On operator-valued spherical functions

We consider the equation integral(K) Phi(x + k (.) y) dk = Phi(x)Phi(y), X, y is an element of 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 Phi : G -> S(H) any bounded solution of (0.1) with Phi(0) = I: (1) Assume G satisfies the second axiom of countability. If Phi is weakly continuous and takes its values in the normal operators, then Phi(x) = integral(K) U (k (.) x) dk, x is an element of 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 is an element of K\{I}, there exists an equivalent inner product on H such that Phi(x) for each x is an element of G is a normal operator with respect to it. Conditions (1) and (2) are partial generalizations of results by Chojnacki on the cosine equation. (c) 2005 Elsevier Inc. All rights reserved.