Finite Rokhlin Dimension Research Articles

Rokhlin dimension: Permanence properties and ideal separation

We study the Rokhlin dimension for actions of residually finite groups on C*-algebras. We give a definition equivalent to the original one due to Szabó, Wu and Zacharias. We then prove a number of permanence properties and discuss actions on C_{0}(X) -algebras and commutative C*-algebras. Finally, we use a theorem of Sierakowski to show that, for an action with finite Rokhlin dimension, every ideal in the associated reduced crossed product C*-algebra arises from an invariant ideal of the underlying algebra.

Chaotic tracial dynamics

Abstract The action on the trace space induced by a generic automorphism of a suitable finite classifiable ${\mathrm {C}^*}$ -algebra is shown to be chaotic and weakly mixing. Model ${\mathrm {C}^*}$ -algebras are constructed to observe the central limit theorem and other statistical features of strongly chaotic tracial actions. Genericity of finite Rokhlin dimension is used to describe $KK$ -contractible stably projectionless ${\mathrm {C}^*}$ -algebras as crossed products.

RATIONAL -STABILITY OF CONTINUOUS -ALGEBRAS

AbstractWe show that the properties of being rationally K-stable passes from the fibres of a continuous $C(X)$ -algebra to the ambient algebra, under the assumption that the underlying space X is compact, metrizable, and of finite covering dimension. As an application, we show that a crossed product C*-algebra is (rationally) K-stable provided the underlying C*-algebra is (rationally) K-stable, and the action has finite Rokhlin dimension with commuting towers.

Strongly outer actions of amenable groups on [formula omitted]-stable nuclear C*-algebras

Let A be a separable, unital, simple, Z-stable, nuclear C⁎-algebra, and let α:G→Aut(A) be an action of a discrete, countable, amenable group. Suppose that the orbits of the action of G on T(A) are finite and that their cardinality is bounded. We show that the following are equivalent:(1)α is strongly outer;(2)α⊗idZ has the weak tracial Rokhlin property. If G is moreover residually finite, the above conditions are also equivalent to(3)α⊗idZ has finite Rokhlin dimension (in fact, at most 2).If ∂eT(A) is furthermore compact, has finite covering dimension, and the orbit space ∂eT(A)/G is Hausdorff, we generalize results by Matui and Sato to show that α is cocycle conjugate to α⊗idZ, even if α is not strongly outer. In particular, in this case the equivalences above hold for α in place of α⊗idZ. In the course of the proof, we develop equivariant versions of complemented partitions of unity and uniform property Γ as technical tools of independent interest.

Rokhlin dimension and equivariant bundles

Given an action of a compact group on a complex vector bundle, there is an induced action of the group on the associated Cuntz--Pimsner algebra. We determine conditions under which this action has finite Rokhlin dimension.

Rokhlin-type properties, approximate innerness and Z-stability

We investigate connections between actions on separable C∗-algebras with Rokhlin-type properties and absorption of the Jiang-Su algebra Z. We show that if A admits an approximately inner group action with finite Rokhlin dimension with commuting towers then A is Z-stable. We obtain analogous results for tracial version of the Rokhlin property and approximate innerness. Going beyond approximate innerness, for actions of a single automorphism which have the Rokhlin property and are almost periodic in a suitable sense, the crossed product absorbs Z even when the original algebra does not.

Rokhlin dimension: duality, tracial properties, and crossed products

We study compact group actions with finite Rokhlin dimension, particularly in relation to crossed products. For example, we characterize the duals of such actions, generalizing previous partial results for the Rokhlin property. As an application, we determine the ideal structure of their crossed products. Under the assumption of so-called commuting towers, we show that taking crossed products by such actions preserves a number of relevant classes of$C^{\ast }$-algebras, including:$D$-absorbing$C^{\ast }$-algebras, where$D$is a strongly self-absorbing$C^{\ast }$-algebra; stable$C^{\ast }$-algebras;$C^{\ast }$-algebras with finite nuclear dimension (or decomposition rank);$C^{\ast }$-algebras with finite stable rank (or real rank); and$C^{\ast }$-algebras whose$K$-theory is either trivial, rational, or$n$-divisible for$n\in \mathbb{N}$. The combination of nuclearity and the universal coefficient theorem (UCT) is also shown to be preserved by these actions. Some of these results are new even in the well-studied case of the Rokhlin property. Additionally, and under some technical assumptions, we show that finite Rokhlin dimension with commuting towers implies the (weak) tracial Rokhlin property. At the core of our arguments is a certain local approximation of the crossed product by a continuous$C(X)$-algebra with fibers that are stably isomorphic to the underlying algebra. The space$X$is computed in some cases of interest, and we use its description to construct a$\mathbb{Z}_{2}$-action on a unital AF-algebra and on a unital Kirchberg algebra satisfying the UCT, whose Rokhlin dimensions with and without commuting towers are finite but do not agree.

