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 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().
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm214-3-6,
author = {L. W. Marcoux and H. Radjavi and A. R. Sourour},
title = {Multiplicative maps that are close to an automorphism on algebras of linear transformations},
journal = {Studia Mathematica},
volume = {215},
year = {2013},
pages = {279-296},
zbl = {1279.15007},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm214-3-6}
}
L. W. Marcoux; H. Radjavi; A. R. Sourour. Multiplicative maps that are close to an automorphism on algebras of linear transformations. Studia Mathematica, Tome 215 (2013) pp. 279-296. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm214-3-6/