**Edit:** According to comment conversations we revise the question.


Let $G$ be a group. We consider the following subset of $G$:
$$\{g\in G \mid e^{\lambda\_g} \in \mathbb{C}\lambda (G)\},$$
where $\lambda\_g\in C^\*\_{\text{red}}(G)$ is the left regular representation of $g$.


Under which conditions is this subset a subgroup of $G$? What is a description of this subset or subgroup?



A subset (or subgroup) associated to a group
Let $(\delta\_g)\_{g\in G}$ be the canonical basis of $\mathbf{C}G$. Define $\exp(g)=\sum\frac1{n!}\delta\_g^n$ as an element of the reduced $C^\*$-algebra.



> 
> I claim that $\exp(g)\in\mathbf{C}G$ iff $g$ has finite order.
> 
> 
> 


It's clear if $g$ has finite order. For the converse, I claim that more generally that for every summable sequence $(c\_g)\_{g\in G}$, the element $\sum\_g c\_g\delta\_g$ has norm $\ge \left(\sum\_g|c\_g|^2\right)^{1/2}$; in particular it belongs to $\mathbf{C}G$ (if and) only if $(c\_g)$ is finitely supported.


Indeed, for $\varepsilon>0$ (say $<1$), there exists a finite subset $I$ of $G$ such that $\sum\_{g\in J}|c\_g|\le\varepsilon$ for every finite subset $J$ of $G$ disjoint of $I$. Hence $\left\|\sum\_g c\_g\delta\_g\right\|\ge \left\|\sum\_{g\in I} c\_g\delta\_g\right\|-\varepsilon$, and also $\left\|\sum\_{g\in I} c\_g\delta\_g\right\|\_2\ge \left\|\sum\_{g} c\_g\delta\_g\right\|\_2-\varepsilon$.


Evaluation at $\delta\_1\in\ell\_2(G)$ shows that the $C^\*$-norm on $\mathbf{C}G$ is $\ge$ than the $\ell^2$ norm. Hence $$\left\|\sum\_g c\_g\delta\_g\right\|\ge \left\|\sum\_{g\in I} c\_g\delta\_g\right\|\_2-\varepsilon\ge \left\|\sum\_{g} c\_g\delta\_g\right\|\_2-2\varepsilon;$$
since this holds for all $\varepsilon>0$ one gets $\|x\|\ge \|x\|\_2$ for $x=\sum\_gc\_g\delta\_g$, which is the desired inequality.


For the conclusion, if $\sum c\_g\delta\_g=\sum c'\_g\delta\_g$ with $(c'\_g)$ finitely supported, one has $\sum (c\_g-c'\_g)\delta\_g=0$, and hence $\sum\_g|c\_g-c'\_g|^2=0$, so $c\_g=c'\_g$ for all $g$. Hence $(c\_g)$ is finitely supported as well.


