**Edit:** According to answer and comments by Prof. Valette we edite the question.

The Kadison Kaplansky conjecture says:

**Kadison-Kaplansky conjecture:** If $G$ is a torsion-free discrete group then $C^*_{\mathrm{red}}(G)$ has no nontrivial projection.

Obviously existence of a non-trivial projection $e$ for a $C^*$-algebra $A$ implies that $A$ has a self-adjoint unitary element $u=1-2e$ which is neither $1$ nor $-1$.

Our main question is the following:

**Question:** Let $G$ be a discrete group and $u$ be a self-adjoint unitary element of $C^*_{\mathrm{red}}(G)$ different from $1$ and $-1$. Let $g\in G \subset C^*_{\mathrm{red}}(G)$ be a nearest element to $u$ among all group elements $h\in G\subset C^*_{\mathrm{red}}(G)$. Here by "nearest" we mean the nearest according to the distance arising from the operator norm on $C^*_{\mathrm{red}}(G)$. **Can one say that such a $g$ is a torsion element of $G$ which is different from the neutral element $e\in G$?**

Or can one find a nearest element $g$ as above and then prove that $g$ is a nontrivial torsion element?

Does the main question has an affirmative answer at least in the abelian case?

**A refinement of the question according to comment discussion:** Assume that $u$ is a non trivial self adjoint unitary element of $C^*_{\text{red}} G$ and $g\in G$ is a group element such that $g$ or $-g$ minimize the quantity $|\pm h-u|,\quad h\in G$. Does this implies that $g$ is of finite order?Can one find such a $g\neq e$?

**Note:** By emphasising on the word "non-neutral element" one can easily check that this nearest element $\pm g$ is always non neutral for the simple case $G=\mathbb{Z}/3\mathbb{Z}$.

How can this very interesting existing answer be generalized to a group which is not necessarilly abelian?