2-local Lie isomorphisms of operator algebras on Banach spaces
Volume 229 / 2015
Studia Mathematica 229 (2015), 1-11 MSC: Primary 47B49; Secondary 16S50. DOI: 10.4064/sm7864-12-2015 Published online: 3 December 2015
Let $X$ and $Y$ be complex Banach spaces of dimension greater than 2. We show that every 2-local Lie isomorphism $\phi $ of $B(X)$ onto $B(Y)$ has the form $\phi =\varphi +\tau $, where $\varphi $ is an isomorphism or the negative of an anti-isomorphism of $B(X)$ onto $B(Y)$, and $\tau $ is a homogeneous map from $B(X)$ into $\mathbb CI$ vanishing on all finite sums of commutators.