[Paper Review] The Mackey Machine for Crossed Products by Regular Groupoids. I

Geoff Goehle|arXiv (Cornell University)|Aug 11, 2009

Advanced Operator Algebra Research14 references3 citations

TL;DR

This paper develops the Mackey Machine for groupoid crossed products by introducing a Rieffel-type induction system, proving that every irreducible representation of a crossed product algebra $A \rtimes G$ arises from inducing a representation of a stabilizer crossed product $A(u) \rtimes S_u$ for some $u \in G^{(0)}$. The key contribution is a complete induction-theoretic classification of irreducible representations under regularity assumptions on the groupoid $G$. The result extends the Mackey machine to groupoid dynamical systems using Hilbert modules and Morita equivalence, generalizing prior results on group and groupoid $C^*$-algebras.

ABSTRACT

We first describe a Rieffel induction system for groupoid crossed products. We then use this induction system to show that, given a regular groupoid $G$ and a dynamical system $(A,G,α)$, every irreducible representation of $A times G$ is induced from a representation of the group crossed product $A(u) times S_u$ where $u\in G\unit$, $A(u)$ is a fibre of $A$, and $S_u$ is a stabilizer subgroup of $G$.

Motivation & Objective

• To extend the Mackey Machine to groupoid crossed products using Rieffel's Hilbert module framework.
• To address the representation theory of $A \rtimes G$ for a regular groupoid $G$ and $C_0(G^{(0)})$-algebra $A$.
• To establish that every irreducible representation of $A \rtimes G$ is induced from a representation of a stabilizer crossed product $A(u) \rtimes S_u$.
• To identify the spectrum of $A \rtimes G$ as a quotient of the spectrum of $A \rtimes S$, where $S$ is the stabilizer subgroupoid.
• To generalize existing results on group crossed products and groupoid $C^*$-algebras to the setting of groupoid dynamical systems.

Proposed method

• Introduces a Rieffel induction system for groupoid crossed products using Hilbert $A \rtimes S_u$-modules.
• Uses the structure of the unit space $G^{(0)}$ and the action of $G$ to define stabilizer subgroups $S_u$ for $u \in G^{(0)}$.
• Applies the Mackey-Glimm Dichotomy to assume $G$ is regular, ensuring $G^{(0)}/G$ is almost Hausdorff.
• Constructs a composition series $\{I_\beta\}$ for $A \rtimes G$ indexed by ordinals, using open $G$-invariant sets $U_\beta \subset G^{(0)}$.
• Proves that each quotient $I_{\beta+1}/I_\beta$ is isomorphic to a crossed product over a Hausdorff quotient space $G|_{U_{\beta+1} \setminus U_\beta}$.
• Applies induction from $S_u$ to $G|_{U_{\beta+1} \setminus U_\beta}$ to show that irreducible representations arise via induction from $A(u) \rtimes S_u$.

Experimental results

Research questions

• RQ1 Can the Mackey Machine be generalized to groupoid crossed products using induction from stabilizer subgroups?
• RQ2 Under what conditions is every irreducible representation of $A \rtimes G$ induced from a representation of $A(u) \rtimes S_u$?
• RQ3 How does the spectrum of $A \rtimes G$ relate to the spectra of the stabilizer crossed products $A(u) \rtimes S_u$?
• RQ4 What role does the regularity of the groupoid $G$ play in ensuring the induction machinery works?
• RQ5 Is the induced representation $\operatorname{Ind}_{S_u}^G R$ irreducible when $R$ is irreducible on $A(u) \rtimes S_u$?

Key findings

• Every irreducible representation of $A \rtimes G$ is induced from a representation of $A(u) \rtimes S_u$ for some $u \in G^{(0)}$, under the assumption that $G$ is regular.
• The canonical extension of the induced representation $\operatorname{Ind}_{S_u}^{G|_{U_{\beta+1} \setminus U_\beta}} R$ to $A \rtimes G$ coincides with $\operatorname{Ind}_{S_u}^G R$, ensuring consistency across induction levels.
• The quotient $I_{\beta+1}/I_\beta$ is isomorphic to $A(U_{\beta+1} \setminus U_\beta) \rtimes G|_{U_{\beta+1} \setminus U_\beta}$, enabling localization of the representation theory.
• The representation $\operatorname{Ind}_{S_u}^{G|_{U_{\delta} \setminus U_{\sigma}}} R$ is irreducible when $R$ is irreducible on $A(u) \rtimes S_u$, provided the orbit space is Hausdorff.
• If $\operatorname{Ind}_{S_u}^G R \simeq \operatorname{Ind}_{S_u}^G L$, then $R \simeq L$, showing that induction is injective on irreducible representations.
• The spectrum of $A \rtimes G$ is homeomorphic to a quotient of the disjoint union $\coprod_{u \in G^{(0)}} (A(u) \rtimes S_u)^\wedge$, reflecting the induction structure.