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arXiv · 2412.06031

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

Tattwamasi Amrutam·David Gao·Srivatsav Kunnawalkam Elayavalli·Gregory Patchell
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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.

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BibTeXRIS

Tattwamasi Amrutam, David Gao, Srivatsav Kunnawalkam Elayavalli, Gregory Patchell. 2024-12-08. Strict comparison in reduced group $C^*$-algebras. https://arxiv.org/abs/2412.06031

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