THE RANGE OF A STRUCTURAL PROJECTION

Let A be a JBW-triple. A linear subspace J of A is called an inner id eal in A provided that the subspace {J A J} is contained in J. A subtr iple B in A is said to be complemented if A = B + Ker(B), where Ker(B) = {a is an element of A : {B a B} = 0}. A complemented subtriple in A is a weak-closed inner ideal. A linear projection on A is said to be structural if, for all elements a, b and c in A, {P a b Pc} = P{a Pb c}. The range of a structural projection is a complemented subtriple a nd, conversely, a complemented subtriple is the range of a unique stru ctural 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 co mpatible tripotents. This result is then used to show that every weak closed inner ideal J in a JBW-triple A is a complemented subtriple i n A and therefore the range of a unique structural projection on A. As an application structural projections on W-algebras are considered. (C) 1996 Academic Press, Inc.