Category UJord of unital Jordan algebras with arbitrary maps

For a field $k$ of characteristic different from $2$ , $\mathbf{UJord}_k$ is the full subcategory of $\mathbf{Jord}_k$ whose objects possess a multiplicative unit. Morphisms remain all $k$ -linear Jordan-product-preserving maps; they are not required to preserve units. The identity and composite maps are those of Jord.

Full subcategory means the same Hom-sets

For unital $J,K$ ,

\operatorname{Hom}_{\mathbf{UJord}_k}(J,K) =\operatorname{Hom}_{\mathbf{Jord}_k}(J,K).

The unit of each object is unique, but its image under a homomorphism may be a proper idempotent or zero. An inclusion of a matrix corner is a basic example. This use of UJord is an explicit catalogue convention rather than a universal notation.

Automorphisms

A surjective homomorphism between unital Jordan algebras preserves the unit: if $b=f(a)$ , then $f(1)\circ b=f(1\circ a)=b$ . Thus bijective homomorphisms agree with the automorphisms in the unit-preserving category.