Jordan idempotent - Knowlpedia

Definition

An element $e$ of a Jordan algebra $J$ is a Jordan idempotent when

e\circ e=e.

In a special Jordan algebra whose product is $x\circ y=(xy+yx)/2$ , this is exactly the usual equation $e^2=e$ .

Role in the structure theory

Left Jordan multiplication by $e$ , $L_e(x)=e\circ x$ , organizes $J$ into its Peirce spaces. In a unital Euclidean Jordan algebra, idempotents behave like orthogonal projections: they can be decomposed into primitive idempotents, and maximal compatible decompositions form Jordan frames.

The comparison with projections is exact for Hermitian-matrix Jordan algebras. There an element is a Jordan idempotent precisely when it is a self-adjoint projection, and its Jordan rank agrees with the ordinary matrix rank.

References

Kevin McCrimmon, A Taste of Jordan Algebras, Springer, 2004, §§5.1 and 13.1. Publisher record.
Jacques Faraut and Adam Korányi, Analysis on Symmetric Cones, Oxford University Press, 1994, Chapter III. Publisher record.