# functional calculus

Suppose $$A$$ is a non-unital $$C^*$$ algebra,$$B$$ is another $$C^*$$ algebra.Suppose $$\phi:A\rightarrow B$$ is a non-zero $$*$$ homomorphism and $$x_0$$ is a normal elememt in $$A$$,by 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 $$|\phi(f(x_0))\|>1$$?

• If you can find a function $f$ such that $\phi(f)\neq0$, can you see why this is true? – Aweygan Dec 3 '18 at 18:30
• If $\phi(x_0)\neq0$,you mean that $f(z)=z,z\in C_0（\sigma_{A}(x_0))$ is suitable?But how to ensure that $|\phi(f(x_0))\|\geq1$? – math112358 Dec 4 '18 at 2:16

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, $$\|\phi(f(x_0))\|=\|f(\phi(x_0))\|=\|f\|>9999$$.
• Probably, the OP wants that $f(\phi(x_0)) \in B$ and so one should also require that $f(0) = 0$. This however is no problem, since the spectrum of $\phi(x_0)$ contains points other than zero. E.g. $f(z) = \lambda z$ for some large $\lambda > 0$. – user42761 Dec 4 '18 at 13:11