Computing the Jones index of quadratic permutation endomorphisms of O2

Conti, Roberto; Szymański, Wojciech

doi:10.1063/1.3068415

Cited by 11 publications

(13 citation statements)

References 30 publications

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“…Hence k i = 0 for all i and thus X (k) z = 1. Now, identity (5) implies that U * v k is fixed by the gauge action and hence belongs to F n . Now, we are ready to prove the second main result of this paper.…”

Section: The Relative Commutantsmentioning

confidence: 99%

“…But it can be shown that λ u is irreducible on O 2 (e.g., see [5], where this endomorphism is denoted ρ 142 ), and hence λ u (O 2 ) ′ ∩O 2 = C1. Thus h is a scalar and consequently Ad u • ϕ is identity on λ u (F 2 ) ′ ∩ O 2 .…”

Section: The Relative Commutantsmentioning

confidence: 99%

“…We remark that the restriction of Ad u • ϕ to (P 11 + P 222 )O 2 (P 11 + P 222 ) is conjugate to endomorphism ρ 1342 from [5]. Let w := S 11 S * 222 + S 222 S * 11 + 1 − P 11 − P 222 .…”

Section: The Criterion and Examplesmentioning

confidence: 99%

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On endomorphisms of the Cuntz algebra which preserve the canonical UHF-subalgebra, II

Hayashi

Hong

Szymański³

2017

Journal of Functional Analysis

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Abstract. It was shown recently by Conti, Rørdam and Szymański that there exist endomorphisms λ u of the Cuntz algebra O n such that λ u (F n ) ⊆ F n but u ∈ F n , and a question was raised if for such a u there must always exist a unitary v ∈ F n with λ u | Fn = λ v | Fn . In the present paper, we answer this question to the negative. To this end, we analyze the structure of such endomorphisms λ u for which the relative commutant λ u (F n ) ′ ∩ F n is finite dimensional.

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Section: The Relative Commutantsmentioning

confidence: 99%

Section: The Relative Commutantsmentioning

confidence: 99%

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On endomorphisms of the Cuntz algebra which preserve the canonical UHF-subalgebra, II

Hayashi

Hong

Szymański³

2017

Journal of Functional Analysis

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“…Permutative endomorphisms of graph C * -algebras and especially of the Cuntz algebras O n have already received considerable attention and have been studied in several different contexts. In particular, localized endomorphisms (a class of endomorphisms that includes permutative ones) of the Cuntz algebras were investigated in the framework of the Jones index theory in [16,22,6,8,7,14]. Similar investigations of localized endomorphisms of the Cuntz-Krieger algebras were carried out in [17].…”

Section: Introductionmentioning

confidence: 99%

Visualizing Automorphisms of Graph Algebras

Avery

Johansen

Szymański³

2018

Proceedings of the Edinburgh Mathematical Society

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Graph C * -algebras have been celebrated as C * -algebras that can be seen, because many important properties may be determined by looking at the underlying graph. This paper introduces the permutation graph for a permutative endomorphism of a graph C * -algebra as a labeled directed multigraph that gives a visual representation of the endomorphism and facilitates computations. Combinatorial criteria have previously been developed for deciding when such an endomorphism is an automorphism, but here the question is reformulated in terms of the permutation graph and new proofs are given. Furthermore, it is shown how to use permutation graphs to efficiently generate exhaustive collections of permutative automorphisms. Permutation graphs provide a natural link to the textile systems representing induced endomorphisms on the edge shift of the given graph, and this allows the powerful tools of the theory of textile systems developed by Nasu to be applied to the study of permutative endomorphisms.

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“…We adopt the same notation as in [CS09], where the monomial s i s j s * k is denoted by s ij,k . The following table displays not only the definitions of the endomorphisms we are going to consider but also says in advance which ones extend and which ones do not extend.…”

mentioning

confidence: 99%

Permutative representations of the 2-adic ring C∗-algebra

Aiello¹,

Conti²,

Rossi³

2019

J. Operator Theory

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The notion of permutative representation is generalized to the 2-adic ring C * -algebra Q 2 . Permutative representations of Q 2 are then investigated with a particular focus on the inclusion of the Cuntz algebra O 2 ⊂ Q 2 . Notably, every permutative representation of O 2 is shown to extend automatically to a permutative representation of Q 2 provided that an extension whatever exists. Moreover, all permutative extensions of a given representation of O 2 are proved to be unitarily equivalent to one another. Irreducible permutative representations of Q 2 are classified in terms of irreducible permutative representations of the Cuntz algebra. Apart from the canonical representation of Q 2 , every irreducible representation of Q 2 is the unique extension of an irreducible permutative representation of O 2 . Furthermore, a permutative representation of Q 2 will decompose into a direct sum of irreducible permutative subrepresentations if and only if it restricts to O 2 as a regular representation in the sense of Bratteli-Jorgensen. As a result, a vast class of pure states of O 2 is shown to enjoy the unique pure extension property with respect to the inclusion O 2 ⊂ Q 2 .

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Computing the Jones index of quadratic permutation endomorphisms of O2

Abstract: We compute the index of the inclusions of type III 1/2 factors arising from endomorphisms of the Cuntz algebra O 2 associated to the rank-two permutation matrices.MSC 2000: 46L37, 46L05

Cited by 11 publications

References 30 publications

On endomorphisms of the Cuntz algebra which preserve the canonical UHF-subalgebra, II

On endomorphisms of the Cuntz algebra which preserve the canonical UHF-subalgebra, II

Visualizing Automorphisms of Graph Algebras

Permutative representations of the 2-adic ring C∗-algebra

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