Corollary 34.9.3.label 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 $\Phi|_{B} = A$.
Proof. By the Hahn-Banach Theorem, 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 Theorem 34.9.2.$\square$
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