双重导子的连续性

On the Continuity of Double Derivations

双重导子是导子的一种推广形式.令δ和ε为复线性代数A到自身内的两个映射,称A到自身内的线性映射d是一个（δ,ε）-双重导子,如果对任意a,b∈A,有d（ab）=d（a）b＋ad（b）＋δ（a）ε（b）＋ε（a）δ（b）成立.本文研究Banach代数上双重导子的自动连续性问题,证明如果δ和ε为含单位元C^＊-代数上的两个在0点连续的映射,则该C^＊-代数上的每个（δ,ε）-双重导子都是自动连续的.

The double derivations are the generalized forms ot ordinary derivations.For two mappings δ and ε from a complex linear algebra A into itself,a linear mapping d from A into itself is called a（δ,ε）-double derivation,if d（ab） — d（a）b＋ad（b）＋δ（a）ε（b）＋ε（a）6（b） for all a,b∈ A.We studies the problem on the automatic continuity of double derivations of Banach algebras.We prove that,if δ and ε are two continuous at zero mappings from a unital C^＊-algebra A into itself,then every（δ,ε）-double derivation of A is automatically continuous.