On operator-valued spherical functions

We consider the equation K (x + k · y) dk = (x) (y), x, y ∈ G, (1) in which a compact group K with normalized Haar measure dk acts on a locally compact abelian group (G,+). Let H be a Hilbert space, B(H) the bounded operators on H. Let : G → B(H) any bounded solution of (0.1) with (0) = I : (1) Assume G satisfies the second axiom of countability. If is weakly continuous and takes its values in the normal operators, then (x) = K U(k · x) dk, x ∈ G, where U is a strongly continuous unitary representation of G on H. (2) Assuming G discrete, K finite and the map x → x −k ·x of G into G surjective for each k ∈ K\{I }, there exists an equivalent inner product on H, such that (x) for each x ∈ G is a normal operator with respect to it. Conditions (1) and (2) are partial generalizations of results by Chojnacki on the cosine equation. © 2005 Elsevier Inc. All rights reserved.