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

We introduce and study weak-2-local symmetric maps between C⁎-algebras A and B as not necessarily linear, nor continuous, maps Δ:A→B such that for each a,b∈A and ϕ∈B⁎, there exists a symmetric linear map Ta,b,ϕ:A→B, depending on a, b and ϕ, satisfying ϕΔ(a)=ϕTa,b,ϕ(a) and ϕΔ(b)=ϕTa,b,ϕ(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-Słodkowski theorem for general C⁎-algebras by proving that every 2-local ⁎-homomorphism between C⁎-algebras is a (linear) ⁎-homomorphism.