Exercise 27, § 4

We revert to the notation of the preceding exercise. Suppose that $(G,K)$ is a Gelfand pair.

One says that $\varphi \in \mathscr{H}(G,K)$ is a K-spherical function if the mapping

is a nonzero character of the algebra $\mathscr{H}(G,K)$.

a) Let $\varphi \in \mathscr{H}(G,K)$. Prove that the following conditions are equivalent :

(i) The function $\varphi$ is a K-spherical function ;

(ii) For all $x$ and $y$ in G, one has

$\int_K\varphi (xky)d\mu(k) =\varphi (x)\varphi (y)$ ;

(iii) One has $\varphi (e) = 1$ and for every function $\psi \in \mathscr{H}(G,K)$, there exists $c\in \mathbf{C}$ such that $\psi *\varphi =c\varphi$.

b) Let $\mathscr{H}^1(G,K)$ denote the space of the $f\in L^1(G)$ such that $\boldsymbol{\varrho }_G(k_1, k_2)f=f$ for every $(k_1, k_2)\in K\times K$. It is a Banach subalgebra of the involutive Banach algebra $L^1(G)$; it is commutative.

c) The algebra $\mathscr{H}(G,K)$ is identified with a dense subalgebra of $\mathscr{H}^1(G,K)$. d) The mapping which to $\varphi \in \mathscr{H}(G,K)$ associates the linear form

on $\mathscr{H}^1(G,K)$ defines a bijection of the space of bounded K-spherical functions on G onto the space of nonzero characters of $\mathscr{H}^1(G,K)$. (Note that every character of $\mathscr{H}^1(G,K)$ is continuous and of norm $\leqslant 1$, cf. prop. 2 of I, p. 104.)