# Metric spaces and C*-algebras

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 $$\ell^\infty(Y) \subset C^*_u(Y)$$.

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