Generalised diagonal dimension and applications to large-scale geometry

TL;DR

This paper introduces a generalized diagonal dimension extending existing notions, explores its properties, and applies it to relate noncommutative algebraic structures with large-scale geometric invariants.

Contribution

It defines the generalized diagonal dimension, establishes its properties, and connects it to asymptotic dimension in large-scale geometry applications.

Findings

The generalized diagonal dimension extends the classical diagonal dimension.

Under certain conditions, the generalized and classical diagonal dimensions coincide.

The generalized diagonal dimension of a noncommutative Cartan subalgebra equals the asymptotic dimension of the underlying space.

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.