[Paper Review] Irreducible Representations of Groupoid $C^*$-algebras

Marius Ionescu, Dana P. Williams|ArXiv.org|Dec 4, 2007

Advanced Operator Algebra Research8 references4 citations

TL;DR

This paper establishes that for second countable, locally compact, Hausdorff groupoids with Haar systems, representations induced from irreducible representations of stability (isotropy) groups are themselves irreducible. Using modern Morita equivalence and disintegration theory, the authors generalize Mackey's result for transformation groups to general groupoid C*-algebras, proving irreducibility via a systematic induction process on closed subgroupoids and verifying the compatibility of induction through a unitary equivalence of Hilbert modules.

ABSTRACT

If $G$ is a second countable locally compact Hausdorff groupoid with Haar system, we show that every representation induced from an irreducible representation of a stability group is irreducible.

Motivation & Objective

• To generalize the classical result that induced representations from irreducible representations of stability groups are irreducible, known for transformation group C*-algebras, to general separable groupoid C*-algebras.
• To formalize the theory of inducing representations from closed subgroupoids in groupoid C*-algebras using contemporary tools from Morita equivalence and disintegration theory.
• To establish a rigorous, modern treatment of induction in groupoid C*-algebras, correcting and extending earlier ad hoc approaches in special cases.
• To prove that the induction process preserves irreducibility when starting from irreducible representations of isotropy groups, thereby providing a systematic construction of irreducible representations.

Proposed method

• The authors construct the imprimitivity groupoid $ H^G $ from the space $ G_{H^{(0)}} = s^{-1}(H^{(0)}) $, equipped with a diagonal right $ H $-action.
• They define a Haar system on $ H^G $ via integration over the Haar system of $ G $, using the $ s $-system structure on $ G_{H^{(0)}} $, ensuring compatibility with the groupoid operations.
• The space $ C_c(G_{H^{(0)}}) $ is equipped with actions and inner products to form a $ C_c(H^G,\beta) $-$ C_c(H,\alpha) $-imprimitivity bimodule, which is completed to a Hilbert bimodule $ \mathsf{X}_H^G $.
• Using Rieffel's theory of Morita equivalence and the equivalence theorem from [mrw:jot87], the authors establish a correspondence between representations of $ C^*(H) $ and $ C^*(H^G) $, enabling the induction of representations.
• The induction process is shown to be compatible with composition via a unitary equivalence $ V $, constructed using a map $ \theta $ that identifies $ \mathsf{X}_K^G \otimes_K (\mathsf{X}_H^K \otimes_H \mathcal{H}_L) $ with $ \mathsf{X}_H^G \otimes_H \mathcal{H}_L $, proving transitivity of induction.
• The key technical step involves proving that the map $ \theta $ is isometric and has dense range, using Fubini's theorem and properties of Haar systems, thereby establishing the unitary equivalence of induced representations.

Experimental results

Research questions

• RQ1 Does the induction of representations from irreducible representations of a stability group $ G(u) $ yield an irreducible representation of the full groupoid $ C^* $-algebra $ C^*(G) $, for general second countable, locally compact, Hausdorff groupoids with Haar systems?
• RQ2 Can the classical Mackey induction theory for transformation groups be extended to general groupoid $ C^* $-algebras using modern tools such as Morita equivalence and disintegration?
• RQ3 Is the induction process from subgroupoids transitive, and does it preserve irreducibility when applied through a chain of subgroupoids?
• RQ4 How can one systematically construct irreducible representations of $ C^*(G) $ from irreducible representations of isotropy groups $ G(u) $, and what is the role of the imprimitivity bimodule in this construction?

Key findings

• Every representation induced from an irreducible representation of a stability group $ G(u) $ is irreducible in the groupoid $ C^* $-algebra $ C^*(G) $, for second countable, locally compact, Hausdorff groupoids with Haar systems.
• The induction process from a closed subgroupoid $ H $ to $ G $ is well-defined and respects the structure of Hilbert modules, with the imprimitivity bimodule $ \mathsf{X}_H^G $ providing a Morita equivalence between $ C^*(H^G) $ and $ C^*(H) $.
• The map $ \theta $, which identifies tensor products of induced modules, is an isometric isomorphism of Hilbert modules, ensuring the compatibility of the induction process.
• The unitary equivalence $ V $ between $ \mathsf{X}_K^G \otimes_K (\mathsf{X}_H^K \otimes_H \mathcal{H}_L) $ and $ \mathsf{X}_H^G \otimes_H \mathcal{H}_L $ confirms the transitivity of induction, proving that $ \operatorname{Ind}_K^G(\operatorname{Ind}_H^K L) \cong \operatorname{Ind}_H^G L $.
• The proof relies on the disintegration theorem and modern Morita equivalence, showing that the irreducibility of the induced representation follows from the structure of the groupoid and its Haar system.
• The result holds under the assumption of second countability, which ensures the applicability of the disintegration theorem, though the authors note that separability may not be strictly necessary for the main theorem.