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

Abstract We prove that for every $n\in \mathbb{N}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>n</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>N</mml:mi> </mml:math> such that $n\geq 2$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>n</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>2</mml:mn> </mml:math> , the reduced group $C^{*}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>C</mml:mi> <mml:mo>∗</mml:mo> </mml:msup> </mml:math> -algebras of the countable free groups $C^{*}_{r}(\mathbb{F}_{n})$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>C</mml:mi> <mml:mi>r</mml:mi> <mml:mo>∗</mml:mo> </mml:msubsup> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>F</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo>)</mml:mo> </mml:math> have strict comparison. Our method works in a general setting: for every finitely generated acylindrically hyperbolic group $G$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>G</mml:mi> </mml:math> with trivial finite radical and the rapid decay property, we have $C^{*}_{r}(G)$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>C</mml:mi> <mml:mi>r</mml:mi> <mml:mo>∗</mml:mo> </mml:msubsup> <mml:mo>(</mml:mo> <mml:mi>G</mml:mi> <mml:mo>)</mml:mo> </mml:math> have strict comparison. This work also has several applications in the theory of $C^{*}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>C</mml:mi> <mml:mo>∗</mml:mo> </mml:msup> </mml:math> -algebras including: resolving Leonel Robert’s selflessness problem for $C^{*}_{r}(G)$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>C</mml:mi> <mml:mi>r</mml:mi> <mml:mo>∗</mml:mo> </mml:msubsup> <mml:mo>(</mml:mo> <mml:mi>G</mml:mi> <mml:mo>)</mml:mo> </mml:math> ; uniqueness of embeddings of the Jiang-Su algebra $\mathcal{Z}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Z</mml:mi> </mml:math> up to approximate unitary equivalence into $C^{*}_{r}(G)$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>C</mml:mi> <mml:mi>r</mml:mi> <mml:mo>∗</mml:mo> </mml:msubsup> <mml:mo>(</mml:mo> <mml:mi>G</mml:mi> <mml:mo>)</mml:mo> </mml:math> ; full computations of the Cuntz semigroup of $C^{*}_{r}(G)$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>C</mml:mi> <mml:mi>r</mml:mi> <mml:mo>∗</mml:mo> </mml:msubsup> <mml:mo>(</mml:mo> <mml:mi>G</mml:mi> <mml:mo>)</mml:mo> </mml:math> and future directions in the $C^{*}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>C</mml:mi> <mml:mo>∗</mml:mo> </mml:msup> </mml:math> -classification program.