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arXiv · 2609.38596

$\mathcal{Z}$-stable and selfless inclusions for the infinite braid group

Tron Omland
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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.

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Tron Omland. 2026-10-26. $\mathcal{Z}$-stable and selfless inclusions for the infinite braid group. https://arxiv.org/abs/2609.38596

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