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 and Alessandro Vignati,A Gelfand-type duality for coarse metric spaces with property A, IMRN
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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.