Central kernels of subspaces of JB\(^{*}\)-triples
For any closed subspace \(L\) of a complex Banach space \(A\), there exists a greatest M-ideal \(k_n(L)\) of \(A\) contained in \(L\). The space \(k_n (L)\) is called the norm central kernel of \(L\) in \(A\). When \(A\) is a dual space and \(L\) is a weak*-closed subspace, there exists a greatest M-summand \(k(L)\) of \(A\) contained in \(A\). \(k(L)\) is termed the central kernel of \(L\) in \(A\). The definition of these two central kernels is given by purely geometric statements. The results of the paper under review describe the norm central kernel of an arbitrary closed subspace \(L\) of a JB*-triple \(A\) and the central kernel of a weak*-closed subspace \(M\) of a JBW*-triple \(W\). The main results of the paper establish a purely algebraic description of these two central kernels based on the JB*-triple structure existing in \(E\) and \(W\). It is the interplay between the geometric, holomorphic, and algebraic structure of JB*-triples which makes the latter an appropriate structure to describe central kernels of subspaces. The final section considers some specialisations to the settings of C*-algebras and von Neumann algebras. We should also add that the notion of the central kernel of a weak*-closed subspace \(L\) of a JBW*-triple \(W\) was already introduced and studied in [\textit{C.\,M.\thinspace Edwards} and \textit{G.\,T.\,Rüttimann}, J.~Algebra 250, No.\,1, 90--114 (2002; Zbl 1019.17010)].
scientific article; zbMATH DE number 5161749
scientific article