Remarks on the Index of Endomorphisms of Cuntz Algebras

Conti, Roberto; Pinzari, Claudia

doi:10.1006/jfan.1996.0154

Cited by 28 publications

(63 citation statements)

References 24 publications

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“…The action can be extended to an action on MY and (the normal extension of) 3 is a KMS state with respect to the 1-parameter (periodic) modular automorphism group ' 3 [17,28]. Furthermore ( [3,28]) we have the following upper bound for the index:…”

Section: Preliminariesmentioning

confidence: 99%

“…To conclude this section, we give a sufficient condition for Yang-Baxter endomorphisms of index 4 of o 2 to be braided. Then we exhaust all the cases appearing in o 2 , see [3], Corollary 7.6. As in [3], Section 7 we define…”

Section: Proof (I) It Follows Easily From Lemmamentioning

confidence: 99%

“…Then we exhaust all the cases appearing in o 2 , see [3], Corollary 7.6. As in [3], Section 7 we define…”

Section: Proof (I) It Follows Easily From Lemmamentioning

confidence: 99%

“…However, when our conditions are fulfilled, we can exhibit very explicit examples of braided endomorphisms. Namely, we show that many nontrivial examples of standard braided endomorphisms of the Cuntz algebra o 2 exist, and in particular with index 2, a problem left open in [3].…”

Section: Introductionmentioning

confidence: 97%

“…Later, they were sistematically studied in [3], where many considerations about the Yang-Baxter endomorphisms were also made.…”

Section: Introductionmentioning

confidence: 99%

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Braided Endomorphisms of Cuntz Algebras

Conti¹,

Fidaleo²

2000

MATH. SCAND.

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We discuss sufficient conditions ensuring that certain endomorphisms of infinite factors arising from Cuntz algebras are braided. We analyze some explicit non-trivial examples associated to unitary solutions of the quantum Yang-Baxter equation on a Hilbert space of dimension 2X In particular we show the existence of endomorphisms of index 2 associated to representations of Hecke algebras at a primitive fourth root of unity. In this case we compute the associated fusion rules. These fusion rules define a finitely generated Ã -semiring which is not finite. Such a picture seems to be closely related to the description of (the dual of) a deformation, at a fourth root of unity, of some compact matrix group. This could be of some interest for the investigation of quantum symmetries naturally appearing in low-dimensional Quantum Field Theory.

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Section: Preliminariesmentioning

confidence: 99%

Section: Proof (I) It Follows Easily From Lemmamentioning

confidence: 99%

“…Then we exhaust all the cases appearing in o 2 , see [3], Corollary 7.6. As in [3], Section 7 we define…”

Section: Proof (I) It Follows Easily From Lemmamentioning

confidence: 99%

Section: Introductionmentioning

confidence: 97%

“…Later, they were sistematically studied in [3], where many considerations about the Yang-Baxter endomorphisms were also made.…”

Section: Introductionmentioning

confidence: 99%

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Braided Endomorphisms of Cuntz Algebras

Conti¹,

Fidaleo²

2000

MATH. SCAND.

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Endomorphisms of On which preserve the canonical UHF-subalgebra

Conti¹,

Rørdam²,

Szymański³

2010

Journal of Functional Analysis

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Unital endomorphisms of the Cuntz algebra O n which preserve the canonical UHF-subalgebra F n ⊆ O n are investigated. We give examples of such endomorphisms λ = λ u for which the associated unitary element u in O n (which satisfies λ(S j ) = uS j for all j ) does not belong to F n . One such example, in the case where n = 2, arises from a construction of a unital endomorphism on O 2 which preserves the canonical UHF-subalgebra and where the relative commutant of its image in O 2 contains a copy of O 2 .

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Endomorphisms of graph algebras

Conti

Hong

Szymański

2012

Journal of Functional Analysis

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We initiate a systematic investigation of endomorphisms of graph C*-algebras C* (E), extending several known results on endomorphisms of the Cuntz algebras O-n. Most but not all of this study is focused on endomorphisms which permute the vertex projections and globally preserve the diagonal MASA DE of C* (E). Our results pertain both automorphisms and proper endomorphisms. Firstly, the Weyl group and the restricted Weyl group of a graph C*-algebra are introduced and investigated. In particular, criteria of outerness for automorphisms in the restricted Weyl group are found. We also show that the restriction to the diagonal MASA of an automorphism which globally preserves both D-E and the core AF-subalgebra eventually commutes with the corresponding one-sided shift. Secondly, we exhibit several properties of proper endomorphisms, investigate invertibility of localized endomorphisms both on C* (E) and in restriction to D-E, and develop a combinatorial approach to analysis of permutative endomorphisms. (c) 2012 Elsevier Inc. All rights reserved

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Remarks on the Index of Endomorphisms of Cuntz Algebras

Abstract: We consider the class of``localized endomorphisms'' of the Cuntz algebras and we make some computations on the index of the associated endomorphisms of type III * factors.

Cited by 28 publications

References 24 publications

Braided Endomorphisms of Cuntz Algebras

Braided Endomorphisms of Cuntz Algebras

Endomorphisms of On which preserve the canonical UHF-subalgebra

Endomorphisms of graph algebras

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