C*-algebra generated by a non invertible normal element

Question

Let $A$ be a C*-algebra and $x\in A$ be a non-invertible normal element. By functional calculus, we know $$C^*(x,1)\simeq C(\sigma(x))$$ Where $\sigma(x)$ means the spectrum of $x$.

I need to construct non-unital C*-algebra generated by $x$ means $C^*(x)$. If $0$ is an accumulation point of $\sigma(x)$, then I think $C*(x)\simeq C_0(\sigma(x)/\{0\})$.

But If $0$ is an isolated point, for which subset of $\sigma(x)$, there is an isomorphism?

Thanks in advance

Answer 1

If $0$ is an isolated point, the only new thing is that $\sigma(x)\setminus\{0\}$ is still compact. So $$ C_0(\sigma(x)\setminus\{0\})=C(\sigma(x)\setminus\{0\}). $$