Linear maps preserving quasi-commutativity
Let X and Y be Banach spaces and B(X) and B(Y ) the algebras of all bounded linear operators on X and Y , respectively. We say that A,B 2 B(X) quasi- commute if there exists a nonzero scalar ! such that AB = !BA. We characterize bijective linear maps � : B(X) ! B(Y ) preserving quasi-commutativity. In fact, such a character- ization can be proved for much more general algebras. In the finite-dimensional case the same result can be obtained without the bijectivity assumption.