Roe bimodules as morphisms of discrete metric spaces (Q2304178)

Let \(X\) and \(Y\) be metric spaces. We consider metrics on their disjoint union \(X\sqcup Y\) that extend the given metrics on \(X\) and \(Y\). The uniform Roe \(C^*\)-algebra of \(X\sqcup Y\) with such a metric contains complementary projections \(p\) and \(q\) such that the corners \(p C^*(X\sqcup Y) p\) and \(q C^*(X\sqcup Y) q\) are naturally isomorphic to the uniform Roe \(C^*\)-algebras of \(X\) and \(Y\), respectively, and the off-diagonal pieces \(p C^*(X\sqcup Y) q\) and \(q C^*(X\sqcup Y) p\) are Hilbert bimodules over the uniform Roe \(C^*\)-algebras of \(X\) and \(Y\). There is a bicategory that has \(C^*\)-algebras as objects and Hilbert bimodules as arrows, with the tensor product of Hilbert bimodules as composition of arrows. Through the construction above, a metric on \(X\sqcup Y\) gives arrows between the uniform Roe \(C^*\)-algebras of \(X\) and \(Y\) in this bicategory. This is the topic of this paper. It is shown that, under a mild condition, two metrics on \(X\sqcup Y\) give the same Hilbert bimodule if and only if the metrics are coarsely equivalent. Metrics on \(X\sqcup Y\) and \(Y\sqcup Z\) are composed to a metric on \(X\sqcup Z\) in such a way that this matches the tensor product composition for the resulting Hilbert bimodules. A metric on \(X\sqcup Y\) is associated to an almost isometry \(f\colon X\to Y\), and it is shown that this construction is functorial up to equivalence. Conversely, if a metric on \(X\sqcup Y\) has a left inverse up to equivalence, then it comes from an almost isometry. Analogous constructions work for non-uniform Roe \(C^*\)-algebras of metric spaces.