PhyslibAlpha.AlgebraicFramework.JordanOrderUnit.JB.GeneratedByOne.Spectrum

For an element $a$ in a normed Jordan algebra $E$, this definition represents $a$ viewed specifically as an element of the closed subalgebra it generates, denoted as $\text{ClosedGeneratedByOne } a$. In other words, it is the element $a$ bundled with the proof that it belongs to the closure of the subalgebra generated by $\{a\}$.

For any element $a$ in a normed Jordan algebra $E$, let $\text{closedGenerator } a$ be the element $a$ viewed as a member of the closed subalgebra generated by $\{a\}$. The value of $\text{closedGenerator } a$ when mapped back to the original algebra $E$ is equal to $a$.

For an element $a$ in a normed Jordan algebra $E$, the Jordan spectrum is the spectrum (over $\mathbb{R}$) of $a$ when viewed as an element of the closed commutative associative real subalgebra generated by $a$ (and the identity). This is denoted as the set of real numbers $\sigma(a)$ within this generated subalgebra.

In a nontrivial normed Jordan algebra $E$, the Jordan spectrum of the zero element $0 \in E$ (also referred to as the canonical spectrum of the zero observable) is the singleton set $\{0\}$.

Let $E$ be a nontrivial normed Jordan algebra. For any real scalar $r \in \mathbb{R}$, the Jordan spectrum of the scalar multiple of the identity element $r \cdot 1$ is the singleton set $\{r\}$. That is, $\sigma(r \cdot 1) = \{r\}$, where $\sigma(a)$ denotes the Jordan spectrum of an element $a \in E$.

Let $E$ be a nontrivial normed Jordan algebra. For any scalar $r \in \mathbb{R}$ and any real polynomial $p$, the image of the Jordan spectrum $\sigma(r \cdot 1)$ under the polynomial mapping $x \mapsto p(x)$ is the singleton set $\{p(r)\}$, where $\sigma(r \cdot 1)$ denotes the Jordan spectrum of the scalar multiple of the identity element.

Let $E$ be a complete normed Jordan algebra. For any element $a \in E$, its Jordan spectrum $\sigma(a) \subseteq \mathbb{R}$ is a compact set.

Let $E$ be a normed Jordan algebra and $a \in E$. Let $\sigma(a)$ be the Jordan spectrum of $a$, which is the spectrum of $a$ as an element of the closed commutative associative real subalgebra generated by $a$. For any real polynomial $p \in \mathbb{R}[X]$, the image of the Jordan spectrum under $p$ is a subset of the spectrum of the element $p(a)$ (evaluated in that same subalgebra): $$\{ p(x) \mid x \in \sigma(a) \} \subseteq \sigma(p(a)).$$

Let $E$ be a normed Jordan algebra and $x$ be an element in the closed subalgebra generated by an element $a \in E$. Let $p, q, r \in \mathbb{R}[X]$ be real polynomials and $\mu, l \in \mathbb{R}$ be real numbers such that $p(X) - \mu = q(X)(X - l)r(X)$. If the algebraic evaluations $q(x)$ and $r(x)$ are units (invertible elements) and $\mu$ is in the spectrum $\sigma(p(x))$, then $l$ is in the spectrum $\sigma(x)$.

Let $E$ be a Jordan-Banach (JB) algebra with an Archimedean order unit. Let $a \in E$, and let $x$ be an element in the closed real subalgebra generated by $a$. If $q \in \mathbb{R}[X]$ is a monic irreducible polynomial of degree $2$, then its algebraic evaluation at $x$, denoted $q(x)$, is a unit (i.e., it is invertible).

Let $x$ be an element of the closed real subalgebra generated by a single element $a$ in a normed Jordan algebra. If $q$ is a monic real polynomial of degree $1$ such that its evaluation at $x$, denoted $q(x)$, is not a unit (not invertible), then there exists a real number $l$ in the spectrum of $x$ (denoted $\text{spectrum}_{\mathbb{R}} \ x$) such that $q(l) = 0$.

