PhyslibAlpha.AlgebraicFramework.JordanOrderUnit.JB.GeneratedByOne.CFC

Let $E$ be a normed Jordan algebra and $a \in E$. The Jordan spectrum of $a$, denoted by $\sigma(a)$, is a compact topological space.

Let $E$ be a nontrivial normed Jordan algebra that is also an archimedean order unit space (specifically a JB algebra). For any element $a \in E$, let $\sigma(a) \subseteq \mathbb{R}$ denote its Jordan spectrum. For any polynomial $p \in \mathbb{R}[x]$, let $p(a)$ denote the evaluation of $p$ at $a$ within the norm-closed Jordan subalgebra generated by $1$ and $a$. Then the norm of the element $p(a)$ is equal to the supremum norm of $p$ viewed as a continuous function on the spectrum $\sigma(a)$: $$\|p(a)\| = \sup_{\lambda \in \sigma(a)} |p(\lambda)|.$$ This equality establishes that polynomial evaluation is an isometry between the algebra of polynomials on $\sigma(a)$ and the polynomial core of the functional calculus in $E$.

Let $a$ be an element of a normed Jordan algebra $E$. Let $\sigma(a) \subseteq \mathbb{R}$ denote the Jordan spectrum of $a$. The set of polynomial functions on $\sigma(a)$ is dense in the algebra $C(\sigma(a), \mathbb{R})$ of continuous real-valued functions on the spectrum.

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

For an element $a$ in a normed Jordan algebra $E$ and a function $f$ belonging to the algebra of polynomial functions on the Jordan spectrum $\sigma(a) \subseteq \mathbb{R}$, this function selects a 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 $a \in E$. Let $\sigma(a) \subseteq \mathbb{R}$ denote the Jordan spectrum of $a$. For any function $f$ in the algebra of polynomial functions on $\sigma(a)$, the continuous function on $\sigma(a)$ induced by the polynomial representative of $f$ is equal to $f$.

Let $E$ be a nontrivial normed Jordan algebra that is a partially ordered Archimedean additive monoid with a JB-order unit. For an element $a \in E$, let $\sigma(a) \subseteq \mathbb{R}$ denote its Jordan spectrum. Given two polynomials $p, q \in \mathbb{R}[x]$, if their induced continuous functions on $\sigma(a)$ are equal (i.e., $p(x) = q(x)$ for all $x \in \sigma(a)$), then their evaluations at $a$ within the norm-closed subalgebra $C_J(a)$ are also equal: $p(a) = q(a)$.

Let $E$ be a nontrivial normed Jordan algebra with a JB-order unit, and let $a \in E$. Let $\sigma(a) \subseteq \mathbb{R}$ denote the Jordan spectrum of $a$, and let $C_J(a) = \overline{J[a]}$ be the norm-closed subalgebra generated by $1$ and $a$. Let $P(\sigma(a))$ be the algebra of polynomial functions on the spectrum $\sigma(a)$. The linear map $\text{polynomialCfcLinearMap } a : P(\sigma(a)) \to C_J(a)$ maps a function $f \in P(\sigma(a))$ to the element $p(a) \in C_J(a)$, where $p \in \mathbb{R}[x]$ is any polynomial whose restriction to $\sigma(a)$ is $f$.

Let $E$ be a nontrivial normed Jordan algebra with a JB-order unit. For any element $a \in E$, let $\sigma(a)$ be its Jordan spectrum and $P(\sigma(a))$ be the algebra of polynomial functions on $\sigma(a)$. Let $\phi_a : P(\sigma(a)) \to C_J(a)$ be the polynomial functional calculus map that sends a function $f$ to the element $f(a)$. Then for any $f \in P(\sigma(a))$, the norm of $f(a)$ in the algebra is equal to the supremum norm of $f$ on the spectrum: $$\|f(a)\| = \|f\|_\infty$$

Let $E$ be a nontrivial normed Jordan algebra with a JB-order unit, and let $a \in E$. Let $\sigma(a) \subseteq \mathbb{R}$ denote the Jordan spectrum of $a$, and let $C_J(a) = \overline{J[a]}$ be the norm-closed Jordan subalgebra generated by $1$ and $a$. Let $P(\sigma(a))$ be the space of polynomial functions on $\sigma(a)$ equipped with the supremum norm. The map $\text{polynomialCfcLinearIsometry } a : P(\sigma(a)) \to C_J(a)$ is a real linear isometry that sends a polynomial function $f \in P(\sigma(a))$ to the element $p(a) \in C_J(a)$, where $p \in \mathbb{R}[x]$ is any polynomial whose restriction to $\sigma(a)$ is $f$.

