# Amenability and uniform Roe algebras

## TL;DR

This paper explores the concept of amenability in metric spaces and uniform Roe algebras, establishing their equivalence through various algebraic and analytical conditions, and demonstrating the properties of tracial states within this framework.

## Contribution

It unifies different notions of amenability in metric spaces and operator algebras, providing new equivalences and insights into the structure of uniform Roe algebras.

## Key findings

- Amenability of the metric space is equivalent to algebraic amenability of the translation algebra.
- The uniform Roe algebra admits a tracial state if and only if the space is amenable.
- All tracial states of the uniform Roe algebra are amenable.

## Abstract

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.