Pureness of Certain Crossed Product C*-Algebras

Abstract: We establish comparison and divisibility properties for crossed product C*-algebras arising from automorphisms of algebras C (X, D) which lie over minimal homeomorphisms, from actions of compact groups which have finite Rokhlin dimension with commuting towers, and from actions of compact groups which have the restricted tracial Rokhlin property with comparison. We deduce that these crossed products we consider are pure, and conclude they have stable rank one, and in certain cases have real rank zero. We give examples in which these properties do not follow from previous results, in the case of C (X, D) due to the lack of Z-stability of D, the underlying topological spaces not being finite dimensional, or both.

• The paper establishes permanence of pureness for crossed product C*-algebras under minimal homeomorphisms and finite Rokhlin dimension actions.
• It employs techniques such as large subalgebras and sequentially split homomorphisms to transfer strict comparison and almost divisibility properties.
• The work provides new examples of pure crossed products in nonnuclear and non-Z-stable settings, ensuring stable rank one and enhanced regularity.

Pureness in Crossed Product C*-Algebras: New Permanence Properties and Examples

Introduction

The paper "Pureness of Certain Crossed Product C*-Algebras" (2604.10080) develops new results on the permanence and transfer of the pureness property for C*-algebras, particularly in the context of crossed product constructions that arise from group actions and automorphisms. By focusing on actions on algebras of the form $C(X, D)$, where $X$ is a compact metric space and $D$ is a simple unital C*-algebra, along with certain structural and dynamical conditions, the authors demonstrate a broad range of circumstances in which the resulting crossed products possess the pureness property. This property has been identified as a key regularity feature in recent classification and structure theory, particularly in the context of nonnuclear or non-${\mathcal{Z}}$-stable algebras.

The pureness property means that the C*-algebra has strict comparison of positive elements and is almost divisible in the sense of the Cuntz semigroup. The main contributions of the paper are the establishment of permanence of pureness under:

• Crossed products by automorphisms lying over minimal homeomorphisms,
• Crossed products by compact group actions with finite Rokhlin dimension with commuting towers,
• Crossed products by actions satisfying the restricted tracial Rokhlin property with comparison.

The authors also address the inheritance of stable and real rank regularity properties, and provide new examples of pure crossed product C*-algebras which are not covered by previous structural results.

Theoretical Background

Pureness, Comparison, and Divisibility

A unital C*-algebra $A$ is called pure if it satisfies strict comparison of positive elements (0-comparison in the Cuntz semigroup) and is almost divisible (0-almost divisibility). The Cuntz semigroup encapsulates the structure of positive elements up to Cuntz equivalence, and strict comparison is a crucial regularity property that often coincides, in the nuclear and simple case, with absorption of the Jiang-Su algebra, i.e., $\mathcal{Z}$-stability. However, in more generality or in the absence of nuclearity, these properties diverge, and pureness constitutes a plausible extension of $\mathcal{Z}$-stability in nonnuclear situations.

Crossed Products and Group Actions

The algebraic objects under consideration are crossed products $C^*(G, A, \alpha)$, where $G$ is a locally compact group, $A$ is a C*-algebra, and $X$0 is an action of $X$1 on $X$2. When $X$3 is a commutative C*-algebra $X$4 and the action lifts to a minimal homeomorphism $X$5, much is known due to deep connections with topological dynamics and classification results. For noncommutative $X$6 or more complicated group actions, much less is generally understood, especially when the resulting crossed products fall outside the classifiable cases.

The authors emphasize cases where either $X$7 is a "bundle algebra" $X$8 with non-$X$9-stable or nonnuclear fiber algebra $D$0, or the underlying space $D$1 is not zero- or one-dimensional, presenting obstacles to standard classification theory.

Rokhlin Properties and Large Subalgebras

A central role is played by group actions with strong regularity assumptions:

• Actions with finite Rokhlin dimension with commuting towers (Gardella) yield crossed products which are $D$2-algebras whose fibers retain many structural features.
• The restricted tracial Rokhlin property (Mohammadkarimi-Phillips) is a weaker variant introduced for preserving trace-related and comparison-theoretic properties.
• The notion of a large subalgebra (Phillips) is used as a technical device to transfer comparison and divisibility properties, exploiting the fact that large subalgebras share significant "semantic" invariants with their ambient algebras.

