A note on 2-local representations of C${}^{\ast }$-algebras

Abstract

We survey the results on linear local and 2-local homomorphisms and zero products preserving operators between C${}^{\ast }$-algebras, and we incorporate some new precise observations and results to prove that every bounded linear 2-local homomorphism between C${}^{\ast }$-algebras is a homomorphism. Consequently, every linear 2-local ${}^{\ast }$-homomorphism between C${}^{\ast }$-algebras is a ${}^{\ast }$-homomorphism.