Rokhlin dimension: duality, tracial properties, and crossed products

Eusebio Gardella,Ilan Hirshberg,Luis Santiago

doi:10.1017/etds.2019.68

Abstract

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.

Highlights

We study compact group actions with finite Rokhlin dimension, in relation to crossed products
The concept of Rokhlin dimension for actions of finite groups and actions of the integers was introduced by the second author, Winter and Zacharias in [36] as a generalization of the well-studied Rokhlin property for group actions. (See [29], [38], and [53] for finite group actions with the Rokhlin property, and [35], [18], and [22] for compact group actions.) The Rokhlin property can be viewed as a regularity condition for the group action, which can be used to show that various structural properties pass from a C∗-algebra to its crossed product; see for example [35], [53], [63], [18]
In the case of finite group actions, the Rokhlin property is a very restrictive hypothesis to place on the action, as it implies that the unit of the algebra can be written nontrivially as a sum of projections indexed by the group

Summary

Finite Rokhlin dimension and duality

We begin by introducing some notation and terminology. Definition 1.1. (Recall that ρ2j is defined using functional calculus for order zero maps.) The assignment γ → ρ2j (vγ) is an order zero representation of Γ in the sense of Definition 1.7, and by condition (2.c), it satisfies the covariance condition ρ2j (vγ )b = βγ (b)ρ2j (vγ ) for all γ ∈ Γ and b ∈ B (it suffices to multiply the identity in (2.c) by ρj(1) on both sides) It follows from Proposition 1.9 that ψj is a completely positive, contractive order zero map. (a) dimRok(α) ≤ d; (b) For every α-invariant σ-unital subalgebra D ⊆ A, there exist G-equivariant completely positive contractive order zero maps θ0, . DimRok(α) ≤ d is equivalent to the existence, for every α-invariant σ-unital subalgebra D ⊆ A, of equivariant completely positive contractive maps φ0, . This is in stark contrast with the fact, proved in [32], that the restrictions of γ and β to finite subgroups of T have finite Rokhlin dimension with commuting towers. (See Proposition 2.9 for a more general result.)

Tracial properties

Structure of the crossed product