On ideals in the semilattice of coarse equivalence classes of metrics

Abstract

For a Hausdorff topology on the set of ideals of the semilattice $M(X)$ of coarse equivalence classes of metrics on a set $X$, the space $I(M(X))$ of ideals is the closure of the set of principal ideals, thus allowing to view non-principal ideals as generalizations of coarse equivalence classes of metrics. Some ideals arise from coarse structures on $X$. We define a map $Φ$ from $I(M(X))$ to the set $CS(X)$ of coarse structures on $X$, and a map $Ψ$ backwards, and show that $Ψ\circΦ$ is the identity map, thus allowing to identify coarse structures with some ideals of $M(X)$. We show that there are ideals that do not come from $CS(X)$. For any ideal $F$ we define the generalized uniform Roe algebra as the direct limit $C^*$-algebra of the uniform Roe algebras for the equivalence classes of metrics in the ideal, and show that it coincides with the uniform Roe algebra of $Φ(F)$.