The annihilator of a subgroup

Let $G$ be an abelian topological group (Topological group: multiplication and inversion are continuous) written additively, with Pontryagin dual $\widehat{G}$ carrying the compact-open topology and pointwise multiplication (The Pontryagin dual with the compact-open topology), and let $H\le G$ be a subgroup (Subgroup).

The annihilator of $H$ is the set of characters trivial on $H$: $H^{\perp }:=\{\gamma \in \widehat{G}:\gamma (h)=1\text{ for every }h\in H\}\mathrm{.}$ It is a subgroup of $\widehat{G}$: it contains the identity character $x\mapsto 1$; if ${\gamma }_1(h)={\gamma }_2(h)=1$ for all $h\in H$ then $({\gamma }_1{\gamma }_2)(h)=1$ and ${\gamma }_1^{-1}(h)=1$ for all $h\in H$, because multiplication and inversion in $\widehat{G}$ are pointwise (The compact-open character group is a Hausdorff topological abelian group). Moreover $H^{\perp }$ is exactly the kernel of the restriction homomorphism $\widehat{G}\to \widehat{H}$, $\gamma \mapsto \gamma {\mathrm{\mid }}_H$, which is a continuous group homomorphism by the functoriality of the dual under pullback along the inclusion $H\hookrightarrow G$ (Dual homomorphisms: continuity, and the annihilator of a closed subgroup, Monoid homomorphism and group homomorphism). Consequently $H^{\perp }$ is a closed subgroup of $\widehat{G}$: it is the kernel of a continuous homomorphism between Hausdorff topological groups, and $\widehat{G}$ is Hausdorff (The compact-open character group is a Hausdorff topological abelian group). In particular the closedness of $H^{\perp }$ holds whenever $H$ is closed in $G$, and no closedness of $H$ is needed for it.

Annihilators in the dual and in the bidual. Let $L\le \widehat{G}$ be a subgroup of the dual. Its annihilator is $L^{\perp }:=\{x\in G:\lambda (x)=1\text{ for all }\lambda \in L\}\le G\mathrm{.}$ The definition uses only the evaluation pairing and makes no isomorphism claim. Two conventions are recorded and used throughout this page.

1. $(H{)}^{\perp }=(\bar{H}{)}^{\perp }$ for every subgroup $H\le G$. Indeed a character $\gamma$ is continuous, so it is trivial on $H$ if and only if it is trivial on the closure of $H$; equivalently, $\gamma$ is trivial on $H$ exactly when its kernel, a closed subgroup, contains $\bar{H}$. The convention lets every annihilator be computed with closed subgroups.
2. For a closed subgroup $H\le G$ the subgroup $H^{\perp }\le \widehat{G}$ is closed by the kernel argument above. If $G$ is locally compact Hausdorff abelian and the Axiom of Choice is assumed (The Axiom of Choice), then $\widehat{G}$ is LCA by The dual of a locally compact abelian group is locally compact abelian, and its closed subgroup $H^{\perp }$ is LCA by A locally compact subgroup of a Hausdorff topological group is closed. Its own annihilator in the bidual is written $H^{\perp \perp }\le \widehat{\widehat{G}}$ and is identified with a subgroup of $G$ through the evaluation map $\mathrm{\Phi }$ of Pontryagin biduality: the evaluation map is a topological isomorphism when that identification is available.

Forming the annihilator and proving its subgroup and closedness properties use no choice principle. The additional local-compactness assertion in convention 2 assumes AC as stated.