Let $A_0$ be a C*-subalgebra in a C*-algebra $A$. Let $\phi_0$ be a bounded linear functional on $A_0$ and assume $\phi$ is an extension of $\phi_0$ on $A$. I mean $\phi\in A^*$ with $\phi_{|_{A_0}}=\phi_0$ ($\phi$ and $\phi_0$ may have different norm).
True or false: We have $|\phi_0|\leq |\phi|$ On $A_0$.