The central hull and central kernel in JBW*-triples - Edwards, C; Ruttimann, G (Elsevier, 2002)
The complete lattice J(A) of weak*-closed inner ideals in a JBW*-triple A has as its centre the complete Boolean algebra LJ(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 J(A), J⊥ also lies in J(A), and A enjoys the generalized Peirce decomposition A=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 LJ(A) containing L and the largest element of LJ(A) contained in L. For any element J of J(A), the relationships that exist between the central hull and central kernel of J and J⊥ are examined and it is shown that (J1,/sub>)⊥ ∩ 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⊥)).