Let $E$ be a nontrivial normed Jordan algebra with a JB-order unit, and let $a \in E$. Let $\sigma(a)$ denote the Jordan spectrum of $a$, and let $P(\sigma(a))$ be the algebra of polynomial functions restricted to $\sigma(a)$. The polynomial functional calculus map $\Phi_a : P(\sigma(a)) \to C_J(a)$ (mapping a function $f$ to the element $f(a)$ in the closed subalgebra $C_J(a)$) is a homomorphism with respect to multiplication. That is, for any $f, g \in P(\sigma(a))$, the identity $$\Phi_a(f \cdot g) = \Phi_a(f) \cdot \Phi_a(g)$$ holds, where the multiplication on the right is the Jordan product in $E$ (which is associative when restricted to $C_J(a)$).

Let $E$ be a nontrivial normed Jordan algebra with a JB-order unit, and let $a \in E$. Let $\sigma(a) \subseteq \mathbb{R}$ denote the Jordan spectrum of $a$, and let $C_J(a) = \overline{J[a]}$ be the norm-closed Jordan subalgebra generated by $1$ and $a$. The continuous linear map $C(\sigma(a), \mathbb{R}) \to C_J(a)$ is the canonical extension of the polynomial functional calculus to the space of continuous real-valued functions on the spectrum. It is constructed as the unique continuous extension of the linear isometry that maps a polynomial function $p|_{\sigma(a)}$ to the element $p(a) \in C_J(a)$, utilizing the fact that polynomial functions are dense in $C(\sigma(a), \mathbb{R})$ by the Stone–Weierstrass theorem.

Let $E$ be a nontrivial normed Jordan algebra with a JB-order unit. For any element $a \in E$, let $\sigma(a) \subseteq \mathbb{R}$ denote its Jordan spectrum, and let $C_J(a)$ denote the norm-closed Jordan subalgebra generated by $1$ and $a$. For any polynomial function $f \in P(\sigma(a))$ (where $P(\sigma(a))$ is the space of polynomial functions restricted to the spectrum), the continuous linear functional calculus map and the polynomial functional calculus map agree on $f$: $$\text{jordanCfcLinear}_a(f) = \text{polynomialCfcLinearMap}_a(f)$$ In other words, the continuous extension of the functional calculus is consistent with the initial definition for polynomial functions.

For an element $a$ in a normed Jordan algebra $E$, let $\sigma(a) \subseteq \mathbb{R}$ be its Jordan spectrum. The coordinate polynomial function is the function $f: \sigma(a) \to \mathbb{R}$ defined by $f(x) = x$, regarded as an element of the algebra of polynomial functions on $\sigma(a)$. It is obtained by restricting the polynomial indeterminate $X$ to the set $\sigma(a)$.

Let $E$ be a nontrivial normed Jordan algebra with a JB-order unit. For any element $a \in E$, let $\sigma(a) \subseteq \mathbb{R}$ denote its Jordan spectrum and let $C_J(a)$ be the norm-closed Jordan subalgebra generated by the identity and $a$. If $f: \sigma(a) \to \mathbb{R}$ is the coordinate function defined by $f(x) = x$, then the continuous linear functional calculus map $\Psi: C(\sigma(a), \mathbb{R}) \to C_J(a)$ satisfies $\Psi(f) = a$.

Let $E$ be a nontrivial normed Jordan algebra equipped with a JB-order unit structure. For any element $a \in E$, let $\sigma(a) \subseteq \mathbb{R}$ denote the Jordan spectrum of $a$ and $C(\sigma(a), \mathbb{R})$ denote the space of continuous real-valued functions on $\sigma(a)$. Let $\text{cfc}_a : C(\sigma(a), \mathbb{R}) \to C_J(a)$ be the continuous functional calculus map which sends a function $f$ to the element $f(a)$ in the closed Jordan subalgebra generated by $1$ and $a$. Then for any $f \in C(\sigma(a), \mathbb{R})$, the norm of the image is equal to the supremum norm of the function: $$\|f(a)\| = \|f\|_\infty.$$ In other words, the continuous functional calculus map is a linear isometry.

