Jump transformations and an embedding of ${\mathcal{O}}_{\infty }$ into ${\mathcal{O}}_2$

Abstract

A measurable map $T$ on a measure space induces a representation ${\Pi }_T$ of a Cuntz algebra ${\mathcal{O}}_N$ when $T$ satisfies a certain condition. For such two maps $\tau$ and $\sigma$ and representations ${\Pi }_{\tau }$ and ${\Pi }_{\sigma }$ associated with them, we show that ${\Pi }_{\tau }$ is the restriction of ${\Pi }_{\sigma }$ when $\tau$ is a jump transformation of $\sigma$. Especially, the Gauss map ${\tau }_1$ and the Farey map ${\sigma }_1$ induce representations ${\Pi }_{{\tau }_1}$ of ${\mathcal{O}}_{\infty }$ and that ${\Pi }_{{\sigma }_1}$ of ${\mathcal{O}}_2$, respectively, and ${\Pi }_{{\tau }_1}={\Pi }_{{\sigma }_1}{|}_{{\mathcal{O}}_{\infty }}$ with respect to a certain embedding of ${\mathcal{O}}_{\infty }$ into ${\mathcal{O}}_2$.

15 pages 6 figures

References in corpus (3)

• Classification and realizations of type III factor representations of Cuntz-Krieger algebras associated with quasi-free states
• Unitary isomorphism of Fock spaces of bosons and fermions arising from a representation of the Cuntz algebra $\co{2}$
• Large deviation asymptotics for continued fraction expansions

Cited by in corpus (1)

• Universal fermionization of bosons on permutative representations of the Cuntz algebra ${\mathcal{O}}_2$