Examples of essentially non-Banach representations (Q1180788)

Let \(G\) be a locally compact unimodular group and \(K\) a compact subgroup of \(G\). Let \(T\) be a topological irreducible representation of \(G\) on a locally convex complete Hausdorff topological vector space \(\mathcal H\). If there exists an equivalence class \(\delta\) of an irreducible representation of \(K\) which is contained finitely may times in \(T\), then the spherical function \(\Phi_ \delta(x)\), \(x\in G\), is defined according to \textit{R. Godement} [Trans. Am. Math. Soc. 73, 496--556 (1952; Zbl 0049.20103)]. A function \(\rho(x)\), \(x\in G\), is called a seminorm if it is positive- valued, lower semicontinuous and satisfies \(\rho(xy)\leq \rho(x)\rho(y)\), \(x,y\in G\). \(\Phi_ \delta(x)\) is called quasi-bounded if there exists a seminorm \(\rho(x)\) such that \(| \Phi_ \delta(x)|\leq \rho(x)\) for \(x\in G\). When \(\mathcal H\) is a Banach space then \(\Phi_ \delta(x)\) is quasi-bounded. A topological irreducible representation which defines non-quasi-bounded spherical functions is called an essentially non-Banach representation. Examples of essentially non-Banach representations are given, where \(G\) is a semidirect product of a free group with infinitely many generators and a compact abelian group.