear continuous Jordan \(*\)-homomorphism provided that \(0\in \mathrm{Ran}\ \Phi \). Also, \(\Phi \) is the sum of a linear Jordan \(*\)-homomorphism and a conjugate-linear Jordan \(*\)-homomorphism. This result also presents an application of maps which preserve the square absolute value.