Let $G$ be a discrete group.



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> Do you know characterizations of amenable groups which use the space $\ell\_1(G)$ and convolution?
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I only know Johnson's theorem:



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> A group is amenable if and only if the Banach algebra $\ell\_1(G)$ is amenable.
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Different characterizations are welcome.



Characterizations of amenable groups which use the space $\ell_1(G)$ and convolution
I like the following characterization, due to Kaimanovich and Vershik (conjectured by Furstenberg):



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> A (countable) group is amenable if and only if there is an everywhere positive $\ell^{1}$-function $\mu$ of norm $1$ on $G$ such that $\|g\mu^{\ast n} - \mu^{\ast n}\|\_{1} \to 0$ for all $g \in G$.
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As a consequence, a group is amenable if and only if some Poisson boundary $\Gamma{(G,\mu)}$ is reduced to a point. The neat thing is that you can always choose a Reiter sequence formed by the convolution powers of a single probability.


