Nonunital Hermitian Jordan corner embedding

Statement

For integers $1\leq r<s$ , the upper-left corner map is

\operatorname{Herm}_r(A)\longrightarrow\operatorname{Herm}_s(A), \qquad X\longmapsto\begin{pmatrix}X&0\\0&0\end{pmatrix}.

Here both Hermitian spaces must be Jordan algebras over the same field; for octonionic coefficients require $s\leq3$ . The map is an injective Jordan homomorphism, but it does not preserve the unit.

Product and unit check

Multiplication of two displayed block matrices has the source product in the upper-left corner and zero elsewhere. This proves preservation of the symmetrized product. The source unit maps to $\operatorname{diag}(I_r,0)$ , which differs from $I_s$ .

Thus the map belongs to $\mathbf{UJord}$ under the catalogue convention and also to $\mathbf{Jord}$ ; it is not a morphism in $\mathbf{Jord}_1$ . This is a change in the permitted arrows, while the source and target remain unital Jordan objects.

Example

The corner embedding from $\operatorname{Herm}_2(\mathbb C)$ into $\operatorname{Herm}_3(\mathbb C)$ sends the unit to $\operatorname{diag}(1,1,0)$ . Both objects are real Jordan algebras.