I want to proof that for a discrete group $G$ and $H$ a normal subgroup of $G$ the following holds:
$G$ and $H$ are amenable $\Rightarrow$ the quotient $G/H$ is amenable (using one of the definitions below).
Def: 1) A discrete group $G$ satisfies the Følner condition if for every finite subset $E\subseteq G$ and for every $\epsilon >0$ there exists a non-empty finite set $F\subseteq G$ such that $$\max\limits_{s\in E}\frac{|F\triangle sF|}{|F|}<\epsilon,$$ where $F\triangle sF$ denotes the symmetric difference of the sets $F$ and $sF$.
2) A discrete group $G$ is called amenable if it satisfies the Følner condition.
Def: A discrete group $G$ is called amenable if there is a left invariant mean, i.e. there exists a state $\mu\in S(l^{\infty}(G))$ such that $\mu(g\cdot f)=\mu(f)$ for all $g\in G,\; f\in l^{\infty}(G)$ (where $g\cdot f(h)=f(g^{-1}h)$ for all $h\in H$).
I have seen a proof of this fact using other definitions of amenability, but I should know a proof of this fact with one of the above stated definitions.
And I only consider discrete groups from now (since I know a proof that these definitions are equivalent if $G$ is discrete).
Let $G$ be a amenable group and $H$ an normal amenable subgroup ($H$ is automatically amenable if $G$ is amenable since if you have a left invariant mean $\mu\in S(l^{\infty}(G))$ then you can construct a left invariant mean $\mu\circ \iota \in S(l^{\infty}(H))$, where $\iota :l^{\infty}(H)\to l^{\infty}(\bigsqcup_{i\in I} Hg_i)\; f\mapsto f_i$ with $f_i(hg_i)=f(h)$ for $h\in H$, $\bigsqcup_{i\in I} Hg_i=G$).
I want to contruct a left inavriant mean $\tau :l^{\infty}(G/H)\to\mathbb{C}$. However, I'm not sure how to define $\tau$. One of my attempts: Consider the canonical projection $\pi:G\to G/H$ and the induced map $\pi^*:l^{\infty}(G/H)\to l^{\infty}(G),\; \pi^*(f)=f\circ \pi$. Let $\mu \in S(l^{\infty}(G))$ be a left invariant mean and define $\tau (f)=\mu ((\pi^*)^{-1}(f))$. I'm not sure if this definition does make sense.. And I guess to define $\tau$ it is necessary that $H$ is amenable as well..
My question is: How to define $\tau$ correctly?
(A proof of this fact using the Følner condition is welcome as well)
Regards
Remember that $\tau$ needs to be defined on functions $f \in l^\infty(G/H)$. So, using your notation, we can try taking $\tau(f) = \mu(\pi^*(f))$.
Clearly $\tau$ is unital and contractive, so it's a state. We just need to check that it's left invariant. Note that $g \pi^*(f) = \pi^*(gH f)$, since $$g \pi^*(f)(h) = \pi^*(f)(g^{-1}h) = f(g^{-1}hH) = f(g^{-1}HhH) = \pi^*(gHf)(h).$$
Let $f \in l^\infty(G/H)$, $gH \in G/H$ (with $g \in G$). Then we have: $$\tau(gHf) = \tau(f(g^{-1}H \cdot)) = \mu(\pi^*(f(g^{-1}H \cdot)))= \mu(g\pi^*(f)) = \mu(\pi^*(f)) = \tau(f),$$ so $\tau$ is left invariant.
So your intuition was right. You just don't want an inverse in your definition of $\tau$.