On Complemented Subsystems of Commutative Jordan Quadruple Systems

Abstract

We extend the notions of commutativity, ideals, anisotropy, and complemented subtriples of Jordan triple systems to those of Jordan quadruple systems. We show that if [Formula: see text] is a complemented subsystem of an anisotropic commutative Jordan quadruple system [Formula: see text], then [Formula: see text] and its annihilator [Formula: see text] are orthogonal ideals and [Formula: see text]. We also prove that the range of a structural projection on an anisotropic commutative Jordan quadruple system is a complemented ideal and, conversely, a complemented subsystem of an anisotropic commutative Jordan quadruple system is the range of a unique structural projection.