The Facial and Inner Ideal Structure of a Real JBW*-Triple

Let B be 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 shown to be order isomorphic to the complete lattice ℱn(B*1 of norm–closed faces of the unit ball B*1 in B* and anti–order isomorphic to the complete lattice ℱw*(B1) of weak*–closed faces of the unit ball B1 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 closed unit ball in the predual of the hermitification of B. Furthermore, every weak*–closed face of B1 is weak*–semi–exposed, and, if non–empty, of the form u + B0(u)1 where u is a tripotent in B and B0(u)1 is the closed unit ball in the zero Peirce space B0(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 subspace J of B is said to be an inner ideal if {J B J}B is contained in J and J is said to be complemented if B is the direct sum of J and the subspace Ker(J) defined to be the set of elements b in B such that, for all elements a in J, {a b a}B is equal to zero. It is shown that every weak*–closed inner ideal in B is complemented or, equivalently, the range of a structural projection. The results are applied to JBW–algebras, real W*–algebras and certain real Cartan factors.

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