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The unitary dual to primitive ideal map is continuous and surjective

Statement

Assume the Axiom of Choice. Let $G$ be an LCH group and let $κ : \hat{G} \to Prim (C^{*} (G))$ be the kernel map $κ ([π]) = C^{*} ⁣ \ker π$ (The primitive ideal space of a group C star algebra, The unitary dual of a locally compact group). Then $κ$ is continuous for the Fell topology on $\hat{G}$ (The Fell topology on the unitary dual) and the Jacobson topology on $Prim (C^{*} (G))$ , and $κ$ is surjective. The induced map $\hat{G} / \sim ⟶ Prim (C^{*} (G))$ on the weak equivalence classes of irreducible representations (Weak containment of unitary representations) is a homeomorphism onto $Prim (C^{*} (G))$ .

Facts & Assumptions

Given: AC; an LCH group $G$ ; the kernel map $κ$ ; the Fell and Jacobson topologies.

[F1]

The kernel-map theorem proves that $κ$ is continuous and surjective, that its fibres are exactly the weak equivalence classes, and that the induced bijection from the quotient by weak equivalence with the quotient Fell topology to the primitive ideal space is a homeomorphism; its internal proof first establishes the Fell/Jacobson closure identity by family selection and only then the topology statement, so the homeomorphism is available in full (The induced kernel map on weak equivalence classes is a homeomorphism).

[F2]

The closure identity used in that proof is also recorded separately: the Fell closure of any subset of the dual consists of the classes whose kernels contain the intersection of the kernels of the subset (Fell closure is characterized by weak containment).

Proof

technique · direct

Given: AC, an LCH group $G$ and the kernel map $κ$ .

1.1F1

Continuity and surjectivity: [F1] states that $κ$ is continuous for the two topologies and surjective, and identifies the fibres of $κ$ with the weak equivalence classes.

1.2F1F2

The induced map on weak equivalence classes is the bijection of [F1] from the quotient Fell topology to the Jacobson topology; [F1] proves it is a homeomorphism, and [F2] records the closure identity on which that proof is based.

2.1step 1.1step 1.2

Steps 1.1 and 1.2 are exactly the assertions of the statement, so the kernel map is a continuous surjection and the induced map is a homeomorphism.

3.1given∎

The Axiom of Choice is inherited from the kernel-map homeomorphism theorem; the specialization to the present statement adds no further choice (The Axiom of Choice).

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

24 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)