Extension problems and non-abelian duality for $C^{\ast }$-algebras

Suppose that $H$ is a closed subgroup of a locally compact group $G$. We show that a unitary representation $U$ of $H$ is the restriction of a unitary representation of $G$ if and only if a dual representation $\widehat{U}$ of a crossed product $C^{\ast }(G)⋊(G\mathrm{/}H)$ is regular in an appropriate sense. We then discuss the problem of deciding whether a given representation is regular; we believe that this problem will prove to be an interesting test question in non-abelian duality for crossed products of $C^{\ast }$-algebras.