Abstract
A linear mapping \(\delta \) from a \({*}\)-algebra \(\mathcal {A}\) into a \({*}\)-\(\mathcal {A}\)-bimodule \(\mathcal {M}\) is a \({*}\)-derivable mapping at \(G\in \mathcal {A}\) if \(A\delta (B)^{*}+\delta (A)B=\delta (G)\) for each A, B in \(\mathcal {A}\) with \(AB^{*}=G\). We prove that every (continuous) \({*}\)-derivable mapping at G from a (unital \(C^{*}\)-algebra) factor von Neumann algebra into its Banach \({*}\)-bimodule is a \({*}\)-derivation if and only if G is a left separating point. A linear mapping \(\delta \) from a \({*}\)-algebra \(\mathcal {A}\) into a \({*}\)-left \(\mathcal {A}\)-module \(\mathcal {M}\) is a \({*}\)-left derivable mapping at \(G\in \mathcal {A}\) if \(A\delta (B)^{*}+B\delta (A)=\delta (G)\) for each A, B in \(\mathcal {A}\) with \(AB^{*}=G\). We prove that every continuous \({*}\)-left derivable mapping at a left separating point from a unital \(C^{*}\)-algebra or von Neumann algebra into its Banach \({*}\)-left \(\mathcal {A}\)-module is identical with zero under certain conditions.
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Acknowledgements
The authors thank the referee for his or her suggestions. This research was supported by the National Natural Science Foundation of China (Grant Nos. 11801342, 11801005, 11871021); Shaanxi Provincial Education Department (Grant No. 19JK0130).
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Communicated by Zinaida Lykova.
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An, G., He, J. & Li, J. Characterizations of \({*}\) and \({*}\)-left derivable mappings on some algebras. Ann. Funct. Anal. 11, 680–692 (2020). https://doi.org/10.1007/s43034-019-00047-8
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DOI: https://doi.org/10.1007/s43034-019-00047-8