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.