Hermitian Jordan embeddings from coefficient inclusions

Statement

Let $A\hookrightarrow B$ be a unital real-algebra inclusion that preserves conjugation, and suppose both Hermitian constructions in size $m$ are Jordan algebras. Applying the inclusion to every matrix entry defines an injective unital Jordan map

\operatorname{Herm}_m(A)\hookrightarrow\operatorname{Herm}_m(B).

The standard coefficient inclusions give $\mathbb R\to\mathbb C\to\mathbb H$ for every positive size. The inclusion $\mathbb H\to\mathbb O$ is used only for $m\in\{1,2,3\}$ .

Product and unit

Conjugation preservation sends self-adjoint matrices to self-adjoint matrices. Entrywise application commutes with each finite sum and two-factor product in matrix multiplication; hence it preserves $(XY+YX)/2$ . Injectivity is checked entrywise, and the identity matrix maps to the identity matrix. These are therefore arrows in $\mathbf{Jord}_1$ , as well as in the two categories with weaker unit policies.

Choice of coefficients

The complex-to-quaternionic map uses a chosen imaginary quaternion, and the quaternion-to-octonionic map uses a chosen copy of $\mathbb H$ . These are specific embeddings, not a classification of every embedding.