## Multi-interval subfactors and modularity of representations in conformal field theory

Venue: | Commun. Math. Phys |

Citations: | 63 - 26 self |

### BibTeX

@ARTICLE{Kawahigashi_multi-intervalsubfactors,

author = {Yasuyuki Kawahigashi and Roberto Longo and Michael Müger},

title = {Multi-interval subfactors and modularity of representations in conformal field theory},

journal = {Commun. Math. Phys},

year = {},

volume = {219},

pages = {631--669}

}

### Years of Citing Articles

### OpenURL

### Abstract

Dedicated to John E. Roberts on the occasion of his sixtieth birthday We describe the structure of the inclusions of factors A(E) ⊂A(E ′ ) ′ associated with multi-intervals E ⊂ R for a local irreducible net A of von Neumann algebras on the real line satisfying the split property and Haag duality. In particular, if the net is conformal and the subfactor has finite index, the inclusion associated with two separated intervals is isomorphic to the Longo-Rehren inclusion, which provides a quantum double construction of the tensor category of superselection sectors of A. As a consequence, the index of A(E) ⊂A(E ′ ) ′ coincides with the global index associated with all irreducible sectors, the braiding symmetry associated with all sectors is non-degenerate, namely the representations of A form a modular tensor category, and every sector is a direct sum of sectors with finite dimension. The superselection structure is generated by local data. The same results hold true if conformal invariance is replaced by strong additivity and there exists a modular PCT symmetry.

### Citations

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Citation Context ...clusion associated with the corresponding irreducible sectors {[ρi]}i. 30sWe note that in this case the 2-interval inclusion is not the asymptotic inclusion of the corresponding Jones-Wenzl subfactor =-=[24, 48]-=-, even up to tensoring by a common injective III1 factor. Consider SU(2)k as an example. The net has k + 1 sectors and if we choose the standard generator, we get a corresponding subfactor of Jones wi... |

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Citation Context ... π acts on a separable Hilbert space. Moreover every representation of A on a separable Hilbert space is automatically locally normal [45], thus localizable. Denote by C∗ (A) the universal C∗-algebra =-=[14]-=- associated with A (see also [16]). For each I ∈Ithere is a canonical embedding ιI : A(I) → C∗ (A) and ιĨ |A(I) = ιI if I ⊂ Ĩ; we identify A(I) with ιI(A(I)) if no confusion arises. There is a one-to-... |

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Citation Context ...clusion associated with the corresponding irreducible sectors {[ρi]}i. 30sWe note that in this case the 2-interval inclusion is not the asymptotic inclusion of the corresponding Jones-Wenzl subfactor =-=[24, 48]-=-, even up to tensoring by a common injective III1 factor. Consider SU(2)k as an example. The net has k + 1 sectors and if we choose the standard generator, we get a corresponding subfactor of Jones wi... |

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Citation Context ...d in the interval I, its restriction ρ|A(I) is an endomorphism of A(I), thus it gives rise to a sector of the factor A(I) (i.e. a normal unital endomorphism of A(I) modulo inner automorphisms of A(I) =-=[25]-=-) and it will be clear from the context which sense will be attributed to the term sector. The reader unfamiliar with the sector strucure is referres to [25, 28, 17] and to the Appendix B. Let E = I1 ... |

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Citation Context ...reducible nets A ⊂ B, where A = B G is the fixedpoint of B with respect to the action of a finite group G and µB = 1. Then [B : A] = |G|, thus by Prop. 21, Iglobal(A) = µA = |G| 2 . Now A has the DHR =-=[8]-=- irreducible sectors [ρπ] associated with π ∈ ˆ G and ∑ d(ρπ) 2 = |G|, π∈ ˆG therefore A has extra irreducible sectors [σi] with ∑ i d(σi) 2 = |G| 2 − |G|. For example, in the case of Ising model, we ... |

