Strict comparison in reduced group C∗-algebras

We prove that for every n∈N such that n≥2, the reduced group C∗-algebras of the countable free groups Cr∗(Fn) have strict comparison. Our method works in a general setting: for every finitely generated acylindrically hyperbolic group G with trivial finite radical and the rapid decay property, we have Cr∗(G) have strict comparison. This work also has several applications in the theory of C∗-algebras including: resolving Leonel Robert’s selflessness problem for Cr∗(G); uniqueness of embeddings of the Jiang-Su algebra Z up to approximate unitary equivalence into Cr∗(G); full computations of the Cuntz semigroup of Cr∗(G) and future directions in the C∗-classification program.