Partially defined σ-derivations on semisimple Banach algebras
Partially defined σ-derivations on semisimple Banach algebras
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DOI：
10.4064/sm190-2-7
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发表时间：
2009-03
影响因子：
0.8
通讯作者：
Tsiu-Kwen Lee;Cheng–Kai Liu
中科院分区：
文献类型：
--
作者：
Tsiu-Kwen Lee;Cheng–Kai Liu
设A是具有线性自同构σ的半单Banach代数，δ：I → A是σ-导子，其中I是A的理想.则Φ（δ）（I <$σ（I））= 0，其中Φ（δ）是δ的分离空间。因此，如果I是本质理想，则σ-导子δ是可闭的。在素C-代数中，我们证明了定义在非零理想上的每个σ-导子都是连续的.最后，证明了具有非平凡幂等元的素半单Banach代数上的任何线性映射是连续的，如果它满足关于零积的σ-导子展开公式。1.结果在本文中，A始终是复域C上的单位Banach代数，σ是A的线性自同态.设1A表示A的单位自同构。所谓A的σ-导子，是指线性映射δ：A → A使得对任意x，y ∈ A，δ（xy）= σ（x）δ（y）+ δ（x）y。显然，映射σ-1A是一个σ-导子，而1A-导子只是普通导子。因此σ-导子的概念可以看作是导子和自同态的推广。设I是A的非零理想。一个线性映射δ：I → A称为定义在I上的σ-导子，如果对所有x，y ∈ I，δ（xy）= σ（x）δ（y）+δ（x）y. A的理想I称为本质的，如果I与A的任何非零理想有非平凡交。对于一个半单代数A，这等价于说，其中a ∈ A蕴涵a = 0。一个σ-导子δ：I → A称为本质定义在一个理想I上，如果I是A的本质理想。Kaplansky证明了C-代数上的每个导子都是连续的[16]，半单Banach代数上的每个导子都是连续的[17]。Sakai在[22]中证实了Kaplansky关于C-代数的猜想。第二个猜想由约翰逊和辛克莱在[15]中证实。2000年数学学科分类：46 H40、47 B47、46 H15。
Let A be a semisimple Banach algebra with a linear automorphism σ and let δ : I → A be a σ-derivation, where I is an ideal of A. Then Φ(δ)(I ∩ σ(I)) = 0, where Φ(δ) is the separating space of δ. As a consequence, if I is an essential ideal then the σ-derivation δ is closable. In a prime C∗-algebra, we show that every σ-derivation defined on a nonzero ideal is continuous. Finally, any linear map on a prime semisimple Banach algebra with nontrivial idempotents is continuous if it satisfies the σ-derivation expansion formula on zero products. 1. Results. Throughout the paper, A is always a unital Banach algebra over the complex field C and σ is a linear endomorphism of A. Let 1A denote the identity automorphism of A. By a σ-derivation of A we mean a linear map δ : A → A such that δ(xy) = σ(x)δ(y) + δ(x)y for all x, y ∈ A. Clearly, the map σ − 1A is a σ-derivation and 1A-derivations are just ordinary derivations. Thus the concept of σ-derivations can be regarded as a generalization of both derivations and endomorphisms. Let I be a nonzero ideal of A. A linear map δ : I → A is called a σ-derivation defined on I if δ(xy) = σ(x)δ(y)+δ(x)y for all x, y ∈ I. An ideal I of A is called essential if I has nontrivial intersection with any nonzero ideal of A. For a semisimple algebra A, this is equivalent to saying that aI = 0 where a ∈ A implies a = 0. A σ-derivation δ : I → A is called essentially defined on an ideal I if I is an essential ideal of A. Kaplansky conjectured that every derivation on a C∗-algebra is continuous [16] and that every derivation on a semisimple Banach algebra is continuous [17]. Sakai confirmed Kaplansky’s conjecture for C∗-algebras in [22]. The second conjecture was confirmed by Johnson and Sinclair in [15]. 2000 Mathematics Subject Classification: 46H40, 47B47, 46H15.