Let $p$ be an element of a unital $C^*$-algebra $A$. One can show that if $p^2=p$, then $e^{2\pi i p}=1$.
Question 1: What is the largest $\delta>0$ (independent of $A$) such that whenever
$$\|p^2-p\|<\delta,$$
we have $\|e^{2\pi ip}-1\|<1$?
It's possible that that answer to Question 1 depends on $\|p\|$ and/or $\|1-p\|$. If so let's say that both of these quantities are less than some constant $C$. Then the relevant question would be:
Question 2: What is the largest $\delta_C>0$ (independent of $A$) such that whenever
$$\|p^2-p\|<\delta_C,$$
we have $\|e^{2\pi ip}-1\|<1$?
Remark: Using the holomorphic functional calculus, one can construct an idempotent $e$ somewhat close to $p$, but with possibly larger norm. One can show using this and the fact that $$\|e^a-e^b\|\leq e^{\|a-b\|}\|a-b\|,$$ that if $\|p\|$ and $\|1-p\|$ are small enough, then $\delta$ can be taken to be $\frac{1}{4}$. But this number seems a bit mysterious to me. I feel like there might be something simpler I am missing, or perhaps something like this is known in the literature?