Strict comparison in reduced group $C^{\ast }$-algebras

Abstract

We prove that for every $n\ge 2$, the reduced group $C^{\ast }$-algebras of the countable free groups $C_r^{\ast }({\mathbb{F}}_n)$ 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 $C_r^{\ast }(G)$ have strict comparison. This work also has several applications in the theory of $C^{\ast }$-algebras including: resolving Leonel Robert's selflessness problem for $C_r^{\ast }(G)$; uniqueness of embeddings of the Jiang-Su algebra $\mathcal{Z}$ up to approximate unitary equivalence into $C_r^{\ast }(G)$; full computations of the Cuntz semigroup of $C_r^{\ast }(G)$ and future directions in the $C^{\ast }$-classification program.

Final version, to appear in Invent. Math. Dedicated to the memory of Dr. S. Balachander