Dimension, comparison, and almost finiteness

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 $\mathcal{Z}$-stability in the Toms–Winter conjecture in its relation to dynamical comparison, and also that it implies $\mathcal{Z}$-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 a 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.