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.

中文翻译：

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Roe双模作为离散度量空间的形态

对于两个离散度量空间X和Ÿ，我们认为在指标X ⊔ Ÿ与度量兼容的X和Y ^。作为从X到Y的射态，我们考虑Roe双模，即从l ²（X）到l ²（Y）的有限有限传播算子的范数闭包。我们研究相应的类别\（\ mathcal {M} \），它也是一个2类。我们证明，几乎等轴测图确定\（\ mathcal {M} \）中的态射。我们还考虑Y =X，当\（\ mathcal {M} \）的射态集上有更丰富的代数结构时：它是具有中立元素，有对合和许多幂等性的部分有序半群。我们还给出了一个态射素为C *-代数的条件。