The Range of a Structural Projection

LetAbe a JBW*-triple. A linear subspaceJofAis called aninner idealinAprovided that the subspace {J A J} is contained inJ. A subtripleBinAis said to becomplementedifA=B⊕Ker(B), where Ker(B)={a∈A : {B a B}=0}. A complemented subtriple inAis a weak*-closed inner ideal. A linear projection onAis said to bestructuralif, for all elementsa,bandcinA,[formula]The range of a structural projection is a complemented subtriple and, conversely, a complemented subtriple is the range of a unique structural projection. We analyze the structure of the weak*-closed inner ideal generated by two arbitrary tripotents in a JBW*-triple in terms of the simultaneous Peirce spaces of three suitably chosen pairwise compatible tripotents. This result is then used to show that every weak* closed inner idealJin a JBW*-tripleAis a complemented subtriple inAand therefore the range of a unique structural projection onA. As an application structural projections on W*-algebras are considered.