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Maps preserving \(\varvec{A}^{\varvec{*}}\varvec{A+AA}^{\varvec{*}}\) on \(\varvec{C}^{\varvec{*}}\)-algebras

  • Ali TaghaviEmail author
Article
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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.

Keywords

\(C^*\)-algebra \({\mathbb {C}}\)-linear \(\mathbb C\)-antilinear homomorphism linear preserver problem real rank zero 

2010 Mathematics Subject Classification

47B48 46L10 

Notes

Acknowledgements

The author is thankful to the anonymous reviewer(s) for their careful reading of this paper and for their valuable suggestions in rewriting the paper in the present form.

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Copyright information

© Indian Academy of Sciences 2019

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of Mathematical SciencesUniversity of MazandaranBabolsarIran

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