This problem is derived from [this post](https://mathoverflow.net/questions/403797/a-variation-of-the-ryll-nardzewski-fixed-point-theorem).


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...).



Trans-amenability of group actions
This is precisely what was called **$L$-amenability** by Kaimanovich and **lamenability** by Bartholdi (who were inspired by [Infinitely supported Liouville measures of Schreier graphs](https://mathscinet.ams.org/mathscinet-getitem?mr=3844999) of Juschenko and Zheng; "L" here stands for Liouville). In a sense, the aforementioned paper also provides a non-trivial example you are asking about. As you say, most of the usual definitions of group amenability can be adapted to this situation as well; in particular, for $\mu\_n$ one can take the sequence of convolution powers of a single measure, and one can reformulate condition (ii) in "Følner" terms. However, I don't think there is an "easy to check equivalent condition" - what is an easy to check condition for the usual group amenability?