Let $E$ be a nontrivial normed Jordan algebra equipped with a JB-order unit structure and let $a \in E$. Let $\sigma(a) \subseteq \mathbb{R}$ denote the Jordan spectrum of $a$ and $C_J(a) = \overline{J[a]}$ denote the norm-closed Jordan subalgebra generated by the identity element $1$ and $a$. The continuous intrinsic Jordan functional calculus is the linear isometry $$\Psi_a : C(\sigma(a), \mathbb{R}) \to C_J(a)$$ which maps a continuous real-valued function $f$ on the spectrum to the element $f(a)$. As a linear isometry, it preserves the norm such that $\|\Psi_a(f)\| = \|f\|_\infty$ for all $f \in C(\sigma(a), \mathbb{R})$.

Let $E$ be a nontrivial normed Jordan algebra with a JB-order unit. For any element $a \in E$, let $\sigma(a) \subseteq \mathbb{R}$ denote its Jordan spectrum, and let $\Phi_a : C(\sigma(a), \mathbb{R}) \to C_J(a)$ be the continuous linear functional calculus map, where $C_J(a)$ is the norm-closed Jordan subalgebra generated by $1$ and $a$. For any continuous real-valued 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 $f \cdot g$ is the pointwise product of functions and the product on the right is the Jordan product in $C_J(a)$.

Let $E$ be a nontrivial normed Jordan algebra with a JB-order unit, and let $a \in E$. Let $\sigma(a) \subseteq \mathbb{R}$ denote the Jordan spectrum of $a$, and let $C_J(a)$ be the norm-closed Jordan subalgebra generated by the identity element $1$ and $a$. The continuous linear functional calculus map $\text{jordanCfcLinear}_a : C(\sigma(a), \mathbb{R}) \to C_J(a)$ maps the constant function $1$ on $\sigma(a)$ to the identity element $1$ of $C_J(a)$, i.e., $$\text{jordanCfcLinear}_a(1) = 1.$$

Let $E$ be a nontrivial normed Jordan algebra with a JB-order unit, and let $a \in E$. Let $\sigma(a) \subseteq \mathbb{R}$ denote the Jordan spectrum of $a$, and let $C_J(a)$ be the norm-closed Jordan subalgebra generated by $1$ and $a$. The continuous functional calculus for $a$ is the unital $\mathbb{R}$-algebra homomorphism $$\Phi_a : C(\sigma(a), \mathbb{R}) \to C_J(a)$$ which maps a continuous real-valued function $f$ on the spectrum to the element $f(a)$ in $C_J(a)$. This map is the unique continuous extension of the polynomial functional calculus $p \mapsto p(a)$.

Let $E$ be a nontrivial normed Jordan algebra with a JB-order unit, and let $a \in E$. Let $\sigma(a) \subseteq \mathbb{R}$ denote the Jordan spectrum of $a$, and let $C_J(a)$ be the norm-closed Jordan subalgebra generated by the identity $1$ and $a$. The continuous Jordan functional calculus $\mathbb{R}$-algebra homomorphism $\Phi_a : C(\sigma(a), \mathbb{R}) \to C_J(a)$ is an isometry. That is, for any continuous function $f \in C(\sigma(a), \mathbb{R})$, we have $\|\Phi_a(f)\| = \|f\|_\infty$.

Let $E$ be a nontrivial normed Jordan algebra with a JB-order unit, and let $a \in E$. Let $\sigma(a) \subseteq \mathbb{R}$ denote the Jordan spectrum of $a$, and let $C_J(a)$ be the norm-closed Jordan subalgebra generated by $1$ and $a$. The continuous Jordan functional calculus $\Phi_a : C(\sigma(a), \mathbb{R}) \to C_J(a)$ maps the identity function $\text{id} : \sigma(a) \to \mathbb{R}$ (where $\text{id}(x) = x$) to the element $a$. That is, $\Phi_a(\text{id}) = a$.

