Abstract
Let R be a ring, M be a R-bimodule and m, n be two fixed nonnegative integers with m + n ≠ 0. An additive mapping δ from R into M is called an (m, n)-Jordan derivation if (m + n)δ(A2) = 2mAδ(A) + 2nδ(A)A for every A in R. In this paper, we prove that every (m, n)-Jordan derivation with m ≠ n from a C* -algebra into its Banach bimodule is zero. An additive mapping δ from R into M is called a (m, n)-Jordan derivable mapping at W in R if (m + n)δ(AB + BA) = 2mδ(A)B + 2mδ(B)A + 2nAδ(B) + 2nBδ(A) for each A and B in R with AB = BA = W. We prove that if M is a unital A-bimodule with a left (right) separating set generated algebraically by all idempotents in A, then every (m, n)-Jordan derivable mapping at zero from A into M is identical with zero. We also show that if A and B are two unital algebras, M is a faithful unital (A, B)-bimodule and \(\mathcal{U}=\begin{bmatrix}\mathcal{A} & \mathcal{M} \\\mathcal{N} & \mathcal{B} \end{bmatrix}\) is a generalized matrix algebra, then every (m, n)-Jordan derivable mapping at zero from U into itself is equal to zero.
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Supported by the National Natural Science Foundation of China (Grant Nos. 11801342 and 11801005)
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An, G.Y., He, J. Characterizations of (m,n)-Jordan Derivations and (m,n)-Jordan Derivable Mappings on Some Algebras. Acta. Math. Sin.-English Ser. 35, 378–390 (2019). https://doi.org/10.1007/s10114-018-7495-x
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DOI: https://doi.org/10.1007/s10114-018-7495-x