The central hull and central kernel in JBW*-triples

Journal article

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 (J_{1,/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^⊥)).}

Authors: Edwards, C; Ruttimann, G

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: Elsevier
• Journal: Journal of Algebra
• Volume: 250
• Issue: 1
• Pages: 90-114
• Publication date: 2002-04-01
• DOI: 10.1006/jabr.2001.9097
• ISSN: 0021-8693