Commutativity preserving linear maps and Lie automorphisms of triangular matrix algebras

Abstract

In this article we classify linear maps ϕ from the algebra T n of n × ? upper triangular matrices into itself satisfying ϕ( ab − ba) = 0 if and only if ab − ba = 0. In particular, we show that for n > 2, any such map is either the sum of an algebra automorphism and a map into the centre, or the sum of the negative of an algebra anti-automorphism and a map into the centre. As a consequence, Lie automorphisms of the algebra T n are also classified.