Question

Let $G$ be a discrete group.

Do you know characterizations of amenable groups which use the space $\ell_1(G)$ and convolution?

I only know Johnson's theorem: A group is amenable if and only if the Banach algebra $\ell_1(G)$ is amenable. Different characterizations are welcome.

Answer 1

I like the following characterization, due to Kaimanovich and Vershik (conjectured by Furstenberg):

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 $g \in G$.

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.

Answer 2

To get the ball rolling: the convolution algebra $\ell^1(G)$ always supports a character $\varepsilon$ (=homomorphism from the algebra to the complex numbers) defined by sending $\delta_x$ to 1 for each $x\in G$ - this is usually called the augmentation character - and its kernel is called the augmentation ideal. Denoting this ideal by $I_0(G)$, we have the following result:

Theorem. The following are equivalent:

1. $G$ is amenable
2. $I_0(G)$ has a bounded approximate identity
3. $I_0(G)$ has an approximate identity

Another variant of this result is that $G$ is amenable if and only if the one-dimensional $\ell^1(G)$-module corresponding to $\varepsilon$ is flat in the sense of Helemskii et al.