Lie Weak Amenability of Triangular Banach Algebra

DOI：10.3770/j.issn:2095-2651.2017.05.010

 作者 单位 陈琳 安顺学院数理学院, 贵州 安顺 561000 陆芳言 苏州大学数学科学学院, 江苏 苏州 215006

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.

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 $\mathcal M$.