The central hull and central kernel in JBW\(^*\)-triples
Let \(A\) be a JBW*-triple. The authors introduce the concepts of central hull \(c(L)\) and central kernel \(k(L)\) of an arbitrary subspace \(L\) of \(A.\) These are, respectively, the smallest element of the complete Boolean algebra \({\mathcal ZI}(A)\) of the weak*-closed ideals in \(A\) containing \(L\) and the largest element in \({\mathcal ZI}(A)\) contained in \(L.\) For every subset \(L\) of \(A\) we have the weak*-closed inner ideal \(L^{\bot },\) which is said to be the annihilator of \(L,\) whose elements are the \(b\in A\) for which \(\{ L b A\} = 0.\) On the other hand, the \(\ker (L)\) of \(L\) is the weak*-closed subspace of the elements \(b\) in \(A\) such that \(\{ L b L \} = 0.\) If \(J\) is a weak*-closed inner ideal in \(A\) then \(A\) has a generalized Peirce decomposition \(A = J\oplus J_1 \oplus J^{\bot },\) with \(J_1 = \ker (J)\cap J^{\bot }.\) The Peirce subspace \(J_1\) is a weak*-closed subspace but in general not a subtriple of \(A.\) The main results of this paper assert: (i) \((J_1)^{\bot }\cap J = k(J),\) (ii) \((J_1)^{\bot }\cap J^{\bot } = k(J^{\bot }),\) (iii) \((J_1)^{\bot}\) is the weak*-closed ideal \(k(J)\oplus k(J^{\bot })\) when \(J\) is a Peirce inner ideal.
scientific article; zbMATH DE number 1762092
scientific article