On uniformly bounded spherical functions in Hilbert space

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 (sic) a bar right arrow ka is an element of G, k is an element of 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) vertical bar K vertical bar(-1) Sigma(k is an element of K) u(a +kb) - u(a)u(b) for any a,b is an element of G, where vertical bar K vertical bar 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 parallel to u parallel to(infinity) = sup(a is an element of G) parallel to u(a)parallel to(L(h)) < infinity, there is an invertible operator S in L(h) with parallel to S parallel to parallel to S(-1)parallel to <= vertical bar K vertical bar parallel to u parallel to(2)(infinity) such that the K-spherical function (u) over bar; G -> L(h) defined by (u) over bar (a) = Su(a)S(-1), a is an element of G, satisfies (u) over bar(-a) = (u) over bar (a)* for each a is an element of G. It is shown that this last condition is equivalent to insisting that (u) over bar (a) be normal for each a is an element of G.