Linear maps preserving quasi-commutativity

Volume 184 / 2008

Heydar Radjavi, Peter Šemrl Studia Mathematica 184 (2008), 191-204 MSC: 15A04, 15A27, 47B49. DOI: 10.4064/sm184-2-7


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.


  • Heydar RadjaviDepartment of Pure Mathematics
    University of Waterloo
    200 University Avenue West
    Waterloo, ON, Canada N2L 3G1
  • Peter ŠemrlDepartment of Mathematics
    University of Ljubljana
    Jadranska 19
    SI-1000 Ljubljana, Slovenia

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