Rokhlin Dimension and C*-Dynamics

Abstract

Abstract:We develop the concept of Rokhlin dimension for integer and for finite group actions on C * -algebras. Our notion generalizes the so-called Rokhlin property, which can be thought of as Rokhlin dimension 0. We show that finite Rokhlin dimension is prevalent and appears in cases in which the Rokhlin property cannot be expected: the property of having finite Rokhlin dimension is generic for automorphisms of Z-stable C * -algebras, where Z denotes the Jiang-Su algebra. Moreover, crossed products by automorphisms with finite Rokhlin dimension preserve the property of having finite nuclear dimension, and under a mild additional hypothesis also preserve Z-stability. In topological dynamics our notion may be interpreted as a topological version of the classical Rokhlin lemma: automorphisms arising from minimal homeomorphisms of finite dimensional compact metrizable spaces always have finite Rokhlin dimension. The latter result has by now been generalized by Szabó to the case of free and aperiodic Z d -actions on compact metrizable and finite dimensional spaces.

Rokhlin Dimension for Flows

“…The situation for D-absorption (D strongly self-absorbing) is similar, just as in [HWZ15]: D-absorption is preserved under forming crossed products by flows with finite Rokhlin dimension with commuting towers; see Theorem 6.3, which generalizes [HW07,Theorem 5.2] to the case of finite Rokhlin dimension.…”

“…We can establish a connection between the nuclear dimension A α G and that of A α| H H , where H is a closed, cocompact subgroup. If one has sufficient information about the restricted action α| H : H → Aut(A) or its crossed product, the method discussed in this section can have some advantages that global Rokhlin dimension, in the sense of the previous section or [HWZ15,SWZ15], does not have in the non-compact case. For example, obtaining bounds concerning decomposition rank of crossed products by non-compact groups appears to be difficult, and would require different techniques and more severe constraints on the action than just finite Rokhlin dimension.…”

“…In fact, it is often generic or equivalent to outerness; cf. [HWZ15,BEM+15]. The concept makes direct contact with the striking recent developments in the structure and classification theory of nuclear C * -algebras.…”

“…Even though the latter carry over the Rokhlin lemma to a C * -algebra context in a very elegant manner, their scope is limited by (sometimes obvious, sometimes hidden) topological obstructions. In [HWZ15] this problem was circumvented by the notion of Rokhlin dimension, which may be regarded as a higher dimensional version of the Rokhlin lemma and is much more prevalent than the Rokhlin property. In fact, it is often generic or equivalent to outerness; cf.…”

“…This variant is much more common, since its K -theoretic obstructions are minimized; of course it still requires the existence of nontrivial projections. The higher dimensional, or colored, version of [HWZ15] bypasses even this latter restriction, and indeed it was shown in [HWZ15] and in [Sza15a] that it occurs in stunning generality. The idea is to replace projections by positive elements (and thus, intuitively speaking, generalize from a clopen partition to an open cover) and group them into finitely many sets of pairwise orthogonal towers, with the action still shifting within each one.…”