functional calculus

Question

Suppose $A$ is a non-unital $C^*$ algebra, $B$ is another $C^*$ algebra. Suppose $\phi: A \to B$ is a non-zero $*$-homomorphism and $x_0$ is a normal elememt in $A$, by the continuous functional calculus, we have $\phi(f(x_0))=f(\phi(x_0))$ for any $f\in C_0（\sigma_{A}(x_0))$. My question is: can we choose a function $f\in C_0（\sigma_{A}(x_0))$ such that $|\phi(f(x_0))\|>1$?

Answer 1

By Urysohn's lemma, we can always choose a continuous function that vanishes at infinity with $f(z)=10000$ at a fixed point $z$. So, $\|\phi(f(x_0))\|=\|f(\phi(x_0))\|=\|f\|>9999$.