2-local triple homomorphisms on von Neumann algebras and JBW$^*$-triples

TL;DR

The paper proves that all 2-local triple homomorphisms from JBW*-triples and von Neumann algebras are necessarily linear and triple homomorphisms, extending the understanding of these mappings in operator algebra theory.

Contribution

It establishes the linearity and triple homomorphism property of 2-local triple homomorphisms on JBW*-triples and von Neumann algebras, regardless of linearity or continuity assumptions.

Findings

All 2-local triple homomorphisms from JBW*-triples are linear.

Such mappings from von Neumann algebras are also linear.

These mappings are proven to be triple homomorphisms.

Abstract

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