Atomic decomposition of real JBW\(^*\)-triples (Q2716949)

A real \(\text{JB}^*\)-triple is a closed real subtriple of a (complex) \(\text{JB}^*\)-triple, and a real \(\text{JBW}^*\)-triple is a real \(\text{JB}^*\)-triple the canonical hermitification of which is a (complex) \(\text{JBW}^*\)-triple. In the complex case, \(\text{JBW}^*\)-triples split into the direct sum of two \(w^*\)-closed ideals, one of them is free of atoms and the other (the \(w^*\)-closed linear hull of all atoms) is a direct sum of a family of Cartan factors. In the present paper the authors provide an analogue of this decomposition for real \(\text{JBW}^*\)-triples.NEWLINENEWLINENEWLINEReviewer's remark: Actually, the atomic part of any (real or complex) \(\text{JBW}^*\)-triple coincides with the \(w^*\)-closure of its socle, and the non-atomic part is precisely the annihilator of the socle.