Structural Projections on JBW*-Triples

A linear projection R on a Jordan*-triple A is said to be structural provided that, for all elements a, b and c in A, the equality {Rab Rc} = R{a Rbc} holds. A subtriple B of A is said to be complemented if A = B + Ker(B), where Ker(B) = {a∈A: {B a B} = 0}. It is shown that a subtriple of a JBW*-triple is complemented if and only if it is the range of a structural projection. A weak* closed subspace B of the dual E* of a Banach space E is said to be an N*-ideal if every weak* continuous linear functional on B has a norm preserving extension to a weak* continuous linear functional on E* and the set of elements in E which attain their norm on the unit ball in B is a subspace of E. It is shown that a subtriple of a JBW*-triple A is complemented if and only if it is an N*-ideal, from which it follows that complemented subtriples of A are weak* closed, and structural projections on A are weak* continuous and norm non-increasing. It is also shown that every N*-ideal in A possesses a triple product with respect to which it is a JBW*-triple which is isomorphic to a complemented subtriple of