Metrics on doubles as an inverse semigroup

Publication date: 18 September 2019

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. We characterize the metrics that are idempotents, find a minimal projection in

M (X)

and give examples of metric spaces, for which the semigroup

M (X)

is commutative. We show that if the Gromov-Hausdorff distance between two metric spaces,

X

and

Y

, is finite then

M (X)

and

M (Y)

are isomorphic. 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

.

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