Trans-amenability of group actions

Question

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...).

Answer 1

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 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?