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, ô€¡¯ô€¢ ô€—›

ôˆºô€¡³ôˆ». The reduced ô€¡¯ô€—› ô€µ† algebra

􀡯􀢘 􀗛

ôˆºô€¡³ôˆ» is naturally contained in ô€¡¯ô€¢ ô€—›

ôˆºô€¡³ôˆ». In this paper, we

will characterize ô€¡¯ô€¢ ô€—›

ôˆºô€¡³ôˆ» as a crossed product.