Duality of partial Rokhlin dimension

Abstract: We extend the notion of representability dimension to partial actions and introduce a notion of dual representability dimension for global actions by finite abelian groups. We show that the Rokhlin dimension of a partial action by a finite abelian group agrees with the dual representability dimension of the dual action on the partial crossed product, while the representability dimension of a partial action agrees with the Rokhlin dimension of its dual.

• The paper establishes a duality between the Rokhlin dimension of partial actions and the dual representability dimension of their dual actions.
• It extends classical Rokhlin theory to partial actions by finite abelian groups using order zero maps and central sequence algebra techniques.
• It translates dynamical invariants between fixed point algebras and crossed products, providing new insights for noncommutative topological dynamics.

Duality of Partial Rokhlin Dimension in Finite Abelian Group Actions on $C^*$-Algebras

Introduction

The theory of finite group actions with the Rokhlin property on $C^*$-algebras is central within the structure and classification theory of operator algebras. The Rokhlin property and its generalizations, such as Rokhlin dimension, ensure that several desirable algebraic and topological properties transfer from the underlying $C^*$-algebra to associated dynamical invariants such as the fixed point algebra and the crossed product. Historically, this theory has been gradually extended to compact groups, non-separable $C^*$-algebras, and more recently, to partial group actions and higher Rokhlin dimensions.

The paper "Duality of partial Rokhlin dimension" (2604.09380) expands this framework by establishing duality results for Rokhlin and representability dimensions in the context of partial actions by finite abelian groups on separable $C^*$-algebras. The main results provide a precise correspondence between the Rokhlin dimension of a partial action and the (dual) representability dimension of its dual action, thus generalizing the relationship known in the global action setting (notably, Izumi's result connecting the Rokhlin property with approximate representability for finite abelian group actions).

Rokhlin and Representability Dimensions for Partial Actions

Partial group actions, extensions of global actions wherein group elements act only on ideals of the algebra, have become significant in noncommutative dynamical systems and the study of Fell bundles. The Rokhlin dimension for such actions, following the approach in [AbaGefGar, Ergodic Theory Dynam. Systems, 2022], is formulated using "Rokhlin towers," which are families of positive contractions in the corresponding ideals, satisfying commutation, orthogonality, approximate partition of unity, and intertwining conditions.

Formally, the Rokhlin dimension $\dim_{\mathrm{Rok}}(\alpha)$ of a partial action $\alpha$ is the minimal $d$ for which, for every finite set and $\varepsilon > 0$, there exist $d+1$ families of towers that satisfy detailed analytic criteria (see Definition~2.3 in the paper). This definition extends to non-unital algebras via central sequence algebra techniques, translating the tower conditions into the existence of completely positive contractive order zero maps from function algebras $C^*$0 to appropriate central sequence subalgebras.

Conversely, the representability dimension $C^*$1—which generalizes approximate representability from global to partial actions—quantifies the minimal number of "colors" (order zero representations of the group) needed to nearly implement the dynamics by elements in the domain ideals, up to the specified tolerance.

Duality via Crossed Products and Dual Actions

Given a partial action $C^*$2 of an abelian group $C^*$3 on $C^*$4, the algebraic structure of the resulting crossed product $C^*$5 admits a canonical dual action $C^*$6 by the dual group $C^*$7, acting globally. The properties of $C^*$8 are tightly linked to those of $C^*$9, and vice versa, paralleling Landstad duality for global actions.

The paper shows, through an intricate analysis of order zero maps, central sequence algebras, and the structure of crossed products, that the Rokhlin dimension of a partial action is equal to the dual representability dimension of its dual action. Moreover, the representability dimension of the partial action coincides with the Rokhlin dimension (in the global sense) of its dual action.

Main Formal Results

Let $C^*$0 be a finite abelian group and $C^*$1 a partial action of $C^*$2 on a separable $C^*$3-algebra $C^*$4. Denote by $C^*$5 the dual action on $C^*$6. The main results are:

• The Rokhlin dimension of $C^*$7 equals the dual representability dimension of $C^*$8:

$C^*$9

• The representability dimension of $C^*$0 equals the Rokhlin dimension of $C^*$1:

$C^*$2

These equivalences are established by constructing explicit equivalences between the sets of tower/representation data for $C^*$3 and for $C^*$4 within the respective central sequence algebras and multiplier algebras and by careful matching of order zero representation structures. The paper further clarifies the distinction between representability and dual representability dimensions, showing that in general:

$C^*$5

with an explicit example demonstrating strict inequality may occur outside the dual-of-global-action setting.

Structural and Theoretical Implications

These duality theorems significantly clarify the landscape of partial group actions and the noncommutative topological dynamics of $C^*$6-algebras. Specifically:

• Transfer of Structural Properties: Finite Rokhlin dimension strongly constrains the structure of both the fixed point algebras and the crossed products, paralleling the global case. The established dualities enable systematic transfer of these results between the realms of (partial) group actions and global dual actions.
• Central Role of Order Zero Maps: The translation of Rokhlin and representability criteria into the language of order zero completely positive maps (as in Winter–Zacharias’ structure theory) enables new connections with noncommutative dimension theory and could facilitate further generalizations (e.g., to compact or locally compact groups).
• Spectral Subspaces and Landstad Duality for Partial Dynamics: The analysis demonstrates how the classical Landstad duality morphs in the partial context, involving the careful use of spectral subspaces and multiplier ideals that are sensitive to the non-global nature of the action.

Numerical and Qualitative Claims

The results are primarily qualitative, equating various dynamical invariants under duality, rather than making quantitative claims in terms of explicit values for the invariants. However, the theoretical consequences are strong: for any partial action of a finite abelian group, these Sui generis duality theorems permit the computation of one invariant given the other, and they precisely characterize when global dual actions admit the Rokhlin property in terms of the (globality and) approximate representability of the original action.

Outlook and Future Directions

The duality theory developed here provides a robust foundation for extending equivariant classification of $C^*$7-algebras (in the sense of Kirchberg–Phillips) to settings involving partial group actions and higher Rokhlin dimensions. The correspondence between dynamical invariants for partial actions and their duals may also inform the development of new invariants or dualities for non-abelian group actions, compact quantum group actions, or more general groupoid actions.

Practically, the methods used suggest that Rokhlin-type arguments applicable in the global case can, via this duality and under the umbrella of order zero techniques, be generalized to much wider dynamical systems, thus broadening the technical toolbox for future progress in noncommutative topology and the structure theory of $C^*$8-algebras.

Conclusion

This paper rigorously establishes a duality between Rokhlin and representability dimensions in the context of partial actions of finite abelian groups on separable $C^*$9-algebras, filling a notable gap in the noncommutative dynamics literature. The techniques and results substantially extend the classical framework, clarifying the interplay between partial and global actions, and laying groundwork for further advances in operator algebraic dynamics, classification theory, and applications to noncommutative geometry.