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Citation Context ...ed in I1. Therefore the isomorphism class of {A(E),λE} is independent of E ∈I2. Hence the LR inclusions based on that are isomorphic. � 24sIndeed, by using the uniqueness of the III1 injective factor =-=[6, 19]-=- and the classification of its finite depth subfactors [40] we have the following. Corollary 35. Let A be completely rational and conformal. The isomorphism class of the inclusion A(E) ⊂ Â(E), E ∈I2, ... |

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Citation Context ...tructure of sectors associated with the LR net, an analysis mostly based on the braiding symmetry, the work of Izumi [22] and the α-induction, which has been introduced in [28] and further studied in =-=[49, 2, 3]-=-. Section 5 combines and develops the previous analysis to obtain our main results for the 2-interval inclusion. These results are extended to the case of n-interval inclusions in Section 6. We then w... |

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Citation Context ...E) is the crossed product of A(E) by a finite-dimensional Hopf algebra) is equivalent to the innerness of the sector λ extending λE (because λE is implemented by a Hilbert space of isometries in Â(E) =-=[26]-=-), hence it is equivalent to the the property that all irreducible sectors of A have dimension 1 by Lemma 31. The following is the main result of this paper. Theorem 33. Let A be completely rational w... |

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Citation Context ...tructure of sectors associated with the LR net, an analysis mostly based on the braiding symmetry, the work of Izumi [22] and the α-induction, which has been introduced in [28] and further studied in =-=[49, 2, 3]-=-. Section 5 combines and develops the previous analysis to obtain our main results for the 2-interval inclusion. These results are extended to the case of n-interval inclusions in Section 6. We then w... |

65 | Operator algebras and conformal field theory. III. Fusion of positive energy representations of LSU(N) using bounded operators
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(Show Context)
Citation Context ...onsidered in the past years, seemingly with different motivations. The most detailed study of this inclusion so far has been done by Xu [50] for the models given by loop group construction for SU(n)k =-=[47]-=-. In this case Xu has computed the index and the dual principal graph of the inclusions. A suggestion to study this inclusion has been made also in [43, Section 3]. Our analysis is model independent, ... |

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(Show Context)
Citation Context ...ic case I1 = I, I2 = −I. Then A(−I) =j(A(I)), where j is the anti-linear PCT automorphism, hence we may identify A(−I) with A(I) opp . Moreover the formula ¯ρi = j·ρi·j holds for the conjugate sector =-=[17]-=-, thus by the split property we may identify {A(E),ρi¯ρi|A(E)} with {A(I)⊗ A(I) opp ,ρi ⊗ ρ opp i }. Now there is an isometry Vi that intertwines the identity and ρi ¯ρi and belongs to Â(E). We then h... |

61 |
A theory of dimension, K-theory 11
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(Show Context)
Citation Context .... Clearly, λ is localized in (a, d), acts trivially on A(b, c) and is transportable. Moreover, λ has finite index as the operators R, ¯R ∈ (i, λ2 ) in the standard solution for the conjugate equation =-=[25, 29]-=- ¯R ∗¯ λ(R) =1, R ∗ λ( ¯R) =1, on Â(E) give the same relation on A(I) for any I ⊃ E, I ∈I. If A is conformal, then ρ is covariant with respect to translations and dilations by [17]. As we may vary the... |

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Citation Context ...rtant consequence is that the braiding symmetry associated with all sectors is always non-degenerate, in other words the localizable representations form a modular tensor category. As shown by Rehren =-=[41]-=-, this implies the existence and non-degeneracy of Verlinde’s matrices S and T , thus the existence of a unitary representation of the modular group SL(2, Z), which plays a role in topological quantum... |

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Citation Context ...e interior of I2. This implies (indeed it is equivalent to e.g. if the A(I)’s are factors) that A(I1)∨A(I ′ 2) is naturally isomorphic to the tensor product of von Neumann algebras A(I1)⊗A(I ′ 2) (cf.=-=[10]-=-) . For a conformal net, the split property holds if Tr(e −βL0 ) < ∞ for all β>0, cf. [8]. Notice that if A is split and A(I) is a factor for I ∈I, then A(E) is a factor for E ∈In for any n. Propositi... |

