On uniformly bounded spherical functions in Hilbert space

Abstract

Let G be a commutative group, written additively, with a neutral element 0, and let K be a finite group. Suppose that K acts on G via group automorphisms G ∋ a {rightwards two-headed} ka ∈ G, k ∈ K. Let H be a complex Hilbert space and let L(H) be the algebra of all bounded linear operators on H. A mapping u: G → L(H) is termed a K-spherical function if it satisfies (1) {pipe}K{pipe}-1 ∑k∈K u(a + kb) = u(a)u(b) for any a, b ∈ G, where {pipe}K{pipe} denotes the cardinality of K, and (2) u(0) = id(H), where id(H) designates the identity operator on H. The main result of the paper is that for each K-spherical function u: G → L(H) such that {double pipe}u{double pipe}∞ = supa∈G {double pipe}u(a){double pipe}L(H) < ∞, there is an invertible operator S in L(H) with {double pipe}S{double pipe} {double pipe}S-1{double pipe} ≤ {pipe}K{pipe} {double pipe}u{double pipe}2∞ such that the K-spherical function ũ: G → L(H) defined by ũ(a) = Su(a)S-1, a ∈ G, satisfies ũ(-a) = ũ(a)* for each a ∈ G. It is shown that this last condition is equivalent to insisting that ũ(a) be normal for each a ∈ G. © 2010 Springer Basel AG.