Bulletin of the Iranian Mathematical Society ( IF 0.7 ) Pub Date : 2021-04-21 , DOI: 10.1007/s41980-020-00499-y Jiankui Li , Shan Li
Let \(\mathcal {A}\) be a unital algebra and \(\mathcal {M}\) be a unital \(\mathcal {A}\)-bimodule. We characterize the linear mappings \(\delta \) and \(\tau \) from \(\mathcal {A}\) into \(\mathcal {M}\), satisfying \(\delta (A)B+A\tau (B)=0\) for every \(A,B \in \mathcal {A}\) with \(AB=0\) when \(\mathcal {A}\) contains a separating ideal \(\mathcal {T}\) of \(\mathcal {M}\), which is in the algebra generated by all idempotents in \(\mathcal {A}\). We apply the result to \(\mathcal {P}\)-subspace lattice algebras, completely distributive commutative subspace lattice algebras, and unital standard operator algebras. Furthermore, suppose that \(\mathcal {A}\) is a unital Banach algebra and \(\mathcal {M}\) is a unital Banach \(\mathcal {A}\)-bimodule, we give a complete description of linear mappings \(\delta \) and \(\tau \) from \(\mathcal A\) into \(\mathcal M\), satisfying \(\delta (A)B+A\tau (B)=0\) for every \(A,B\in \mathcal {A}\) with \(AB=I\).
中文翻译:
线性映射的特征在于对零产品或单位产品的作用
假设\(\ mathcal {A} \)为单位代数,\(\ mathcal {M} \)为单位\(\ mathcal {A} \)- bimodule。我们将线性映射\(\ delta \)和\(\ tau \)从\(\ mathcal {A} \)到\(\ mathcal {M} \)进行表征,满足\(\ delta(A)B + A \ tau蛋白(B)= 0 \)每\(A,B \在\ mathcal {A} \)与\(AB = 0 \)时\(\ mathcal {A} \)含有分离理想\(\ mathcal【T} \)的\(\ mathcal {M} \) ,这是在由所有幂等所产生的代数\(\ mathcal {A} \) 。我们将结果应用于\(\ mathcal {P} \)-子空间格代数,完全分布式可交换子空间格代数和unit元标准算子代数。此外,假设\(\ mathcal {A} \)是一个单位Banach代数,\(\ mathcal {M} \)是一个单位Banach \(\ mathcal {A} \)- bimodule,我们给出完整的描述线性映射\(\ delta \)和\(\ tau \)从\(\ mathcal A \)到\(\ mathcal M \),满足\(\ delta(A)B + A \ tau(B)= 0 \)与\(AB = I \)中的每个\(A,B \在\ mathcal {A} \)中。