We introduce the concept of Rokhlin dimension for actions of residually finite groups on C * -algebras, extending previous such notions for actions of finite groups and the integers by Hirshberg, Winter and the third author. We are able to extend most of their results to a much larger class of groups: those admitting box spaces of finite asymptotic dimension. This latter condition is a refinement of finite asymptotic dimension and has not been considered previously. In a detailed study we show that finitely generated, virtually nilpotent groups have box spaces with finite asymptotic dimension, providing a large class of examples. We show that actions with finite Rokhlin dimension by groups with finite dimensional box spaces preserve the property of having finite nuclear dimension when passing to the crossed product C * -algebra. We then establish a relation between Rokhlin dimension of residually finite groups acting on compact metric spaces and amenability dimension of the action in the sense of Guentner, Willett and Yu. We show that for free actions of infinite, finitely generated, nilpotent groups on finite dimensional spaces, both these dimensional values are finite. In particular, the associated transformation group C * -algebras have finite nuclear dimension. This extends an analogous result about Z m -actions by the first author to a significantly larger class of groups, showing that a large class of crossed products by actions of such groups fall under the remit of the Elliott classification programme. We also provide results concerning the genericity of finite Rokhlin dimension, and permanence properties with respect to the absorption of a strongly self-absorbing C * -algebra.

We introduce a notion of Rokhlin dimension for one parameter automorphism groups of C * -algebras. This generalizes Kishimoto's Rokhlin property for flows, and is analogous to the notion of Rokhlin dimension for actions of the integers and other discrete groups introduced by the authors and Zacharias in previous papers. We show that finite nuclear dimension and absorption of a strongly self-absorbing C * -algebra are preserved under forming crossed products by flows with finite Rokhlin dimension, and that these crossed products are stable. Furthermore, we show that a flow on a commutative C * -algebra arising from a free topological flow has finite Rokhlin dimension, whenever the spectrum is a locally compact metrizable space with finite covering dimension. For flows that are both free and minimal, this has strong consequences for the associated crossed product C * -algebras: Those containing a non-zero projection are classified by the Elliott invariant (for compact manifolds this consists of topological K -theory together with the space of invariant probability measures and a natural pairing given by the Ruelle-Sullivan map).

Amenability for groups can be extended to metric spaces, algebras over commutative fields and C * -algebras by adapting the notion of Følner nets. In the present article we investigate the close ties among these extensions and show that these three pictures unify in the context of the uniform Roe algebra C * u (X) over a metric space (X, d) with bounded geometry. In particular, we show that the following conditions are equivalent: (1) (X, d) is amenable;(2) the translation algebra generating C

In this article we analyze the notions of amenability and paradoxical decomposition from an algebraic perspective. We consider this dichotomy for locally finite Communicated by Efim Zelmanov. extended metric spaces and for general algebras over fields. In the context of algebras we also study the relation of amenability with proper infiniteness. We apply our general analysis to two important classes of algebras: the unital Leavitt path algebras and the translation algebras on locally finite extended metric spaces. In particular, we show that the amenability of a metric space is equivalent to the algebraic amenability of the corresponding translation algebra.

We show that if X is a finite dimensional locally compact Hausdorff space, then the crossed product of C0(X) by any automorphism has finite nuclear dimension. This generalizes previous results, in which the automorphism was required to be free. As an application, we show that group C * -algebras of certain non-nilpotent groups have finite nuclear dimension.Nuclear dimension for C * -algebras was introduced by Winter and Zacharias in [WZ10], as a noncommutative generalization of covering dimension. This is a variant of the previous notion of decomposition rank ([KW04]), and is also applicable to non-quasidiagonal C * -algebras. Since then, it has come to play a major role in structure and classification of C * -algebras. It was shown in [WZ10] that if X is a locally compact metrizable space, then dim nuc (C 0 (X)) coincides with the covering dimension of X, and the property of having finite nuclear dimension is preserved under various constructions: forming direct sums and tensor products, passing to quotients and hereditary subalgebras, and forming extensions. An important problem which was left open in [WZ10] is to understand the behavior of finite nuclear dimension under forming crossed products. It was shown in [TW13] that finite nuclear dimension passes to crossed products by minimal homeomorphisms: if X is a compact metric space with finite covering dimension and h : X → X is a minimal homeomorphism, then denoting α(f ) = f • h, we have dim nuc (C(X) ⋊ α Z) < ∞. This was re-proved in a different way in [HWZ15]. The paper [HWZ15] develops a notion of Rokhlin dimension for an automorphism of a C * -algebra (extended in [HP15] to the non-unital setting). It was shown there that in general, if A has finite nuclear dimension and α ∈ Aut(A) has finite Rokhlin dimension, then A ⋊ α Z has finite nuclear dimension as well, and furthermore, for a minimal homeomorphism as above, the induced automorphism on C(X) always has finite Rokhlin dimension. Szabó ([Sza15b]) then showed that the minimality condition can be weakened to freeness: if X is as above and h : X → X has no periodic points, then α has finite Rokhlin dimension (and therefore, by [HWZ15, Theorem 4.1], the crossed product has finite nuclear dimension). In fact, Szabó's result works for actions of Z m as well. This uses the marker property, introduced by Gutman in [Gut15b]. Those results were further extended to free actions of finitely generated nilpotent groups in [SWZ14].For the case of integer actions arising from homeomorphisms, this leaves the case of actions which also have periodic points. Those include important examples. For instance, suppose G is a countable abelian group and G has finite covering dimension, and suppose α is an automorphism of G. The group C * -algebra C * (G⋊ α Z) is isomorphic to a crossed product C( G) ⋊ Z, and such actions are never free: an