Amenability and uniform Roe algebras

Abstract

Amenability for groups can be extended to metric spaces, algebras over commutative fields and $C^*$-algebras by adapting the notion of Følner 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 Følner $C^*$-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.