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

Let be a compact abelian group having the property that its character group is totally ordered by a semigroup . We prove that every operator-valued function on of the form , such that the Hankel operator is bounded, has an essentially bounded extension with . The proof is based on Arveson's Extension Theorem for completely positive functions on -algebras. Among the corollaries we have a Carathéodory-Fejér type result for analytic operator-valued functions defined on such groups.