Prove the existence of $\Psi\in(\ell_\infty(G))^\ast$ satisfying the amenability conditions

Question

Let $G$ be a finitely generated group satisfying the Folner condition and let $S\subset G$ be a finite generating set of $G$. Denote by $\rho=\{\rho_g\}_{g\in G}$ the right translation action of $G$ on $\ell_{\infty}(G)$ defined by $$[\rho_g(\zeta)](k)=\zeta(kg), \ \ \zeta\in\ell_\infty(G),k\in G.$$ Denote by $\lambda$ the left translation action of $G$ on $\ell_\infty(G)$. Prove that there exists $\Psi\in(\ell_\infty(G))^\ast$ satisfying the following conditions:

1. $\Psi(1)=1$, where $1$ on the left hand side is the constant function $1$ on $G$,
2. $\left\|\Psi\right\|=1$,
3. $0\leq\Psi(\zeta)$ for every non-negative real-valued bounded function $\zeta$ on $G$,
4. $\Psi(\lambda_g\zeta)=\Psi(\zeta)$ for every $g\in G$ and every $\zeta\in\ell_\infty(G)$,
5. $\Psi(\rho_g\zeta)=\Psi(\zeta)$ for every $g\in G$ and every $\zeta\in\ell_\infty(G)$.

Attempt: Already proved that there exists $\Phi\in(\ell_\infty(G))^\ast$ satisfying $1,2,3, 4$.

Problem: I want to show that for an arbitrary non-empty finite subset $F\subset G$, the linear functional $\Psi_F\in(\ell_\infty(G))^\ast$ defined by $$\Psi_F(\zeta)=\frac{1}{\#F}\sum_{h\in F}\Phi(\rho_h\zeta), \ \ \zeta\in\ell_\infty(G)$$ also satisfies 1,2,3,4. If $F$ is an $(\varepsilon,S)$-Folner set, then $\Psi_F$ satisfies an inquality. By Banach-Alaoglu theorem, we can obtain something that is essential for the proof.

Answer 1

Here is one way to prove (4):

$$ \Psi _F(\lambda _g\zeta ) = \frac{1}{\#F}\sum _{h\in F}\Phi (\rho _h\lambda _g\zeta ) {\ \buildrel {(i)} \over {=}\ } \frac{1}{\#F}\sum _{h\in F}\Phi (\lambda _g\rho _h\zeta ) {\ \buildrel {(ii)} \over {=}\ } \frac{1}{\#F}\sum _{h\in F}\Phi (\rho _h\zeta ) = \Psi _F(\zeta ). $$

Notes:

• (i) Use that $\rho _h$ and $\lambda _g$ commute. This is a simple consequence of the associativity axiom of $G$.

• (ii) Use that $\Phi $ is invariant under $\lambda _g$.