A Kowalski-Słodkowski theorem for 2-local ${}^{\ast }$-homomorphisms on von Neumann algebras

Abstract

It is established that every (not necessarily linear) 2-local ${}^{\ast }$-homomorphism from a von Neumann algebra into a C${}^{\ast }$-algebra is linear and a ${}^{\ast }$-homomorphism. In the setting of (not necessarily linear) 2-local ${}^{\ast }$-homomorphism from a compact C${}^{\ast }$-algebra we prove that the same conclusion remains valid. We also prove that every 2-local Jordan ${}^{\ast }$-homomorphism from a JBW${}^{\ast }$-algebra into a JB${}^{\ast }$-algebra is linear and a Jordan ${}^{\ast }$-homomorphism.