PhyslibAlpha.ProbabilisticTheory.JordanOrderUnit.JB.GeneratedByOne.CFC

Let $E$ be a normed Jordan algebra (such as a JB-algebra). For any element (observable) $a \in E$, the Jordan spectrum of $a$, denoted by $\text{jordanSpectrum}(a)$, is a compact topological space.

Let $E$ be a nontrivial normed Jordan algebra and $a \in E$ be an element. Let $\sigma(a)$ denote the Jordan spectrum of $a$. For any real polynomial $p \in \mathbb{R}[x]$, the norm of the polynomial evaluation $p(a)$ (viewed as an element of the closed subalgebra $C(a)$ generated by $a$ and the identity) is equal to the supremum norm of the continuous function induced by $p$ on the spectrum $\sigma(a)$. Mathematically, $$\|p(a)\| = \sup_{\lambda \in \sigma(a)} |p(\lambda)|$$

For any element $a$ in a normed Jordan algebra $E$, the set of polynomial functions restricted to its Jordan spectrum $\sigma(a)$ is dense in the algebra $C(\sigma(a), \mathbb{R})$ of continuous real-valued functions on the spectrum.

Let $E$ be a normed Jordan algebra and $a \in E$ be an element. Let $\sigma(a)$ denote the Jordan spectrum of $a$. For every function $f$ in the space of polynomial functions on the spectrum $\sigma(a)$, there exists a real polynomial $p \in \mathbb{R}[x]$ such that the continuous function induced by $p$ on $\sigma(a)$ is equal to $f$.

Let $E$ be a normed Jordan algebra and $a \in E$ be an element with Jordan spectrum $\sigma(a)$. Given a function $f$ in the space of polynomial functions on $\sigma(a)$, this definition selects a representative polynomial $p \in \mathbb{R}[x]$ such that the continuous function induced by $p$ on $\sigma(a)$ is equal to $f$.

Let $E$ be a normed Jordan algebra and $a \in E$ be an element with Jordan spectrum $\sigma(a)$. For any function $f$ in the space of polynomial functions on $\sigma(a)$, the representative polynomial $p \in \mathbb{R}[x]$ associated with $f$ (denoted as `polynomialRepresentative a f`) has the property that the continuous function it induces on $\sigma(a)$ is equal to $f$.

Let $E$ be a nontrivial normed Jordan algebra and let $a \in E$. Let $p, q \in \mathbb{R}[x]$ be polynomials with real coefficients. If the continuous functions induced by $p$ and $q$ on the Jordan spectrum $\sigma(a)$ are identical, then their evaluations at $a$ are equal, i.e., $p(a) = q(a)$ in the closed Jordan subalgebra $C(a)$ generated by $1$ and $a$.

Let $E$ be a nontrivial normed Jordan algebra and $a \in E$. Let $\sigma(a)$ denote the Jordan spectrum of $a$, and let $\mathcal{P}(\sigma(a))$ be the space of polynomial functions on $\sigma(a)$ (that is, functions $f: \sigma(a) \to \mathbb{R}$ that are restrictions of polynomials in $\mathbb{R}[x]$). This definition defines the linear map from $\mathcal{P}(\sigma(a))$ to the closed Jordan subalgebra $C(a)$ generated by $1$ and $a$, which sends a polynomial function $f$ to the element $f(a)$. Specifically, for any $f \in \mathcal{P}(\sigma(a))$, the map is defined by choosing a representative polynomial $p \in \mathbb{R}[x]$ such that $p(x) = f(x)$ for all $x \in \sigma(a)$ and evaluating it to $p(a) \in C(a)$. This map is well-defined and independent of the choice of representative polynomial $p$.

