On ideals in the semilattice of coarse equivalence classes of metrics

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 $\Phi$ from $I(M(X))$ to the set $CS(X)$ of coarse structures on $X$, and a map $\Psi$ backwards, and show that $\Psi\circ\Phi$ 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 $\Phi(F)$.