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 $(\operatorname{SL}_2(\Qp), \operatorname{SL}_2(\Zp))$ 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 $(\operatorname{SL}_n(\Qp), \operatorname{SL}_n(\Zp))$ for $n\geq 3$ we show, invoking property (T) of $\operatorname{SL}_n(\Qp)$, that the $C^*$-completion of the $L^1$-Banach algebra and the corner of $C^*(\operatorname{SL}_n(\Qp))$ 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 $\mathfrak{p}$-adic field with a good choice of a maximal compact open subgroup.