The spectrally bounded linear maps on operator algebras

Jianlian Cui¹ ; Jinchuan Hou²

¹ Jianlian 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
² Jinchuan Hou Department of Mathematics Shanxi Teachers University Linfen 041004, P.R. China Current address Department of Mathematics Shanxi University Taiyuan 030000, P. R. China

Studia Mathematica, Tome 150 (2002) no. 3, pp. 261-271

Résumé

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.

Détail

DOI : 10.4064/sm150-3-4

Keywords: every spectrally bounded linear map mit phi banach algebra standard operator algebra acting complex banach space square zero preserving result mit phi spectrally bounded mit phi homomorphism multiplied nonzero complex number another application hilbert space classification theorem obtained which states every spectrally bounded linear bijection mit phi cal cal where infinite dimensional complex hilbert spaces either isomorphism anti isomorphism multiplied nonzero complex number mit phi injective mit phi vanishes compact operators