Selfless reduced $C^{*}$-algebras of linear groups

TL;DR

This work proves that for any nontrivial linear group $\Gamma\le\mathrm{GL}_d(k)$ with trivial amenable radical, the reduced group C*-algebra $C_r^*(\Gamma)$ is selfless, strengthening the known equivalence with simplicity and unique trace. The authors build on Ozawa's $\mathrm{P}_{\mathrm{PHP}}$ criterion by establishing $\mathrm{P}_{\mathrm{PHP}}$ for $S$-Zariski dense subgroups acting on $S$-adic flag spaces, using a detailed geometric theory of flag varieties and proximal dynamics. They reduce to finitely generated subgroups, prove PHP for each, and then pass to directed unions to obtain selflessness for the whole group; they also extend the result to twisted reduced C*-algebras $C_r^*(\Gamma,\omega)$. Consequences include strict comparison and stable rank one, with relevance to K-theory and Baum–Connes-type phenomena for linear groups, and applicability to broad classes such as fundamental groups of locally symmetric manifolds and other linear groups. The methods unify geometric, dynamical, and operator-algebraic techniques to derive regularity properties of reduced group C*-algebras in a broad linear-group setting.

Abstract

It is shown that the reduced $C^{*}$-algebra of a nontrivial linear group $Γ\leq\mathrm{GL}_{d}(k)$ with trivial amenable radical is selfless. Similar results are obtained for twisted reduced $C^{*}$-algebra.

Theorems & Definitions (35)

• Definition 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Lemma 2.1: Zariski density of $k$-points
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• proof