Endomorphisms of the Cuntz Algebras and the Thompson Groups

Abstract

We investigate the relationship between endomorphisms of the Cuntz algebra ${\mathcal{O}}_2$ and endomorphisms of the Thompson groups $F$, $T$ and $V$ represented inside the unitary group of ${\mathcal{O}}_2$. For an endomorphism ${\lambda }_u$ of ${\mathcal{O}}_2$, we show that ${\lambda }_u(V)\subseteq V$ if and only if $u\in V$. If ${\lambda }_u$ is an automorphism of ${\mathcal{O}}_2$ then $u\in V$ is equivalent to ${\lambda }_u(F)\subseteq V$. Our investigations are facilitated by introduction of the concept of modestly scaling endomorphism of ${\mathcal{O}}_n$, whose properties and examples are investigated.

arXiv commentv1: 10 pages. v2: minor changes, updated reference list, 11 pages; to appear in Studia Mathematica