Weak-2-local symmetric maps on C^⁎-algebras

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• 作者：Juan Carlos Cabello ^{jcabello@ugr.es" class="auth_mail" title="E-mail the corresponding author} ; Antonio M. Peralta ; ^{aperalta@ugr.es" class="auth_mail" title="E-mail the corresponding author}
• 关键词：47B49 ; 46L05 ; 46L40 ; 46T20 ; 47L99
• 刊名：Linear Algebra and its Applications
• 出版年：2016
• 出版时间：1 April 2016
• 年：2016
• 卷：494
• 期：Complete
• 页码：32-43
• 全文大小：326 K

文摘

We introduce and study weak-2-local symmetric maps between C^⁎${\mathrm{C}}^{⁎}$-algebras A and B   as not necessarily linear, nor continuous, maps Δ:A→B$\mathrm{\Delta }:A\to B$ such that for each a,b∈A$a,b\in A$ and 059778d9e532216c2ab5c3" title="Click to view the MathML source">ϕ∈B^⁎$\varphi \in B^{⁎}$, there exists a symmetric linear map T_a,b,ϕ:A→B$T_{a,b,\varphi }:A\to B$, depending on a, b and ϕ  , satisfying ϕΔ(a)=ϕT_a,b,ϕ(a)$\varphi \mathrm{\Delta }(a)=\varphi T_{a,b,\varphi }(a)$ and ϕΔ(b)=ϕT_a,b,ϕ(b)$\varphi \mathrm{\Delta }(b)=\varphi T_{a,b,\varphi }(b)$. We prove that every weak-2-local symmetric map between C^⁎${\mathrm{C}}^{⁎}$-algebras is a linear map. Among the consequences we show that every weak-2-local ^⁎-derivation on a general C^⁎${\mathrm{C}}^{⁎}$-algebra is a (linear) ^⁎-derivation. We also establish a 2-local version of the Kowalski–Słodkowski theorem for general C^⁎${\mathrm{C}}^{⁎}$-algebras by proving that every 2-local ^⁎-homomorphism between C^⁎${\mathrm{C}}^{⁎}$-algebras is a (linear) ^⁎-homomorphism.