PhyslibAlpha.ProbabilisticTheory.JordanOrderUnit.CStarAlgebra.CFC

For a self-adjoint element $a$ in a $C^*$-algebra $A$, the function `jordanCfc a` is the $\mathbb{R}$-linear map from the space of continuous real-valued functions on the spectrum of $a$, denoted $C(\sigma(a), \mathbb{R})$, to the space of self-adjoint elements (observables) of $A$. This map assigns to each continuous function $f \in C(\sigma(a), \mathbb{R})$ the element $f(a)$ as defined by the continuous functional calculus.

For a self-adjoint element $a$ in a $C^*$-algebra $A$, let $\text{id}$ denote the identity function $x \mapsto x$ restricted to the spectrum $\sigma(a) \subseteq \mathbb{R}$. The Jordan continuous functional calculus maps this identity function to the original element $a$, such that $\text{jordanCfc}(a)(\text{id}) = a$.

Let $A$ be a $C^*$-algebra and $a \in A$ be a self-adjoint element. For any continuous real-valued function $f$ defined on the spectrum $\sigma(a)$ of $a$, the norm of the element $f(a)$ (the image of $f$ under the Jordan continuous functional calculus) is equal to the supremum norm of $f$: $$\|f(a)\| = \|f\|$$

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 two continuous real-valued functions $f, g \in C(\sigma(a), \mathbb{R})$, the Jordan continuous functional calculus satisfies: $$(fg)(a) = f(a) * g(a)$$ where $f(a)$ denotes the image of $f$ under the functional calculus, and $*$ denotes the Jordan product (anticommutator) on the self-adjoint elements of $A$, defined by $x * y = xy + yx$.

Let $A$ be a $C^*$-algebra and $a$ be a self-adjoint element of $A$. For any continuous real-valued function $f \in C(\sigma(a), \mathbb{R})$ defined on the spectrum of $a$, the element $f(a)$ (obtained via the Jordan continuous functional calculus) commutes with $a$ in the algebra $A$.

Let $a$ be a self-adjoint element of a $C^*$-algebra $A$, and let $\sigma(a)$ denote its spectrum. For any continuous real-valued function $f \in C(\sigma(a), \mathbb{R})$, if $f \ge 0$ (meaning $f(x) \ge 0$ for all $x \in \sigma(a)$), then the element $f(a)$ obtained via the Jordan continuous functional calculus is non-negative ($f(a) \ge 0$).

Let $a$ be a self-adjoint element of a $C^*$-algebra $A$. The Jordan continuous functional calculus map $f \mapsto f(a)$, which sends a continuous real-valued function $f \in C(\sigma(a), \mathbb{R})$ to a self-adjoint element in $A$, is monotone. That is, for any $f, g \in C(\sigma(a), \mathbb{R})$, if $f \leq g$ pointwise on the spectrum $\sigma(a)$, then $f(a) \leq g(a)$ in the standard ordering of self-adjoint elements.