PhyslibAlpha.ProbabilisticTheory.StarAlgebra.Restrict

Given a unital positive linear map $f : E_1 \to E_2$ and submodules $F_1 \subseteq E_1$ and $F_2 \subseteq E_2$ over a ring $S$, such that the multiplicative identity elements $1_{E_1}$ and $1_{E_2}$ are contained in $F_1$ and $F_2$ respectively, and $f$ maps $F_1$ into $F_2$ (i.e., $f(x) \in F_2$ for all $x \in F_1$), this definition constructs the restriction $f|_{F_1} : F_1 \to F_2$ as a unital positive linear map.

Let $A_1$ and $A_2$ be star-algebras. Given a unital positive $\mathbb{C}$-linear map $f : A_1 \to A_2$, this definition constructs its restriction to the submodules of self-adjoint elements. The resulting map is a unital positive $\mathbb{R}$-linear map from $\text{selfAdjoint}(A_1)$ to $\text{selfAdjoint}(A_2)$, which preserves the identity and the property of positivity.

Let $A_1$ and $A_2$ be complex star-algebras. Let $f: A_1 \to A_2$ be a unital positive $\mathbb{C}$-linear map, and let $f_{\text{res}}: \text{selfAdjoint}(A_1) \to \text{selfAdjoint}(A_2)$ be its restriction to the submodules of self-adjoint elements. For any self-adjoint element $x \in \text{selfAdjoint}(A_1)$, the value of the restricted map $f_{\text{res}}(x)$ is equal to the value of the original map $f(x)$ when both are viewed as elements of $A_2$.

Let $A_1$ be a star-algebra. Given a unital positive $\mathbb{C}$-linear functional $f : A_1 \to \mathbb{C}$, this definition constructs its restriction to the submodule of self-adjoint elements $\text{selfAdjoint}(A_1) = \{x \in A_1 \mid x^* = x\}$. Since the image of a self-adjoint element under a positive map is real, the resulting map is a unital positive $\mathbb{R}$-linear functional from $\text{selfAdjoint}(A_1)$ to the real numbers $\mathbb{R}$.

Let $A_1$ be a star-algebra and $f : A_1 \to \mathbb{C}$ be a unital positive $\mathbb{C}$-linear functional. Let $f_{\text{res}} : \text{selfAdjoint}(A_1) \to \mathbb{R}$ be the restriction of $f$ to the submodule of self-adjoint elements $\text{selfAdjoint}(A_1) = \{x \in A_1 \mid x^* = x\}$. For any $x \in \text{selfAdjoint}(A_1)$, the value of the restricted functional $f_{\text{res}}(x)$, when embedded into $\mathbb{C}$, is equal to the value of the original functional $f(x)$.