  Amenability for groups can be extended to metric spaces, algebras over
commutative fields and $C^*$-algebras by adapting the notion of F{\o}lner nets.
In the present article we investigate the close ties among these extensions and
show that these three pictures unify in the context of the uniform Roe algebra
$C_u^*(X)$ over a metric space $(X,d)$ with bounded geometry. In particular, we
show that the following conditions are equivalent: (1) $(X,d)$ is amenable; (2)
the translation algebra generating $C_u^*(X)$ is algebraically amenable (3)
$C_u^*(X)$ has a tracial state; (4) $C_u^*(X)$ is not properly infinite; (5)
$[1]_0\neq [0]_0$ in the $K_0$-group $K_0(C_u^*(X))$; (6) $C_u^*(X)$ does not
contain the Leavitt algebra as a unital $*$-subalgebra; (7) $C_u^*(X)$ is a
F{\o}lner $C^*$-algebra in the sense that it admits a net of unital completely
positive maps into matrices which is asymptotically multiplicative in the
normalized trace norm. We also show that every possible tracial state of the
uniform Roe algebra $C_u^*(X)$ is amenable.
