Jump transformations and an embedding of ${\cal O}_{\infty}$ into ${\cal O}_{2}$

Abstract

A measurable map $T$ on a measure space induces a representation $Π_{T}$ of a Cuntz algebra ${\cal O}_{N}$ when $T$ satisfies a certain condition. For such two maps $τ$ and $σ$ and representations $Π_τ$ and $Π_σ$ associated with them, we show that $Π_τ$ is the restriction of $Π_σ$ when $τ$ is a jump transformation of $σ$. Especially, the Gauss map $τ_1$ and the Farey map $σ_1$ induce representations $Π_{τ_1}$ of ${\cal O}_{\infty}$ and that $Π_{σ_1}$ of ${\cal O}_{2}$, respectively, and $Π_{τ_1}=Π_{σ_1}|_{{\cal O}_{\infty}}$ with respect to a certain embedding of ${\cal O}_{\infty}$ into ${\cal O}_{2}$.

15 pages 6 figures