Added states.

src/op/c-star/positive.tex

@@ -29,4 +29,13 @@
 \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}
+
+