Let $E$ be a nontrivial normed Jordan algebra with a JB-order unit. For an element $a \in E$ and a continuous real-valued function $f \in C(\sigma(a), \mathbb{R})$ defined on the Jordan spectrum $\sigma(a)$ of $a$, the Jordan continuous functional calculus $f(a)$ is the element of $E$ obtained by applying the unital $\mathbb{R}$-algebra homomorphism $\Phi_a : C(\sigma(a), \mathbb{R}) \to C_J(a)$ to $f$ and viewing the result as an element of the ambient algebra $E$. Here $C_J(a)$ denotes the norm-closed Jordan subalgebra generated by $1$ and $a$.

Let $E$ be a nontrivial normed Jordan algebra with a JB-order unit. For any element $a \in E$ and any continuous real-valued function $f \in C(\sigma(a), \mathbb{R})$ defined on the Jordan spectrum $\sigma(a)$, the norm of the element $f(a)$ obtained through the Jordan continuous functional calculus is equal to the supremum norm of the function $f$: $$\|f(a)\| = \|f\|$$

Let $E$ be a nontrivial normed Jordan algebra with a JB-order unit, and let $a \in E$. Let $\sigma(a) \subseteq \mathbb{R}$ denote the Jordan spectrum of $a$, and let $C_J(a)$ be the norm-closed Jordan subalgebra generated by the identity element $1$ and $a$. The range of the continuous functional calculus homomorphism $$\Phi_a : C(\sigma(a), \mathbb{R}) \to C_J(a)$$ is a closed set.

Let $E$ be a normed Jordan algebra and $a \in E$. Let $J[a]$ denote the Jordan subalgebra generated by the identity element $1$ and the element $a$, and let $\overline{J[a]}$ denote its norm-closure. This definition is the canonical inclusion map from $J[a]$ into $\overline{J[a]}$, treated as a linear map over $\mathbb{R}$.

Let $E$ be a normed Jordan algebra and $a \in E$. Let $J[a]$ be the Jordan subalgebra generated by the identity element $1$ and $a$, and let $\overline{J[a]}$ denote its norm-closure. The range of the canonical inclusion map $J[a] \hookrightarrow \overline{J[a]}$ is dense in $\overline{J[a]}$.

Let $E$ be a normed Jordan algebra and $a \in E$. Let $J[a]$ be the Jordan subalgebra generated by the identity $1$ and $a$, and let $\overline{J[a]}$ be its norm-closure. Let $\iota : J[a] \to \overline{J[a]}$ be the canonical inclusion map. For any $n \in \mathbb{N}$, the image of the $n$-th Jordan power $a^{[n]}$ under $\iota$ is equal to the $n$-th power of $a$ in the commutative ring $\overline{J[a]}$, i.e., $$\iota(a^{[n]}) = a^n.$$

Let $E$ be a nontrivial normed Jordan algebra with a JB-order unit and $a \in E$. Let $\overline{J[a]}$ be the norm-closed Jordan subalgebra generated by $1$ and $a$, and let $\Phi_a : C(\sigma(a), \mathbb{R}) \to \overline{J[a]}$ be the continuous functional calculus for $a$. Then for every $n \in \mathbb{N}$, the $n$-th Jordan power $a^{[n]}$, when viewed as an element of $\overline{J[a]}$, lies in the range of $\Phi_a$.

Let $E$ be a nontrivial normed Jordan algebra with a JB-order unit. For any $a \in E$, let $J[a]$ be the Jordan subalgebra generated by the identity element $1$ and $a$, and let $C_J(a) = \overline{J[a]}$ be its norm-closure. Let $\Phi_a : C(\sigma(a), \mathbb{R}) \to C_J(a)$ be the continuous functional calculus map, where $\sigma(a)$ is the Jordan spectrum of $a$. For every $x \in J[a]$, its canonical inclusion in $C_J(a)$ lies in the range of $\Phi_a$.

