The generalised diagonal dimension of a noncommutative Cartan subalgebra in the C*-algebra of finite-propagation operators on a uniformly locally finite metric space equals the asymptotic dimension of the space.
Braga, Yeong Chyuan Chung, and Kang Li,Coarse Baum–Connes conjecture and rigidity for Roe algebras, Journal of Functional Analysis279(2020), no
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Generalised diagonal dimension and applications to large-scale geometry
The generalised diagonal dimension of a noncommutative Cartan subalgebra in the C*-algebra of finite-propagation operators on a uniformly locally finite metric space equals the asymptotic dimension of the space.