Automatic continuity of biorthogonality preservers between weakly compact JB*-triples and atomic JBW*-triples

Abstract

We prove that every biorthogonality preserving linear surjection from a weakly compact JB∗triple containing no infinite dimensional rank-one summands onto another JB∗-triple is automatically continuous. We also show that every biorthogonality preserving linear surjection between atomic JBW∗triples containing no infinite dimensional rank-one summands is automatically continuous. Consequently, two atomic JBW∗-triples containing no rank-one summands are isomorphic if, and only if, there exists a (non necessarily continuous) biorthogonality preserving linear surjection between them.