Let $E$ be a nontrivial normed Jordan algebra and let $a \in E$. Let $\sigma(a) \subset \mathbb{R}$ denote the Jordan spectrum of $a$. For any polynomial function $f \in \mathcal{P}(\sigma(a))$ (a function $f: \sigma(a) \to \mathbb{R}$ that is the restriction of a polynomial $p \in \mathbb{R}[x]$), the norm of the element $f(a)$ in the algebra is equal to the supremum norm of $f$ on the spectrum, i.e., $$\|f(a)\| = \|f\|_\infty = \sup_{\lambda \in \sigma(a)} |f(\lambda)|.$$

Let $E$ be a nontrivial normed Jordan algebra and $a \in E$. Let $\sigma(a) \subseteq \mathbb{R}$ denote the Jordan spectrum of $a$. Let $\mathcal{P}(\sigma(a))$ be the space of polynomial functions on $\sigma(a)$ (functions $f: \sigma(a) \to \mathbb{R}$ that are restrictions of real polynomials) equipped with the supremum norm $\|f\|_\infty = \sup_{x \in \sigma(a)} |f(x)|$. This definition defines the real linear isometry from $\mathcal{P}(\sigma(a))$ to the closed Jordan subalgebra $C(a)$ generated by $1$ and $a$. The map sends each polynomial function $f$ to the element $f(a) \in C(a)$ and satisfies $\|f(a)\| = \|f\|_\infty$.

Let $E$ be a nontrivial normed Jordan algebra and $a \in E$. Let $\sigma(a) \subseteq \mathbb{R}$ denote the Jordan spectrum of $a$, and let $\mathcal{P}(\sigma(a))$ be the space of polynomial functions on the spectrum. Let $\Phi_a: \mathcal{P}(\sigma(a)) \to C(a)$ be the linear map of the polynomial functional calculus that sends a polynomial function $f$ to the element $f(a)$ in the closed Jordan subalgebra $C(a)$ generated by $1$ and $a$. Then for any $f, g \in \mathcal{P}(\sigma(a))$, it holds that: $$\Phi_a(f \cdot g) = \Phi_a(f) \cdot \Phi_a(g)$$ where the product on the right is the Jordan product restricted to $C(a)$.

Let $E$ be a nontrivial normed Jordan algebra and $a \in E$. Let $\sigma(a) \subseteq \mathbb{R}$ be the Jordan spectrum of $a$, and let $C(\sigma(a), \mathbb{R})$ be the space of continuous real-valued functions on the spectrum equipped with the supremum norm. This definition denotes the continuous linear map $\Phi_a: C(\sigma(a), \mathbb{R}) \to C(a)$, where $C(a)$ is the closed associative Jordan subalgebra generated by $1$ and $a$. This map is the unique continuous linear extension of the polynomial evaluation map (the polynomial functional calculus). It is constructed by extending the isometric map from the dense subspace of polynomial functions $\mathcal{P}(\sigma(a))$ to the complete space $C(a)$.

Let $E$ be a normed Jordan algebra and let $a \in E$. Let $\sigma(a) \subseteq \mathbb{R}$ denote the Jordan spectrum of $a$. The set of polynomial functions on $\sigma(a)$ is dense in the space $C(\sigma(a), \mathbb{R})$ of continuous real-valued functions on the spectrum with respect to the supremum norm.

Let $E$ be a nontrivial normed Jordan algebra and $a \in E$. Let $\sigma(a) \subseteq \mathbb{R}$ be the Jordan spectrum of $a$. Let $C(\sigma(a), \mathbb{R})$ be the space of continuous real-valued functions on the spectrum, and let $\mathcal{P}(\sigma(a))$ be the space of polynomial functions on the spectrum. Let $\Phi_a: C(\sigma(a), \mathbb{R}) \to C(a)$ denote the continuous functional calculus and let $\Psi_a: \mathcal{P}(\sigma(a)) \to C(a)$ denote the polynomial functional calculus. For any polynomial function $f \in \mathcal{P}(\sigma(a))$, the continuous functional calculus agrees with the polynomial functional calculus, such that $\Phi_a(f) = \Psi_a(f)$.

Given an element $a$ in a normed Jordan algebra $E$, let $\sigma(a) \subseteq \mathbb{R}$ denote its Jordan spectrum. This definition is the coordinate function $x \mapsto x$ for $x \in \sigma(a)$, regarded as an element of the space of real-valued polynomial functions on $\sigma(a)$.

