Compatible subtriples of Jordan \(^*\)-triples
In the structure theory and coordinatization of Jordan triples, mixed Peirce decompositions for sets of compatible tripotents are a major tool. Compatibility for tripotents in Jordan triple systems was introduced by K. McCrimmon: two tripotents \(e\) and \(f\) are \textsl{compatible} if the operators \(Q(e)^2\), \(D(e,e)\) commute with \(Q(f)^2\), \(D(f,f)\). For compatible tripotents \(e\) and \(f\) in a Jordan triple system \(J\) there exists a mixed Peirce decomposition \(J=\bigoplus_{i,j=0,1,2}J_i(e)\cap J_j(f)\). In this paper, the authors extend this notion to that of subtriples of anisotropic Jordan \(*\)-triples. For a subtriple \(S\) of a Jordan \(*\)-triple \(J\), the \textsl{Peirce subspaces} \(S_i\) (\(i=0,1,2\)) are defined as \(S_2=S\), \(S_1=\ker (S)\cap\ker (S^\bot)\) and \(S_0=S^\bot\), where \(\ker (S)=\{ x\in J: \{ SxS\} =0\}\) and \(S^\bot=\{ x\in J:\{ SxJ\} =0\}\). The subtriples \(S\) and \(T\) are said to be compatible if \(J=\bigoplus_{i,j=0,1,2}S_i\cap T_j\). In section three it is proved that this concept generalizes the one above, since two tripotents \(e\) and \(f\) in a Jordan \(*\)-triple are compatible iff their Peirce \(2\)-spaces are compatibles. Moreover, it is shown that any tripotent \(e\in J\) in a self-compatible subtriple of \(J\) is compatible with itself. A self-compatible subtriple \(S\) of \(J\) is complemented in the sense that \(J=S\oplus\ker (B)\), which implies that \(S\) is an inner ideal of \(J\). Several applications to JBW\(^*\)-triples, abelian JBW\(^*\)-triples, W\(^*\)-algebras and Hilbert spaces are given, for instance, a subtriple \(S\) of a JBW\(^*\)-triple \(J\) is a complemented ideal in \(J\) iff it is compatible with any self-compatible subtriple \(T\).
scientific article; zbMATH DE number 1340597
scientific article