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Relativistic invariance and charge conjugation in quantum field theory
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Citation Context ...f the energy does not necessarily holds. Note that eq. (4) implies Haag duality for half-lines A(−∞,a) ′ = A(a, ∞), a ∈ R . Setting j ≡ AdJ, the conjugate sector exists and it is given by the formula =-=[16]-=- ¯ρ = j · ρ · j. Corollary 10. If A is completely rational with modular PCT, then A is rational, namely there are only finitely many irreducible sectors [ρ0], [ρ1],...,[ρn] with finite dimension and w... |

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Citation Context ...line (Reeh-Schlieder property), and the modular conjugation J of A(a, ∞) with respect to Ω has the geometric property JA(I + a)J = A(−I + a), I ∈I, a ∈ R. (4) This is automatic if A is conformal, see =-=[4, 15]-=-. It easy to see that the modular PCT property implies translation covariance, where the translation unitaries are products of modular conjugations, but positivity of the energy does not necessarily h... |

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Citation Context ...gebra generated. If E ⊂ S1 is any set, we put and set A(E) ≡ � {A(I) : I ∈I,I⊂ E} Â(E) ≡A(E ′ ) ′ with E ′ ≡ S1 � E. 3 We shall assume Haag duality on S1 , which automatically holds if A is conformal =-=[4]-=-, namely, A(I) ′ = A(I ′ ), I ∈I, thus Â(I) =A(I), I ∈I, but for a disconnected set E ⊂ S1 , A(E) ⊂ Â(E) is in general a non-trivial inclusion. We shall say that E ⊂ S 1 is an n-interval if both E and... |

39 | Orbifold subfactors from Hecke algebras II —Quantum doubles and braiding
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(Show Context)
Citation Context ...p][α j + ρi⊗id ] as M-M sectors. The following proposition is originally due to Izumi [22] (with a different proof) and first due to Ocneanu [37] in the setting of the asymptotic inclusion. (Also see =-=[13]-=-.) 17sProposition 21. Each [βij] is an irreducible M-M sector and these are mutually different for different pairs of i, j. Each irreducible M-M sector arising from N⊗ N opp ⊂Mis of this form. Proof W... |

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Citation Context ...ectation E : M→N⊗N opp . Setting ρ0 = id, the expectation E may be defined by E(X) =x0 for X given by (16), once we show that this is well-defined. To this end we will apply the averaging argument in =-=[23]-=-. Let J be the set of all x0 ∈N⊗N opp such that there exist xi ∈N⊗N opp , i>0, with � i≥0 xiRi = 0. Clearly J is a two-sided ideal of N⊗N opp , hence J = 0 (as we want to show) or J = N⊗N opp (we may ... |

38 |
Quantum symmetry, differential geometry of finite graphs, and classification of subfactors, Univ
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Citation Context ...s of A⊗A opp that are implemented by isometries in B. Indeed B is a the crossed product of A⊗A opp by the tensor category of all its sectors. 3sAs shown by Masuda [30], Ocneanu’s asymptotic inclusion =-=[35]-=- and the LongoRehren inclusion in [28] are, from the categorical viewpoint, essentially the same constructions. The construction of the asymptotic inclusion gives a new subfactor M∨(M ′ ∩M∞) ⊂M∞ from ... |

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Citation Context ... the sectors and the index of the 2-interval subfactor. In Section 4 we study the structure of sectors associated with the LR net, an analysis mostly based on the braiding symmetry, the work of Izumi =-=[22]-=- and the α-induction, which has been introduced in [28] and further studied in [49, 2, 3]. Section 5 combines and develops the previous analysis to obtain our main results for the 2-interval inclusion... |

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Classification of subfactors and their endomorphisms
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Citation Context ...pendent of E ∈I2. Hence the LR inclusions based on that are isomorphic. � 24sIndeed, by using the uniqueness of the III1 injective factor [6, 19] and the classification of its finite depth subfactors =-=[40]-=- we have the following. Corollary 35. Let A be completely rational and conformal. The isomorphism class of the inclusion A(E) ⊂ Â(E), E ∈I2, dependes only on the tensor category of the sectors of A, n... |

