# Wydawnictwa / Czasopisma IMPAN / Studia Mathematica / Wszystkie zeszyty

## The spectrally bounded linear maps on operator algebras

### Tom 150 / 2002

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
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