Year: 2026
Type: preprint
Abstract: Let $B_\infty$ be the braid group on infinitely many strands and $P_\infty$ its pure braid group. We show that $C^*_r(P_\infty)\subseteq C^*_r(B_\infty)$ and all its intermediate inclusions are $\mathcal{Z}$-stable and completely selfless. This gives natural examples of simple nonnuclear $\mathcal{Z}$-stable $C^*$-algebras. Further examples come from $\mathrm{SL}_\infty(\mathbb{Z})$ and $\mathrm{GL}_\infty(\mathbb{Z})$. We also show that $C^*_r(\mathbb{F}_\infty)\subseteq C^*_r(B_\infty)$ is completely selfless for the free group $\mathbb{F}_\infty$ of braids in which only the first strand moves. All results hold for twisted group $C^*$-algebras. In contrast, $\mathcal{Z}$-stability fails for $B_n$ with $n$ finite and for groups that are not inner amenable.
Source: arXiv (Cornell University)
Author: Omland, Tron
Institutions: University of Oslo, National Security Authority
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7215108119