Additive maps on standard operator algebras preserving invertibilities or zero divisors.
The work under review is part of a series of works from the authors and other mathematicians, where they characterize linear maps on operator algebras over Banach spaces that preserve certain properties of the operators in the algebra. In the present paper, the setting is that of a unital additive surjection \(\Phi\) between two standard operator algebras (closed subalgebras of the algebra of all bounded operators acting on a Banach space that contain the identity and the ideal of finite rank operators). It is proved that if \(\Phi\) preserves (in both directions) any of the following properties/sets: invertibility, zero divisors, topological divisors of zero, quasi-affinity, injectivity, surjectivity, range density, lower-boundedness and left maximal ideals; then \(\Phi\) is either an isomorphism, conjugate isomorphism, anti-isomorphism, or conjugate anti-isomorphism. By preserving in ``both directions'' it is meant that ``\(t\) has property \(X\) if and only \(\Phi(t)\) has property \(X\)''. The technique used consists essentially of proving that a map \(\Phi\) in the above conditions should necessarily be injective and should preserve rank one operators in both directions. These two properties permit to show that \(\Phi\) is either a similarity or a conjugate similarity.
scientific article; zbMATH DE number 1886838