Abstract
For two discrete metric spaces X and Y, we consider metrics on X ⊔ Y compatible with the metrics on X and Y. As morphisms from X to Y, we consider Roe bimodules, i.e., the norm closures of bounded finite propagation operators from l2(X) to l2(Y). We study the corresponding category \(\mathcal{M}\), which is also a 2-category. We show that almost isometries determine morphisms in \(\mathcal{M}\). We also consider the case Y = X, when there is a richer algebraic structure on the set of morphisms of \(\mathcal{M}\): it is a partially ordered semigroup with the neutral element, with involution, and with a lot of idempotents. We also give a condition when a morphism is a C*-algebra.
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The author acknowledges partial support by the RFBR grant 19-01-00574.
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Manuilov, V.M. Roe Bimodules as Morphisms of Discrete Metric Spaces. Russ. J. Math. Phys. 26, 470–478 (2019). https://doi.org/10.1134/S1061920819040058
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DOI: https://doi.org/10.1134/S1061920819040058