Annihilator duality

Remarks

Proof idea: a continuous character of $G/H$ composed with the quotient map is a character of $G$ trivial on $H$, and every such character arises this way, which gives 2. The inclusion $H \subseteq (H^\perp)^\perp$ is clear. For the reverse, if $x \notin H$, then $xH$ is not the identity of the locally compact abelian group $G/H$, and the characters of $G/H$ separate points, so some $\chi \in H^\perp$ has $\chi(x) \ne 1$. Statement 3 follows from 2 applied to the subgroup $H^\perp$ of $\widehat{G}$, together with duality.

Example: let $G = \R$ with characters $\chi_\xi(x) = e^{2\pi i \xi x}$, and let $H = \Z$. Then $\chi_\xi$ is trivial on $\Z$ exactly when $\xi \in \Z$, so $H^\perp \cong \Z$, which is the dual of the circle $\R/\Z$. Also $\widehat{\R}/H^\perp \cong \R/\Z$, which is the dual of $\Z$.

The subgroup must be closed. For the dense subgroup $\Q$ of $\R$, a continuous character trivial on $\Q$ is trivial on $\R$, so $\Q^\perp$ is trivial and $(\Q^\perp)^\perp = \R \ne \Q$.

Going deeper

For a student

Annihilator duality is the group form of the duality between subspaces and their annihilators in linear algebra. It is used to compute duals of quotients and subgroups, for example $\widehat{\R/\Z} = \Z$ from $\Z^\perp$ in $\widehat{\R}$. It is the setting of the Poisson summation formula, where a sum over $H$ equals a sum over $H^\perp$. It also shows that a closed subgroup $H$ is determined by the characters that are trivial on it. This makes it a tool in the structure theory of locally compact abelian groups.