Let $E$ be a nontrivial normed Jordan algebra and let $a \in E$. Let $\sigma(a) \subseteq \mathbb{R}$ be the Jordan spectrum of $a$, and let $\Phi_a : C(\sigma(a), \mathbb{R}) \to C(a)$ be the continuous linear map representing the functional calculus, where $C(a)$ is the closed Jordan subalgebra generated by $1$ and $a$. If $f \in C(\sigma(a), \mathbb{R})$ is the coordinate function defined by $f(x) = x$ for all $x \in \sigma(a)$, then $\Phi_a(f) = a$, where $a$ is regarded as an element of $C(a)$.

Let $E$ be a nontrivial normed Jordan algebra and $a \in E$. Let $\sigma(a) \subseteq \mathbb{R}$ denote the Jordan spectrum of $a$, and let $C(\sigma(a), \mathbb{R})$ be the space of continuous real-valued functions on $\sigma(a)$ equipped with the supremum norm. For any function $f \in C(\sigma(a), \mathbb{R})$, the continuous linear functional calculus $\Phi_a$ is an isometry, meaning $$\|\Phi_a(f)\| = \|f\|,$$ where the norm on the left is the norm in $E$ and the norm on the right is the supremum norm $\|f\| = \sup_{x \in \sigma(a)} |f(x)|$.

Let $E$ be a nontrivial normed Jordan algebra and $a \in E$. Let $\sigma(a) \subseteq \mathbb{R}$ be the Jordan spectrum of $a$, and let $C(\sigma(a), \mathbb{R})$ be the space of continuous real-valued functions on $\sigma(a)$ equipped with the supremum norm. Let $C(a)$ be the norm-closed associative Jordan subalgebra generated by the identity element $1$ and $a$. This definition represents the continuous functional calculus $\Phi_a : C(\sigma(a), \mathbb{R}) \to C(a)$ as a linear isometry, meaning that for any $f \in C(\sigma(a), \mathbb{R})$, $\|\Phi_a(f)\| = \|f\|_{\infty}$.

Let $E$ be a nontrivial normed Jordan algebra and $a \in E$. Let $\sigma(a) \subseteq \mathbb{R}$ be the Jordan spectrum of $a$, and $C(\sigma(a), \mathbb{R})$ be the space of continuous real-valued functions on the spectrum. Let $\Phi_a : C(\sigma(a), \mathbb{R}) \to C(a)$ be the continuous linear map of the functional calculus, where $C(a)$ is the closed associative Jordan subalgebra generated by $1$ and $a$. For any functions $f, g \in C(\sigma(a), \mathbb{R})$, the map $\Phi_a$ is multiplicative: $$\Phi_a(f \cdot g) = \Phi_a(f) \cdot \Phi_a(g)$$ where the product on the left is the pointwise multiplication of functions and the product on the right is the Jordan product in $C(a)$.

Let $E$ be a nontrivial normed Jordan algebra and $a \in E$. Let $\sigma(a) \subseteq \mathbb{R}$ be the Jordan spectrum of $a$, and let $\Phi_a: C(\sigma(a), \mathbb{R}) \to C(a)$ be the continuous linear map of the functional calculus. Then $\Phi_a(1) = 1$, where the $1$ on the left is the constant function $f(x) = 1$ for all $x \in \sigma(a)$ and the $1$ on the right is the identity element of the closed Jordan subalgebra $C(a)$.

Let $E$ be a nontrivial normed Jordan algebra and $a \in E$. The continuous functional calculus for $a$ is the unital $\mathbb{R}$-algebra homomorphism $\Phi_a : C(\sigma(a), \mathbb{R}) \to C(a)$, where $\sigma(a) \subseteq \mathbb{R}$ is the Jordan spectrum of $a$ and $C(a)$ is the closed associative Jordan subalgebra of $E$ generated by $1$ and $a$. This homomorphism extends the polynomial functional calculus to all continuous functions on the spectrum.

Let $E$ be a nontrivial normed Jordan algebra and $a \in E$. Let $\sigma(a) \subset \mathbb{R}$ denote the Jordan spectrum of $a$, and let $C(a)$ be the closed associative Jordan subalgebra generated by $1$ and $a$. The continuous functional calculus algebra homomorphism $\Phi_a : C(\sigma(a), \mathbb{R}) \to C(a)$ is an isometry. That is, for every continuous function $f \in C(\sigma(a), \mathbb{R})$, the norm of its image under the functional calculus satisfies $\|\Phi_a(f)\| = \|f\|_\infty$, where $\|f\|_\infty = \sup_{x \in \sigma(a)} |f(x)|$.

