\section{Positive Linear Functionals}
\label{section:cstar-positive}

\begin{definition}[Positive Linear Functional]
\label{definition:cstar-positive-functional}
    Let $A$ be a unital $C^*$-algebra and $\phi \in A^*$, then $\phi$ is \textbf{positive} if $\dpn{x, \phi}{A} \ge 0$ for all positive elements $x \in A$. 
\end{definition}

\begin{theorem}
\label{theorem:cstar-positive-algebraic}
    Let $A$ be a unital $C^*$-algebra and $\phi \in \hom(A; \complex)$, then the following are equivalent:
    \begin{enumerate}
        \item $\phi$ is a positive linear functional.
        \item $\phi \in A^*$ with $\normn{\phi}_{A^*} = \dpn{1, \phi}{A}$. 
    \end{enumerate}
\end{theorem}
\begin{proof}
    (1) $\Rightarrow$ (2): For any $x \in A_{sa}$ with $\norm{x}_A \le 1$, $\sigma_A(1 - x) \subset 1 - [-1, 1] = [0, 2]$ by \autoref{proposition:commutative-spectrum-gymnastics}. Thus $1 - x \ge 0$ by \autoref{proposition:positive-spectrum}, and $\dpn{x, \phi}{A} \le \dpn{1, \phi}{A}$. By \autoref{proposition:hermitian-functional-norm}, $\norm{\phi}_{A^*} \le \dpn{1, \phi}{A}$, so $\norm{\phi}_{A^*} = \dpn{1, \phi}{A}$. 

    (2) $\Rightarrow$ (1): For each $x \in A_{sa}$, $\sigma_A(x) = \sigma_{A[x]}(x)$ by \autoref{corollary:c-star-algebra-preserve-spectrum}, so $x \ge 0$ in $A$ if and only if $x \ge 0$ in $A[x]$ by \autoref{proposition:positive-spectrum}. In addition, the \hyperref[Gelfand-Naimark Theorem]{theorem:gelfand-naimark} implies that $x \ge 0$ if and only if $\Gamma_{A[x]}(x) \ge 0$. Moreover, since $1 \in A[x]$, the norm of $\phi$ alongside (2) is preserved by restricting to $A[x]$. By considering each commutative subalgebra, assume without loss of generality that $A$ is commutative. 
    
    By the \hyperref[Riesz Representation Theorem]{theorem:riesz-radon-c0}, $\phi$ takes the form of a complex Radon measure $\mu$ on $\Omega(A)$, and $\norm{\phi}_{A^*} = \norm{\mu}_{\text{var}}$. For each Borel set $E \in \cb_{\Omega(A)}$, 
    \begin{align*}
        \norm{\mu}_{\text{var}} &= \int_{\Omega(A)} 1 d\mu = \mu(E) + \mu(\Omega(A) \setminus E) \\
        &\le |\mu(E)| + |\mu(\Omega(A) \setminus E)| \le \norm{\mu}_{\text{var}}
    \end{align*}

    which is only possible if $\mu(E), \mu(\Omega(A) \setminus E) \ge 0$. As this holds for all $E \in \cb_{\Omega(A)}$, $\mu$ is positive, and $\phi$ then is a positive linear functional. 
\end{proof}


\begin{corollary}
\label{corollary:positive-linear-functional-extension}
    Let $A$ be a unital $C^*$-algebra, $B \subset A$ be a closed subspace with $1_A \in B$, and $\phi \in B^*$ with $\norm{\phi}_{B^*} = \dpn{1_A, \phi}{B}$, then there exists a positive linear functional $\Phi \in A^*$ such that $\Phi|_B = A$. 
\end{corollary}
\begin{proof}
    By the \hyperref[Hahn-Banach Theorem]{theorem:hahn-banach}, there exists $\Phi \in A^*$ such that $\Phi|_B = A$ and $\norm{\Phi}_{A^*} = \norm{\phi}_{B^*}$. In which case, $\norm{\Phi}_{A^*} = \dpn{1_A, \Phi}{A}$, and $\Phi$ is also positive by \autoref{theorem:cstar-positive-algebraic}. 
\end{proof}