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Characters of a closed subgroup extend to the ambient LCA group

Statement

Assume the Axiom of Choice (The Axiom of Choice) and Dependent Choice (The axiom of dependent choice: a relation in which every element is related to something admits an $N$ -indexed chain). Let $G$ be a locally compact Hausdorff abelian group and let $H \leq G$ be a closed subgroup. Then the restriction homomorphism $R : \hat{G} \to \hat{H}, R (γ) : = γ ∣_{H},$ is surjective: every continuous character of $H$ extends to a continuous character of $G$ .

Facts & Assumptions

Given: A locally compact Hausdorff abelian group $G$ , a closed subgroup $H \leq G$ , and the restriction map $R : \hat{G} \to \hat{H}$ .

[F1]

$R (γ) = γ ∣_{H}$ is a continuous group homomorphism with kernel $H^{⊥} = {γ \in \hat{G} : γ (h) = 1 for all h \in H}$ , which is a closed subgroup of $\hat{G}$ . (The annihilator of a subgroup, Dual homomorphisms: continuity, and the annihilator of a closed subgroup, The Pontryagin dual with the compact-open topology)

[F2]

In a locally compact Hausdorff abelian group, $(L^{⊥})^{⊥} = \overline{L}$ for every subgroup $L$ of its dual, and $(H^{⊥})^{⊥} = H$ for a closed subgroup $H$ . (Annihilators reverse inclusions and the double annihilator closes the subgroup)

[F3]

For a closed subgroup $B$ of a locally compact Hausdorff abelian group $A$ , the quotient $A / B$ is locally compact Hausdorff abelian and the pullback of the quotient map is a topological group isomorphism of $(A / B) \hat{}$ onto $B^{⊥}$ . (The dual of a quotient is the annihilator, The quotient of an LCA group by a closed subgroup is LCA, The quotient group $G / N$ and coset product $(g N) (h N) = g h N$ , The quotient topology of a surjection, quotient maps, saturated sets, and the quotient of a space by an equivalence relation with its canonical projection)

[F4]

Continuous characters separate points: for $h \neq 0$ in $G$ there is $γ \in \hat{G}$ with $γ (h) \neq 1$ . (Continuous characters separate points of an LCA group)

[F5]

The closed subgroup $H$ is LCA by A locally compact subgroup of a Hausdorff topological group is closed, and the dual of any LCA group is LCA under AC by The dual of a locally compact abelian group is locally compact abelian. The evaluation maps $Φ_{A} : A \to \hat{\hat{A}}$ are isomorphisms of topological groups; pullback along a continuous homomorphism of abelian topological groups is a continuous homomorphism, composition of pullbacks reverses order, and the dual of a topological isomorphism is a topological isomorphism. (Pontryagin biduality: the evaluation map is a topological isomorphism, Dual homomorphisms: continuity, and the annihilator of a closed subgroup, Homeomorphism, open map, closed map, embedding, and what it means for a property to be topological)

[F6]

A subgroup which is locally compact in the subspace topology is closed in a Hausdorff topological group. (A locally compact subgroup of a Hausdorff topological group is closed, Topological group: multiplication and inversion are continuous, Subgroup)

Proof

1.1F1

$R$ is a continuous group homomorphism with kernel $H^{⊥}$ by [F1], so $H^{⊥}$ is a closed subgroup of $\hat{G}$ ; let $L : = R (\hat{G}) \leq \hat{H}$ be its image.

1.2F1F4

The annihilator of $L$ inside $H$ is trivial: $L^{⊥} = {h \in H : γ (h) = 1 for all γ \in \hat{G}} = {0}$ , because a nonzero $h$ is separated from $0$ by some character of $G$ by [F4].

2.1F2step 1.2

Applying the double-annihilator identity [F2] in the locally compact Hausdorff abelian group $\hat{H}$ gives $\overline{L} = (L^{⊥})^{⊥} = {0}^{⊥} = \hat{H}$ ; that is, $L$ is dense in $\hat{H}$ .

2.2F1F3F5step 1.1

Because $\ker R = H^{⊥}$ , the map $R$ factors as $R = ψ \circ q$ with $q : \hat{G} \to \hat{G} / H^{⊥}$ the quotient homomorphism and $ψ : \hat{G} / H^{⊥} \to \hat{H}$ the injective continuous homomorphism $ψ (γ H^{⊥}) = γ ∣_{H}$ ; the group $\hat{G} / H^{⊥}$ is locally compact Hausdorff abelian by [F3]. Thus [F5] applies to this quotient.

3.1F2F3step 2.2

The quotient-dual theorem [F3], applied to the group $\hat{G}$ and its closed subgroup $H^{⊥}$ , gives a topological isomorphism $Ξ : (\hat{G} / H^{⊥}) \hat{} \to (H^{⊥})^{⊥}$ , $Ξ (ξ) = ξ \circ q$ , onto the annihilator of $H^{⊥}$ inside $\hat{\hat{G}}$ ; by the double-annihilator identity [F2] in the group $G$ and closedness of $H$ , this annihilator is $Φ_{G} (H)$ , the image of $H$ under the biduality identification. Composing $Ξ$ with $Φ_{G}^{- 1}$ therefore identifies $(\hat{G} / H^{⊥}) \hat{}$ topologically with $H$ itself.

4.1F5step 3.1

The transpose $\hat{ψ} : \hat{\hat{H}} \to (\hat{G} / H^{⊥}) \hat{}$ is a topological isomorphism. Indeed for $h \in H$ and $γ \in \hat{G}$ one computes $\hat{ψ} (Φ_{H} (h)) (γ H^{⊥}) = Φ_{H} (h) (ψ (γ H^{⊥})) = γ (h) = Φ_{G} (h) (γ)$ , so $\hat{ψ} (Φ_{H} (h)) = Ξ^{- 1} (Φ_{G} (h))$ for every $h$ , that is $\hat{ψ} = Ξ^{- 1} \circ Φ_{G} ∣_{H} \circ Φ_{H}^{- 1}$ ; here $Φ_{H} : H \to \hat{\hat{H}}$ , $Φ_{G} ∣_{H} : H \to Φ_{G} (H)$ and $Ξ^{- 1} : Φ_{G} (H) \to (\hat{G} / H^{⊥}) \hat{}$ are topological isomorphisms by [F5] and step 3.1.

5.1F5F6step 4.1

Since $\hat{ψ}$ is a topological isomorphism, so is its dual $\hat{\hat{ψ}}$ , and naturality of evaluation $Φ_{\hat{H}} \circ ψ = \hat{\hat{ψ}} \circ Φ_{\hat{G} / H^{⊥}}$ (a direct computation from $Φ_{A} (a) (λ) = λ (a)$ ) exhibits $ψ$ as the composite $Φ_{\hat{H}}^{- 1} \circ \hat{\hat{ψ}} \circ Φ_{\hat{G} / H^{⊥}}$ of topological isomorphisms; hence $ψ$ is a homeomorphism onto its image $L$ . Therefore $L$ is locally compact in the subspace topology and, being a subgroup of the Hausdorff group $\hat{H}$ , is closed in $\hat{H}$ by [F6].

6.1step 2.1step 5.1∎

The image $L$ is dense in $\hat{H}$ by step 2.1 and closed by step 5.1, so $L = \hat{H}$ : the restriction map $R$ is surjective, that is, every continuous character of $H$ extends to a continuous character of $G$ .

Depends on

Used by

Dependency tree · two levels

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Sources

T. W. Koerner, Topological Groups (author lecture notes) (standard reference, not scraped)