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Citation Context ...ed in I1. Therefore the isomorphism class of {A(E),λE} is independent of E ∈I2. Hence the LR inclusions based on that are isomorphic. � 24sIndeed, by using the uniqueness of the III1 injective factor =-=[6, 19]-=- and the classification of its finite depth subfactors [40] we have the following. Corollary 35. Let A be completely rational and conformal. The isomorphism class of the inclusion A(E) ⊂ Â(E), E ∈I2, ... |

27 |
Drinfel ′ d, Quantum groups
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Citation Context ...inclusion gives a new subfactor M∨(M ′ ∩M∞) ⊂M∞ from a hyperfinite II1 subfactor N⊂Mwith finite index and finite depth and it is a subfactor analogue of the quantum double construction of Drinfel ′ d =-=[11]-=-, as noted by Ocneanu. That is, the tensor category of the M∞-M∞ bimodules arising from the new subfactor is regarded a “quantum double” of the original category of M-M (or N -N ) bimodules. On the ot... |

26 | Jones-Wassermann subfactors for Disconnected Intervals, Preprint 97, see also q-alg/9704003
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(Show Context)
Citation Context ...ated to two or more separated intervals. This problem has been considered in the past years, seemingly with different motivations. The most detailed study of this inclusion so far has been done by Xu =-=[50]-=- for the models given by loop group construction for SU(n)k [47]. In this case Xu has computed the index and the dual principal graph of the inclusions. A suggestion to study this inclusion has been m... |

25 |
Subalgebras of infinite C ∗ -algebras with finite Watatani indices II: Cuntz-Krieger algebras
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(Show Context)
Citation Context ... is localized in I. We shorten our notation by setting N ≡ A(I) and M = B(I). We thus have λ(x) = ∑ i Vi(ρi ⊗ ρ opp i )(x)V ∗ i , where Vi’s are isometries in N ⊗ N opp ∑ with ∗ i ViVi = 1. We follow =-=[21]-=- for the terminology of (N ⊗N opp )-M sectors, and so on, and study the sector structure of the subfactor N ⊗ N opp ⊂ M in this section. In other words we study the sector structure of a single subfac... |

21 |
An analogue of Longo’s canonical endomorphism for bimodule theory and its application to asymptotic inclusions
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(Show Context)
Citation Context ...here is a generating family of sectors of A⊗A opp that are implemented by isometries in B. Indeed B is a the crossed product of A⊗A opp by the tensor category of all its sectors. 3sAs shown by Masuda =-=[30]-=-, Ocneanu’s asymptotic inclusion [35] and the LongoRehren inclusion in [28] are, from the categorical viewpoint, essentially the same constructions. The construction of the asymptotic inclusion gives ... |

20 |
Algebraic and modular structure of von Neumann algebras of
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Citation Context ...tensor category of the sectors of A, not on its model realization. Proof If A is non-trivial and I is an interval, the A(I) isaIII1 factor and, as the split property hols, A(I) is injective (see e.g. =-=[27]-=-). Thus A(I) is the unique injective III1 factor [19]. By Popa’s theorem [40], if N is a III1 injective factor and T ⊂ End(N ) a rational tensor category isomorphic to the tensor category of sectors o... |

18 |
On α-induction, chiral projectors and modular invariants for subfactors
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(Show Context)
Citation Context ...tructure of sectors associated with the LR net, an analysis mostly based on the braiding symmetry, the work of Izumi [22] and the α-induction, which has been introduced in [28] and further studied in =-=[49, 2, 3]-=-. Section 5 combines and develops the previous analysis to obtain our main results for the 2-interval inclusion. These results are extended to the case of n-interval inclusions in Section 6. We then w... |

