The facial and inner ideal structure of a real JBW*-triple

Let B a real JBW*-triple with predual B* and canonical hermitification the JBW*-triple A. It is shown that the set U(B)(-) consisting of the partially ordered set U(B) of tripotents in B with a greatest element adjoined forms a sub-complete lattice of the complete lattice U(A)(-) of tripotents in A with the same greatest element adjoined. The complete lattice U(B)(-) is sh own to be order isomorphic to the complete lattice F-n(B*(1)) of norm-close d faces of the unit ball B*(1) in B* and anti-order isomorphic to the compl ete lattice F-omega*(B-1) of weak*-closed faces of the unit bail B-1 in B. Consequently, every proper norm-closed face of B*(1) is norm-exposed (by a tripotent) and has the property that it is also a norm-closed face of the c losed unit ball in the predual of the hermitification of B. Furthermore, ev ery weak* -closed face of B-1 is weak*-semi-exposed, and, if non-empty, of the form u + B-0(u)(1), where u is a tripotent in B and B-0(u)(1) is the cl osed unit ball in the zero Peirce space B-0(u) corresponding to u. A structural projection on B is a real linear projection R on B such that, for all elements a and b in B, {Ra b Ra}(B) is equal to R{a Rb a}(B). A sub space J of B is said to be an inner ideal if {J B J}(B) is contained in J a nd J is said to be complemented if B is the direct sum of J and the subspac e Ker(J) defined to be the set of elements b in B such that, for all elemen ts a in J, {a b a}(B) is equal to zero. It is shown that every weak* -close d inner ideal in B is complemented or, equivalently, the range of a structu ral projection. The results are applied to JEW-algebras, real W*-algebras and certain real Cartan factors.