Weak Containment Property in C*-Systems

• Weak Containment Property (WCP) is a tensorial criterion that ensures the isomorphism between full and reduced crossed products in C*-dynamical systems and Fell bundles.
• It unifies amenability and positive approximation properties by linking various equivalence conditions and approximation criteria in operator algebras.
• WCP's stability under restrictions, quotients, and tensor products aids in the structural analysis and classification of noncommutative dynamical systems.

The Weak Containment Property (WCP) is a pivotal concept in the theory of operator algebras and noncommutative dynamics. In the context of C*-dynamical systems and Fell bundles over locally compact groups, WCP functions as a tensorial criterion that precisely characterizes amenability and certain positive approximation properties. The property establishes when canonical quotient maps between full and reduced crossed products or cross-sectional C*-algebras are isomorphisms, and its stability under various natural algebraic operations is essential for structural analysis and classification.

1. Definition and Context of WCP

For a C*-dynamical system, that is, a triple $(A, G, \alpha)$ with $A$ a C*-algebra, $G$ a locally compact group, and $\alpha$ an action of $G$ on $A$, the weak containment property asserts that for every other C*-system $(B, G, \beta)$, the diagonal system $$(A \otimes_{\max} B, G, \alpha \otimes^d_{\max} \beta)$$ admits the equality

$$(A \otimes_{\max} B) \rtimes_{\alpha \otimes^d_{\max} \beta} G \cong (A \otimes_{\max} B) \rtimes_{\text{red},\,\alpha \otimes^d_{\max} \beta} G,$$

where $\otimes_{\max}$ denotes the maximal tensor product and $\otimes^d_{\max}$ the diagonal tensor product defined via the restriction of the tensor product to the diagonal subgroup. The quotient map involved becomes an isomorphism precisely when WCP holds.

For a Fell bundle $\mathcal{A} = \{A_t\}_{t\in G}$, the property requires that the canonical map between the full cross-sectional algebra $C^*(\mathcal{A})$ and its reduced counterpart $C^*_{\text{red}}(\mathcal{A})$ is an isomorphism: $$C^*(\mathcal{A}) \cong C^*_{\text{red}}(\mathcal{A}).$$ The diagonal tensor product bundle $$\mathcal{A}^d \mathcal{B} = \{A_t \otimes B_t\}_{t\in G}$$ also becomes central in tensor-testing for amenability and approximation properties.

2. Characterizations and Equivalence Conditions

The paper demonstrates several strong equivalence results:

• Amenability of a C*-dynamical system in the sense of Anantharaman-Delaroche is equivalent to the WCP for every diagonal tensor product system with another action $(B, G, \beta)$ over $G$. That is, amenability is characterized by the requirement that

$$(A \otimes_{\max} B) \rtimes_{\alpha \otimes^d_{\max} \beta} G \cong (A \otimes_{\max} B) \rtimes_{\text{red},\,\alpha \otimes^d_{\max} \beta} G$$

for all $(B, \beta)$.

• For Fell bundles, the positive approximation property (AP) introduced by Exel and Ng is equivalent to WCP for all diagonal tensor product bundles: $$C^*(\mathcal{A} \otimes^d_{\max} \mathcal{B}) \cong C^*_{\text{red}}(\mathcal{A} \otimes^d_{\max} \mathcal{B})$$ for every Fell bundle $\mathcal{B}$ over $G$.
• The amenability of the natural action of $G$ on the kernel algebra (associated to a Fell bundle) implies AP, and vice versa.
• Various previously formulated approximation properties (Exel-Ng's AP, Abadie's, Bedos-Conti's BCAP) are proved to be equivalent to AP and thus to WCP.

3. Mathematical Formulations

Some central formulas underlying the paper's results are:

• Diagonal tensor product system quotient:

$$(A \otimes_{\max} B) \rtimes_{\alpha \otimes^d_{\max} \beta} G \rightarrow (A \otimes_{\max} B) \rtimes_{\text{red},\,\alpha \otimes^d_{\max} \beta} G$$

is an isomorphism if and only if WCP holds.

• Fell bundle cross-sectionals:

$$C^*(\mathcal{A}) \cong C^*_{\text{red}}(\mathcal{A})$$

• Diagonal tensor product bundle:

$$\mathcal{A}^d \mathcal{B} = \{ A_t \otimes B_t \}_{t \in G }$$

The regular (or left regular) representations and conditional expectations inherent in these constructions are essential to the proof techniques.

4. Applications and Structural Implications

The tensorial characterization of WCP has notable applications:

• Amenability Testing: Amenability can be checked via the WCP for diagonal tensor products, providing a concrete and uniform criterion that does not require supplementary hypotheses such as group exactness.
• Unification of Approximation Properties: The identification of AP, BCAP, and similar axioms with WCP unifies disparate strands of the theory and streamlines amenability characterization for both C*-systems and Fell bundles.
• Nuclearity Results: Amenability (i.e., WCP) of a C*-system or Fell bundle often implies nuclearity of the crossed product or cross-sectional algebra, a critical property for classification and analysis.
• Structural Transfer: The WCP provides a means for transferring amenability, nuclearity, and AP from a system to its subgroups, quotients, or tensor products.

5. Permanence Properties

WCP, AP, and nuclearity are shown to be stable under various natural operations:

• Restrictions to Subgroups: Given $(A, G, \alpha)$ or a Fell bundle $\mathcal{A}$, restriction to a closed subgroup $H$ preserves amenability and WCP. Existence of conditional expectations ensures transfer of nuclearity.
• Quotient Constructions: For normal closed subgroup $H \triangleleft G$, corresponding quotient bundles inherit properties under suitable hypotheses.
• Tensor Products: Diagonal tensor product formation preserves amenability and the weak containment property when one factor is amenable.
• Other Reductions: Conditional expectations and expectations onto subalgebras guarantee the passage of nuclearity and AP under these operations.

These permanence results facilitate the structural analysis and classification of complex systems built from simpler building blocks.

6. Unified Framework and Further Directions

The framework established in this work refines classical perspectives by grounding amenability and approximation properties in the language of weak containment for diagonal tensor products. This paradigm not only generalizes prior characterizations but also enables sharper analysis of nuclearity, exactness, and classification in the context of noncommutative dynamical systems and Fell bundles.

The approach developed is applicable to a broad class of systems, and the unification provided by the weak containment property suggests further directions for research on structural invariants, permanence phenomena, and classification via tensorial techniques in operator algebras. The equivalence of WCP with amenability and AP offers a robust foundation for analyzing the interplay between algebraic and analytic properties in noncommutative harmonic analysis.

Conclusion

The weak containment property serves as a central and unifying tool for characterizing amenability and positive approximation properties in C*-dynamical systems and Fell bundles. Through tensor product testing, it connects structural features such as nuclearity with combinatorial and functional analytic criteria, and its permanence under reduction, quotient, and tensor product formation consolidates its role as a fundamental invariant in the theory of noncommutative dynamical systems. The results established in (Buss et al., 17 Oct 2025) extend and unify prior approaches and open expanded possibilities for operator algebraic classification and analysis.