An example of a continuous field of Roe algebras

Abstract

The Roe algebra $C^*(X)$ is a non-commutative $C^*$-algebra reflecting metric properties of a space $X$, and it is interesting to understand relation between the Roe algebra of $X$ and the (uniform) Roe algebra of its discretization. Here we do a minor step in this direction in the simplest non-trivial example $X=\mathbb R$ by constructing a continuous field of $C^*$-algebras over $[0,1]$ with the fibers over non-zero points the uniform $C^*$-algebra of the integers, and the fiber over 0 a $C^*$-algebra related to $\mathbb R$.