Weak-2-local symmetric maps on C*-algebras

Abstract

We introduce and study weak-2-local symmetric maps between C${}^{\ast }$-algebras $A$ and $B$ as non necessarily linear nor continuous maps $\Delta :A\to B$ such that for each $a,b\in A$ and $\varphi \in B^{\ast }$, there exists a symmetric linear map $T_{a,b,\varphi }:A\to B$, depending on $a$, $b$ and $\varphi$, satisfying $\varphi \Delta (a)=\varphi T_{a,b,\varphi }(a)$ and $\varphi \Delta (b)=\varphi T_{a,b,\varphi }(b)$. We prove that every weak-2-local symmetric map between C${}^{\ast }$-algebras is a linear map. Among the consequences we show that every weak-2-local ${}^{\ast }$-derivation on a general C${}^{\ast }$-algebra is a (linear) ${}^{\ast }$-derivation. We also establish a 2-local version of the Kowalski-Słodkowski theorem for general C${}^{\ast }$-algebras by proving that every 2-local ${}^{\ast }$-homomorphism between C${}^{\ast }$-algebras is a (linear) ${}^{\ast }$-homomorphism.