This is a continuation of the previous two posts I and II. In the first one I defined the uniform Roe algebra \(C_u^*(X)\) of a metric space \(X\) of bounded geometry and mentioned the following interesting result: Two metric spaces \(X\) and \(Y\) are coarsely equivalent if and only if their uniform Roe algebras \(C_u^*(X)\) and \(C_u^*(Y)\) are Morita equivalent.

In a recent preprint (arXiv:2403.13624) Martínez and Vigolo propose a complete solution to the rigidity problem of such Roe-like algebras. For example, one of their results reads as follows: *Let \(X\) and \(Y\) be proper, separable metric spaces. Then \(X\) and \(Y\) are coarsely equivalent if and only if \(C^*(X)\) and \(C^*(Y)\) are \(C^*\)-isomorphic (here \(C^*(-)\) denotes the classical, i.e. non-uniform Roe algebra).* One of the interesting things here to note is that there is now no bounded geometry assumption anymore.

The paper also provides a lot of useful intermediate results and quite a few interesting corollaries. One of them is the following: *Let \(X\) be a proper metric space. Then the group of coarse self-equivalences of \(X\) up to closeness is canonically isomorphic to the group of outer automorphisms of \(C^*(X)\).*