Let $E$ be a nontrivial normed Jordan algebra equipped with a JB-order unit. For any element $a \in E$, let $\sigma(a) \subseteq \mathbb{R}$ denote the Jordan spectrum of $a$ and let $C_J(a)$ be the norm-closed Jordan subalgebra generated by $1$ and $a$. The continuous Jordan functional calculus, represented by the unital $\mathbb{R}$-algebra homomorphism $\Phi_a : C(\sigma(a), \mathbb{R}) \to C_J(a)$, is surjective. That is, the range of $\Phi_a$ is the entirety of $C_J(a)$.

Let $E$ be a nontrivial normed Jordan algebra with a JB-order unit (formally, a partially ordered, Archimedean ordered additive monoid with monotone positive scalar multiplication). For any element $a \in E$, let $\sigma(a) \subseteq \mathbb{R}$ denote the Jordan spectrum of $a$, and let $C_J(a)$ be the norm-closed Jordan subalgebra generated by the identity element $1$ and $a$. The continuous Jordan functional calculus is defined as the unital $\mathbb{R}$-algebra homomorphism $$\Phi_a : C(\sigma(a), \mathbb{R}) \to C_J(a)$$ which maps a continuous real-valued function $f$ on the spectrum to an element $f(a) \in C_J(a)$. This theorem states that the range of the homomorphism $\Phi_a$ is the entire subalgebra $C_J(a)$.

Let $E$ be a nontrivial normed Jordan algebra with a JB-order unit, and let $a \in E$. Let $\sigma(a) \subseteq \mathbb{R}$ denote the Jordan spectrum of $a$, and let $C_J(a)$ be the norm-closed Jordan subalgebra generated by $1$ and $a$. The continuous Jordan functional calculus is the unital $\mathbb{R}$-algebra isomorphism $$\Phi_a : C(\sigma(a), \mathbb{R}) \cong C_J(a)$$ which identifies the algebra of continuous real-valued functions on the spectrum of $a$ with the closed associative subalgebra generated by $a$. This map is obtained from the continuous functional calculus homomorphism by proving it is bijective (specifically, it is an isometry and its range is the entirety of $C_J(a)$).

Let $E$ be a nontrivial normed Jordan algebra that is a JB-algebra with an order unit. For any element $a \in E$, let $\sigma(a) \subseteq \mathbb{R}$ be its Jordan spectrum, and let $\Phi_a : C(\sigma(a), \mathbb{R}) \cong C_J(a)$ be the continuous Jordan functional calculus isomorphism. For any continuous real-valued function $f$ on the spectrum of $a$, the norm of the element $\Phi_a(f)$ is equal to the supremum norm of $f$, i.e., $$\|\Phi_a(f)\| = \|f\|.$$

Let $E$ be a nontrivial normed Jordan algebra with a JB-order unit. For any element $a \in E$, let $\text{id} \in C(\sigma(a), \mathbb{R})$ denote the restriction of the identity function $x \mapsto x$ to the Jordan spectrum $\sigma(a)$. The Jordan continuous functional calculus applied to this identity function is $a$ itself, i.e., $\text{id}(a) = a$. This represents the coordinate law for the functional calculus.

Let $E$ be a nontrivial normed Jordan algebra equipped with a JB-order unit. For any element $a \in E$ and any continuous real-valued functions $f, g \in C(\sigma(a), \mathbb{R})$ defined on the Jordan spectrum $\sigma(a)$, the Jordan continuous functional calculus satisfies $$(f + g)(a) = f(a) + g(a).$$ Here, $f(a)$ (represented as `jordanCfc a f`) denotes the result of applying the functional calculus to the element $a$.

Let $E$ be a nontrivial normed Jordan algebra with a JB-order unit. For any element $a \in E$ and any continuous real-valued functions $f, g \in C(\sigma(a), \mathbb{R})$ defined on the Jordan spectrum $\sigma(a)$ of $a$, the Jordan continuous functional calculus preserves subtraction: $$(f - g)(a) = f(a) - g(a)$$ where $f(a)$ denotes the element of $E$ obtained by applying the functional calculus to the function $f$ at $a$.

Let $E$ be a nontrivial normed Jordan algebra with a JB-order unit and let $a \in E$. Let $\sigma(a) \subseteq \mathbb{R}$ be the Jordan spectrum of $a$. The Jordan continuous functional calculus applied to the constant zero function $0 \in C(\sigma(a), \mathbb{R})$ is equal to the zero element $0 \in E$.

