Maps completely preserving commutativity and maps completely preserving Jordan zero-product
Let \(X\) and \(Y\) infinite-dimensional Banach spaces over the real or complex field \(\mathbb{F}\), and let \({\mathcal A}\) and \({\mathcal B}\) be standard operator algebras on \(X\) and \(Y\), respectively. A map from \({\mathcal A}\) into \({\mathcal B}\) is said to be \(n\)-commutativity preserving if the induced map from \({\mathcal A}\otimes M_n(\mathbb F)\) into \({\mathcal B}\otimes M_n(\mathbb F)\) preserves the commutativity, and is said to be completely commutativity preserving if it is \(n\)-commutativity preserving map for all \(n\). In this paper, the authors show that every map completely preserving commutativity from \({\mathcal A}\) onto \({\mathcal B}\) is a scalar multiple of either an isomorphism or (in the complex case) a conjugate isomorphism. They also obtain the complete description of all surjective maps between \({\mathcal A}\) and \({\mathcal B}\) that completely preserve the Jordan zero-product of operators.
scientific article; zbMATH DE number 6343725
scientific article