18 |
Takesaki: “Theory of operator algebras I
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Citation Context ...I. As the A(I)’s are properly infinite the two notions coincide if π acts on a separable Hilbert space. Moreover every representation of A on a separable Hilbert space is automatically locally normal =-=[45]-=-, thus localizable. Denote by C∗ (A) the universal C∗-algebra [14] associated with A (see also [16]). For each I ∈Ithere is a canonical embedding ιI : A(I) → C∗ (A) and ιĨ |A(I) = ιI if I ⊂ Ĩ; we iden... |

17 |
An invariant coupling between 3-manifolds and subfactors, with connections to topological and conformal quantum field theory. Unpublished manuscript. Ca
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(Show Context)
Citation Context ...p ∈N⊗N opp in an intertwiner from βijβkl to βmn. Proof By a direct computation. � Then we easily get the following from the above lemma. (The quantum 6j-symbols for subfactors have been introduced in =-=[36]-=- as a generalization for classical 6j-symbols. See [12, Chapter 12] for details.) Theorem 23. In the above setting, the tensor categories of (N ⊗N opp )-(N ⊗N opp ) sectors and M-M sectors with quantu... |

16 |
Nuclear Maps and Modular Structures I
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Citation Context ...nd A to the dual net A , which is conformal and satisfies Haag duality. 2 −βL0 This general property is satisfied, in particular, if Tr(e ) < ∞ for all β>0, where L0 is the conformal Hamiltonian, cf. =-=[5, 8]-=-. 2sfication A(I2) �A(I1) opp ). In particular, [ Â(E) :A(E)] = � i d(ρi) 2 , the global index of the superselection sectors. In fact A will turn out to be rational in an even stronger sense, namely t... |

16 |
Space-time fields and exchange
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(Show Context)
Citation Context ...h ρ0 = id), namely {[ρi]}i is a family of finitely many different irreducible sectors of A with finite dimension stable under conjugation and irreducible components of compositions. One may construct =-=[42, 28]-=- a net of subfactors A⊗A opp ⊂Bso that the corresponding canonical endomorphism restricted on A⊗Aopp is given by � i ρi ⊗ ρ opp i . We call this B the LR net. For Aopp , we use εopp (ρ opp k ,ρ opp l ... |

15 | F.: Conformal nets, maximal temperature and models from free probability
- D’Antoni, Longo, et al.
(Show Context)
Citation Context ... A to the dual net A , which is conformal and satisfies Haag duality. 2 −βL0 This general property is satisfied, in particular, if Tr(e ) < ∞ for all β > 0, where L0 is the conformal Hamiltonian, cf. =-=[5, 8]-=-. 2fication A(I2) ≃ A(I1) opp ). In particular, ∑ [ Â(E) : A(E)] = d(ρi) 2 , the global index of the superselection sectors. In fact A will turn out to be rational in an even stronger sense, namely t... |

13 | On charged fields with group symmetry and degeneracies of Verlinde’s matrix - Müger - 1999 |

11 | An algebraic formulation of level one Wess-Zumino-Witten models
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(Show Context)
Citation Context ... half of that of the subfactor arising from 4 intervals and the net for SU(2)k. ForSU(n)k, this ratio of the two indices is n. Finally we notice that there are models like the SO(2N)1 WZW models, see =-=[1]-=- or [34], where all irreducible sectors have dimension one, yet the superselection category C is modular in agreement with our results. In these cases the fusion graph is disconnected, therefore the e... |

11 |
Symmetric enveloping algebras, amenability and AFD properties for subfactors
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(Show Context)
Citation Context ...18) both in M and in M1. � 33 i (19)sNote that the above Proposition gives an alternative construction of the LR inclusion, which is similar to Popa’s construction of the symmetric enveloping algebra =-=[39]-=-, as follows. Let N act standardly on L 2 (N ) and Vi be the standard isometry implementing ρi. The ∗ -algebra A generated by N and N ′ is naturally isomorphic to the algebraic tensor product N ⊚ N op... |

