Metrics on Doubles as an Inverse Semigroup

Abstract

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, which is not necessarily commutative. 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.