Positive definite *-spherical functions, property (T) and C*-completions of Gelfand pairs

Abstract

The study of existence of a universal C * -completion of the * -algebra canonically associated to a Hecke pair was initiated by Hall, who proved that the Hecke algebra associated to (SL2(Qp), SL2(Zp)) does not admit a universal C * -completion. Kaliszewski, Landstad and Quigg studied the problem by placing it in the framework of Fell-Rieffel equivalence, and highlighted the role of other C * -completions. In the case of the pair (SLn(Qp), SLn(Zp)) for n ≥ 3 we show, invoking property (T) of SLn(Qp), that the C * -completion of the L 1 -Banach algebra and the corner of C * (SLn(Qp)) determined by the subgroup are distinct. In fact, we prove a more general result valid for a simple algebraic group of rank at least 2 over a p-adic field with a good choice of a maximal compact open subgroup.

$L^2$-Betti numbers of $C^*$-tensor categories associated with totally disconnected groups

“…In this context, it commonly happens that analytic properties of Λ < Γ (and by extension P Λ ⊂ P Γ) correspond to analytic properties of the Schlichting completion G of Λ < Γ, see e.g. [AD12;LP14;Pop01]. Since the Schlichting completion G of Λ < Γ is unimodular whenever the inclusion P Λ ⊂ P Γ is, one can ask the same question about the L 2 -Betti numbers β…”

Length functions and property (RD) for locally compact Hecke pairs

“…We started the study of property (RD) (Rapid Decay) for Hecke pairs in [25], see also [26]. More recently, amenability, weak amenability, Haagerup property (a-T-menability) and property (T) of Hecke pairs have also been studied in [1,20,27]. In all these works, Hecke pairs have been considered as a generalization of quotient groups, and so capable of many notions and constructions that had been originally invented for groups.…”