PhyslibAlpha.AlgebraicFramework.JordanOrderUnit.JB.GeneratedByOne.Inherited

Let $E$ be a normed Jordan algebra and let $a \in E$. The norm-closed Jordan subalgebra $C_J(a) = \overline{J[a]}$ generated by the identity element $1$ and $a$ is a normed Jordan algebra. It inherits the norm, addition, scalar multiplication, and the Jordan product $x \cdot y$ from the ambient algebra $E$. Furthermore, while the multiplication in $E$ is generally non-associative, the multiplication restricted to $C_J(a)$ is associative.

Let $E$ be a nontrivial normed Jordan algebra. For any element $a \in E$, the norm-closed Jordan subalgebra $C_J(a)$ generated by the identity $1$ and $a$ is also nontrivial.

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

Let $E$ be a JB algebra and let $a \in E$. The norm-closed Jordan subalgebra $C_J(a)$ (also denoted as $\overline{J[a]}$) generated by the identity element $1$ and $a$ is itself a JB algebra.

In the norm-closed Jordan subalgebra $C_J(a)$ generated by the identity and an element $a$ of a normed Jordan algebra, scalar multiplication by non-negative real numbers preserves the partial order. That is, for any $c \in \mathbb{R}$ such that $c \ge 0$ and any elements $x, y \in C_J(a)$, if $x \le y$, then $c \cdot x \le c \cdot y$.

In a JB algebra $E$, for any element $a \in E$, the identity element $1$ is an order unit for the norm-closed Jordan subalgebra $C_J(a)$ generated by $1$ and $a$. This means that for every element $x \in C_J(a)$, there exists a real number $c > 0$ such that $-c \cdot 1 \le x \le c \cdot 1$.

Let $E$ be a JB algebra and let $a \in E$. 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 implies that the identity $1$ is an order unit for $C_J(a)$ and the partial order inherited by $C_J(a)$ satisfies the Archimedean property.

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 a Jordan order unit algebra. This means that $C_J(a)$ is a Jordan algebra where the identity element $1$ acts as an order unit and the square of any element $x \in C_J(a)$ is non-negative ($x^2 \ge 0$).

Let $E$ be a JB algebra and let $a \in E$. The norm-closed Jordan subalgebra $C_J(a) = \overline{J[a]}$ generated by the identity element $1$ and $a$ is a JB-order unit space. This signifies that $C_J(a)$ inherits the compatibility between the norm and the order structure from the ambient algebra, where the norm is the order unit norm and the Jordan-specific norm-order laws are satisfied.