PhyslibAlpha.ProbabilisticTheory.JordanOrderUnit.JB.GeneratedByOne.SqrtUniqueness

For an element $x$ of a Jordan algebra $E$ and a real polynomial $p(X) = \sum_{i=0}^n c_i X^i \in \mathbb{R}[X]$, this function computes the evaluation $p(x) = \sum_{i=0}^n c_i x^i \in E$. Here, $x^i$ denotes the $i$-th Jordan power of $x$, defined recursively by $x^0 = 1$ and $x^{k+1} = x \circ x^k$ where $\circ$ is the Jordan product. This definition is "ambient-independent," meaning it relies solely on the Jordan multiplication of $x$ and does not require reference to a specific closed one-generator subalgebra.

Let $E$ be a normed Jordan algebra and $u \in E$. Let $C(u)$ be the closed Jordan subalgebra generated by the identity $1$ and $u$. For any polynomial $p \in \mathbb{R}[X]$ and any element $x \in C(u)$, the standard polynomial evaluation $p(x)$ (as defined in the associative algebra $C(u)$) is equal to the Jordan polynomial evaluation $p(x)$ computed in the ambient algebra $E$.

Let $E$ be a Jordan algebra and $x \in E$ be an element. For any two real polynomials $p, q \in \mathbb{R}[X]$, the Jordan polynomial evaluation of their sum at $x$ is equal to the sum of their individual evaluations at $x$. That is, $$(p + q)(x) = p(x) + q(x)$$ where $p(x)$ denotes the evaluation $\sum_{i=0}^n c_i x^i$ using Jordan powers of $x$.

Let $E$ be a Jordan algebra and $v \in E$. For any real polynomial $p \in \mathbb{R}[X]$, let $p(v)$ denote the Jordan polynomial evaluation of $p$ at $v$ (where powers $v^n$ are computed using the Jordan product $\circ$). Then, evaluating the composed polynomial $p(X^2)$ at $v$ is equivalent to evaluating $p$ at $v^2 = v \circ v$. That is, $(p \circ X^2)(v) = p(v^2)$.

Let $E$ be a nontrivial normed Jordan algebra. For any element $a \in E$ and any real polynomial $p \in \mathbb{R}[X]$, let $f_p: \sigma(a) \to \mathbb{R}$ be the continuous function defined on the Jordan spectrum $\sigma(a)$ of $a$ by the evaluation of $p$. Then the Jordan continuous functional calculus of $a$ applied to $f_p$ is equal to the Jordan polynomial evaluation $p(a)$: $$\text{jordanCfc}(a, f_p) = p(a)$$ where $p(a)$ denotes the evaluation of the polynomial at $a$ using the Jordan product structure.

Let $E$ be a nontrivial normed Jordan algebra and $a \in E$ be an element with Jordan spectrum $\sigma(a) \subseteq \mathbb{R}$. Let $p \in \mathbb{R}[X]$ be a real polynomial and $F: \sigma(a) \to \mathbb{R}$ be a continuous function. Suppose there exists a non-negative real number $\delta \ge 0$ such that for all $x \in \sigma(a)$, the inequality $|p(x) - F(x)| \le \delta$ holds. Then the norm of the difference between the Jordan polynomial evaluation $p(a)$ and the Jordan continuous functional calculus of $F$ at $a$ is bounded by $\delta$: $$\|p(a) - \text{jordanCfc}(a, F)\| \le \delta$$

Let $s \subseteq \mathbb{R}$ be a set and $C(s, \mathbb{R})$ denote the space of continuous real-valued functions on $s$. If a function $g \in C(s, \mathbb{R})$ is a polynomial function on $s$ (meaning it belongs to the set `polynomialFunctions s`), then there exists a real polynomial $p \in \mathbb{R}[x]$ such that the evaluation of $p$ on $s$ is equal to $g$.

For any real number $M$ and any positive real number $\epsilon > 0$, there exists a polynomial $p$ with real coefficients such that for all $s \in [0, M]$, the inequality $|p(s) - \sqrt{s}| < \epsilon$ holds.

Let $E$ be a nontrivial normed Jordan algebra. For any non-negative element $a \in E$ and any non-negative element $b \in E$, if $b^2 = a$ (where the square is taken with respect to the Jordan product), then $b$ is equal to the canonical positive square root of $a$, denoted by $\sqrt{a}$.