Split-complex Hermitian matrices as full-matrix Jordan algebras

Statement

For $F=\mathbb R$ or $\mathbb C$ , equip $D=F\times F$ with conjugation $\overline{(a,b)}=(b,a)$ . Projection onto the first matrix component defines a unital Jordan isomorphism

\operatorname{Herm}_n(D)\ \cong\ M_n(F)^+, \qquad (X,X^T)\longmapsto X,

for every integer $n\geq1$ . On the right, the product is $X\circ Y=(XY+YX)/2$ .

Verification

The equation $(X,Y)^*=(X,Y)$ is exactly $Y=X^T$ . Thus the inverse map is $X\mapsto(X,X^T)$ . Symmetrization of the product in the second component gives $(XY+YX)^T/2$ , so projection preserves the Jordan product and the unit.

Catalogue instances

Over $\mathbb R$ , the coefficient algebra is $\mathbb C_s$ . Over $\mathbb C$ , it is $\mathbb C\otimes_{\mathbb R}\mathbb C$ , whose composition conjugation exchanges the two factors. Consequently the complexification of the ordinary real Jordan algebra $\operatorname{Herm}_n(\mathbb C)$ is $M_n(\mathbb C)^+$ . It is not the ordinary conjugate-self-adjoint subspace over $\mathbb C$ .