Multiplicative maps that are close to an automorphism on algebras of linear transformations
Volume 214 / 2013
Abstract
Let ${\mathcal H}$ 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 $\mathcal B(\mathcal H)$.
