On the primitive ideal space of radial representations of free groups
Shigeru Yamagami
TL;DR
The paper analyzes radial representations of a finitely generated free group $G$ by building the radial C*-algebra $C^*_{rad}(G)$ through a universal radial bimodule, situating it between the full and reduced group C*-algebras. It characterizes the primitive ideal space Prim$(C^*_{rad}(G))$ as the set of kernels of spherical representations, specifically $\\\ker \\\pi_t$ for $t \\ rightarrow \\sigma_r$ and the kernels $\\ker\\varepsilon_{\pm1}$, with $\\ker \\\pi$ matching the kernel of the regular representation. The hull–kernel topology is made explicit by identifying $\\Delta$ with a quotient of $[-1,1]$ that collapses the regular spectrum interval $\\\sigma_r$ to a single point, and the analysis shows that $C^*_{rad}(G)$ is CCR and that its center is trivial. By leveraging the Haagerup framework, the spectral analysis of radial functions, and the Pytlik–Szwarc deformation $(\\lambda_z)$, the work provides a concrete, topologically described primitive ideal structure for radial representations, linking spherical and principal/complementary series in a precise C*-algebraic setting.
Abstract
Radial representations of finitely generated free groups are studied. The associated C*-algebra is located between the reduced and full group C*-algebras and its primitive ideal space is described concretely as a topological space.