Definition 6.4.10.label Let $A$ be a non-unital $C^{*}$-algebra then $\qspec{A}{x}\ni 0$ and we consider the unital $C^{*}$-algebra $A_{I}$ obtained by adjunction of an identity to $A$. For a normal $x\in A$ and $f\in C(\qspec{A_I}{x})$, if $f(0)=0$ then we define $f(x)\define\p(f)$ where $f(x)\in A$.
Proof. Let $\om_{0}$ be the homomorphism of $A_{I}$ with kernel $A$ then $\om_{0}(f(x))=f(\om_{0}(x))=f(0)=0$ so $f(x)\in \ker(\om_{0})=A$$\square$