PEIRCE INNER IDEALS IN JORDAN-ASTERISK-TRIPLES

A subspace J of an anisotropic Jordan-triple A is said to be an inner ideal if the subspace {J A J} is contained in J. An inner ideal J in A is said to be complemented if A is equal to the sum of J and the ker nel Ker(J) of J, defined to be the subspace of A consisting of element s a in A for which (J a J) is equal to {O}. The annihilator J(perpendi cular to) of an inner ideal J in A is the inner ideal consisting of el ements a in A such that {J a A} is equal to {O}. When both J and J per pendicular to are complemented, A can be decomposed into the direct su m of J, Ker(J) boolean AND Ker(J(perpendicular to)) and J(perpendicula r to). Module six of the generalized Peirce relations this decompositi on is a grading of A of Peirce type. Since an inner ideal in a JBW-tr iple is complemented if and only if it is weak-closed, the result des cribed above applies to all weak-closed inner ideals J in a JBW*-trip le A. Furthermore, it can be shown that in this case all except five o f the generalized Peirce relations hold, and an example is given of a weak-closed inner ideal in a JBW*-triple for which all five fail to h old, thereby showing that the result is the best possible. It is also shown that the condition that a weak-closed inner ideal in a JBW*-tri ple A leads to a grading of A which is of Peirce type is equivalent to several other conditions, all of a topological, rather than algebraic , nature. These results are applied to W-algebras, spin triples, and the bi-Cayley triple. (C) 1996 Academic Press, Inc.