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.

中文翻译：

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维数、比较和几乎有限性

我们针对简单的可分离核 C*-代数开发了围绕 Toms-Winter 猜想的一些理论的动力学版本，并通过交叉积研究了它与 C*-代数侧的联系。我们为紧凑空间上的服从群体的自由行动引入了超有限性的类比，并表明它在与动力学比较相关的 Toms-Winter 猜想中扮演 Z 稳定性的角色，并且它暗示了交叉的 Z 稳定性产品。我们称之为几乎有限性的这个属性从零维设置中概括了 Matui 的同名概念。我们还引入了塔维数作为核维数的部分类比的概念，并研究了它与动力学比较和几乎有限性的关系，以及与 Guentner 的动力学渐近维数和顺应性维数的关系，