Suppose $G$ is a finitely generated group, with given generating set $S={g_1, \dots, g_n}$. (Assume that if $g\in S$, then $g^{-1}\notin S$). Given complex numbers $a_1, \dots , a_n$, we can form the element $\sum a_j g_j$ of the group ring $\mathbb C[G]$ and consider its norm as an element of the full group C*-algebra $C^\ast(G)$: $$ \|\sum a_j g_j \| =\sup_\pi \|\sum a_j \pi(g_j)\| $$ where the supremum is over all unitary representation $\pi$ of $G$ on Hilbert space. I am interested in the quantity $$ \alpha(S) = \inf_{\|a\|_1=1} \|\sum a_j g_j\|. $$ (Here $\|a\|_1 =\sum |a_j|$).

Question: Are there known examples of groups $G$ and generating sets $S$ for which $\alpha(S)<1$?

(I am mostly interested in the case where all the generators have infinite order, so I haven't spent much time looking at finite groups. So, in case there are finite examples, a follow-up question would be for examples with all $g$ of infinite order.)

Some background and discussion: The question arose from an attempt to analyze the convex set of all points in $\mathbb C^n$ of the form $$ (\langle \pi(g_1)x,y\rangle, \dots \langle \pi(g_n)x,y\rangle ) $$ where $\pi$ ranges over all representations of $G$ on Hilbert spaces $\mathcal H$ and $x,y$ range over all vectors of the unit ball of $\mathcal H$. I am interested in how large a polydisk centered at the origin such a set can contain; estimating the norm of $\sum a_j g_j$ is essentially the dual problem to this one. In this setting, cases where $\alpha(S)$ is small are the "enemy", so I'm trying to understand more about when this happens.

It is not hard to prove the bounds $$ \frac{1}{\sqrt{n}} \leq \alpha(S) \leq 1 $$ (indeed the upper bound is trivial, and the lower bound comes from considering the regular representation of $G$ on $\ell^2(G)$. ) As simple examples, it can be shown that the free abelian group $\mathbb Z^n$ and the free group $\mathbb F^n$ (with their standard $n$-element generating sets) both have $\alpha(S)=1$. Now, one can try to get better lower bounds on $\alpha(S)$ in particular cases by considering particular representations $\pi$ (such as the regular representation), since then one is dealing with particular concrete unitaries, but it seems much harder to improve the upper bound (when it can be improved) unless one has a handle on a faithful representation of $C^\ast(G)$. Of course this is the case when $G$ is amenable, though at the same time it's not clear that amenability is relevant, given the free (non-amenable) and free abelian (amenable) examples just mentioned.

Suppose $G$ is a finitely generated group, with given generating set $S={g_1, \dots, g_n}$. (Assume that if $g\in S$, then $g^{-1}\notin S$. (EDIT: Also assume that $S$ is minimal in the sense that no proper subset of $S$ is generating.)) Given complex numbers $a_1, \dots , a_n$, we can form the element $\sum a_j g_j$ of the group ring $\mathbb C[G]$ and consider its norm as an element of the full group C*-algebra $C^\ast(G)$: $$ \|\sum a_j g_j \| =\sup_\pi \|\sum a_j \pi(g_j)\| $$ where the supremum is over all unitary representation $\pi$ of $G$ on Hilbert space. I am interested in the quantity $$ \alpha(S) = \inf_{\|a\|_1=1} \|\sum a_j g_j\|. $$ (Here $\|a\|_1 =\sum |a_j|$).

Question: Are there known examples of groups $G$ and generating sets $S$ for which $\alpha(S)<1$?

(I am mostly interested in the case where all the generators have infinite order, so I haven't spent much time looking at finite groups. So, in case there are finite examples, a follow-up question would be for examples with all $g$ of infinite order.) (EDIT: Since the C*-norm will be strictly smaller than $\ell^1$ norm, there should be lots of examples, see answers & comments below. We can still ask for examples when the generating set is minimal, and then there is the harder question of just how small $\alpha(S)$ can be.)

Some background and discussion: The question arose from an attempt to analyze the convex set of all points in $\mathbb C^n$ of the form $$ (\langle \pi(g_1)x,y\rangle, \dots \langle \pi(g_n)x,y\rangle ) $$ where $\pi$ ranges over all representations of $G$ on Hilbert spaces $\mathcal H$ and $x,y$ range over all vectors of the unit ball of $\mathcal H$. I am interested in how large a polydisk centered at the origin such a set can contain; estimating the norm of $\sum a_j g_j$ is essentially the dual problem to this one. In this setting, cases where $\alpha(S)$ is small are the "enemy", so I'm trying to understand more about when this happens.

It is not hard to prove the bounds $$ \frac{1}{\sqrt{n}} \leq \alpha(S) \leq 1 $$ (indeed the upper bound is trivial, and the lower bound comes from considering the regular representation of $G$ on $\ell^2(G)$. ) As simple examples, it can be shown that the free abelian group $\mathbb Z^n$ and the free group $\mathbb F^n$ (with their standard $n$-element generating sets) both have $\alpha(S)=1$. Now, one can try to get better lower bounds on $\alpha(S)$ in particular cases by considering particular representations $\pi$ (such as the regular representation), since then one is dealing with particular concrete unitaries, but it seems much harder to improve the upper bound (when it can be improved) unless one has a handle on a faithful representation of $C^\ast(G)$. Of course this is the case when $G$ is amenable, though at the same time it's not clear that amenability is relevant, given the free (non-amenable) and free abelian (amenable) examples just mentioned.