# characterizations of amenable groups which use the space $\ell_1(G)$ and convolution

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.

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I think this needs to be community wiki – Yemon Choi Dec 6 '10 at 0:48
What is community wiki? Can you explain this in a few lines? – BigBill Dec 10 '10 at 9:57

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.

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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.

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@Yemon: Can the approximate unit of $I_0(G)$ be taken to be two-sided and of norm at most one? – Theo Buehler Dec 6 '10 at 1:18
Theo: I think that if $G$ is amenable then the BAI can indeed be chosen to have the properties you describe, but I would have to check this. Conversely, the existence of any approximate identity (possibly unbounded) does force $G$ to be amenable; if I recall correctly this is due to George Willis but I will have to check later. – Yemon Choi Dec 6 '10 at 5:37
Good point, Andreas. I wasn't thinking clearly when I said that one could get a BAI of norm 1, this seems to fail for any finite group with cardinality $\geq 3$. (What I meant in my head was that 1 minus the BAI has norm 1, via Folner sets as you say.) – Yemon Choi Dec 6 '10 at 15:13
Yemon: I suspect Willis proved 3 $\Rightarrow$ 2, right? – Theo Buehler Dec 6 '10 at 21:32
I've now found a reference: Delaporte-Derighetti, ams.org/mathscinet-getitem?mr=1301019 Essentially they proceed as I outlined an hour ago. Their Theorem 3 can be applied in our situation with $H=G$ and shows that a two-sided approximate unit of norm two can indeed be found. If $G$ is infinite, their Theorem 5 shows that $2$ is the best possible bound. – Theo Buehler Dec 7 '10 at 8:32