Continued fraction expansions and permutative representations of the Cuntz algebra ${\cal O}_{\infty}$

TL;DR

This paper establishes a link between continued fraction expansions of irrationals and permutative representations of the Cuntz algebra ${ m O}_ {infty}$, revealing a deep connection between number theory and operator algebras.

Contribution

It introduces a novel correspondence between continued fractions and permutative representations of ${ m O}_ {infty}$, relating modular transformations to unitary equivalence of representations.

Findings

Equivalence of real numbers under modular transformations corresponds to unitary equivalence of representations.

Quadratic irrationals are associated with permutative representations with cycles.

A new bridge between number theory and operator algebra representations is established.

Abstract

We show a correspondence between simple continued fraction expansions of irrational numbers and irreducible permutative representations of the Cuntz algebra ${\cal O}_{\infty}$. With respect to the correspondence, it is shown that the equivalence of real numbers with respect to modular transformations is equivalent to the unitary equivalence of representations. Furthermore, we show that quadratic irrationals are related to irreducible permutative representations of ${\cal O}_{\infty}$ with a cycle.