Proper actions, fixed-point algebras and naturality in nonabelian duality

Suppose a locally compact group G acts freely and properly on a locally compact Hausdorff space X, and let gamma be the induced action on C_0(X). We consider a category in which the objects are C*-dynamical systems (A, G, alpha) for which there is an equivariant homomorphism of (C_0(X), gamma) into the multiplier algebra M(A). Rieffel has shown that such systems are proper and saturated, and hence have a generalized fixed-point algebra A^alpha which is Morita equivalent to A times_{alpha,r} G. We show that the assignment (A, alpha) maps to A^alpha is functorial, and that Rieffel’s Morita equivalence is natural in a suitable sense. We then use our results to prove a categorical version of Landstad duality which characterizes crossed products by coactions, and to prove that Mansfield imprimitivity for crossed products by homogeneous spaces is natural.

💡 Research Summary

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The paper investigates C*‑dynamical systems that admit an equivariant embedding of the commutative system (C₀(X), γ) arising from a free and proper action of a locally compact group G on a locally compact Hausdorff space X. If ι : C₀(X) → M(A) is a G‑equivariant ‑homomorphism into the multiplier algebra of a C‑algebra A, then (A, α) is automatically a proper and saturated system in the sense of Rieffel. Consequently, A possesses a generalized fixed‑point algebra A^α, and Rieffel’s theory guarantees a Morita equivalence between A^α and the reduced crossed product A ⋊_{α,r} G.

The authors’ first contribution is to place this construction inside a categorical framework. They define a category 𝔇 whose objects are triples (A, α, ι) as above and whose morphisms are G‑equivariant ‑homomorphisms φ : A → B that intertwine the embeddings of C₀(X). Within this category they introduce the fixed‑point functor
 F : 𝔇 → 𝔠, F(A, α, ι) = A^α,
where 𝔠 denotes the ordinary category of C‑algebras. The paper proves that F is indeed a functor: any morphism φ induces a *‑homomorphism φ|_{A^α} : A^α → B^β, and the assignment respects composition and identities. This establishes that the passage from a proper system to its generalized fixed‑point algebra is functorial.

The second major result concerns the naturality of Rieffel’s Morita equivalence. For each object (A, α, ι) there is a canonical imprimitivity bimodule X_A implementing a Morita equivalence between A^α and A ⋊{α,r} G. The authors show that the family {X_A} constitutes a natural transformation
 η : F ⇒ G,
where G : 𝔇 → 𝔠 is the crossed‑product functor (A, α, ι) ↦ A ⋊{α,r} G. In other words, for any morphism φ the diagram

A^α ──η_A──► A⋊G
│                │
φ|_{A^α}      φ⋊G
│                │
▼                ▼
B^β ──η_B──► B⋊G

commutes. This demonstrates that Rieffel’s equivalence is not merely an object‑wise statement but a coherent, functorial relationship throughout the whole category.

Having set up these categorical tools, the authors turn to Landstad duality. Classical Landstad duality characterises reduced crossed products by coactions of G via three algebraic conditions (the “Landstad conditions”). By interpreting a coaction (B, δ) as an object of a suitable coaction category 𝔏, the paper constructs a functor C : 𝔏 → 𝔇 that sends (B, δ) to the associated proper system (B ⋊_δ Ĝ, \hat{δ}, ι). Using the fixed‑point functor F and the natural transformation η, they obtain a natural isomorphism
 θ : C ≅ F ∘ H,
where H is the standard passage from a coaction to its dual action. This yields a categorical Landstad duality: the reduced crossed product by a coaction is naturally isomorphic to the generalized fixed‑point algebra of the dual action, and the isomorphism respects all morphisms in the respective categories.

The final part of the paper addresses Mansfield imprimitivity for crossed products by homogeneous spaces G/H (with H a closed subgroup). Mansfield’s theorem provides an imprimitivity bimodule linking A ⋊{α} G and A ⋊{α|_H} H when A carries an action α that is “compatible” with the quotient. The authors embed this situation into the previously defined category 𝔇_H (objects where the embedding ι factors through C₀(G/H)). They then define a composite functor M that first takes a proper system, forms the crossed product by G, and then restricts to the homogeneous space. By constructing a natural transformation μ between M and the functor that directly produces the crossed product by the homogeneous space, they prove that Mansfield’s imprimitivity bimodule is natural: it yields a commuting diagram for every morphism in 𝔇_H, exactly as in the case of Rieffel’s equivalence.

In summary, the paper achieves three intertwined goals: (1) it shows that the passage from a proper G‑action to its generalized fixed‑point algebra is functorial; (2) it proves that Rieffel’s Morita equivalence, Landstad duality, and Mansfield imprimitivity are all natural transformations between appropriate functors. By doing so, the authors provide a unified categorical perspective on several cornerstone results in non‑abelian duality, opening the way for further structural investigations and potential generalisations to quantum groups, groupoid actions, and beyond.