On finite free Fisher information for eigenvectors of a modular operator

Nelson, Brent

doi:10.1016/j.jfa.2017.06.014

Cited by 7 publications

(2 citation statements)

References 22 publications

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“…In this section, we will show that for an irreducible inclusion (N ⊆ M, E), discreteness is equivalent to quasi-regularity. Moreover, if N = M φ , the centralizer of φ, then these conditions are equivalent to M being generated by eigenoperators for σ φ [Nel16]. First, we show that these properties imply that ∆ φ is almost periodic [Con74].…”

Section: Equivalencesmentioning

confidence: 77%

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Realizations of algebra objects and discrete subfactors

Jones¹,

Penneys²

2019

Advances in Mathematics

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We give a characterization of extremal irreducible discrete subfactors (N ⊆ M, E) where N is type II 1 in terms of connected W*-algebra objects in rigid C*-tensor categories. We prove an equivalence of categories where the morphisms for discrete inclusions are normal N − N bilinear ucp maps which preserve the state τ • E, and the morphisms for W*-algebra objects are categorical ucp morphisms.As an application, we get a well-behaved notion of the standard invariant of an extremal irreducible discrete subfactor, together with a subfactor reconstruction theorem. Thus our equivalence provides many new examples of discrete inclusions (N ⊆ M, E), in particular, examples where M is type III coming from non Kac-type discrete quantum groups and associated module W*-categories. Finally, we obtain a Galois correspondence between intermediate subfactors of an extremal irreducible discrete inclusion and intermediate W*-algebra objects. Corollary B. The realization |A| H of the connected W*-algebra object A coming from a discrete quantum group G = (C, F) together with a fully faithful representation H : C → Bim sp bf (N ) is type III if and only if G is not Kac-type. (See Section 6.3 for more details.) This corollary is related to [Vae05, Thm. 3.5] in the case of compact quantum groups. De Commer and Yamashita's classification of T LJ (δ)-module W*-categories [DCY15] gives many more such examples (see Section 6.4 for more details). BackgroundWe refer the reader to [HP17b, §2.1-2.2] and [JP17a, §2.1-2.3] for background on rigid C*tensor categories. Of particular importance is their bi-involutive structure, consisting of two commuting involutions (the adjoint * and conjugate · ) together with coherence data. As usual, we will suppress all associators, unitors, and the involutive structure morphisms.We fix a rigid C*-tensor category C. As in [JP17a, §2.4], Vec(C) denotes the involutive tensor category of linear functors C op → Vec with linear transformations. Note that Vec(C) = ind(C ), where C denotes the involutive tensor category obtained from C by forgetting the adjoint. As in [JP17a, §2.6], Hilb(C) denotes the bi-involutive tensor W*-category of linear dagger functors C op → Hilb with bounded natural transformations. Note that Hilb(C) is the Neshveyev-Yamashita unitary ind-category of C from [NY16]. 6Proposition 5.42. The map κ gives a natural isomorphism id ⇒ | · | H .

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Section: Equivalencesmentioning

confidence: 77%

“…Example 3.25. We thank Brent Nelson for pointing out to us examples of irreducible inclusions where M is type III and generated by eigenoperators, and N = M φ is type II 1 [Nel16]. In particular, we can take M to be one of Shlyakhtenko's non-trivial almost periodic free Araki-Woods factors and φ to be the free quasi-free state where…”

Section: Equivalencesmentioning

confidence: 99%

Realizations of algebra objects and discrete subfactors

Jones¹,

Penneys²

2019

Advances in Mathematics

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Non-tracial Free Graph von Neumann Algebras

Hartglass

Nelson

2020

Commun. Math. Phys.

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Full Solution of the Factoriality Question for q-Araki-Woods von Neumann Algebras Via Conjugate Variables

Kumar

Skalski

Wasilewski

2023

Commun. Math. Phys.

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We establish factoriality of q-Araki-Woods von Neumann algebras (with the number of generators at least two) in full generality, exploiting the approach via conjugate variables developed recently in the tracial case by Akihiro Miyagawa and Roland Speicher, and abstract results of Brent Nelson. We also establish non-injectivity and determine the type of the factors in question. The factors are solid and full when the number of generators is finite.

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On finite free Fisher information for eigenvectors of a modular operator

Cited by 7 publications

References 22 publications

Realizations of algebra objects and discrete subfactors

Realizations of algebra objects and discrete subfactors

Non-tracial Free Graph von Neumann Algebras

Full Solution of the Factoriality Question for q-Araki-Woods von Neumann Algebras Via Conjugate Variables

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