Model theory and Rokhlin dimension for compact quantum group actions

We show that, for a given compact or discrete quantum group $G$, the class of actions of $G$ on C\*-algebras is first-order axiomatizable in the logic for metric structures. As an application, we extend the notion of Rokhlin property for $G$-C\*-algebra, introduced by Barlak, Szabó, and Voigt in the case when $G$ is second countable and coexact, to an arbitrary compact quantum group $G$. All the preservation and rigidity results for Rokhlin actions of second countable coexact compact quantum groups obtained by Barlak, Szabó, and Voigt are shown to hold in this general context. As a further application, we extend the notion of equivariant order zero dimension for equivariant \*-homomorphisms, introduced in the classical setting by the first and third authors, to actions of compact quantum groups. This allows us to define the Rokhlin dimension of an action of a compact quantum group on a C\*-algebra, recovering the Rokhlin property as Rokhlin dimension zero. We conclude by establishing a preservation result for finite nuclear dimension and finite decomposition rank when passing to fixed point algebras and crossed products by compact quantum group actions with finite Rokhlin dimension.

Group actions on Smale space -algebras

Group actions on a Smale space and the actions induced on the$\text{C}^{\ast }$-algebras associated to such a dynamical system are studied. We show that an effective action of a discrete group on a mixing Smale space produces a strongly outer action on the homoclinic algebra. We then show that for irreducible Smale spaces, the property of finite Rokhlin dimension passes from the induced action on the homoclinic algebra to the induced actions on the stable and unstable$\text{C}^{\ast }$-algebras. In each of these cases, we discuss the preservation of properties (such as finite nuclear dimension,${\mathcal{Z}}$-stability, and classification by Elliott invariants) in the resulting crossed products.

Rokhlin dimension: absorption of model actions

In this paper, we establish a connection between Rokhlin dimension and the absorption of certain model actions on strongly self-absorbing C*-algebras. Namely, as to be made precise in the paper, let $G$ be a well-behaved locally compact group. If $\mathcal D$ is a strongly self-absorbing C*-algebra, and $\alpha: G\curvearrowright A$ is an action on a separable, $\mathcal D$-absorbing C*-algebra that has finite Rokhlin dimension with commuting towers, then $\alpha$ tensorially absorbs every semi-strongly self-absorbing $G$-actions on $\mathcal D$. This contains several existing results of similar nature as special cases. We will in fact prove a more general version of this theorem, which is intended for use in subsequent work. We will then discuss some non-trivial applications. Most notably it is shown that for any $k\geq 1$ and on any strongly self-absorbing Kirchberg algebra, there exists a unique $\mathbb R^k$-action having finite Rokhlin dimension with commuting towers up to (very strong) cocycle conjugacy.

Quasidiagonal traces and crossed products

Let $A$ be a simple, exact, separable, unital $C^*$-algebra and let $\alpha \colon G \rightarrow Aut(A)$ be an action of a finite group $G$ with the weak tracial Rokhlin property. We show that every trace on $A \rtimes_{\alpha} G$ is quasidiagonal provided that all traces on $A$ are quasidiagonal. As an application, we study the behavior of finite decomposition rank under taking crossed products by finite group actions with the weak tracial Rokhlin property. Moreover, we discuss the stability of the property that all traces are quasidiagonal under taking crossed products of finite group actions with finite Rokhlin dimension with commuting towers.

The tracial Rokhlin property for actions of amenable groups on $C^*$-algebras

In this paper, we present a definition of the tracial Rokhlin property for (cocyclic) actions of countable discrete amenable groups on simple $C^*$-algebras, which generalize Matui and Sato's definition. We show that generic examples, like Bernoulli shift on the tensor product of copies of the Jiang-Su algebra, has the weak tracial Rokhlin property, while it is shown in {Hirshberg 2014} that such an action does not have finite Rokhlin dimension. We further show that forming a crossed product from actions with the tracial Rokhlin property preserves the class of $C^*$-algebras with real rank $0$, stable rank $1$ and has strict comparison for pro\-jec\-tions, generalizing the structural results in {Osaka 2006}. We use the same idea of the proof with significant simplification. In another joint paper with Chris Phillips and Joav Orovitz, we shall show that pureness and $\mathcal{Z} $-stability could be preserved by crossed product of actions with the weak tracial Rokhlin property. The combination of these results yields an application to the classification program, which is discussed in the aforementioned paper. These results indicate that we have the correct definition of tracial Rokhlin property for actions of general countable discrete amenable groups.

