三角Banach代数的Lie弱顺从性

设$mathcal {A, B}$ 是含单位元的Banach代数, $mathcal M$ 是一个Banach $mathcal {A, B}$-双模. $mathcal {T}=left ( begin{array}{cc} mathcal {A} & mathcal M & mathcal {B} end{array} right )$按照通常矩阵加法和乘法,范数定义为$|left( begin{array}{cc} a & m & b end{array} right)|=|a|_{mathcal A}+|m|_{mathcal M}+|b|_{mathcal B}$,构成三角Banach 代数.如果从$mathcal T$到其$n$次对偶空间$mathcal T^{n}$上的Lie导子都是标准的,则称$mathcal T$是Lie $n$弱顺从的.本文研究了三角Banach代数$mathcal T$上的Lie $n$弱顺从性,证明了有限维套代数是Lie $n$弱顺从的.

Lie Weak Amenability of Triangular Banach Algebra

Let $mathcal {A}$ and $mathcal B$ be unital Banach algebra and $mathcal M$ be Banach $mathcal A, mathcal B$-module. Then $mathcal T=big( begin{smallmatrix} mathcal {A} & mathcal M & mathcal {B} end{smallmatrix}big)$ becomes a triangular Banach algebra when equipped with the Banach space norm $|big( begin{smallmatrix} a & m & b end{smallmatrix} big)|=|a|_{mathcal A}+|m|_{mathcal M}+|b|_{mathcal B}$. A Banach algebra $mathcal T$ is said to be Lie $n$-weakly amenable if all Lie derivations from $mathcal T$ into its $n^{text{th}}$ dual space ${mathcal T}^{(n)}$ are standard. In this paper we investigate Lie $n$-weak amenability of a triangular Banach algebra $mathcal T$ in relation to that of the algebras $mathcal A, mathcal B$ and their action on the module $mathcal M$.