A Kowalski-S{\l}odkowski theorem for 2-local $^*$-homomorphisms on von Neumann algebras

TL;DR

This paper proves that 2-local *-homomorphisms from von Neumann algebras and certain C*-algebras are necessarily linear and *-homomorphisms, extending classical results to broader non-linear settings.

Contribution

It establishes the linearity and *-homomorphism property of 2-local *-homomorphisms on von Neumann and compact C*-algebras, generalizing previous theorems.

Findings

2-local *-homomorphisms are linear and *-homomorphisms on von Neumann algebras

The same linearity result holds for compact C*-algebras

2-local Jordan *-homomorphisms are linear and Jordan *-homomorphisms on JBW*- and JB*-algebras

Abstract

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