# Trans-amenability of group actions

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 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?
• Thank you for the reference. The examples given in Juschenko and Zheng's paper are that $X$ is a directed union of some orbits of some amenable subgroups. Sep 23 at 22:42