Leptin's theorem

Remarks

The idea of the proof in one direction: if $G$ is amenable, it satisfies the Følner or Reiter condition. For a compact set $K$ and $\varepsilon > 0$ there is a set or function that is almost invariant under translation by $K$. For a Følner set $F$, the function $u = \mu(F)^{-1} \mathbf{1}_F * \check{\mathbf{1}}_F$, where $\check{\mathbf{1}}_F(x) = \mathbf{1}_F(x^{-1})$, lies in $A(G)$ with norm at most $1$ and is close to $1$ on $K$. These functions form an approximate identity bounded by $1$. The converse direction uses a bounded approximate identity to build an invariant mean.

Example: for a compact group $G$ the constant function $1$ lies in $A(G)$ and is a unit, so the theorem is immediate. For $G = \R$, the functions $u_n = \frac{1}{2n} \mathbf{1}_{[-n, n]} * \mathbf{1}_{[-n, n]}$ are the triangle functions $u_n(x) = \max(0, 1 - \abs{x}/(2n))$. They satisfy $\norm{u_n}_A \le 1$ and tend to $1$ uniformly on compact sets.

Non-example: for the free group $F_2$ on two generators, which is not amenable, $A(F_2)$ has no bounded approximate identity.

Going deeper

For a student

The theorem translates amenability of $G$ into a Banach algebra property of $A(G)$. It is dual to the fact that $L^1(G)$ always has a bounded approximate identity: on the Fourier side, this property characterizes amenability. It is the starting point of the study of $A(G)$ as a Banach algebra. For instance, Forrest and Runde showed that $A(G)$ is an amenable Banach algebra if and only if $G$ has an abelian subgroup of finite index. Weaker approximation properties of $A(G)$, such as weak amenability in the sense of Cowling and Haagerup, give invariants of non-amenable groups.