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Characterizing linear mappings through zero products or zero Jordan products

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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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Authors and Affiliations

  1. Department of Mathematics, Shaanxi University of Science and Technology, Xi’an, 710021, China

    Guangyu An

  2. Department of Mathematics, Anhui Polytechnic University, Wuhu, 241000, China

    Jun He

  3. Department of Mathematics, East China University of Science and Technology, Shanghai, 200237, China

    Jiankui Li

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  1. Guangyu An
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  2. Jun He
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  3. Jiankui Li
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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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