2-local triple homomorphisms on von Neumann algebras and JBW${}^{\ast }$-triples

Abstract

We prove that every (not necessarily linear nor continuous) 2-local triple homomorphism from a JBW${}^{\ast }$-triple into a JB${}^{\ast }$-triple is linear and a triple homomorphism. Consequently, every 2-local triple homomorphism from a von Neumann algebra (respectively, from a JBW${}^{\ast }$-algebra) into a C${}^{\ast }$-algebra (respectively, into a JB${}^{\ast }$-algebra) is linear and a triple homomorphism.