# Metric spaces and C*-algebras II

In the previous post about metric spaces and C*-algebras we saw the definition of the uniform Roe algebra $$C^*_u(X)$$ of a metric space $$X$$ and discussed its remarkable property that two metric spaces (of bounded geometry) $$X$$ and $$Y$$ are coarsely equivalent if and only if their uniform Roe algebras $$C^*_u(X)$$ and $$C^*_u(Y)$$ are Morita equivalent.

This remarkable property led people to believe that all the coarse information about a metric space $$X$$ is stored in its uniform Roe algebra $$C^*_u(X)$$ and the problem is only to find a way to retrieve it (i.e., to define the corresponding invariant of C*-algebras).

One such example is the asymptotic dimension of $$X$$. One of its possible definitions is as follows: $$X$$ has asymptotic dimension at most $$d$$ if for every $$R \gg 1$$ there is a cover $$\mathcal{U}$$ of $$X$$ such that the members of $$\mathcal{U}$$ have uniformly bounded diameter and every $$R$$-ball in $$X$$ intersects at most $$d+1$$ members of $$\mathcal{U}$$. A candidate for the corresponding invariant of C*-algebras is nuclear dimension as introduced by Winter-Zacharias (arXiv:0903.4914), which we are not going to define here, and they proved the inequality $\mathrm{dim}_{\mathrm{nuc}}(C^*_u(X)) \le \textrm{as-dim}(X)\,.$

But the reverse inequality is still open. And in this post I actually want to express my opinion that, even if one can prove the reverse inequality, it would not be a natural statement for me. The reason is that the nuclear dimension is an invariant defined for any C*-algebra (though it might be infinite), but by far not every C*-algebra is the uniform Roe algebra of a (bounded geometry) metric space.

In my previous post about this topic I explained the following result by White-Willett: Let $$A$$ be a C*-algebra and assume that it contains a Cartan subalgebra $$B$$ with nice’ properties. Then there exists a metric space $$Y$$ of bounded geometry such that the pair $$B \subset A$$ is isomorphic as C*-algebras to $$\ell^\infty(Y) \subset C^*_u(Y)$$.

Therefore, a more natural theorem (than the above proposed equality of nuclear dimension of $$C^*_u(X)$$ with asymptotic dimension of $$X$$) would be a comparison of asymptotic dimension of $$X$$ with an invariant defined for pairs $$B \subset A$$ of a Cartan subalgebra $$B$$ inside a C*-algebra $$A$$. And this was actually achieved at the beginning of this year by Li-Liao-Winter (arXiv:2303.16762): They defined for any pair $$D \subset A$$, where $$D$$ is an abelian sub-C*-algebra in $$A$$, an invariant they called the diagonal dimension and proved $\mathrm{dim}_{\mathrm{diag}}(B \subset C^*_u(X)) = \textrm{as-dim}(X)$ for every Cartan subalgebra $$B \subset C^*_u(X)$$ with nice’ properties as laid out by White-Willett.

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