Peirce Inner Ideals in Jordan*-Triples Original Research Article

A subspaceJof an anisotropic Jordan*-tripleAis said to be aninner idealif the subspace {J A J} is contained inJ. An inner idealJinAis said to becomplementedifAis equal to the sum ofJand thekernelKer(J) ofJ, defined to be the subspace ofAconsisting of elementsainAfor which {J a J} is equal to {0}. TheannihilatorJperpendicularof an inner idealJinAis the inner ideal consisting of elementsainAsuch that {J a A} is equal to {0}. When bothJandJperpendicularare complemented,Acan be decomposed into the direct sum ofJ, Ker(J)∩Ker(Jperpendicular) andJperpendicular. Modulo six of the generalized Peirce relations this decomposition is a grading ofAof Peirce type. Since an inner ideal in a JBW*-triple is complemented if and only if it is weak*-closed, the result described above applies to all weak*-closed inner idealsJin a JBW*-tripleA. Furthermore, it can be shown that in this case all except five of 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 hold, 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*-tripleAleads to a grading ofAwhich 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.