摘要
设R是含非平凡幂等元P的素环,C∈R,C=PC.本文证明可加映射△：R→R在C可导,即△（AB）=△（A）B＋A△（B）,A,B∈R,AB=C当且仅当存在导子δ：R→R,使得△（A）=δ（A）＋△（I）A,A∈R.没有I1型中心直和项的von Neumann代数上的可导映射也有类似结论.利用该结论证明了,若非零算子C∈B（X）,使得ran（C）或ker（C）在X中可补,则可加映射△：B（X）→B（X）在C可导当且仅当它是导子.特别地,证明了因子von Neumann代数上的可加映射在任意但固定的非零算子可导当且仅当它是导子.
Let R be a prime ring containing a nontrivial idempotent P. Suppose C ∈R satisfies C = PC. It is shown that an additive map △ ：R→Ris derivable at C, that is, △（AB） = △（A）B ＋ A△（B） for every A,B ∈R with AB = C if and only if there exists a derivation δ： R→R such that △（A） = δ（A） ＋ △（I）A for all A ∈ R. Similar results are obtained for yon Neumann algebras with no central abelian projections. As its application, we obtain that, if nonzero operator C ∈B（X） such that ran（C） or ker（C） is complementary in X, then an additive map△： B（X） →B（X） is derivable at C if and only if it is a derivation. In particular, we show that an additive map from a factor von Neumann algebra into itself is derivable at an arbitrary but fixed nonzero operator if and only if it is a derivation.
作者
郭玉琴
安润玲
Yu Qin GUO;Run Ling AN(Department of Mathematics, Taiyuan University of Technology, Taiyuan 030024, P. R. Chin)
出处
《数学学报（中文版）》
CSCD
北大核心
2018年第4期631-640,共10页
Acta Mathematica Sinica：Chinese Series
基金
国家自然科学基金资助项目(11001194)
