Multiplicative maps that are close to an automorphism on algebras of linear transformations
Volume 214 / 2013
Abstract
Let be a complex, separable Hilbert space of finite or infinite dimension, and let \mathcal B(\mathcal H) be the algebra of all bounded operators on {\mathcal H}. It is shown that if \varphi:\mathcal B(\mathcal H) \to \mathcal B(\mathcal H) is a multiplicative map (not assumed linear) and if \varphi is sufficiently close to a linear automorphism of \mathcal B(\mathcal H) in some uniform sense, then it is actually an automorphism; as such, there is an invertible operator S in \mathcal B(\mathcal H) such that \varphi(A) = S^{-1} AS for all A in \mathcal B(\mathcal H). When {\mathcal H} is finite-dimensional, similar results are obtained with the mere assumption that there exists a linear functional f on \mathcal B(\mathcal H) so that f\circ \varphi is close to f \circ \mu for some automorphism \mu of \mathcal B(\mathcal H).
