The Central Hull and Central Kernel in JBW*-Triples,

The complete lattice (A) of weak*-closed inner ideals in a JBW*-triple A has as its centre the complete Boolean algebra (A) of weak*-closed ideals in A. The annihilator L of the subset L of A consists of elements b of A for which {L b A} is equal to zero, and the kernel Ker(L) of L consists of those elements b in A for which {L b L} is equal to zero. For each element J of (A), J also lies in (A), and A enjoys the generalized Peirce decompositionA = J MJ J1,where J1 is the intersection of the kernels of J and J . To investigate the properties of the weak*-closed subspace J1 of A, which is not, in general, a subtriple, the notions of the central hull c(L) and central kernel k(L) of a subspace L are introduced. These are, respectively, the smallest element of (A) containing L and the largest element of (A) contained in L. For any element J of (A), the relationships that exist between the central hull and central kernel of J and J are examined and it is shown that (J1) ∩ J is the weak*-closed ideal k(J), that (J1) ∩ J is the weak*-closed ideal k(J ), and, when J is a Peirce inner ideal, that (J1) is the weak*-closed ideal (k(J) Mk(J )).