LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generated

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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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Fell closure is characterized by weak containment

Statement

Assume the Axiom of Choice. Let $G$ be an LCH group, let $S \subseteq \hat{G}$ and let $π \in \hat{G}$ (The unitary dual of a locally compact group). Then $π$ lies in the Fell closure of $S$ (The Fell topology on the unitary dual) if and only if $π ≺ {⨁^{^}}_{σ \in S} σ$ (Weak containment of unitary representations, Hilbert direct sums of unitary representations). Equivalently, the Fell closure of $S$ is the set of all $π \in \hat{G}$ whose $C^{*}$ -kernel contains the intersection of the kernels of the classes in $S$ : $\overline{S} = {π \in \hat{G} : ⋂_{σ \in S} C^{*} ⁣ \ker σ \subseteq C^{*} ⁣ \ker π}$ (The primitive ideal space of a group C star algebra).

Facts & Assumptions

Given: AC; an LCH group $G$ ; a subset $S \subseteq \hat{G}$ ; a class $π \in \hat{G}$ ; the kernel map $κ$ .

[F1]

The proof of The induced kernel map on weak equivalence classes is a homeomorphism establishes, before using any homeomorphism statement, the closure identity $\overline{S} = κ^{- 1} ({\overline{κ (S)}}^{J a c})$ for every $S \subseteq \hat{G}$ , the Jacobson closure being taken in the sense of The primitive ideal space of a group C star algebra; explicitly $\overline{S} = {π \in \hat{G} : ⋂_{σ \in S} C^{*} ⁣ \ker σ \subseteq C^{*} ⁣ \ker π}$ , with $⋂_{\emptyset} C^{*} ⁣ \ker σ = C^{*} (G)$ so that $\overline{\emptyset} = \emptyset$ .

[F2]

For unitary representations of $G$ , $π ≺ ρ$ if and only if $\ker_{C^{*} (G)} ρ \subseteq \ker_{C^{*} (G)} π$ ; moreover the kernel of a Hilbert direct sum is the intersection of the kernels of its summands (Weak containment is equivalent to kernel inclusion, Hilbert direct sums of unitary representations).

Proof

technique · direct

Given: AC, an LCH group $G$ , a subset $S \subseteq \hat{G}$ and a class $π \in \hat{G}$ .

1.1F1

By [F1] the Fell closure of $S$ is ${π \in \hat{G} : ⋂_{σ \in S} C^{*} ⁣ \ker σ \subseteq C^{*} ⁣ \ker π}$ .

1.2F2

For $π \in \hat{G}$ , the weak containment $π ≺ {⨁^{^}}_{σ \in S} σ$ holds if and only if $\ker ({⨁^{^}}_{σ \in S} σ) \subseteq C^{*} ⁣ \ker π$ , by [F2]; and $\ker ({⨁^{^}}_{σ \in S} σ) = ⋂_{σ \in S} C^{*} ⁣ \ker σ$ by the direct-sum computation of [F2] (for $S = \emptyset$ the direct sum is the zero representation with kernel $C^{*} (G)$ , and no $π \in \hat{G}$ is contained in it, matching the empty intersection convention).

2.1step 1.1step 1.2

Comparing steps 1.1 and 1.2, $π \in \overline{S}$ is equivalent to $π ≺ {⨁^{^}}_{σ \in S} σ$ , which is the first claim, and the displayed description of $\overline{S}$ is exactly step 1.1.

3.1givenF1∎

The Axiom of Choice is inherited from the kernel-map theorem and the weak-containment suppliers; the comparison of the closure identity with the direct-sum kernel uses no further choice (The Axiom of Choice).

Depends on

Used by

The unitary dual to primitive ideal map is continuous and surjective Proposition

Dependency tree · two levels

31 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Bachir Bekka and Pierre de la Harpe, Unitary Representations of Groups, Duals, and Characters (arXiv:1912.07262v1, 16 December 2019) (standard reference, not scraped)
Bachir Bekka, Pierre de la Harpe and Alain Valette, Kazhdan's Property (T) (Cambridge University Press 2008; author-hosted complete text) (standard reference, not scraped)