Lie homomorphisms of operator algebras

Abstract

A mapping : M-+N between *-algebras M, N which is *-linear, and which preserves the Lie bracket [X, Y] = XY -YX of elements X, Y in M is called a Lie *-homomorphism or just a Lie homomorphism. The main result of this paper states that if : A -> B is a uniformly continuous Lie *-homomorphism of the C*-algebra A onto the C*-algebra B then there exists a central projection D in the weak closure of B such that modulo a center-valued *-linear map which annihilates brackets, is a *-homomorphism and (/ -D) is the negative of a *anti-homomorphism.