Metrics on Doubles as an Inverse Semigroup
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Metrics on Doubles as an Inverse Semigroup V. Manuilov1 Received: 7 December 2019 © Mathematica Josephina, Inc. 2020
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. Keywords Metric · Inverse semigroup Mathematics Subject Classification Primary: 20M18 · 54E35
1 Introduction Given metric spaces X and Y , a metric d on X Y that extends the metrics on X and Y depends only on the values of d(x, y), x ∈ X , y ∈ Y , but it may be hard to check which functions d : X × Y → (0, ∞) determine a metric on X Y : one has to check the triangle inequality too many times. The problem of description of all such extended metrics is difficult due to the lack of a nice algebraic structure on the set of metrics. It was a surprise for us to discover that in the case Y = X , there is a nice
The author acknowledges partial support by the RFBR Grant No. 18-01-00398.
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V. Manuilov [email protected] Moscow Center for Fundamental and Applied Mathematics and Moscow State University, Leninskie Gory 1, Moscow, Russia 119991
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V. Manuilov
algebraic structure on the set M(X ) of quasi-isometry classes of extended metrics on the double X X : it is an inverse semigroup. Recall that a semigroup S is an inverse semigroup if for any u ∈ S there exists a unique v ∈ S such that u = uvu and v = vuv [1]. Philosophically, inverse semigroups describe local symmetries in a similar way as groups describe global symmetries, and technically, the construction of the (reduced) group C ∗ -algebra of a group generalizes to that of the (reduced) inverse semigroup C ∗ -algebra [3]. Thus, one can associate a new (non-commutative) C ∗ -algebra to any metric space. In particular, all quasi-isometry classes of metrics on the double of X are partial isometries. We characterize the metrics that are idempotents in M(X ) and show that any two idempotents commute (which proves that M(X ) is an inverse semigroup). We show that 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 the corresponding C ∗ -algebras) 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 and the class of inverti
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