10 |
Subalgebras of infinite C∗-algebras with finite Watatani indices. II. Cuntz-Krieger algebras, preprint
- Izumi
(Show Context)
Citation Context ...also λ is localized in I. We shorten our notation by setting N≡A(I) and M = B(I). We thus have λ(x) = � i Vi(ρi ⊗ ρ opp i )(x)V ∗ i , where Vi’s are isometries in N⊗Nopp � with ∗ i ViVi =1. We follow =-=[21]-=- for the terminology of (N⊗Nopp )-M sectors, and so on, and study the sector structure of the subfactor N⊗Nopp ⊂Min this section. In other words we study the sector structure of a single subfactor, no... |

6 |
Inclusioni di algebre di von Neumann e teoria algebrica dei campi. Unpublished Ph.D. thesis. Università di Roma “Tor Vergata
- Conti
- 1996
(Show Context)
Citation Context ...g are equivalent: (i) The braiding of the net A is non-degenerate. 22s(ii) B has no non-trivial localized endomorphism (localized in a bounded interval, finite index). Proof We use now an argument in =-=[7]-=-. Let σ be a non-trivial irreducible localized endomorphism of B localized in an interval, with d(σ) < ∞. By Frobenius reciprocity σ ≺ α + σ rest, σ ≺ α − σ rest, where σrest = γ · σ|A⊗Aopp and γ : B→... |

6 | Extensions of conformal nets and superselection - Guido, Longo, et al. - 1998 |

6 | Recent developments of algebraic methods in quantum field theory - Schroer - 1992 |

5 |
Conformal nets, maximal temperature and models from free probability
- D’Antoni, Longo, et al.
- 2001
(Show Context)
Citation Context ...nd A to the dual net A , which is conformal and satisfies Haag duality. 2 −βL0 This general property is satisfied, in particular, if Tr(e ) < ∞ for all β>0, where L0 is the conformal Hamiltonian, cf. =-=[5, 8]-=-. 2sfication A(I2) �A(I1) opp ). In particular, [ Â(E) :A(E)] = � i d(ρi) 2 , the global index of the superselection sectors. In fact A will turn out to be rational in an even stronger sense, namely t... |

5 |
Winnink: Local normality in quantum statistical mechanics
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Citation Context ...nd of DHR localized endomorphisms of A0 are equivalent, but we do not need this form of the above statement. 38sC Disintegration of locally normal representations and of sectors. Takesaki and Winnink =-=[44]-=- have shown that a locally normal state decomposes into locally normal states, if the split property holds. We shall show here analogous results for localizable representations (sectors). Our argument... |

4 |
Type III Factors and Index Theory”, Res
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Citation Context ... with E(x) ≥ λx, for all x ∈M+, and denote the index by [M : N ]E = λ −1 with λ the best constant for the inequality to hold and [M : N ]=[M : N ]min = inf E [M : N ]E denotes the minimal index, (see =-=[20]-=- for an overview). Recall that A is split if there exists an intermediate type I factor between A(I1) and A(I2) whenever I1, I2 are intervals and the closure Ī1 is contained in the interior of I2. Thi... |

4 |
Chirality for operator algebras (Notes recorded by Y
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(Show Context)
Citation Context ... ρi⊗id ][α− id⊗ρ opp]=[α j − ρj⊗id ][α+ ρi⊗id ]=[α− id⊗ρ opp][α j + ρi⊗id ] as M-M sectors. The following proposition is originally due to Izumi [22] (with a different proof) and first due to Ocneanu =-=[37]-=- in the setting of the asymptotic inclusion. (Also see [13].) 17sProposition 21. Each [βij] is an irreducible M-M sector and these are mutually different for different pairs of i, j. Each irreducible ... |

3 |
Categorical approach to paragroup theory II. The quantum double of tensor categories and subfactors
- Müger
(Show Context)
Citation Context ... That is, the tensor category of the M∞-M∞ bimodules arising from the new subfactor is regarded a “quantum double” of the original category of M-M (or N -N ) bimodules. On the other hand, as shown in =-=[33]-=-, the Longo-Rehren construction gives the quantum double of the original tensor category of endomorphisms. (See also [12, Chapter 12] for a general theory of asymptotic inclusions and their relations ... |