The spectrally bounded linear maps on operator algebras

Jianlian Cui, and Jinchuan Hou. "The spectrally bounded linear maps on operator algebras." Studia Mathematica 150.3 (2002): 261-271. <http://eudml.org/doc/285091>.

@article{JianlianCui2002,
abstract = {We show that every spectrally bounded linear map Φ 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 Φ₂ is spectrally bounded, then Φ 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 Φ from ℬ(H) onto 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 Φ is not injective, then Φ vanishes at all compact operators.},
author = {Jianlian Cui, Jinchuan Hou},
journal = {Studia Mathematica},
keywords = {spectral radius; Jordan homomorphism; Banach algebra},
language = {eng},
number = {3},
pages = {261-271},
title = {The spectrally bounded linear maps on operator algebras},
url = {http://eudml.org/doc/285091},
volume = {150},
year = {2002},
}

TY - JOUR
AU - Jianlian Cui
AU - Jinchuan Hou
TI - The spectrally bounded linear maps on operator algebras
JO - Studia Mathematica
PY - 2002
VL - 150
IS - 3
SP - 261
EP - 271
AB - We show that every spectrally bounded linear map Φ 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 Φ₂ is spectrally bounded, then Φ 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 Φ from ℬ(H) onto 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 Φ is not injective, then Φ vanishes at all compact operators.
LA - eng
KW - spectral radius; Jordan homomorphism; Banach algebra
UR - http://eudml.org/doc/285091
ER -