Symmetric 1-by-1 complex Jordan algebra

The Symmetric 1-by-1 complex Jordan algebra is the $\mathbb C$ -vector space

\operatorname{Herm}_{1}((\mathbb R\otimes_{\mathbb R}\mathbb C))=\{X\in M_{1}((\mathbb R\otimes_{\mathbb R}\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. That conjugation is extended complex-linearly, so it fixes the newly adjoined complex scalars.

Coordinates and dimension

The diagonal contributes $1$ scalar coordinates, and the off-diagonal pairs contribute $0$ , giving dimension $1$ over $\mathbb C$ . Here the coefficient involution is the identity, so the condition is $X^T=X$ , not $\overline X^T=X$ with ordinary complex conjugation.

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 $\mathbb C1$ , so $[a]\mapsto a$ identifies this one-dimensional Jordan algebra with $\mathbb C$ .

Scalar convention

This is the complexification of the corresponding real Hermitian Jordan algebra. Its complex-bilinear product and real restriction give different category views; it is not the space of ordinary self-adjoint matrices on a complex Hilbert space.