Multiplicative maps that are close to an automorphism on algebras of linear transformations

L. W. Marcoux, H. Radjavi, and A. R. Sourour. "Multiplicative maps that are close to an automorphism on algebras of linear transformations." Studia Mathematica 214.3 (2013): 279-296. <http://eudml.org/doc/286135>.

@article{L2013,
abstract = {Let be a complex, separable Hilbert space of finite or infinite dimension, and let ℬ() be the algebra of all bounded operators on . It is shown that if φ: ℬ() → ℬ() is a multiplicative map(not assumed linear) and if φ is sufficiently close to a linear automorphism of ℬ() in some uniform sense, then it is actually an automorphism; as such, there is an invertible operator S in ℬ() such that $φ(A) = S^\{-1\} AS$ for all A in ℬ(). When is finite-dimensional, similar results are obtained with the mere assumption that there exists a linear functional f on ℬ() so that f ∘ φ is close to f ∘ μ for some automorphism μ of ℬ().},
author = {L. W. Marcoux, H. Radjavi, A. R. Sourour},
journal = {Studia Mathematica},
keywords = {multiplicative maps; simultaneous similarity; automorphisms; Hilbert space},
language = {eng},
number = {3},
pages = {279-296},
title = {Multiplicative maps that are close to an automorphism on algebras of linear transformations},
url = {http://eudml.org/doc/286135},
volume = {214},
year = {2013},
}

TY - JOUR
AU - L. W. Marcoux
AU - H. Radjavi
AU - A. R. Sourour
TI - Multiplicative maps that are close to an automorphism on algebras of linear transformations
JO - Studia Mathematica
PY - 2013
VL - 214
IS - 3
SP - 279
EP - 296
AB - Let be a complex, separable Hilbert space of finite or infinite dimension, and let ℬ() be the algebra of all bounded operators on . It is shown that if φ: ℬ() → ℬ() is a multiplicative map(not assumed linear) and if φ is sufficiently close to a linear automorphism of ℬ() in some uniform sense, then it is actually an automorphism; as such, there is an invertible operator S in ℬ() such that $φ(A) = S^{-1} AS$ for all A in ℬ(). When is finite-dimensional, similar results are obtained with the mere assumption that there exists a linear functional f on ℬ() so that f ∘ φ is close to f ∘ μ for some automorphism μ of ℬ().
LA - eng
KW - multiplicative maps; simultaneous similarity; automorphisms; Hilbert space
UR - http://eudml.org/doc/286135
ER -