Endomorphisms of the Cuntz Algebras and the Thompson Groups
Sel\c{c}uk Barlak, Jeong Hee Hong, Wojciech Szymanski

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 and endomorphisms of the Thompson groups , and represented inside the unitary group of . For an endomorphism of , we show that if and only if . If is an automorphism of then is equivalent to . Our investigations are facilitated by introduction of the concept of modestly scaling endomorphism of , whose properties and examples are investigated.
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Taxonomy
TopicsAdvanced Operator Algebra Research · Advanced Topics in Algebra · Holomorphic and Operator Theory
