A Kowalski-Słodkowski theorem for 2-local $^*$-homomorphisms on von Neumann algebras

María Burgos, Francisco J. Fernández-Polo, Jorge J. Garcés, Antonio M. Peralta

Published 2014-04-30Version 1

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.