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.

Theorems & Definitions (43)

• Remark 2.1
• Lemma 3.1
• Lemma 3.2
• Proposition 3.3
• proof
• Remark 3.4
• Corollary 3.5
• proof
• Lemma 3.6
• Proposition 3.7