Dimension, comparison, and almost finiteness

TL;DR

This paper introduces the concept of almost finiteness for group actions on compact spaces, linking it to Z-stability and dynamical comparison, and explores its implications for the structure of crossed product C*-algebras.

Contribution

It generalizes Matui's almost finiteness to higher dimensions and establishes its role in the Toms-Winter conjecture for dynamical systems and C*-algebras.

Findings

Almost finiteness implies Z-stability of crossed products.

Introduces tower dimension as an analogue of nuclear dimension.

Connects dynamical comparison with almost finiteness and various dynamical dimensions.

Abstract

We develop a dynamical version of some of the theory surrounding the Toms-Winter conjecture for simple separable nuclear C*-algebras and study its connections to the C*-algebra side via the crossed product. We introduce an analogue of hyperfiniteness for free actions of amenable groups on compact spaces and show that it plays the role of Z-stability in the Toms-Winter conjecture in its relation to dynamical comparison, and also that it implies Z-stability of the crossed product. This property, which we call almost finiteness, generalizes Matui's notion of the same name from the zero-dimensional setting. We also introduce a notion of tower dimension as partial analogue of nuclear dimension and study its relation to dynamical comparison and almost finiteness, as well as to the dynamical asymptotic dimension and amenability dimension of Guentner, Willett, and Yu.