Main Results

Permanence of Pureness for Crossed Products

The paper establishes several assertions regarding the permanence of the pureness property:

1. Automorphisms of $D$3 Over Minimal Homeomorphisms: If $D$4 is a simple unital pure C*-algebra and $D$5 is an automorphism of $D$6 lying over a minimal homeomorphism $D$7, then the crossed product $D$8 is pure. This holds even when $D$9 is nonnuclear or not ${\mathcal{Z}}$0-stable, and even when ${\mathcal{Z}}$1 is infinite dimensional or of positive mean dimension.
2. Compact Group Actions With Finite Rokhlin Dimension: If ${\mathcal{Z}}$2 is simple, separable, stably finite and pure, and ${\mathcal{Z}}$3 is a compact Lie group acting on ${\mathcal{Z}}$4 with finite Rokhlin dimension with commuting towers, the crossed product ${\mathcal{Z}}$5 is again pure.
3. Tracial Rokhlin Property With Comparison: If ${\mathcal{Z}}$6 is simple, separable, stably finite and pure, and ${\mathcal{Z}}$7 is a compact group action with the restricted tracial Rokhlin property with comparison, the fixed point algebra ${\mathcal{Z}}$8 and the crossed product ${\mathcal{Z}}$9 (up to Morita equivalence) are pure.

These results rely critically on:

• Isomorphisms or comparability between the (purely positive part of the) Cuntz semigroups of the algebra and certain large subalgebras (or fixed point subalgebras under group actions);
• Stability of pureness under inductive limits of recursive subhomogeneous algebras over pure fibers (Seth-Vilalta);
• Recent advances in the structure theory of group actions with finite Rokhlin dimension.

Stable and Real Rank Regularity

The authors prove that, under the above conditions, the resulting crossed products always have stable rank one. Real rank zero is obtained under the extra assumption that the image of $A$0 in the affine space of traces is dense. For instance, in the case where $A$1 has a unique trace (e.g., reduced free group C*-algebras), the tracial state spaces of $A$2 and the crossed product can be described explicitly, and density arguments from the commutative case extend to this setting.

Explicit Examples and Nonstandard Cases

The authors construct a suite of new examples where pureness and stable rank one in crossed products hold without nuclearity or $A$3-stability, and sometimes with base spaces of infinite or arbitrary dimension. Notably:

• Crossed products with free group C*-algebra fibers: For instance, $A$4 where $A$5 lifts a minimal shift on $A$6 and $A$7 is the countable free group. Such algebras can be pure yet are not $A$8-stable or classifiable by known results.
• Actions constructed from reduced free products/amalgamated free products and nonexact fibers.
• Noncommutative Furstenberg transformations: By combining irrational rotations and free group automorphisms, the examples encompass both high-mean-dimension dynamics and noncommutative base fibers.

These examples demonstrate that the regularity properties (comparison, divisibility, stable and real rank one) can be secured far beyond the previously established boundaries in the structure theory of C*-algebras.

Technical Innovations

A technical highlight of the paper is the careful extension and adaptation of results concerning large subalgebras, subequivalence in Cuntz semigroups, and divisibility/comparison arguments across various contexts (including hereditary and non-hereditary subalgebras, $A$9-bundles with nonunital fibers, and continuous fields over higher-dimensional spaces). The authors additionally develop transfer techniques via sequentially split homomorphisms for approximating C*-algebras by those with better understood comparison and divisibility properties.

Implications and Future Directions

These results substantially expand the known territory where regularity properties crucial for the Elliott classification program persist in crossed product constructions, especially in nonnuclear and nonsimple settings. The techniques may generalize further to other regularity properties—notably nuclear dimension, decomposition rank, quasidiagonality, and absorption of other strongly self-absorbing C*-algebras.

On the dynamical side, the interplay between mean topological dimension, Rokhlin-type properties, and the structure of crossed products is now clarified for new classes of noncommutative bundles and group actions. The methods lay groundwork for:

• Further study of classification for nonnuclear and nonsimple crossed products,
• The extension to groupoid C*-algebras and nonminimal dynamics,
• Automated and explicit construction of irregular, yet comparably regular, C*-algebras for testing conjectures in structure theory.

Conclusion

"Pureness of Certain Crossed Product C*-Algebras" provides a robust conceptual and technical framework for demonstrating the permanence of pureness and associated regularity properties in a wide range of crossed product situations, well beyond the nuclear, $\mathcal{Z}$0-stable, or classifiable paradigms. The paper's results and techniques offer versatile tools for ongoing and future study of regularity properties in the structure and classification of C*-algebras arising from complex dynamical and group action settings, and significantly broaden the landscape of natural examples exhibiting comparison and divisibility beyond prior boundaries.