Let $G$ be a finite group and $\mathbb{C}G$ its group ring.
Now taking the approach of orangeskid, consider the space $\mathbb{C}G$ as a Hilbert space with orthonormal basis $\delta^g$. $G$ acts on it by left multiplication which are isometries. $$L_g \colon \delta^h \mapsto \delta^{g h}.$$ Hence the adjoint of the map $L_g$ is $L_{g}^{-1} = L_{g^{-1}}$.
Now $\mathbb{C}G$ acts on $\mathbb{C}G$ by left multiplication so we can look at $\mathbb{C}G$ as an algebra of linear operators on the Hilberts spaced $\mathbb{C}G$. This algebra is closed under $*$, since $$\left(\sum a_g g\right)^* = \sum \bar a_g g^{-1}$$ so it's a $*$-subalgebra of $End(\mathbb{C}G)$, hence a $\mathrm{C}^*$-algebra.
I am interested in the states of $\mathbb{C}G$. Now the states of $\mathbb{C}G$ lie in the second dual of $F(G)$ in other words they lie in $F(G)$.
$\mathbb{C}G$ is unital with unit $1_{\mathbb{C}G}=\delta^e$ and we have (e.g. Murphy, Corollary 3.3.4):
If $\tau$ is a bounded linear functional on a unital $\mathrm{C}^*$-algebra, then $\tau$ is positive if and only if $\tau(1)=\|\tau\|$.
Now... I have no idea what the $\mathrm{C}^*$-norm on $\mathbb{C}G$ actually is... this is something that I just get very confused about (any information here also greatly appreciated).
If I knew what the norm on $\mathbb{C}G$ was I could use this... so instead what I want to do is find the condition on linear functional on $\mathbb{C}G$ being positive and from there I can say that, well, we must have $\|\tau\|=\tau(\delta^e)$ and this means that for $$\tau=\sum_{g\in G}\alpha_g\delta_g$$ to be a state we must have $\alpha_e=1$.
For a state to be positive we must have $$\tau(v)\geq 0$$ for all $v\in \mathbb{C}G^+$. Now an element of $\mathbb{C}G$ is positive if it is of the form $\phi^*\star \phi$... that is for $\varphi$ in to be positive there must exist complex numbers $\{\beta_g:g\in G\}$ such that $$\varphi=\sum_g\left(\sum_{t\in G}\overline{\beta_t}\beta_{tg}\right)\delta^g,$$
and if $\tau=\sum_{s\in G}\alpha_s\delta_s$ a linear functional is to be positive we must have
$$\sum_{g\in G}\alpha_g\left(\sum_{t\in G}\overline{\beta_t}\beta_{tg}\right)\geq 0,$$
for all possible selections of $\{\beta_g:g\in G\}\subset \mathbb{C}$.
Are things as (computationally) awkward as this or is there another way to characterise/recognise the states?
Context: I want to study states on $\mathbb{C}G$ that have the property that
$$\lim_{k\rightarrow \infty} \sum_{g\in G}a_g\nu(\delta^g)^k=a_e;$$ for $\nu$ a state and $\{a_g\}\subset\mathbb{C}$.
These are states on the (algebra of functions on the) quantum group $\mathbb{C}G$ such that their convolution powers converges to the Haar state.
The states on $\mathbb{C}G$ are given by positive definite functions in $F(G)$ such that $u(e)=1$.
Therefore, states on $\mathbb{C} G$ are given by unitary representations $\rho:G\rightarrow \hom(V)$ and unit vectors $\xi\in V$ according to
$$u(g)=\langle \rho(g)\xi,\xi\rangle.$$