Herm_1(C) Jordan algebra

The Herm_1(C) Jordan algebra is the $\mathbb R$ -vector space

\operatorname{Herm}_{1}(\mathbb C)=\{X\in M_{1}(\mathbb C):X^*=X\}, \qquad X\circ Y=\tfrac12(XY+YX),

with unit $I_{1}$ . Here $X^*=\overline X^T$ , using the standard conjugation of the coefficient algebra.

Coordinates and dimension

The diagonal contributes $1$ scalar coordinates, and the off-diagonal pairs contribute $0$ , giving dimension $1$ over $\mathbb R$ . In particular, the usual Hermitian complex matrices form a real vector space; multiplication by $i$ does not preserve self-adjointness.

Jordan identity and range

The fixed subspace is closed under $X\circ Y$ , because $(XY)^*=Y^*X^*$ . Associativity of the ambient matrix algebra proves the Jordan identity, so the construction works in every positive size. Conjugation-fixed coefficients are exactly $\mathbb R1$ , so $[a]\mapsto a$ identifies this one-dimensional Jordan algebra with $\mathbb R$ .