[Paper Review] The stabilization theorem for proper groupoids

Alan L. T. Paterson|ArXiv.org|Sep 22, 2009

Advanced Operator Algebra Research13 references3 citations

TL;DR

This paper establishes a stabilization theorem for proper groupoids, generalizing Kasparov's and Phillips's equivariant stabilization theorems to the setting of locally compact, Hausdorff, second countable, proper groupoids. It proves that for any $G$-Hilbert module $P$, the direct sum $P \oplus P_G^\infty$ is isomorphic to $P_G^\infty$, where $P_G$ is the completion of $C_c(G)$, extending foundational results in equivariant KK-theory and groupoid index theory.

ABSTRACT

The stabilization theorem for $A$-Hilbert modules was established by G. G. Kasparov. The equivariant version, in which a locally compact group $H$ acts properly on a locally compact space $Y$, was proved by N. C. Phillips. This equivariant theorem involves the Hilbert $(H,C_{0}(Y))$-module $C_{0}(Y,L^{2}(H)^{\infty})$. It can naturally be interpreted in terms of a stabilization theorem for proper groupoids, and the paper establishes this theorem within the general proper groupoid context. The theorem has applications in equivariant KK-theory and groupoid index theory.

Motivation & Objective

• To generalize the equivariant stabilization theorem of Phillips for group actions to the broader context of proper groupoids.
• To provide a natural framework for groupoid equivariant KK-theory and noncommutative index theory by formulating the stabilization theorem in terms of groupoid Hilbert modules.
• To address the technical challenge that $G$-Hilbert bundles over proper groupoids are not generally locally trivial, requiring new constructions beyond classical bundle theory.
• To establish a groupoid analog of the standard stabilization theorem for $C^*$-algebras, replacing $A^\infty$ with $P_G^\infty$ for a proper groupoid $G$.
• To support future developments in equivariant index theory and the generalized Green-Rosenberg theorem within the groupoid framework.

Proposed method

• Define $P_G$ as the completion of the pre-Hilbert $C_0(Y)$-module $C_c(G)$, where $G$ is a proper groupoid with unit space $Y$.
• Construct a $G$-invariant, isometric, $C_0(Y)$-linear map $W: C_0(Y,E) \to C_0(Y,E \otimes L^2(G)^\infty)$ using a partition of unity and a $G$-invariant section $\phi$.
• Lift $W$ to a $G$-equivariant, adjointable, isometric map $V: P \to P \otimes_{C_0(Y)} P_G^\infty$ via a groupoid version of the Mingo-Phillips construction.
• Define a unitary operator $U$ on $P \oplus (P \otimes_{C_0(Y)} P_G^\infty)$ using $V$ and $1 - VV^*$, preserving the $G$-action and inner product.
• Use the non-equivariant stabilization theorem and module isomorphisms to derive the key isomorphism $P \oplus P_G^\infty \cong P_G^\infty$.
• Verify that all maps are $G$-equivariant, continuous, and preserve the $C_0(Y)$-module structure, ensuring the isomorphism is in the category of $G$-Hilbert modules.

Experimental results

Research questions

• RQ1 How can the equivariant stabilization theorem for group actions be generalized to the setting of proper groupoids?
• RQ2 What is the appropriate analog of the Hilbert module $L^2(H) \otimes C_0(Y)^\infty$ in the groupoid framework?
• RQ3 How can one construct a $G$-equivariant isomorphism $P \oplus P_G^\infty \cong P_G^\infty$ when the associated $G$-Hilbert bundle is not locally trivial?
• RQ4 What technical tools are needed to extend the Mingo-Phillips approach to groupoid-equivariant Hilbert modules?
• RQ5 How does the stabilization theorem for proper groupoids support the development of groupoid equivariant KK-theory and index theory?

Key findings

• The stabilization theorem for proper groupoids is established: for any $G$-Hilbert module $P$, there exists a $G$-equivariant isomorphism $P \oplus P_G^\infty \cong P_G^\infty$.
• The module $P_G$ is defined as the completion of $C_c(G)$, forming the universal stabilizing module for $G$-Hilbert modules.
• The construction relies on a $G$-invariant section $\phi$ and a partition of unity to define a $G$-equivariant isometric map $W$, which lifts to a $G$-equivariant isomorphism $V$.
• The key isomorphism is constructed via a unitary operator $U$ on $P \oplus (P \otimes_{C_0(Y)} P_G^\infty)$, which is shown to be unitary and $G$-equivariant.
• The proof uses the non-equivariant stabilization theorem and module isomorphisms to derive $P_G^\infty \cong (P \otimes_{C_0(Y)} P_G^\infty) \oplus P_G^\infty$, enabling the final isomorphism.
• The result confirms that $P_G^\infty$ stabilizes all $G$-Hilbert modules in the category of $G$-equivariant Hilbert modules, generalizing classical results to the groupoid setting.