Generalised diagonal dimension and applications to large-scale geometry

Abstract: In this paper, we introduce a generalised diagonal dimension. We explain why the generalised diagonal dimension extends the notion of diagonal dimension defined by Li, Liao, and Winter, and under which conditions these dimensions coincide. We prove permanence properties for the generalised diagonal dimension and compare it with the nuclear dimension. We investigate applications of the generalised diagonal dimension in large-scale geometry; specifically, we show that the generalised diagonal dimension of a noncommutative Cartan subalgebra in the C*-algebra of finite-propagation operators on a uniformly locally finite metric space is equal to the asymptotic dimension of the space.