2-local Jordan automorphisms on operator algebras (Q2891267)

Let \(\mathcal A\) be an algebra over a field \(F\) and let \(\phi: \mathcal A \to \mathcal A\) be a map. The author calls \(\phi\) a 2-local Jordan automorphism if, for every \(a,b \in \mathcal A\), there exists a Jordan automorphism \(\phi_{a,b}: \mathcal A \to \mathcal A\) such that \( \phi (a)= \phi_{a,b} (a)\) and \(\phi (b)= \phi_{a,b} (b)\). This notion was introduced by \textit{P. Šemrl} [Proc. Am. Math. Soc. 125, No. 9, 2677--2680 (1997; Zbl 0887.47030)].NEWLINENEWLINELet \(X\) be a real or complex separable Banach space and let \(\mathcal B(X)\) stand for the algebra of all bounded linear operators on \(X\). In the present paper, the author shows that, if \(\mathcal A\) is a subalgebra of \(\mathcal B (X)\) containing the ideal of all compact operators on \(X\), then every 2-local Jordan automorphism of \(\mathcal A\) is either an automorphism or an anti-automorphism of \(\mathcal A\).