Invariant subalgebras of the reduced group $C^*$-algebra

TL;DR

The paper introduces the C$^*$-ISR property for countable groups and demonstrates it for broad classes, notably torsion-free, non-amenable hyperbolic groups with property-AP and finite products thereof. It develops an averaging framework, leveraging selective and finitely supported averaging, to show that every $oldsymbol{\Gamma}$-invariant unital subalgebra of $C_r^*(oldsymbol{\Gamma})$ must arise as $C_r^*(N)$ for a normal subgroup $N riangleleft oldsymbol{\Gamma}$. The authors extend these results to torsion-free acylindrically hyperbolic groups and to finite products, establishing ISR for those as well, and derive a strong dichotomy: an infinite group with C$^*$-ISR is either $C^*$-simple or simple amenable. They further explore the relationship between ISR and C$^*$-ISR, showing the two notions can diverge, as illustrated by examples, and discuss implications for the ideal structure in $C_r^*(oldsymbol{\Gamma})$ in the amenable case.

Abstract

Let $Γ$ be a countable discrete group. We say that $Γ$ has $C^*$-invariant subalgebra rigidity (ISR) property if every $Γ$-invariant $C^*$-subalgebra $\mathcal{A}\le C_r^*(Γ)$ is of the form $C_r^*(N)$ for some normal subgroup $N\triangleleftΓ$. We show that all torsion-free, non-amenable (cylindrically) hyperbolic groups with property-AP and a finite direct product of such groups have this property. We also prove that an infinite group $Γ$ has the C$^*$-ISR property only if $Γ$ is simple amenable or $C^*$-simple.

Theorems & Definitions (50)

• Definition 1.1: $C^*$-ISR property
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Proposition 2.1
• proof
• Proposition 2.2
• proof
• Lemma 2.3