I am working on a problem and one of the intermediate steps I want to prove is that in a unital $C^*$-algebra, given a selfadjoint element $x$, there exists a non-trivial projection $y \in \mathcal{A}_x$, where $\mathcal{A}_x$ denotes the $C^*$-algebra generated by $x$. If not mistaken, one has an isometric $^*$-isomorphism $$ \Psi: \mathcal{A}_x \rightarrow \mathcal{C}(\sigma(x)), $$ which let's me believe there has to be such element given a disconnected spectrum. Still, I fail to see how a mere algebra guarantees such existence.
I am curious also if such existence is conditional also on the general $C^*$-algebra, i.e. existence of a non-trivial projection $e \neq 0,1$ in the bigger algebra implies existence of a projection in other generated subalgebras.
I consider the algebra generated by $x$ to be $$ \mathcal{A}_x = \overline{\{ \sum_{m,n=1}^N \alpha_{m,n}\;x^{m}(x^*)^n : \alpha_{i,j} \in \mathbb{C} \}} $$
In response to the comments, the original problem requires to show that for any $\varepsilon > 0$ there exists $\delta > 0$ such that for all projections $a$ in the $C^*$-algebra and $x$ self-adjoint with $\|a - x\| < \delta$, there exists a projection $g \in C(\sigma(x))$ with the property $\|a - \Psi^{-1}(g)| < \varepsilon.$
My initial thought was to first identify the projections of $C(\sigma(x))$ and from there to map them back to the algebra and gain more insight afterwards.