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.