Compatible Subtriples of Jordan-Triples

The purpose of this paper is to extend the notion of compatibility of tripotents in an anisotropic Jordan*-triple A to that of subtriples of A. For a subtriple B of A define the kernel Ker(B) and the annihilator B of B to be the subsets {a A: {B a B} = {0}} and {a A: {B a A} = {0}}, respectively. The Peirce spaces Bi, i = 0, 1, 2, of B are defined in terms of the kernel and the annihilator by B0 = B , B1 = Ker(B) ∩ Ker(B ), and B2 = B. A pair (B, C) of subtriples of A is said to be compatible if A coincides with i, j = 0, 1, 2Bi ∩ Cj. The subtriple B is said to be self-compatible if it is compatible with itself. A self-compatible subtriple B is complemented in that B Ker(B) coincides with A, which implies that B is an inner ideal in A. It follows that a pair (u, v) of tripotents in A is compatible if and only if the ranges im P2(u), im P2(v) of the Peirce projections P2(u) and P2(v) form a compatible pair of subtriples. Moreover, for any tripotent u, im P2(u) is a self-compatible subtriple of A. For the special case of a JBW*-triple A it is shown that a subtriple I is compatible with every self-compatible subtriple J if and only if I is a complemented ideal in A. These results are then applied to abelian JBW*-triples, W*-algebras, and Hilbert spaces.