This problem is derived from this post.

Let $G$ be a countable discrete group and $H\le G$ be a subgroup. Consider the $G$-action on $X=G/H$. Then the following amenability-like conditions are equivalent.

(i) Every $G$-invariant weak*-compact convex subset $K$ of $\ell_\infty(X)$ contains a constant function.

(ii) There is a sequence $\mu_n$ in $\mathrm{Prob}(G)$ such that $|| \mu_n x - \mu_n y ||_{\ell_1(X)}\to0$ for every $x,y\in X$.

As in the case of other amenability-like conditions, one can easily give more equivalent conditions. However, none seems easy to check.

Is there an equivalent condition that is easy to check?

I would be happy if the criterion applies to any non-trivial case (if any...).

This problem is derived from this post.

Let $G$ be a countable discrete group and $H\le G$ be a subgroup. Consider the $G$-action on $X=G/H$. Then the following amenability-like conditions are equivalent.

(i) Every $G$-invariant weak*-compact convex subset $K$ of $\ell_\infty(X)$ contains a constant function.

(ii) There is a sequence $\mu_n$ in $\mathrm{Prob}(G)$ such that $\| \mu_n x - \mu_n y \|_{\ell_1(X)}\to0$ for every $x,y\in X$.

As in the case of other amenability-like conditions, one can easily give more equivalent conditions. However, none seems easy to check.

Is there an equivalent condition that is easy to check?

I would be happy if the criterion applies to any non-trivial case (if any...).