PhyslibAlpha.AlgebraicFramework.CStarAlgebra.JordanCFC

Let $A$ be a $C^*$-algebra and $a \in A$ be a self-adjoint element. Let $\sigma(a) \subseteq \mathbb{R}$ denote the spectrum of $a$. The mapping `jordanCfc` is an $\mathbb{R}$-linear map from the space of continuous real-valued functions on the spectrum, denoted $C(\sigma(a), \mathbb{R})$, to the space of self-adjoint elements of $A$. For any $f \in C(\sigma(a), \mathbb{R})$, the map returns the element $f(a)$ defined by the continuous functional calculus.

Let $A$ be a $C^*$-algebra and $a$ be a self-adjoint element of $A$. Let $\sigma(a) \subseteq \mathbb{R}$ denote the spectrum of $a$. Let $\text{id} \in C(\sigma(a), \mathbb{R})$ be the identity function (or coordinate function) restricted to the spectrum, defined by $\text{id}(x) = x$ for all $x \in \sigma(a)$. Then the continuous functional calculus for Jordan observables maps the identity function to the element $a$: $$\text{jordanCfc}(a, \text{id}) = a$$

Let $A$ be a $C^*$-algebra and $a$ be a self-adjoint element of $A$. Let $\sigma(a) \subseteq \mathbb{R}$ denote the spectrum of $a$. For any continuous real-valued function $f \in C(\sigma(a), \mathbb{R})$, the norm of the element $f(a)$ (the image of $f$ under the continuous functional calculus for $a$) is equal to the supremum norm of $f$. That is, $\|f(a)\| = \|f\|$.

Let $A$ be a $C^*$-algebra and $a \in A$ be a self-adjoint element. Let $\sigma(a) \subseteq \mathbb{R}$ denote the spectrum of $a$. For any continuous real-valued functions $f, g \in C(\sigma(a), \mathbb{R})$, the continuous functional calculus for Jordan observables preserves multiplication. Specifically, the image of the pointwise product of functions $f \cdot g$ is equal to the Jordan product of their respective images $f(a)$ and $g(a)$ in the space of self-adjoint elements: $$\text{jordanCfc}(a, f \cdot g) = \text{jordanCfc}(a, f) * \text{jordanCfc}(a, g)$$ Since the functions in $C(\sigma(a), \mathbb{R})$ commute, their images under the functional calculus also commute, ensuring that the Jordan product is preserved exactly.

Let $A$ be a $C^*$-algebra and $a \in A$ be a self-adjoint element (an observable). Let $\sigma(a) \subseteq \mathbb{R}$ denote the spectrum of $a$. For any continuous real-valued function $f \in C(\sigma(a), \mathbb{R})$, the element $f(a)$ obtained via the continuous functional calculus commutes with $a$ in $A$. That is, $f(a)a = af(a)$.

Let $A$ be a $C^*$-algebra and $a$ be a self-adjoint element of $A$ (a Jordan observable). Let $\sigma(a) \subseteq \mathbb{R}$ denote the spectrum of $a$. If $f: \sigma(a) \to \mathbb{R}$ is a continuous function such that $f \geq 0$, then the element $f(a)$ obtained via the continuous functional calculus is non-negative (i.e., $f(a) \geq 0$).

Let $A$ be a $C^*$-algebra and $a$ be a self-adjoint element of $A$. Let $\sigma(a) \subseteq \mathbb{R}$ denote the spectrum of $a$. The continuous functional calculus map $f \mapsto f(a)$ from the space of continuous real-valued functions $C(\sigma(a), \mathbb{R})$ to the space of self-adjoint elements of $A$ is monotone. That is, for any $f, g \in C(\sigma(a), \mathbb{R})$, if $f \leq g$, then $f(a) \leq g(a)$ with respect to the natural partial order on self-adjoint elements.