Finite-dimensional approximation properties for uniform Roe algebras

TL;DR

This paper explores the relationship between property A of metric spaces with bounded geometry and finite-dimensional approximation properties of their associated uniform Roe algebras, extending known equivalences to exactness and local reflexivity.

Contribution

It establishes that exactness and local reflexivity of uniform Roe algebras also characterize property A, broadening the understanding of operator algebra properties linked to metric space geometry.

Findings

Property A is equivalent to nuclearity of C*_u(X).

Exactness of C*_u(X) characterizes property A.

Local reflexivity of C*_u(X) also characterizes property A.

Abstract

We study property A for metric spaces $X$ with bounded geometry introduced by Guoliang Yu. Property A is an amenability-type condition, which is less restrictive than amenability for groups. The property has a connection with finite-dimensional approximation properties in the theory of operator algebras. It has been already known that property A of a metric space $X$ with bounded geometry is equivalent to nuclearity of the uniform Roe algebra C$^*_u(X)$. We prove that exactness and local reflexivity of C$^*_u(X)$ also characterize property A of $X$.