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

Abstract

Let $B_{\infty }$ and $P_{\infty }$ be the braid group and the pure braid group on infinitely many strands. We show that the inclusion $C_r^{\ast }(P_{\infty })\subseteq C_r^{\ast }(B_{\infty })$ is $\mathcal{Z}$-stable and selfless. Hence every intermediate inclusion and every intermediate $C^{\ast }$-algebra is $\mathcal{Z}$-stable and selfless. The intermediate $C^{\ast }$-algebras are natural examples of simple nonnuclear $\mathcal{Z}$-stable $C^{\ast }$-algebras, and they are singly generated. The proof of $\mathcal{Z}$-stability uses a simple relative centralizer criterion. We also show that $C_r^{\ast }({\mathbb{F}}_{\infty })\subseteq C_r^{\ast }(B_{\infty })$ is a selfless inclusion, where ${\mathbb{F}}_{\infty }$ is the free group of braids in which only the first string moves. All results hold for reduced twisted group $C^{\ast }$-algebras. Selflessness of $C_r^{\ast }(B_{\infty })$ itself already follows from work of Ozawa. On the other hand, $\mathcal{Z}$-stability fails for all twisted group $C^{\ast }$-algebras of groups that are not inner amenable. A variant shows that it fails for all twisted group $C^{\ast }$-algebras of $B_n$ with $n$ finite, and for the twisted crossed products from Artin's representation of $B_n$. Finally the method applies to other increasing unions of groups, such as ${\mathrm{SL}}_{\infty }(\mathbb{Z})$, whose reduced group $C^{\ast }$-algebra is $\mathcal{Z}$-stable.

arXiv comment11 pages