[Paper Review] CCR and GCR Groupoid C*-algebras

Lisa Orloff Clark|ArXiv.org|Nov 17, 2005

Advanced Operator Algebra Research6 references4 citations

TL;DR

This paper establishes necessary and sufficient conditions for groupoid C*-algebras to be CCR or GCR, generalizing classical results for transformation group C*-algebras. It proves that C*(G) is CCR iff the orbit space is T₁ and all isotropy groups are CCR, and GCR iff the orbit space is T₀ and all isotropy groups are GCR, under the assumption that isotropy groups are amenable.

ABSTRACT

Suppose $G$ is a second countable, locally compact, Hausdorff groupoid with a fixed left Haar system. Let $\go/G$ denote the orbit space of $G$ and $C^*(G)$ denote the groupoid $C^*$-algebra. Suppose that the isotropy groups of $G$ are amenable. We show that $C^*(G)$ is CCR if and only if $\go/G$ is a $T_1$ topological space and all of the isotropy groups are CCR. We also show that $C^*(G)$ is GCR if and only if $\go/G$ is a $T_0$ topological space and all of the isotropy groups are GCR.

Motivation & Objective

• To generalize CCR and GCR classification theorems from transformation group C*-algebras to general locally compact groupoid C*-algebras.
• To identify topological and algebraic conditions on the groupoid that ensure its C*-algebra is CCR or GCR.
• To establish a continuous, injective map from the orbit space G⁰/G to the spectrum of C*(G), enabling the classification results.
• To extend prior results in the principal groupoid case to non-principal groupoids with potentially discontinuous isotropy.
• To show that the amenability of isotropy groups is a necessary technical assumption for constructing the spectrum map, despite its potential redundancy in theory.

Proposed method

• Define a representation lᵘ on C*(G) by inducing the trivial representation of the isotropy group Gᵘᵘ to the full groupoid C*-algebra.
• Construct a map ω: G⁰/G → C*(G)∧ by sending each orbit [u] to the unitary equivalence class of lᵘ.
• Prove ω is well-defined, continuous, and injective using Renault’s Disintegration Theorem and amenability of isotropy groups.
• Use the existence of this continuous injection to relate topological properties of G⁰/G (T₁ or T₀) to spectral properties of C*(G).
• Leverage known results: every irreducible representation of C*(G) factors through C*(G|ₐ[u]) ≅ C*(Gᵘᵘ) ⊗ K, and C*(G|ₐ[u]) is CCR/GCR iff C*(Gᵘᵘ) is.
• Apply [2, Corollary 3.5] and [8, Theorem 3.1] to reduce the global CCR/GCR property to the isotropy group level.

Experimental results

Research questions

• RQ1 Under what conditions is the groupoid C*-algebra C*(G) CCR?
• RQ2 Under what conditions is the groupoid C*-algebra C*(G) GCR?
• RQ3 Can the spectrum map from the orbit space G⁰/G to C*(G)∧ be constructed in the non-principal groupoid setting with non-abelian, possibly discontinuous isotropy?
• RQ4 Why is the amenability of isotropy groups required in the construction of the spectrum map, and is this condition necessary or merely technical?
• RQ5 How do the T₁ and T₀ properties of the orbit space relate to the CCR and GCR structure of C*(G)?

Key findings

• C*(G) is CCR if and only if the orbit space G⁰/G is a T₁ space and all isotropy groups Gᵘᵘ are CCR.
• C*(G) is GCR if and only if the orbit space G⁰/G is a T₀ space and all isotropy groups Gᵘᵘ are GCR.
• The key technical contribution is the construction of a continuous, injective map ω: G⁰/G → C*(G)∧, which links the topology of the orbit space to the spectrum of the C*-algebra.
• The map ω is defined via induction of the trivial representation from each isotropy group Gᵘᵘ to C*(G), and its well-definedness and continuity rely on the amenability of isotropy groups.
• The proof relies on the fact that every irreducible representation of C*(G) factors through C*(G|ₐ[u]) ≅ C*(Gᵘᵘ) ⊗ K for some u ∈ G⁰, reducing the problem to the isotropy group level.
• The amenability assumption is essential for the continuity of ω via Renault’s Disintegration Theorem, though the authors suspect it may be redundant in the final characterization.