Linear maps preserving quasi-commutativity
Volume 184 / 2008
Studia Mathematica 184 (2008), 191-204
MSC: 15A04, 15A27, 47B49.
DOI: 10.4064/sm184-2-7
Abstract
Let $X$ and $Y$ be Banach spaces and ${\cal B}(X)$ and ${\cal B}(Y)$ the algebras of all bounded linear operators on $X$ and $Y$, respectively. We say that $A,B \in {\cal B}(X)$ quasi-commute if there exists a nonzero scalar $\omega $ such that $AB = \omega BA$. We characterize bijective linear maps $\phi : {\cal B}(X) \to {\cal B}(Y)$ preserving quasi-commutativity. In fact, such a characterization can be proved for much more general algebras. In the finite-dimensional case the same result can be obtained without the bijectivity assumption.
