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, Ilijas Farah, and Alessandro Vignati,Embeddings of uniform Roe algebras, Communications in Mathematical Physics377(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.