Exponentiating an almost idempotent and almost self-adjoint element Let $A$ be a unital $C^*$-algebra, and let $q\in A$ be such that $\|q^2-q\|<\varepsilon$, and $\|q\|<K$.
In answer to another question, s.harp showed that
$$\|e^{2\pi iq}-1_A\| < \varepsilon\ \frac{e^{2\pi K}-e^{2\pi}}{K^3-K^2}.$$
On the other hand, for any $s\in\mathbb{R}$,
$$|e^{2\pi is}-1| < 2\pi|s|.$$
Using this, a better bound should be possible in the case that $q$ is self-adjoint, or, more generally, close to self-adjoint.
Question: Suppose $q$ also satisfies $\|q-q^*\|<\delta$ in addition to the above. Then is it possible to improve the above bound for $\|e^{2\pi iq} - 1_A\|$ by taking $\delta$ into account?
 A: Here is a calculation yielding better bounds, but its probably very far away from optimal. The gain is that there is a uniform bound in $\delta$ and $\epsilon$.
Fix some constants:
$$K:=\|q\|, \quad z:= 2\pi i, \quad C:=\sup_{t\in [0,1]}\|\exp( z \,t \,(q-q^*)/2+ zq)\|$$
(one bound of $C$ is $e^{\pi\delta}e^{2\pi K}$, but thats probably a bad bound).
One bound I can get is the following:

If $\epsilon + (K+1)\frac{\delta}2<\frac{\pi^2}9+\frac{\pi}3$ then:
$$\|e^{i2\pi q}-1\|≤C\frac \delta2+\sqrt{2\left(1-\cos(2\pi R (\epsilon, \delta) )\right)}$$
With $R(\epsilon,\delta)= \sqrt{1/4 + \epsilon +(K+1)\frac\delta2}- 1/2$.

The only hard part is the bookkeeping, the path is conceptually simple start with:
$$\|\exp( z\,q)-1\| ≤ \|\exp(z\,q)-\exp(z\frac{q+q^*}2)\|+\|\exp(z\frac{q+q^*}2)-1\|\tag{$*$}$$
now since $q$ is close to $q^*$ we will see if this makes the first term small and since $\frac{q+q^*}2$ is hermitian we will apply the inequality from the hermitian case.
For the first term note:
$$\exp(z\,q)-\exp(z\frac{q+q^*}2) = \left[\exp(zt\,(q-q^*)/2+ zq)\right]_{t=0}^{t=1}=z\int_0^1dt \frac{q-q^*}2 \exp(zt\,(q-q^*)/2+ zq)$$
taking the norm and pulling it into the integral yields that the first term of $(*)$ is smaller than $\pi\,C\,\delta$.
For the second term compute:
$$\|\frac{q+q^*}2(1-\frac{q+q^*}2)\| ≤ \begin{split}\frac14\left( \|q(1-q)\|+\|q^*(1-q^*)\|+\|q^*(1-q^*)+(q-q^*)(1-q^*)\|+\\ \|q(1-q)+(q^*-q)(1-q)\|\right)\end{split}$$
which you further simplify to $\epsilon +(K+1)\frac\delta2$.
Now you may apply the hermitian case to $\frac{q+q^*}2$ using $\epsilon + (K+1)\frac{\delta}2$ as the parameter. This yields, provided $\epsilon + (K+1)\frac{\delta}2<\frac{\pi^2}9+\frac\pi3$, that
$$\|e^{i2\pi (q+q^*)/2}-1\|≤\sqrt{2\left(1-\cos(2\pi R(\epsilon, \delta))\right)}$$