Let $E$ be a nontrivial normed Jordan algebra with a JB-order unit. For any element $a \in E$ and continuous real-valued functions $f, g \in C(\sigma(a), \mathbb{R})$ defined on the Jordan spectrum $\sigma(a)$, the Jordan continuous functional calculus satisfies $(fg)(a) = f(a)g(a)$, where $fg$ denotes the pointwise product of functions.

Let $E$ be a nontrivial normed Jordan algebra with a JB-order unit. For any element $a \in E$ and 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 the Jordan spectrum $\sigma(a)$. Then the Jordan continuous functional calculus of $f$ at $a$ satisfies $f(a) = r \cdot 1$, where $1$ is the identity element of $E$.

Given an element $a$ of a normed Jordan algebra $E$, this is the continuous function from the Jordan spectrum $\sigma(a) \subseteq \mathbb{R}$ to the real numbers defined by the map $x \mapsto \sqrt{x}$.

Let $E$ be a nontrivial normed Jordan algebra with a JB-order unit. For a non-negative element $a \in E$ ($a \geq 0$), the Jordan square root is the element of $E$ obtained by applying the Jordan continuous functional calculus to the square root function $f(x) = \sqrt{x}$ defined on the Jordan spectrum $\sigma(a)$.

Let $E$ be a normed Jordan algebra (specifically a JB-algebra with an order unit) and let $a \in E$ be a positive element ($a \ge 0$). Let $\sigma(a) \subseteq \mathbb{R}$ denote the Jordan spectrum of $a$. Let $f: \sigma(a) \to \mathbb{R}$ be the continuous square root function defined by $f(x) = \sqrt{x}$. Then the pointwise product of $f$ with itself satisfies $f \cdot f = \text{id}_{\sigma(a)}$, where $\text{id}_{\sigma(a)}: \sigma(a) \to \mathbb{R}$ is the identity function $x \mapsto x$ restricted to the spectrum.

Let $E$ be a nontrivial normed Jordan algebra with a JB-order unit. For any non-negative element $a \in E$ ($a \ge 0$), let $\sqrt{a}$ denote the Jordan square root of $a$ (the element obtained by applying the Jordan continuous functional calculus to the function $f(x) = \sqrt{x}$ on the spectrum of $a$). Then the Jordan product of $\sqrt{a}$ with itself satisfies $\sqrt{a} \cdot \sqrt{a} = a$.

Let $E$ be a nontrivial normed Jordan algebra equipped with a JB-order unit (a JB-algebra). For any non-negative element $a \in E$ (where $a \ge 0$), its Jordan square root $\sqrt{a}$, obtained via the Jordan continuous functional calculus for the function $f(x) = \sqrt{x}$, is also non-negative, i.e., $0 \le \sqrt{a}$.

Let $E$ be a nontrivial normed Jordan algebra with a JB-order unit. For any element $a \in E$ and any continuous real-valued functions $f, g \in C(\sigma(a), \mathbb{R})$ defined on the Jordan spectrum $\sigma(a)$, if $f = g^2$, then the element $f(a)$ obtained via the Jordan continuous functional calculus is non-negative, i.e., $f(a) \ge 0$.

Let $E$ be a nontrivial normed Jordan algebra equipped with a JB-order unit. 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 $f$ is pointwise nonnegative (i.e., $f(x) \ge 0$ for all $x \in \sigma(a)$), then the element $f(a)$ obtained via the Jordan continuous functional calculus is also nonnegative in $E$, so that $0 \le f(a)$.

Let $E$ be a nontrivial normed Jordan algebra with a JB-order unit. For any element $a \in E$ and any continuous real-valued functions $f, g \in C(\sigma(a), \mathbb{R})$ defined on the Jordan spectrum $\sigma(a)$, if $f \le g$ pointwise (i.e., $f(x) \le g(x)$ for all $x \in \sigma(a)$), then $f(a) \le g(a)$ holds in $E$, where $f(a)$ and $g(a)$ denote the elements obtained via the Jordan continuous functional calculus.

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