Let $E$ be a nontrivial normed Jordan algebra and $a \in E$. Let $\sigma(a) \subseteq \mathbb{R}$ denote the Jordan spectrum of $a$, and let $C(a)$ be the norm-closed Jordan subalgebra generated by $a$ and the identity $1$. Let $\Phi_a : C(\sigma(a), \mathbb{R}) \to C(a)$ be the continuous functional calculus homomorphism. Then the image of the identity function $f(x) = x$ (restricted to the spectrum $\sigma(a)$) under $\Phi_a$ is $a$.

For an element $a$ in a nontrivial normed Jordan algebra $E$ and a continuous real-valued function $f \in C(\sigma(a), \mathbb{R})$ defined on the Jordan spectrum $\sigma(a)$ of $a$, this definition denotes the element $f(a) \in E$ obtained via the continuous functional calculus. The result $f(a)$ is the image of $f$ under the isometric $\mathbb{R}$-algebra isomorphism from $C(\sigma(a), \mathbb{R})$ to the closed Jordan subalgebra $C(a)$ generated by $1$ and $a$.

Let $E$ be a nontrivial normed Jordan algebra and let $a \in E$. Let $\sigma(a) \subseteq \mathbb{R}$ denote the Jordan spectrum of $a$. For any continuous real-valued function $f \in C(\sigma(a), \mathbb{R})$, the norm of the element $f(a)$ obtained via the continuous functional calculus is equal to the supremum norm of $f$ on the spectrum, i.e., $$\|f(a)\| = \|f\|$$ where $\|f\| = \sup_{\lambda \in \sigma(a)} |f(\lambda)|$.

Let $E$ be a nontrivial normed Jordan algebra and $a \in E$. Let $\sigma(a) \subseteq \mathbb{R}$ be the Jordan spectrum of $a$, and let $C(a)$ be the norm-closed associative Jordan subalgebra generated by $1$ and $a$. The range of the continuous functional calculus algebra homomorphism $\Phi_a : C(\sigma(a), \mathbb{R}) \to C(a)$ is a closed subset of $C(a)$. This follows from the fact that the domain $C(\sigma(a), \mathbb{R})$ is complete and the map $\Phi_a$ is an isometry.

Given an element $a$ in a normed Jordan algebra $E$ over $\mathbb{R}$, let $J[a]$ be the $\mathbb{R}$-submodule of $E$ spanned by the Jordan powers $\{a^n\}_{n \in \mathbb{N}}$. This definition defines the canonical $\mathbb{R}$-linear inclusion map $\iota: J[a] \to C(a)$, where $C(a)$ is the norm-closed Jordan subalgebra of $E$ generated by $1$ and $a$.

Let $E$ be a normed Jordan algebra and $a \in E$. Let $J[a]$ be the $\mathbb{R}$-submodule of $E$ spanned by the set of Jordan powers $\{a^n\}_{n \in \mathbb{N}}$, where $a^0 = 1$ and $a^{n+1} = a \circ a^n$. Let $C(a)$ be the norm-closed Jordan subalgebra generated by $1$ and $a$. The range of the canonical inclusion map $\iota: J[a] \to C(a)$ is dense in $C(a)$.

Let $E$ be a normed Jordan algebra over $\mathbb{R}$ and $a \in E$. Let $a^n$ denote the $n$-th Jordan power of $a$, defined recursively by $a^0 = 1$ and $a^{n+1} = a \circ a^n$, where $\circ$ is the Jordan product. Let $C(a)$ be the norm-closed Jordan subalgebra generated by $1$ and $a$. While the product in $E$ is generally non-associative, the restriction of the product to $C(a)$ is associative, making $C(a)$ a commutative ring. This theorem states that the inclusion of the Jordan power $a^n$ into $C(a)$ coincides with the $n$-th power of $a$ as calculated using the associative multiplication in $C(a)$.

