Coordinatization of triangulated Jordan systems
Let J be a Jordan triple system. J is called triangulated if there is a Peirce decomposition \(J=J_{11}\oplus J_{12}\oplus J_{22}\) relative to an orthogonal tripotent \((e_ 1,e_ 2)\) and if there is \(u\in J_{12}\) such that \(P(e)u=u\) and \(P(u)e=e\) for \(e=e_ 1+e_ 2\). J is called faithfully triangulated if it is triangulated and if \(\{x_ 1e_ 1u\}=0\) implies that \(x_ 1=0\) for \(x_ 1\in J_ 1\). If J is a faithfully triangulated triple system, the authors determine two subspaces of J, one isomorphic to a diagonal system of \(2\times 2\) Hermitian matrices, and one isomorphic to an ample Clifford system determined by a quadratic form. When J is simple, it equals at least one of these subspaces, and this yields a classification of simple triangulated Jordan triple systems. By coordinatizing non-semisimple systems, the authors describe unital bimodules for diagonal Hermitian matrix systems.
simple triangulated Jordan triple systems
scientific article; zbMATH DE number 4061451