Characterizations of ( m,n )-Jordan Derivations and ( m,n )-Jordan Derivable Mappings on Some Algebras |
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Authors: | Guang Yu An Jun He |
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Institution: | 1. Department of Mathematics, Shaanxi University of Science and Technology, Xi'an 710021, P. R. China;2. Department of Mathematics and Physics, Anhui Polytechnic University, Wuhu 241000, P. R. China |
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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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Keywords: | (m n)-Jordan derivation (m n)-Jordan derivable mapping C*-algebra generalized matrix algebra |
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