Generalised Hecke algebras and C^*-completions

Abstract

For a Hecke pair $(G,H)$ and a finite-dimensional representation $\sigma$ of $H$ on $V_{\sigma }$ with finite range we consider a generalised Hecke algebra $\H_σ(G, H)$, which we study by embedding the given Hecke pair in a Schlichting completion $(G_{\sigma },H_{\sigma })$ that comes equipped with a continuous extension $\sigma$ of $H_{\sigma }$. There is a (non-full) projection $p_σ\in C_c(G_σ, {\cc B}(V_σ))$ such that $\H_σ(G, H)$ is isomorphic to $p_σC_c(G_σ, {\cc B}(V_σ))p_σ$. We study the structure and properties of $C^{\ast }$-completions of the generalised Hecke algebra arising from this corner realisation, and via Morita-Fell-Rieffel equivalence we identify, in some cases explicitly, the resulting proper ideals of $C^*(G_σ, {\cc B}(V_σ))$. By letting $\sigma$ vary, we can compare these ideals. The main focus is on the case with $\dim \sigma =1$ and applications include $ax+b$-groups and the Heisenberg group.

arXiv comment25 pages. Revised version

References in corpus (1)

• Hecke C*-algebras, Schlichting completions, and Morita equivalence