Jordan Algebra - Peirce Decomposition

Peirce Decomposition

If e is an idempotent in a Jordan algebra A (e2=e) and R is the operation of multiplication by e, then

• R(2R−1)(R−1) = 0

so the only eigenvalues of R are 0, 1/2, 1. If the Jordan algebra A is finite-dimensional over a field of characteristic not 2, this implies that it is a direct sum of subspaces A = A₀(e) ⊕ A_1/2(e) ⊕ A₁(e) of the three eigenspaces. This decomposition was introduced by Albert (1947) and is called the Peirce decomposition of A relative to the idempotent e.