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Strong 2-skew Commutativity Preserving Maps on Prime Rings with Involution

  • Jinchuan HouEmail author
  • Wei Wang
Article

Abstract

Let \({\mathcal {R}}\) be a unital prime \(*\)-ring containing a nontrivial symmetric idempotent. For \(A,B\in {\mathcal {R}}\), the skew commutator and 2-skew commutator are defined, respectively, by \({}_*[A,B]=AB-BA^*\) and \({}_*[A,B]_2= {{}_*[A, {{}_*[A,B]}]}\). Let \(\Phi :{\mathcal {R}} \rightarrow {\mathcal {R}}\) be a surjective map. We show that (1) \(\Phi \) satisfies \({}_*[\Phi (A),\Phi (B)] = {{}_*[A,B] }\) for all \(A, B\in {\mathcal {R}}\) if and only if there exists \(\lambda \in \{-1,1\}\) such that \(\Phi (A)=\lambda A\) for all \(A\in {\mathcal {R}}\); (2) \(\Phi \) satisfies \({}_*[\Phi (A),\Phi (B)]_2= {{}_*[A,B]_2}\) for all \(A, B\in {\mathcal {R}}\) if and only if there exists \(\lambda \in {\mathcal {C}}_S \) with \(\lambda ^{3} = I\) such that \(\Phi (A) = \lambda A \) for all \(A \in {\mathcal {R}}\), where I is the unit of \({\mathcal {R}}\) and \({\mathcal {C}}_S \) is the symmetric extend centroid of \({\mathcal {R}}\). This is then applied to prime \(\hbox {C}^*\)-algebras, factor von Neumann algebras and indefinite self-adjoint standard operator algebras to get a complete invariant for the identity map and to symmetric standard operator algebras as well as matrix algebras.

Keywords

Prime rings with involution \(\hbox {C}^*\)-algebras 2-skew commutators Preservers Identity map 

Mathematics Subject Classification

47B49 47B47 16W10 

Notes

Acknowledgements

The authors would like to thank the referees for their careful reading of the original manuscript and for their many helpful comments to improve the paper.

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Copyright information

© Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2017

Authors and Affiliations

  1. 1.Department of MathematicsTaiyuan University of TechnologyTaiyuanPeople’s Republic of China

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