Endomorphisms of the Cuntz Algebras and the Thompson Groups

TL;DR

This paper explores the connection between endomorphisms of the Cuntz algebra ${ m O}_2$ and the Thompson groups $F$, $T$, and $V$, revealing conditions under which these endomorphisms preserve the groups within the algebra.

Contribution

It introduces the concept of modestly scaling endomorphisms of ${ m O}_n$ and characterizes when these endomorphisms map Thompson groups into each other.

Findings

$ ext{Endomorphism } ext{of } { m O}_2 ext{ preserves } V$ iff $u ext{ is in } V$

Automorphisms of ${ m O}_2$ map $F$ into $V$ iff $u$ is in $V$

Introduction of modestly scaling endomorphisms with their properties and examples

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.