三角Banach代数上的对偶模Jordan导子和对偶模广义导子

设A,B是含单位元的Banach代数, M是一个Banach A,B-双模。 T=(^{A M}_B) 按照通常矩阵加法和乘法,范数定义为‖(^{a m}_b)‖=‖a‖_A+‖m‖_M+‖b‖_B,构成三角Banach代数。通过作用(^{f h}_g)(^{a m}_b)=f(a)+h(m)+g(b), T的对偶空间 T^*为(^{A* M*}_B^*)。 在T^*上定义模作用 (^{a m}_b)·(^{f h}_g)=(^{a·f+m·h b·h}_b·g), (^{f h}_g)·(^{a m}_b)=(^{f·a h·a}_h·m+g·b), 使其成为一个对偶Banach T-双模。从T到T^*的映射称为对偶模映射。 本文对T上对偶模Jordan导子和对偶模广义导子进行讨论, 给出了T上对偶模Jordan导子是对偶模导子的一个充分条件并且对T上对偶模广义导子进行了刻画。

Dual module Jordan derivations and dual module generalized derivations of triangular Banach algebra

Let A and B be unital Banach algebras, and let M be a Banach A, B-bimodule. Then T=(^{A M}_B) becomes a triangular Banach algebra when equipped with the usual matrix operation and a Banach space norm ‖(^{a m}_b)‖=‖a‖_A+‖m‖_M+‖b‖_B. T^*=(^{A* M*}_B^*) is the dual space of T by the action (^{f h}_g)(^{a m}_b)=f(a)+h(m)+g(b). T^* becomes a dual Banach T- bimodule with the module action defined by am (^{a m}_b)·(^{f h}_g)=(^{a·f+m·h b·h}_b·g), (^{f h}_g)·(^{a m}_b)=(^{f·a h·a}_h·m+g·b). The map from T into T^* is called dual module map. We investigate the dual module Jordan derivations and dual module generalized derivations on T, giving a condition under which a dual module Jordan derivation is a dual module derivation and a characterization of dual module generalized derivation.