PhyslibAlpha.ProbabilisticTheory.StarAlgebra.ComplexExtension

Given an $\mathbb{R}$-linear functional $\phi$ defined on the set of self-adjoint elements $A_{sa}$ of a $\ast$-algebra $A$ (mapping to $\mathbb{R}$), this definition constructs its unique complex-linear extension to the entire algebra $A$. For any operator $a \in A$, the extension is defined as $\phi(\text{Re } a) + i \phi(\text{Im } a)$, where $\text{Re } a = \frac{a + a^*}{2}$ and $\text{Im } a = \frac{a - a^*}{2i}$ are the self-adjoint real and imaginary parts of $a$.

Let $A$ be a $\ast$-algebra and $\phi: A_{sa} \to \mathbb{R}$ be an $\mathbb{R}$-linear functional on the space of self-adjoint elements of $A$. For any operator $a \in A$, the value of the complex-linear extension of $\phi$ is given by $$(\text{complexExtension } \phi)(a) = \phi(\Re a) + i \phi(\Im a)$$ where $\Re a = \frac{a + a^*}{2}$ and $\Im a = \frac{a - a^*}{2i}$ are the self-adjoint real and imaginary parts of $a$.

Let $A$ be a $\ast$-algebra and $A_{sa}$ denote the set of its self-adjoint elements. Let $\phi : A_{sa} \to \mathbb{R}$ be an $\mathbb{R}$-linear functional (representing an expectation value on observables), and let $\tilde{\phi} : A \to \mathbb{C}$ be its complex-linear extension to the whole algebra. For any self-adjoint operator $a \in A_{sa}$, the extension evaluated at $a$ coincides with the original functional, such that $\tilde{\phi}(a) = \phi(a)$.

Let $A$ be a $\ast$-algebra and let $\phi: A_{sa} \to \mathbb{R}$ be an $\mathbb{R}$-linear functional defined on the self-adjoint elements of $A$. Let $\langle \cdot \rangle$ denote the unique complex-linear extension of $\phi$ to the entire algebra $A$. For any operator $a \in A$, the expectation value of the adjoint $a^*$ is the complex conjugate of the expectation value of $a$: $$\langle a^* \rangle = \overline{\langle a \rangle}$$

Let $f: C \to D$ be a $\mathbb{C}$-linear map between spaces equipped with an adjoint (star) operation. If $f$ commutes with the adjoint, such that $f(a^*) = f(a)^*$ for all $a \in C$, then the real part of the image of an operator $a$ under $f$ is equal to the image of the real part of $a$. That is, $\Re(f(a)) = f(\Re(a))$, where $\Re(a) = \frac{1}{2}(a + a^*)$ denotes the self-adjoint real part of the operator $a$.

Let $C$ and $D$ be complex vector spaces equipped with a star-operation (adjoint). If $f: C \to D$ is a $\mathbb{C}$-linear map that commutes with the adjoint operation (i.e., $f(a^*) = f(a)^*$ for all $a \in C$), then for any $a \in C$, the image of the imaginary part of $a$ under $f$ is equal to the imaginary part of the image of $a$. That is, $$f(\Im a) = \Im(f a)$$ where $\Im a$ denotes the imaginary part of the operator $a$.