Bounding $\|e^{2\pi iq}-1\|$ for $q$ an almost idempotent

Question

Let $A$ be a unital $C^*$-algebra, and let $q\in A$ such that $\|q^2-q\|<\varepsilon$, and $\|q\|<K$.

Question: How small does $\varepsilon$ need to be (possibly in terms of $K$) to guarantee that

$$\|e^{2\pi iq}-1_A\|<1?$$

Answer 1

You can get a numerical condition by bounding the terms in the sum. Like the others I don't know what you mean with homotopic. Start with

$$\|q^n-q\| = \|q^{n-2}(q^2-q)+q^{n-1}-q \| < \epsilon K^{n-2} +\|q^{n-1}-q\| $$ so by induction $$\|q^n-q\| < \epsilon \sum_{k=0}^{n-1} K^{n-2-k} = \epsilon K^{-2} \dfrac{1-K^{n}}{1-K}$$ (for $K\neq 1$). Making use of $\sum_{k=1}^\infty \frac{(i2\pi)^k}{k!}=0$ we may evaluate $$e^{i2\pi q}-1 =\sum_{k=1}^\infty \frac{(i 2\pi)^k q^k}{k!}= \sum_{k=1}^\infty \frac{(i2\pi)^k(q^k-q)}{k!}$$ Applying the bound we have derived for $q^k-q$ retrieves: $$\|e^{i2\pi q}-1\| < \epsilon \frac{K^{-2}}{1-K}\sum_k \frac{(2\pi)^k(1-K^n)}{k!}= \epsilon\ \frac{e^{2\pi K}-e^{2\pi}}{K^3-K^2}$$ which provides you with a condition on $\epsilon$ depending on $K$. This bound is definitely not optimal.