Nehari and Carathéodory-Fejér type extension results for operator-valued functions on groups

Let $G$ be a compact abelian group having the property that its character group $\Gamma$ is totally ordered by a semigroup $P$. We prove that every operator-valued function $k$ on $G$ of the form $k(x)=\sum\limits_{\gamma \in (-P)}\gamma (x)k_{\gamma }$, such that the Hankel operator $H_k$ is bounded, has an essentially bounded extension $K$ with $\vert\vert K\vert\vert _{\infty }=\vert\vert H_k\vert\vert$. The proof is based on Arveson's Extension Theorem for completely positive functions on $C^*$-algebras. Among the corollaries we have a Carathéodory-Fejér type result for analytic operator-valued functions defined on such groups.