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 Delta : A -> B such that for each a, b is an element of A and phi is an element of B*, there exists a symmetric linear map T-a,T-b,T-phi : A -> B, depending on a, b and phi, satisfying phi Delta(a) = phi T-a,T-b,T-phi(a) and phi Delta(b) = phi T-a,T-b,T-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-Slodkowski theorem for general C*-algebras by proving that every 2-local *-homomorphism between C*-algebras is a (linear) *-homomorphism. (C) 2015 Elsevier Inc. All rights reserved.