Automatic continuity of biorthogonality preservers between weakly compact $JB^*$-triples and atomic $JBW^*$-triples
Volume 204 / 2011
Studia Mathematica 204 (2011), 97-121
MSC: Primary 46L05, 46L70; Secondary 17A40, 17C65, 46K70, 46L40, 47B47, 47B49.
DOI: 10.4064/sm204-2-1
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 (not necessarily continuous) biorthogonality preserving linear surjection between them.
