Let \(X\) be a metric space of bounded geometry. The latter means that for all \(r > 0\) exists an \(n_r \in \mathbb{N}\) such that every ball in \(X\) of radius \(r\) has at most \(n_r\) elements. This especially implies that \(X\) is discrete. It might seem a bit strange for some to consider discrete metric spaces (say, if you are used to work with manifolds only); but this post delves into coarse geometry, and there actually any metric space is equivalent to a discrete one (though not necessarily of bounded geometry).
Let us define the main player of this post, the uniform Roe algebra: We consider bounded operators \(a\) on \(\ell^2(X)\) and think of them as \(X\text{-by-}X\)-matrices \(a=(a_{xy})_{x,y \in X})\). We define the propagation of \(a\) by \[\mathrm{prop}(a) := \sup\{d(x,y)\colon a_{xy} \not= 0\} \in [0,\infty]\,.\] The uniform Roe algebra \(C_u^*(X)\) is defined as the norm closure of the \({}^*\)-algebra of all bounded operators on \(\ell^2(X)\) of finite propagation.
Note that the uniform Roe algebra \(C^*_u(X)\) naturally contains \(\ell^\infty(X)\), considered as diagonal operators (i.e. operators with zero propagation). If \(X\) is the metric space of a finitely generated group \(G\) with a word metric, then we have a canonical isomorphism \(C^*_u(X) \cong \ell^\infty(G) \rtimes_r G\) to the corresponding reduced crossed product.
The uniform Roe algebra has the following remarkable property: 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 (a notion slightly more general than being isomorphic as \(C^*\)-algebras). This was finally proven, after several preliminary results by other authors, by Baudier-Braga-Farah-Khukhro-Vignati-Willett (arXiv:2106.11391).
The above reminds one of the classical Gelfand duality stating that, say, two compact Hausdorff spaces \(X\) and \(Y\) are homeomorphic if and only if their \(C^*\)-algebras of continuous functions \(C(X)\) and \(C(Y)\) are isomorphic as \(C^*\)-algebras.
But Gelfand duality actually says more. For example, it also says that every unital, commutative \(C^*\)-algebra \(A\) arises as \(C(Z)\) for some compact Hausdorff space \(Z\). But not every arbitrary (unital) \(C^*\)-algebra is the uniform Roe algebra of a metric space. Therefore, to go towards an honest duality between metric spaces (of bounded geometry) with coarse equivalences and \(C^*\)-algebras with Morita equivalence, one has to characterize those \(C^*\)-algebras which actually do arise as uniform Roe algebras of metric spaces.
This was achieved by White-Willett (arXiv:1808.04410) who recognized the importance of the sub-\(C^*\)-algebra \(\ell^\infty(X)\) inside \(C^*_u(X)\). This is a Cartan subalgebra with certain special properties and abstracting these they arrived at the following theorem: Let \(A\) be a \(C^*\)-algebra and assume that it contains a Cartan subalgebra \(B\) with `nice’ properties (these are of course made precise by White-Willett). Then there exists a metric space \(Y\) of bounded geometry such that the pair \(B \subset A\) is isomorphic as \(C^*\)-algebras to \(\ell^\infty(Y) \subset C^*_u(Y)\).