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Orthogonally a-Jensen mappings on \(C^*\)-modules

  • Ali ZamaniEmail author
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Abstract

We investigate the representation of the so-called orthogonally a-Jensen mappings acting on \(C^*\)-modules. More precisely, let \({\mathfrak {A}}\) be a unital \(C^*\)-algebra with the unit 1, let \(a \in {\mathfrak {A}}\) be fixed such that \(a, 1-a\) are invertible and let \({\mathscr {E}}, {\mathscr {F}}, {\mathscr {G}}\) be inner product \({\mathfrak {A}}\)-modules. We prove that if there exist additive mappings \(\varphi , \psi \) from \({\mathscr {F}}\) into \({\mathscr {E}}\) such that \(\big \langle \varphi (y), \psi (z)\big \rangle =0\) and \(a \big \langle \varphi (y), \varphi (z)\big \rangle a^*= (1 - a)\big \langle \psi (y), \psi (z)\big \rangle (1 - a)^*\) for all \(y, z\in {\mathscr {F}}\), then a mapping \(f: {\mathscr {E}} \rightarrow {\mathscr {G}}\) is orthogonally a-Jensen if and only if it is of the form \(f(x) = A(x) + B(x, x) +f(0)\) for \(x\in {\mathscr {K}} := \varphi ({\mathscr {F}})+\psi ({\mathscr {F}})\), where \(A: {\mathscr {E}} \rightarrow {\mathscr {G}}\) is an a-additive mapping on \({\mathscr {K}}\) and B is a symmetric a-biadditive orthogonality preserving mapping on \({\mathscr {K}}\times {\mathscr {K}}\). Some other related results are also presented.

Keywords

Orthogonality preserving mapping Orthogonally a-Jensen mapping Additive mapping Hilbert \(C^*\)-module 

Mathematics Subject Classification

46L05 47B49 39B55 

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Notes

Acknowledgements

The author would like to thank the referee for her/his valuable suggestions and comments. He would also like to thank Professor M. S. Moslehian and Professor M. Frank for their invaluable suggestions while writing this paper.

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of MathematicsFarhangian UniversityTehranIran

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