Let $E$ be a nontrivial normed Jordan algebra and $a \in E$. For any natural number $n \in \mathbb{N}$, the $n$-th Jordan power $a^n$ (defined recursively by $a^0 = 1$ and $a^{n+1} = a \circ a^n$) belongs to the range of the continuous functional calculus homomorphism $\Phi_a : C(\sigma(a), \mathbb{R}) \to C(a)$, where $C(a)$ is the closed Jordan subalgebra generated by $1$ and $a$.

Let $E$ be a nontrivial normed Jordan algebra and $a \in E$. Let $J[a]$ denote the $\mathbb{R}$-submodule of $E$ spanned by the Jordan powers $\{a^n\}_{n \in \mathbb{N}}$, and let $C(a)$ be the norm-closed Jordan subalgebra generated by $1$ and $a$. Let $\Phi_a : C(\sigma(a), \mathbb{R}) \to C(a)$ be the continuous functional calculus homomorphism, where $\sigma(a)$ is the Jordan spectrum of $a$. Then for any $x \in J[a]$, its inclusion in $C(a)$ lies in the range of $\Phi_a$.

Let $E$ be a nontrivial normed Jordan algebra and $a \in E$. Let $\sigma(a) \subseteq \mathbb{R}$ denote the Jordan spectrum of $a$, and let $C(a)$ denote the norm-closed Jordan subalgebra generated by $1$ and $a$. The continuous functional calculus $\Phi_a : C(\sigma(a), \mathbb{R}) \to C(a)$ is surjective; that is, the range of $\Phi_a$ is the entire subalgebra $C(a)$.

Let $E$ be a nontrivial normed Jordan algebra and $a \in E$. Let $\sigma(a)$ denote the Jordan spectrum of $a$, and let $C(\sigma(a), \mathbb{R})$ be the algebra of continuous real-valued functions on $\sigma(a)$. Let $C(a)$ be the norm-closed Jordan subalgebra of $E$ generated by the identity $1$ and the element $a$. Then the range of the continuous functional calculus algebra homomorphism $\Phi_a : C(\sigma(a), \mathbb{R}) \to C(a)$ is equal to $C(a)$.

Let $E$ be a nontrivial normed Jordan algebra and $a \in E$. The continuous functional calculus for $a$ is the $\mathbb{R}$-algebra isomorphism $\Phi_a : C(\sigma(a), \mathbb{R}) \cong C(a)$, where $\sigma(a) \subseteq \mathbb{R}$ is the Jordan spectrum of $a$ and $C(a)$ is the norm-closed Jordan subalgebra of $E$ generated by $1$ and $a$. This map identifies the algebra of continuous real-valued functions on the spectrum with the associative real Banach algebra generated by the element $a$.

Let $E$ be a nontrivial normed Jordan algebra and $a \in E$. Let $\sigma(a) \subseteq \mathbb{R}$ denote the Jordan spectrum of $a$, and let $C(\sigma(a), \mathbb{R})$ be the algebra of continuous real-valued functions on $\sigma(a)$ equipped with the supremum norm $\|f\|_\infty = \sup_{\lambda \in \sigma(a)} |f(\lambda)|$. Let $\Phi_a : C(\sigma(a), \mathbb{R}) \cong C(a)$ be the continuous functional calculus $\mathbb{R}$-algebra isomorphism onto the norm-closed Jordan subalgebra $C(a)$ generated by $1$ and $a$. For any $f \in C(\sigma(a), \mathbb{R})$, the norm of its image under this isomorphism is equal to its supremum norm: $$\|\Phi_a(f)\| = \|f\|_\infty$$

Let $E$ be a nontrivial normed Jordan algebra and $a \in E$. Let $\text{id} : \sigma(a) \to \mathbb{R}$ be the identity function on the Jordan spectrum of $a$, defined by $\text{id}(x) = x$ for all $x \in \sigma(a)$. Then the continuous functional calculus of $a$ associated with the identity function is $a$ itself, i.e., $\text{id}(a) = a$.

Let $E$ be a nontrivial normed Jordan algebra and $a \in E$. Let $\sigma(a) \subseteq \mathbb{R}$ be the Jordan spectrum of $a$. For any two continuous real-valued functions $f, g \in C(\sigma(a), \mathbb{R})$, the continuous functional calculus is additive, satisfying $(f + g)(a) = f(a) + g(a)$.

