Additive Biderivations and Centralizing Maps on Nest Algebras

DOI：10.3770/j.issn:2095-2651.2013.02.012

 作者 单位 齐霄霏 山西大学数学学院, 山西 太原 030006

令$\mathcal N$是Banach空间$X$上的套, Alg$\mathcal N$是相应的套代数. 本文证明了, 如果套$\mathcal N$中存在非平凡元$N$在$X$中可补, 且$\dim N\not=1$, 则Alg$\mathcal N$上的每个可加双导子是内导子. 作为此定理的应用, 分别给出了套代数上中心化(交换)映射, 斜中心化导子以及斜交换的广义导子的具体刻画.

Let $\mathcal N$ be a nest on a Banach space $X$, and Alg$\mathcal N$ be the associated nest algebra. It is shown that, if there exists a non-trivial element $N$ in $\mathcal N$ which is complemented in $X$ and $\dim N\not=1$, then every additive biderivation from Alg$\mathcal N$ into itself is an inner biderivation. Based on this result, we give characterizations of centralizing (commuting) maps, cocentralizing derivations, and cocommuting generalized derivations on nest algebras.