Stability of derivations under weak-2-local continuous perturbations

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• 作者：Enrique Jordá ; Antonio M. Peralta
• 关键词：Derivation ; 2 ; local linear map ; 2 ; local symmetric map ; 2 ; local * ; derivation ; 2 ; local derivation ; weak ; 2 ; local derivation
• 刊名：Aequationes mathematicae
• 出版年：2017
• 出版时间：February 2017
• 年：2017
• 卷：91
• 期：1
• 页码：99-114
• 全文大小：
• 刊物类别：Mathematics and Statistics
• 刊物主题：Analysis; Combinatorics;
• 出版者：Springer International Publishing
• ISSN：1420-8903
• 卷排序：91

文摘

Let \({\Omega}\) be a compact Hausdorff space and let A be a C*-algebra. We prove that if every weak-2-local derivation on A is a linear derivation and every derivation on \({C(\Omega,A)}\) is inner, then every weak-2-local derivation \({\Delta:C(\Omega,A)\to C(\Omega,A)}\) is a (linear) derivation. As a consequence we derive that, for every complex Hilbert space H, every weak-2-local derivation \({\Delta : C(\Omega,B(H)) \to C(\Omega,B(H))}\) is a (linear) derivation. We actually show that the same conclusion remains true when B(H) is replaced with an atomic von Neumann algebra. With a modified technique we prove that, if B denotes a compact C*-algebra (in particular, when \({B=K(H)}\)), then every weak-2-local derivation on \({C(\Omega,B)}\) is a (linear) derivation. Among the consequences, we show that for each von Neumann algebra M and every compact Hausdorff space \({\Omega}\), every 2-local derivation on \({C(\Omega,M)}\) is a (linear) derivation.