Thanks for your reply. Great answer!

Thanks for the OEIS reference. It's interesting: For the next higher order case $\mathbb{Z}_2^{*3}$ the operator $X_\theta$ is $X_{\theta}:=-12\cdot\tan\left(\theta\right)+T_{r}^{\left(1\right)}+T_{s}^{\left(1\right)}+T_{t}^{\left(1\right)}$. Here I would have also expected that $\left\Vert X_{\theta}\right\Vert \neq\left\Vert X_{\theta}-2\tan\left(\theta\right)P_{\left\{ e\right\} }\right\Vert$ for $\theta \neq 0$. But (using the same methods as above) it seems like that in this case my claim is not true

First of all thanks for your answer! I had to think about it before I respond. You asked about the origin: The problem is related to the question whether or not certain deformations of the group $C^*$-algebra of $\mathbb{Z}_2^{*L}$ are simple or not. This question (and I find this quite remarkable) leads to operators of the type above. The one I mentioned is the easiest non-trivial case of those

Thanks for your response! Under assuming equality of both norms and by using your suggestion I can show that $t \mapsto \left\Vert X_\theta -tP \right\Vert$ is constant on the interval from $[16\text{tan}\left(\theta\right), -2\text{tan}\left(\theta\right)]$ (assuming $\theta \leq 0$). Do you think that could help deducing a contradiction?

I'm interested in any result on virtually abelian groups, so a (maybe partial) answer in the discrete case would already be nice