PhyslibAlpha.AlgebraicFramework.JordanOrderUnit.Quadratic.Order

In a Jordan algebra $E$, for any elements $a, b \in E$, if $b$ is nonnegative ($0 \le b$), then its image under the quadratic representation $U_a$, denoted $U_a(b)$, is also nonnegative ($0 \le U_a(b)$).

For an element $a$ in a Jordan algebra $E$, the quadratic representation $U_a$ is the positive linear map from $E$ to $E$ defined by the formula $U_a(b) = 2 a \circ (a \circ b) - (a \circ a) \circ b$, where $\circ$ denotes the Jordan product. This map is bundled with the property that it preserves the positive cone, meaning that for any $b \in E$, if $b \ge 0$, then $U_a(b) \ge 0$.

For any element $a$ in a Jordan algebra $E$, the underlying $\mathbb{R}$-linear map of the positive linear map `quadRepPositiveLinearMap a` is equal to the quadratic representation $U_a$.

Let $E$ be a Jordan algebra. For any elements $a, b \in E$, the application of the quadratic representation of $a$ bundled as a positive linear map (denoted `quadRepPositiveLinearMap a`) to the element $b$ is equal to the value of the quadratic representation operator $U_a$ applied to $b$. Here, $U_a$ is the linear map defined by $U_a(x) = 2 a \circ (a \circ x) - (a \circ a) \circ x$, where $\circ$ is the Jordan product.

Let $E$ be a Jordan algebra with unit element $1$. The quadratic representation $U_1$, viewed as a positive linear map from $E$ to $E$, is equal to the identity positive linear map $\text{id}_E$.

Let $E$ be a Jordan algebra and let $a$ be an element in $E$. The quadratic representation $U_a: E \to E$, defined as $U_a = 2 L_a^2 - L_{a^2}$ (where $L_a$ is the left multiplication operator), is a monotone operator. Specifically, for any $x, y \in E$ such that $x \le y$ with respect to the natural ordering of the Jordan algebra, it holds that $U_a(x) \le U_a(y)$.