Strict comparison in reduced group $C^*$-algebras

TL;DR

This paper proves that reduced group C*-algebras of certain non-amenable groups, including free groups and hyperbolic groups, have strict comparison, advancing the classification theory of these algebras.

Contribution

It establishes strict comparison for a broad class of non-amenable groups' reduced C*-algebras, extending previous results and enabling new classification applications.

Findings

Reduced C*-algebras of free groups have strict comparison.

Applicable to hyperbolic groups, free products, and mapping class groups.

Enables classification results and solves open problems in C*-algebra theory.

Abstract

We prove that for every $n\geq 2$, the reduced group $C^*$-algebras of the countable free groups $C^*_r(\mathbb{F}_n)$ have strict comparison. Our method works in a general setting: for $G$ in a large family of non-amenable groups, including hyperbolic groups, free products, mapping class groups, right-angled Artin groups etc., we have $C^*_r(G)$ have strict comparison. This work also has several applications in the theory of $C^*$-algebras including: resolving Leonel Robert's selflessness problem for $C^*_r(G)$; uniqueness of embeddings of the Jiang-Su algebra $\mathcal{Z}$ up to approximate unitary equivalence into $C^*_r(G)$; full computations of the Cuntz semigroup of $C^*_r(G)$ and future directions in the $C^*$-classification program.