MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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.

share|cite|improve this question
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
up vote 4 down vote accepted

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.

share|cite|improve this answer

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.

share|cite|improve this answer
@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, 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

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.