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Characterizing linear mappings through zero products or zero Jordan products

Abstract

Let \({\mathcal {A}}\) be a \(*\)-algebra and \({{\mathcal {M}}}\) be a \(*\)-\({\mathcal {A}}\)-bimodule. We study the local properties of \(*\)-derivations and \(*\)-Jordan derivations from \({\mathcal {A}}\) into \({{\mathcal {M}}}\) under the following orthogonality conditions on elements in \({\mathcal {A}}\): \(ab^*=0\), \(ab^*+b^*a=0\) and \(ab^*=b^*a=0\). We characterize the mappings on zero product determined algebras and zero Jordan product determined algebras. Moreover, we give some applications on \(C^*\)-algebras, group algebras, matrix algebras, algebras of locally measurable operators and von Neumann algebras.

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Acknowledgements

The authors thank the referee for his or her suggestions. This research was partly supported by the National Natural Science Foundation of China (Grant Nos. 11801342, 11801005, 11871021); Natural Science Foundation of Shaanxi Province (Grant No. 2020JQ-693); Scientific research plan projects of Shannxi Education Department (Grant No.  9JK0130).

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An, G., He, J. & Li, J. Characterizing linear mappings through zero products or zero Jordan products. Period Math Hung 84, 270–286 (2022). https://doi.org/10.1007/s10998-021-00404-y

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  • DOI: https://doi.org/10.1007/s10998-021-00404-y

Keywords

  • \(*\)-(Jordan) derivation
  • \(*\)-(Jordan) left derivation
  • Zero (Jordan) product determined algebra
  • \(C^*\)-algebra
  • von Neumann algebra

Mathematics Subject Classification

  • 15A86
  • 47A07
  • 47B47
  • 47B49