Applications of model theory to C*-dynamics

We initiate the study of compact group actions on C*-algebras from the perspective of model theory, and present several applications to C*-dynamics. Firstly, we prove that the continuous part of the central sequence algebra of a strongly self-absorbing action is indistinguishable from the continuous part of the sequence algebra, and in fact equivariantly isomorphic under the Continuum Hypothesis. As another application, we present a unified approach to several dimensional inequalities in C*-algebras, which is done through the notion of order zero dimension for an (equivariant) *-homomorphism. Finiteness of the order zero dimension implies that the dimension of the target algebra can be bounded by the dimension of the domain. The dimension can be, among others, decomposition rank, nuclear dimension, or Rokhlin dimension. As a consequence, we obtain new inequalities for these quantities.As a third application we obtain the following result: if a C*-algebra A absorbs a strongly self-absorbing C*-algebra D, and α is an action of a compact group G on A with finite Rokhlin dimension with commuting towers, then α absorbs any strongly self-absorbing action of G on D. This has a number of interesting consequences, already in the case of the trivial action on D. For example, we deduce that D-stability passes from A to the crossed product. Additionally, in many cases of interest, our result restricts the possible values of the Rokhlin dimension to 0,1 and ∞, showing a striking parallel to the behavior of the nuclear dimension for simple C*-algebras. We also show that an action of a finite group with finite Rokhlin dimension with commuting towers automatically has the Rokhlin property if the algebra is UHF-absorbing.

Rokhlin dimension for actions of residually finite groups

We introduce the concept of Rokhlin dimension for actions of residually finite groups on$\text{C}^{\ast }$-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 previously been considered. 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$\text{C}^{\ast }$-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$\text{C}^{\ast }$-algebras have finite nuclear dimension. This extends an analogous result about$\mathbb{Z}^{m}$-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$\text{C}^{\ast }$-algebra.

Rokhlin dimension of ℤm-actions on simple C∗-algebras

We study Rokhlin dimension of [Formula: see text]-actions on simple separable stably finite nuclear [Formula: see text]-algebras. We prove that under suitable assumptions, a strongly outer [Formula: see text]-action has finite Rokhlin dimension. This extends the known result for automorphisms. As an application, we show that for a large class of [Formula: see text]-algebras, the [Formula: see text]-Bernoulli action has finite Rokhlin dimension.

Rokhlin dimension for compact group actions

We introduce and systematically study the notion of Rokhlin dimension (with and without commuting towers) for compact group actions on $C^*$-algebras. This notion generalizes the one introduced by Hirshberg, Winter and Zacharias for finite groups, and contains the Rokhlin property as the zero dimensional case. We show, by means of an example, that commuting towers cannot always be arranged, even in the absence of $K$-theoretic obstructions. For a compact Lie group action on a compact Hausdorff space, freeness is equivalent to finite Rokhlin dimension of the induced action. We compare the notion of finite Rokhlin dimension to other existing definitions of noncommutative freeness for compact group actions. We obtain further $K$-theoretic obstructions to having an action of a non-finite compact Lie group with finite Rokhlin dimension with commuting towers, and use them to confirm a conjecture of Phillips.

Rokhlin Dimension for Flows

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).

A Rokhlin type theorem for simple C⁎-algebras of finite nuclear dimension

We study Z-actions on unital simple separable stably finite C⁎-algebras of finite nuclear dimension. Assuming that the extreme boundary of the trace space is compact and finite dimensional, and that the induced action on the trace space is trivial, we show that strongly outer Z-actions have finite Rokhlin dimension in the sense of Hirshberg, Winter and Zacharias.

The Rokhlin property vs. Rokhlin dimension 1 on unital Kirchberg algebras

We investigate outer symmetries on unital Kirchberg algebras with respect to the Rokhlin property and finite Rokhlin dimension. In stark contrast to the restrictiveness of the Rokhlin property, every such action has Rokhlin dimension at most one. A consequence of these observations is a relationship between the nuclear dimension of an \mathcal O_\infty -absorbing C*-algebra and its \mathcal O_2 -stabilization. We also give a more direct and alternative approach to this result. Several applications of this relationship are discussed to cover a fairly large class of \mathcal O_\infty -absorbing C*-algebras that turn out to have finite nuclear dimension.

Rokhlin Dimension and C*-Dynamics

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 $$\mathcal{Z}$$ -stable C*-algebras, where $$\mathcal{Z}$$ 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 $${\mathcal{Z}}$$ -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 $${\mathbb{Z}^{d}}$$ -actions on compact metrizable and finite dimensional spaces.