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

Abstract

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