Let $E$ be a JB-algebra (a Jordan Banach algebra) with an Archimedean order unit. Let $x$ be an element in the closed real subalgebra generated by an element $a \in E$. If $f$ is a monic real polynomial such that its evaluation at $x$, denoted $f(x)$, is not invertible, then there exists a real number $\lambda$ in the spectrum of $x$ (denoted $\text{spectrum}_{\mathbb{R}} \ x$) such that $f(\lambda) = 0$.

Let $E$ be a nontrivial JB-algebra (a Jordan Banach algebra equipped with an Archimedean order unit and a compatible partial order). For any element $x$ in the closed real subalgebra generated by an element $a \in E$, either the norm $\|x\|$ or its negative $-\|x\|$ belongs to the spectrum of $x$ over the real numbers, denoted $\text{spectrum}_{\mathbb{R}} \ x$.

Let $E$ be a nontrivial ordered JB algebra. For any element $a \in E$, let $x$ be an element in the closed subalgebra generated by $a$. Then the spectral radius of $x$ over the real numbers $\mathbb{R}$, denoted $\rho(x)$, is equal to its norm $\|x\|$.

Let $E$ be a nontrivial ordered JB-algebra (a Jordan Banach algebra equipped with an Archimedean order unit and a compatible partial order). For any element $a \in E$, the Jordan spectrum $\sigma(a)$ is non-empty. The Jordan spectrum $\sigma(a)$ is defined as the spectrum of $a$ over the real numbers $\mathbb{R}$ when $a$ is viewed as an element of the closed commutative associative real subalgebra generated by $a$ and the identity.

Let $E$ be a JB-algebra with an order unit. For any non-negative element $a \in E$ (such that $0 \le a$), every real number $r$ in the Jordan spectrum $\sigma(a)$ is non-negative ($0 \le r$).

Let $E$ be a JB-algebra with an order unit. For any element $a \in E$, let $x$ be an element in the closed subalgebra generated by $a$. For any real polynomial $p$ and any real number $\mu$, if $\mu$ belongs to the spectrum $\sigma(p(x))$, then there exists a real number $\lambda \in \sigma(x)$ such that $p(\lambda) = \mu$.

Let $E$ be a nontrivial normed Jordan algebra satisfying the conditions of a JB-algebra with an order unit. For any element $a \in E$ and any real polynomial $p \in \mathbb{R}[X]$, the image of the Jordan spectrum $\sigma(a)$ under the polynomial mapping $t \mapsto p(t)$ is equal to the spectrum of the polynomial evaluation $p(a)$ (where $p(a)$ is viewed as an element of the closed subalgebra generated by $a$). That is, $$\{ p(\lambda) \mid \lambda \in \sigma(a) \} = \sigma(p(a)).$$

Let $E$ be a nontrivial JB-algebra (a Jordan algebra equipped with a norm and order structure satisfying the JB-algebra axioms). For any element $a \in E$ and any real polynomial $p$, let $p(a)$ denote the evaluation of $p$ at $a$ within the closed subalgebra generated by $a$, and let $\sigma(a) \subseteq \mathbb{R}$ denote the Jordan spectrum of $a$. The norm of $p(a)$ is equal to the supremum of the absolute values of $p$ evaluated on the spectrum $\sigma(a)$: $$\|p(a)\| = \sup_{\lambda \in \sigma(a)} |p(\lambda)|.$$ (Note: In the formal statement, the values are expressed in the extended non-negative real numbers $[0, \infty]$ to accommodate the supremum operator).

In a nontrivial normed Jordan algebra $E$ equipped with an Archimedean order unit (a JB algebra), let $a \in E$ be an element and $p \in \mathbb{R}[x]$ be a polynomial. Let $p(a)$ denote the evaluation of the polynomial at $a$ within the closed subalgebra generated by $a$, and let $\sigma(a)$ denote the Jordan spectrum of $a$. Then, the norm of $p(a)$ is equal to the supremum of the absolute value of $p$ over the Jordan spectrum: $$\|p(a)\|_+ = \sup_{\lambda \in \sigma(a)} |p(\lambda)|_+$$ where the equality is expressed in the extended non-negative real numbers $[0, \infty]$.