1999
DOI: 10.1090/memo/0663
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Iterated function systems and permutation representations of the Cuntz algebra

Abstract: We study a class of representations of the Cuntz algebras O N , N = 2, 3, . . . , acting on L 2 (T) where T = R 2πZ. The representations arise in wavelet theory, but are of independent interest. We find and describe the decomposition into irreducibles, and show how the O N -irreducibles decompose when restricted to the subalgebra UHF N ⊂ O N of gauge-invariant elements; and we show that the whole structure is accounted for by arithmetic and combinatorial properties of the integers Z. We have general results on… Show more

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Cited by 147 publications
(250 citation statements)
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“…We consider a permutative representation of O N in the sense of [5]. By Lemma 9.3.5 [5] every permutative representation of O N on a Hilbert space H has the following form:…”
Section: Cuntz Algebras and Iterated Function Systemsmentioning
confidence: 99%
See 2 more Smart Citations
“…We consider a permutative representation of O N in the sense of [5]. By Lemma 9.3.5 [5] every permutative representation of O N on a Hilbert space H has the following form:…”
Section: Cuntz Algebras and Iterated Function Systemsmentioning
confidence: 99%
“…By Lemma 9.3.5 [5] every permutative representation of O N on a Hilbert space H has the following form:…”
Section: Cuntz Algebras and Iterated Function Systemsmentioning
confidence: 99%
See 1 more Smart Citation
“…A branching function system was introduced by [3] in order to study representation of O N . It is convenient to construct concrete examples of representations easily.…”
Section: Action Of Permutations On Branching Function Systemsmentioning
confidence: 99%
“…This fact disturbs an intention to study an ordinary representation theory of operator algebras like that of semisimple Lie algebras and quantum groups. In spite of this, permutative representations of the Cuntz algebra O N ( [3,5,6]) are completely reducible and their irreducible decompositions are unique up to unitary equivalences. Roughly speaking, there are two kinds of (cyclic)permutative representations, "cycle" and "chain".…”
mentioning
confidence: 99%