Let $G$ be an abelian locally compact group and $H$ be its closed subgroup. It is known from Pontryagin duality theory that every unitary character of $H$ can be extended to $G$. I think this is true for any character. Am I right? What is the reference?
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If you mean that every continuous homomorphism $H\to\mathbf{R}$ can be extended to a continuous homomorphism $G\to\mathbf{R}$, this is contained in Theorème 5 of [1] (the "unitary" version is when $\mathbf{R}$ is replaced twice by $\mathbf{R}/\mathbf{Z}$).
[1] J. Dixmier. Quelques propriétés des groupes abéliens localement compacts. Bull. Sci. Math. (2) 81 (1957) 38-48.
