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

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

2009-01-15

arxiv.org/abs/0901.2186v1

Mathematics

Operator Algebras

17 pages

Scientific paper

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.

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