Euclidean Jordan algebra

Definition

A Euclidean Jordan algebra is a finite-dimensional real unital Jordan algebra $J$ , together with a positive-definite inner product $\langle-,-\rangle$ , such that

\langle x\circ y,z\rangle=\langle y,x\circ z\rangle \qquad(x,y,z\in J).

Equivalently, every multiplication operator $L_x(y)=x\circ y$ is self-adjoint.

Euclidean versus formally real

A real Jordan algebra is formally real if

x_1^2+\cdots+x_m^2=0 \quad\Longrightarrow\quad x_1=\cdots=x_m=0.

Formal reality is a property of the algebra; a Euclidean structure includes a chosen compatible inner product. In finite dimensions, a real unital Jordan algebra is formally real exactly when it admits such a Euclidean inner product. The terms are therefore often used interchangeably when classifying algebras, but they are not literally identical data.

Spectral theorem

Every $x\in J$ admits a spectral decomposition

x=\lambda_1c_1+\cdots+\lambda_rc_r,

where the $c_i$ are pairwise orthogonal primitive idempotents summing to the unit. A maximal such family is a Jordan frame, and its cardinality is the rank. This gives a functional calculus, the Jordan trace $\operatorname{tr}_J(x)=\sum_i\lambda_i$ , and the determinant $\det_J(x)=\prod_i\lambda_i$ .

Cone of squares

The cone

J_+=\{x^2:x\in J\}

is closed, convex, and self-dual, and its interior is homogeneous. Conversely, every homogeneous self-dual open cone arises from a Euclidean Jordan algebra. This is the bridge between Jordan theory and symmetric cones.

Classification

Every Euclidean Jordan algebra is an orthogonal direct sum of simple Euclidean Jordan algebras. The simple factors are the real, complex, and quaternionic Hermitian matrix families, spin factors, and $\mathfrak h_3(\mathbb O)$ .

References

Jacques Faraut and Adam Korányi, Analysis on Symmetric Cones, Oxford University Press, 1994. Publisher record.
Pascual Jordan, John von Neumann, and Eugene Wigner, “On an Algebraic Generalization of the Quantum Mechanical Formalism,” Annals of Mathematics 35 (1934), 29–64. JSTOR record.