Uniform Roe Algebras as Crossed Product

Abstract

We define what a coarse space is, and we

study a number of ways of constructing a coarse

structure on a set so as to make it into a coarse space.

We also consider some of the elementary concepts

associated with coarse spaces. A discrete group 􀡳 has

natural coarse structure which allows us to define the

uniform Roe algebra, 􀡯􀢁 􀗛

􁈺􀡳􁈻. The reduced 􀡯􀗛 􀵆 algebra

􀡯􀢘 􀗛

􁈺􀡳􁈻 is naturally contained in 􀡯􀢁 􀗛

􁈺􀡳􁈻. In this paper, we

will characterize 􀡯􀢁 􀗛

􁈺􀡳􁈻 as a crossed product.