Local automorphisms of standard operator algebras
Let \(\mathcal A\) be an algebra. A linear map \(\phi: \mathcal A \to \mathcal A\) is said to be a local automorphism if, for every \(x \in \mathcal A\), there exists an automorphism \(\phi_x: \mathcal A \to \mathcal A\) such that \(\phi (x)= \phi_x (x)\). Let \(X\) be an infinite-dimensional Banach space and let \({\mathcal B}(X)\) be the algebra of all bounded linear operators on \(X\). It was shown in [\textit{D.\,R.\thinspace Larson} and \textit{A.\,R.\thinspace Sourour}, Proc.\ Symp.\ Pure Math.\ 51, 187--194 (1990; Zbl 0713.47045)] that every surjective local automorphism of \({\mathcal B}(X)\) is an automorphism. In [\textit{M.\,Brešar} and \textit{P.\,Šemrl}, Stud.\ Math.\ 113, 101--108 (1995; Zbl 0835.47020)], it was proved that, if, moreover, \(X\) is a separable Hilbert space, the above conclusion holds true without the surjectivity assumption. In the present paper, the author shows that, if \(\mathcal A\) is a closed standard operator algebra on a separable infinite-dimensional (real or complex) Banach space \(X\), then every local automorphism of \(\mathcal A\) is an automorphism. Recall that a subalgebra of \({\mathcal B}(X)\) is called standard if it contains all finite rank operators. The author also gives examples showing that the assumptions of closedness and separability are essential.
scientific article; zbMATH DE number 5777317