On operator-valued spherical functions

Abstract

We consider the equation (1) ∫ K Φ ( x + k · y ) dk = Φ ( x ) Φ ( y ) , x , y ∈ G , 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 Φ : G → B ( H ) any bounded solution of (0.1) with Φ ( 0 ) = I : (1) Assume G satisfies the second axiom of countability. If Φ is weakly continuous and takes its values in the normal operators, then Φ ( 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 Φ ( 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.