MAPS PRESERVING LIE PRODUCT ON B(X)

Abstract

Banach spaces. Let $\phi$ be a bijection from $B(X)$ onto $B(Y)$ satisfying ${\phi}([A,B])=[\phi(A),\phi(B)]$ for all $A, B\in B(X)$. Then $\phi=\psi+\tau$, where $\psi$ is a ring isomorphism or a negative of a ring anti-isomorphism from $B(X)$ onto $B(Y)$, and $\tau$ is a map from $B(X)$ into $\Bbb CI$ satisfying $\tau([A,B])=0$ for all $A, B\in B(X)$.