Let $E$ be a nontrivial normed Jordan algebra and let $a \in E$. For any continuous real-valued functions $f, g \in C(\sigma(a), \mathbb{R})$ defined on the Jordan spectrum $\sigma(a)$ of $a$, the continuous functional calculus satisfies $(f - g)(a) = f(a) - g(a)$.

Let $E$ be a nontrivial normed Jordan algebra and let $a \in E$. Let $\sigma(a)$ denote the Jordan spectrum of $a$. Then the continuous functional calculus of the zero function $0 \in C(\sigma(a), \mathbb{R})$ is the zero element of $E$. That is, $0(a) = 0$.

Let $E$ be a nontrivial normed Jordan algebra and let $a \in E$. For any continuous real-valued functions $f, g \in C(\sigma(a), \mathbb{R})$ defined on the Jordan spectrum $\sigma(a)$ of $a$, the continuous functional calculus map satisfies $(f \cdot g)(a) = f(a) \cdot g(a)$, where the product on the left is the pointwise multiplication of functions and the product on the right is the Jordan product in $E$.

Let $E$ be a nontrivial normed Jordan algebra and let $a \in E$ be an element with Jordan spectrum $\sigma(a)$. For any scalar $r \in \mathbb{R}$, let $f: \sigma(a) \to \mathbb{R}$ be the constant function defined by $f(x) = r$ for all $x \in \sigma(a)$. Then the continuous functional calculus of $a$ with respect to $f$ is equal to the scalar multiple of the identity element $r \cdot 1$.

For an element $a$ in a normed Jordan algebra $E$, this defines the continuous function $f \in C(\sigma(a), \mathbb{R})$ on the spectrum $\sigma(a)$ such that $f(x) = \sqrt{x}$ for all $x \in \sigma(a)$.

Given a non-negative element $a \geq 0$ in a nontrivial normed Jordan algebra $E$, this definition represents the positive square root $\sqrt{a} \in E$. It is obtained by applying the continuous functional calculus for $a$ to the square root function $f: \sigma(a) \to \mathbb{R}$, defined by $f(x) = \sqrt{x}$ for all $x$ in the Jordan spectrum $\sigma(a)$.

For a non-negative element $a$ in a normed Jordan algebra, let $\sigma(a) \subseteq \mathbb{R}$ denote its spectrum. Let $f: \sigma(a) \to \mathbb{R}$ be the continuous square root function defined by $f(x) = \sqrt{x}$ for all $x \in \sigma(a)$. Then the pointwise product of $f$ with itself is equal to the identity function $\text{id}_{\sigma(a)}$ (where $\text{id}(x) = x$) restricted to the spectrum $\sigma(a)$.

Let $E$ be a nontrivial normed Jordan algebra. For any element $a \in E$ such that $a \geq 0$, the positive square root $\sqrt{a}$ (obtained via the continuous functional calculus) satisfies the identity $(\sqrt{a})^2 = a$.

Let $E$ be a nontrivial normed Jordan algebra. For any element $a \in E$ such that $a \geq 0$, the square root of $a$, denoted $\sqrt{a}$ and obtained via the continuous functional calculus, is also non-negative, i.e., $\sqrt{a} \geq 0$.

Let $E$ be a nontrivial normed Jordan algebra and $a \in E$. Let $\sigma(a)$ denote the Jordan spectrum of $a$. If $f$ and $g$ are continuous real-valued functions on $\sigma(a)$ such that $f = g^2$ (meaning $f$ is a pointwise square), then the element $f(a)$ obtained through the continuous functional calculus is non-negative, written as $f(a) \geq 0$.

Let $E$ be a nontrivial normed Jordan algebra. For any element $a \in E$ and any continuous real-valued function $f \in C(\sigma(a), \mathbb{R})$ defined on the Jordan spectrum $\sigma(a)$ of $a$, if $f(x) \geq 0$ for all $x \in \sigma(a)$, then the element $f(a)$ obtained via the continuous functional calculus is non-negative in $E$ (i.e., $f(a) \geq 0$).