PhyslibAlpha.ProbabilisticTheory.JordanOrderUnit.JB.Dynamics

In a normed Jordan algebra $E$ over $\mathbb{R}$, the Jordan product $(x, y) \mapsto x \cdot y$ is a bounded bilinear map. Specifically, the norm of the product is bounded by the product of the norms of its factors, satisfying $\|x \cdot y\| \le \|x\| \|y\|$ for all $x, y \in E$.

In a normed Jordan algebra $E$, for any fixed element $a \in E$, the left Jordan multiplication map $x \mapsto a \cdot x$ is continuous.

Let $E$ be a normed Jordan algebra. For any fixed element $a \in E$, the right multiplication map $x \mapsto x \cdot a$ is continuous.

For an element $a$ in a normed Jordan algebra $E$ over $\mathbb{R}$, the left Jordan multiplication operator $L_a : E \to E$ is the continuous linear map defined by $L_a(x) = a \cdot x$ for any $x \in E$, where the operator norm $\|L_a\|$ is bounded by $\|a\|$.

Let $E$ be a normed Jordan algebra. For any elements $a, b \in E$, applying the continuous linear left multiplication operator $L_a$ (denoted by `mulLeftCLM a`) to $b$ yields the Jordan product $a \cdot b$. That is, $L_a(b) = a \cdot b$.

In a normed Jordan algebra $E$, for any elements $a, b \in E$, the quadratic representation $U_a$ satisfies the norm inequality: $$\|U_a(b)\| \leq 3 \|a\|^2 \|b\|$$ where $U_a$ is the linear map defined by $U_a = 2L_a^2 - L_{a^2}$, and $L_a$ is the Jordan left-multiplication operator $L_a(x) = a \cdot x$.

Let $E$ be a normed Jordan algebra. For any element $a \in E$, the quadratic representation $U_a$ is the continuous $\mathbb{R}$-linear map from $E$ to $E$ (denoted by $U_a \in \mathcal{L}(E, E)$). It is defined as $U_a = 2L_a^2 - L_{a^2}$, where $L_a$ is the Jordan left-multiplication operator $L_a(x) = a \cdot x$ and $a^2 = a \cdot a$. The continuity of this map is provided by the norm bound $\|U_a(b)\| \le 3\|a\|^2 \|b\|$.

Let $E$ be a normed Jordan algebra. For any elements $a, b \in E$, the continuous linear map version of the quadratic representation, denoted as `quadRepCLM a`, evaluated at $b$ is equal to the quadratic representation $U_a(b)$.

Let $E$ be a normed Jordan algebra. Let $\alpha_t: E \to E$ be a one-parameter family of maps (indexed by $t \in \mathbb{R}$) such that $\alpha_0$ is the identity map ($\alpha_0(a) = a$ for all $a \in E$) and each $\alpha_t$ is a Jordan homomorphism, satisfying $\alpha_t(a \cdot b) = \alpha_t(a) \cdot \alpha_t(b)$ for all $a, b \in E$. If a linear operator $D: E \to E$ is the infinitesimal generator of the family $\alpha_t$, then $D$ is a Jordan derivation, meaning it satisfies the Leibniz rule $D(a \cdot b) = D(a) \cdot b + a \cdot D(b)$ for all $a, b \in E$.