Metrics on doubles as an inverse semigroup

For a metric space $X$ we study metrics on the two copies of $X$. We define composition of such metrics and show that the equivalence classes of metrics are a semigroup $M(X)$. Our main result is that $M(X)$ is an inverse semigroup, therefore, one can define the $C^*$-algebra of this inverse semigroup. This new $C^*$-algebra of a metric space is not necessarily commutative and differs from the Roe algеbra. If the Gromov–Hausdorff distance between two metric spaces, $X$ and $Y$, is finite then their inverse semigroups $M(X)$ and $M(Y)$ (and hence their $C^*$-algebras) are isomorphic. We characterize the metrics that are idempotents, and give examples of metric spaces, for which the semigroup $M(X)$ (and the corresponding $C^*$-algebra) is commutative. We also describe the class of metrics determined by subsets of $X$ in terms of the closures of the subsets in the Higson corona of $X$ and the class of invertible metrics.