Cocycle perturbations and ergodicity for actions on type III factors (2512.12931v1)

Abstract: We study cocycle perturbations of state preserving actions on type $\mathrm{III}_1$ factors. Extending the theorem of Marrakchi and Vaes for type $\mathrm{II}_1$ factors, we show that a state preserving outer $\mathbb Z$-action on a type $\mathrm{III}_1$ factor with trivial bicentralizer admits a unitary cocycle whose perturbation becomes an ergodic action. This partially answers a question of Marrakchi and Vaes. A major difference from the type $\mathrm{II}$ case is that the modular automorphism group naturally appears as part of the action, making the construction of the required cocycle more delicate.

• The paper establishes that for type III₁ factors with trivial bicentralizer, every outer automorphism admits a cocycle perturbation leading to ergodicity.
• It employs ultraproducts, Connes–Stormer transitivity, and free independence techniques to generalize results from type II₁ to the type III₁ regime.
• The study demonstrates that ergodic and weakly mixing cocycle perturbations are generic for amenable group actions with a trivial modular kernel.

Cocycle Perturbations and Ergodicity for Actions on Type III Factors

Overview

This paper investigates the ergodicity of cocycle perturbations of state-preserving automorphism group actions on type $\mathrm{III}_1$ von Neumann factors, particularly those with trivial bicentralizer. The context extends fundamental results about cocycle perturbations and ergodic properties from the setting of type $\mathrm{II}_1$ factors, addressed by Marrakchi and Vaes, to type $\mathrm{III}_1$ factors, resolving open questions about the possibility of inducing ergodicity through cocycle perturbation in this more singular context. The work employs advanced tools from Tomita–Takesaki theory, structure theorems for type III factors, bicentralizer methods, and free independence constructions in ultraproducts.

Main Results

The principal theorem asserts the following equivalence for a type $\mathrm{III}_1$ factor $M$ with separable predual and trivial bicentralizer, and for a state-preserving automorphism $\theta \in \mathrm{Aut}(M)$:

• $\theta$ has infinite order in the outer automorphism group $\mathrm{Out}(M)$.
• There exists a unitary $u \in \mathcal{U}(M)$ such that the perturbed automorphism $Ad(u) \circ \theta$ is ergodic and state-preserving.

This realizes a full ergodicity classification for such automorphisms, mirroring the type $\mathrm{II}_1$ factor case, and provides an explicit procedure for constructing such ergodic cocycles. Furthermore, for the amenable type $\mathrm{III}_1$ factor $R_\infty$, every automorphism of infinite order in $\mathrm{Out}(R_\infty)$ admits an ergodic cocycle, without the state-preserving constraint.

Additionally, for actions of countably infinite amenable discrete groups on type $\mathrm{III}_1$ factors (with trivial bicentralizer), the ergodicity and, in fact, weak mixing of cocycle perturbations is generic in the Polish space of cocycles, provided the so-called “modular kernel” subgroup is trivial.

Technical Approach

The proof strategy divides into two cases, depending on the modular kernel subgroup $$\Gamma_{\mathrm{mod}} = \alpha^{-1}(\mathrm{Mod}(M))$$:

• Trivial Modular Kernel: The argument adapts the approach of Marrakchi–Vaes to the type $\mathrm{III}_1$ setting by employing ultraproducts and the Connes–Stormer transitivity, guaranteeing that faithful normal states become unitarily conjugate in the ultrapower. Free independence results (generalizing Popa's theorems to type III) are exploited to construct cocycles whose perturbations satisfy the required ergodic properties. The existence of ergodic and weakly mixing cocycles forms a dense $G_\delta$ set in the space of state-preserving cocycles.
• Nontrivial Modular Kernel: The scenario reduces to actions by $\mathbb{Z}$, with the modular kernel having the form $p\mathbb{Z} \subset \mathbb{Z}$. Here, the construction leverages the structure of periodic automorphisms modulo modular components, extension of ergodic states to the centralizer, and explicit spectral decompositions. Ergodicity is achieved by analyzing the restriction of the cocycle to the modular part, lifting ergodic cocycles from the modular kernel, and using partial isometries and Connes cocycles to extend to the full group.

The proofs require careful management of centralizer and bicentralizer structures in ultraproducts, applications of cohomology vanishing for state-preserving cocycles, and generalizations of classical independence results to the type III context, such as the free Bernoulli shift and its fixed point properties.

Numerical and Structural Claims

• For amenable groups acting outerly on type $\mathrm{III}_1$ factors with trivial bicentralizer and trivial modular kernel, the sets of ergodic or weakly mixing cocycles are dense $G_\delta$ in the cocycle space.
• When the modular kernel is nontrivial, ergodic cocycle perturbations still exist for automorphisms of infinite order.
• The results hold uniformly for tensor stabilizations by arbitrary type $\mathrm{III}_1$ factors with separable predual.
• In the presence of diffuse centralizer, the fixed point algebra in the state-preserving ultrapower under cocycle-perturbed actions is a type $\mathrm{II}_1$ factor, and the action is not strongly ergodic.
• The modular automorphism group systematically complicates the type III case, necessitating the use of modular inclusion in cocycle constructions, in stark contrast to the type II setting.

Theoretical and Practical Implications

The extension of the existence and genericity of ergodic cocycle perturbations to the type $\mathrm{III}_1$ setting has direct impact on the structure theory of group actions on von Neumann algebras, especially in quantum dynamical systems and in the analysis of non-tracial factors. Demonstrating the ubiquity of ergodic and weakly mixing perturbations for outer actions tightly links the orbit structure of automorphism groups with properties of the algebraic core, especially in the presence of trivial or nontrivial bicentralizer.

For the classification program of group actions on operator algebras, especially those arising from quantum field theory or infinite tensor product systems, these structural insights into ergodicity under cocycle perturbation form a bridge between the well-understood tracial case and the more intricate type III regime. The techniques introduced, particularly the interplay between ultraproduct arguments, Connes–Stormer transitivity, and advanced independence theorems, are likely to have further applications in rigidity and deformation/rigidity theory, cohomological obstructions, and the study of symmetry breaking mechanisms in non-commutative probability.

Future Directions

This work opens the way for several potential research directions:

• Full extension of the generic ergodicity results to actions of non-amenable groups, and to arbitrary (possibly non-outer) actions, contingent on further understanding of the modular kernel in type III factors.
• Deeper analysis of the bicentralizer problem, especially confirmation of its triviality in broader classes of type $\mathrm{III}_1$ factors, which would remove the remaining hypothesis for the main theorem.
• Examination of the structure of fixed point algebras and their invariants under cocycle perturbations for actions on type $\mathrm{III}_\lambda$ or semifinite factors.
• Connections with the classification of full factors, flow of weights, and the invariants for approximately inner or pointwise inner actions.

Conclusion

The paper provides a definitive characterization of when cocycle perturbations of outer state-preserving group actions on type $\mathrm{III}_1$ factors can achieve ergodicity, completing the analogy with the type $\mathrm{II}_1$ situation under the assumption of trivial bicentralizer. The methods establish new links between ergodic theory, cohomology, and the structure of operator algebras, advancing the understanding of dynamics and symmetries in this setting and laying the groundwork for further advances in the study of von Neumann algebra automorphism groups and their actions.