The spectrally bounded linear maps on operator algebras

Tom 150 / 2002

Jianlian Cui, Jinchuan Hou Studia Mathematica 150 (2002), 261-271 MSC: Primary 47B48, 47L10, 47A10. DOI: 10.4064/sm150-3-4

Streszczenie

We show that every spectrally bounded linear map ${\mit \Phi }$ from a Banach algebra onto a standard operator algebra acting on a complex Banach space is square-zero preserving. This result is used to show that if ${\mit \Phi }_{2}$ is spectrally bounded, then ${\mit \Phi }$ is a homomorphism multiplied by a nonzero complex number. As another application to the Hilbert space case, a classification theorem is obtained which states that every spectrally bounded linear bijection ${\mit \Phi }$ from ${\cal B}(H)$ onto ${\cal B}(K)$, where $H$ and $K$ are infinite-dimensional complex Hilbert spaces, is either an isomorphism or an anti-isomorphism multiplied by a nonzero complex number. If ${\mit \Phi }$ is not injective, then ${\mit \Phi }$ vanishes at all compact operators.

Autorzy

  • Jianlian CuiJianlian Cui
    Institute of Mathematics
    Chinese Academy of Sciences
    Beijing 100080, P.R. China
    Current address
    Department of Applied Mathematics
    Taiyuan University of Technology
    Taiyuan 030024, P.R. China
    and
    Department of Mathematics
    Shanxi Teachers University
    Linfen 041004, P.R. China
    e-mail
  • Jinchuan HouJinchuan Hou
    Department of Mathematics
    Shanxi Teachers University
    Linfen 041004, P.R. China
    Current address
    Department of Mathematics
    Shanxi University
    Taiyuan 030000, P. R. China
    e-mail

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