PhyslibAlpha.ProbabilisticTheory.JordanOrderUnit.JB.GeneratedByOne.Inherited

For an element $a$ in a normed Jordan algebra $E$, the closed Jordan subalgebra $C(a)$ generated by $a$ and the identity element $1$ is itself a normed Jordan algebra. The norm, metric, and $\mathbb{R}$-module structures on $C(a)$ are inherited from the ambient algebra $E$. While the Jordan product in $E$ is generally non-associative, the multiplication restricted to $C(a)$ is associative, and $C(a)$ satisfies the submultiplicative property $\|x \cdot y\| \leq \|x\| \cdot \|y\|$ and the Jordan identity $(x^2 \cdot y) \cdot x = x^2 \cdot (y \cdot x)$ for all $x, y \in C(a)$.

If the ambient normed Jordan algebra $E$ is nontrivial, then the norm-closed Jordan subalgebra $C(a)$ generated by the identity element $1$ and an element $a \in E$ is also nontrivial.

Let $E$ be a nontrivial normed Jordan algebra and let $a \in E$. Let $C(a)$ be the norm-closed Jordan subalgebra generated by the identity element $1$ and $a$. Then the norm of the identity element in $C(a)$ is equal to $1$, i.e., $\|1\| = 1$.

Let $E$ be a JB-algebra and let $a \in E$. The norm-closed Jordan subalgebra $C(a)$ generated by the identity element $1$ and the element $a$ is itself a JB-algebra.

Let $E$ be a JB-algebra and $a \in E$. The norm-closed Jordan subalgebra $C_J(a) = \overline{J[a]}$ generated by the identity element $1$ and $a$ is an Archimedean order-unit space. This space inherits its partial order and scalar multiplication from the ambient algebra $E$, and its order unit is the identity element $1$.

Let $E$ be a JB-algebra and let $a \in E$. The norm-closed Jordan subalgebra $C(a)$ generated by the identity element $1$ and the element $a$ is a JB order-unit algebra. This means that $C(a)$ inherits the structure of an Archimedean order-unit space from $E$, where the Jordan product satisfies $x^2 \ge 0$ for all $x \in C(a)$, and the norm on $C(a)$ coincides with the order-unit norm defined by the identity $1$.