Cohomology and Extensions of von Neumann Algebras, II

Sutherland, Colin E.

doi:10.2977/prims/1195187502

Cited by 44 publications

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“…Thus g → v g is a (strongly continuous) 1-cocycle in A(J). Since A(J) is a type III factor and the G-action has full G-spectrum, there exists [57] a unitary w ∈ A(J) such that v g = β g (w)w * for all g ∈ G. Defining ρ = Ad w * • ρ, we have w * u ∈ Hom(ρ, ρ). Now, β g (u)u * = β g (w)w * is equivalent to β g (w * u) = w * u, thus w * u is G-invariant.…”

Section: Propositionmentioning

confidence: 99%

“…[57]. We recall some well known facts: The associativity (ρ g ρ h )ρ k = ρ g (ρ h ρ k ) implies the existence of α g,h,k ∈ T such that…”

Section: Corollarymentioning

confidence: 99%

“…(Actually, since in QFT the algebras A(I) are type III factors with separable predual, the converse is also true: If [α] = 0 then one can find a homomorphism g → ρ g , cf. [57].) ✷ 4.9 For a further analysis it is more convenient to adopt a purely categorical viewpoint.…”

Section: Corollarymentioning

confidence: 99%

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Conformal Orbifold Theories and Braided Crossed G-Categories

Mueger

2005

Commun. Math. Phys.

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The aim of the paper is twofold. First, we show that a quantum field theory A living on the line and having a group G of inner symmetries gives rise to a category G − LocA of twisted representations. This category is a braided crossed G-category in the sense of Turaev [60]. Its degree zero subcategory is braided and equivalent to the usual representation category Rep A. Combining this with [29], where Rep A was proven to be modular for a nice class of rational conformal models, and with the construction of invariants of G-manifolds in [60], we obtain an equivariant version of the following chain of constructions: Rational CFT ❀ modular category ❀ 3-manifold invariant.Secondly, we study the relation between G−LocA and the braided (in the usual sense) representation category Rep A G of the orbifold theory A G . We prove the equivalence Rep A G ≃ (G−LocA) G , which is a rigorous implementation of the insight that one needs to take the twisted representations of A into account in order to determine Rep A G . In the opposite direction we have G− LocA ≃ Rep A G ⋊ S, where S ⊂ Rep A G is the full subcategory of representations of A G contained in the vacuum representation of A, and ⋊ refers to the Galois extensions of braided tensor categories of [44,48].Under the assumptions that A is completely rational and G is finite we prove that A has g-twisted representations for every g ∈ G and that the sum over the squared dimensions of the simple g-twisted representations for fixed g equals dim Rep A. In the holomorphic case (where Rep A ≃ Vect C ) this allows to classify the possible categories G−LocA and to clarify the rôle of the twisted quantum doubles D ω (G) in this context, as will be done in a sequel. We conclude with some remarks on non-holomorphic orbifolds and surprising counterexamples concerning permutation orbifolds. * Supported by NWO.

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Section: Propositionmentioning

confidence: 99%

“…[57]. We recall some well known facts: The associativity (ρ g ρ h )ρ k = ρ g (ρ h ρ k ) implies the existence of α g,h,k ∈ T such that…”

Section: Corollarymentioning

confidence: 99%

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Conformal Orbifold Theories and Braided Crossed G-Categories

Mueger

2005

Commun. Math. Phys.

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“…By [25,Theorem 4.1.3], it is the only obstruction. Suppose it vanishes, then the topological bundle B = X ×φ M is nontrivial.…”

Section: Locally Trivial Bundlesmentioning

confidence: 99%

Locally trivial W∗-bundles

Evington

Pennig

2016

Int. J. Math.

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“…We cite [Brown 1994;Eilenberg and Mac Lane 1947;Mac Lane and Whitehead 1950;Huebschmann 1981;Jones 1980;Ratcliffe 1980] for the general cohomology theory of abstract groups and [Sutherland 1980] for the cohomology theory related to von Neumann algebras. See [Takesaki 1979;2003a;2003b] for the general theory of von Neumann algebras.…”

Section: Introductionmentioning

confidence: 99%

Outer actions of a discrete amenable group on approximately finite-dimensional factors, III: The type III_λcase, 0<λ <1, asymmetrization and examples

Katayama¹,

Takesaki²

2009

Pacific J. Math.

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In this last article of the series on outer actions of a countable discrete amenable group on approximately finite-dimensional factors, we analyze outer actions of a countable discrete free abelian group on an approximately finite-dimensional factor of type III λ with 0 < λ < 1 and compute outer conjugacy invariants. As a byproduct, we discover the asymmetrization technique for coboundary condition on a ‫-ޔ‬valued cocycle of a torsion-free abelian group, which might have been known by group cohomologists. As the asymmetrization technique gives us a very handy criterion for coboundaries, we present it here in detail. IntroductionThis article concludes the series 2004;2007] on the outer conjugacy classification of outer actions of a countable discrete amenable group on an approximately finite-dimensional (AFD) factor, by examining outer actions of a countable discrete abelian group G on an AFD factor R λ of type III λ MSC2000: primary 46L55; secondary 46L35, 46L40, 20J06. Keywords: outer action, outer conjugacy, cohomology of group, asymmetrization. 57 58 YOSHIKAZU KATAYAMA AND MASAMICHI TAKESAKI with 0 < λ < 1. Prior to the outer conjugacy classification theory, the cocycle conjugacy classification theory of actions of a countable discrete amenable group on an AFD factor had been completed through the work of many mathematicians over three decades; see [Connes 1977;1976b;1975;Jones 1980;Jones and Takesaki 1984; Ocneanu 1985;Katayama et al. 1998;1997;Kawahigashi et al. 1992;Sutherland and Takesaki 1985;1989;]. Unlike the general classification program in operator algebras, the outer conjugacy classification of a countable discrete amenable group on R λ is almost smooth, as shown in our series of previous work; see [Katayama and Takesaki 2007]. The only nonsmooth part of the classification theory stems from the classification of subgroups N of G; for instance, the classification of subgroups of a torsionfree abelian group of higher rank is nonsmooth. See [Sutherland 1985] for the Borel parameterization of polish groups. When the modular automorphism part N =α −1 (Cnt r (M)) of the outer actionα of G on R λ is fixed, the set of invariants becomes a compact abelian group. This is a rare case in the theory of operator algebras. So we are encouraged to make a concrete analysis of outer conjugacy classes of a countable discrete amenable group. Of course, without having a concrete data on the group G involved, we cannot make a fine analysis. So we take a countable discrete free abelian group G and study its outer actions on R λ and identify the invariants completely. The justification of this restriction rests on the fact that all outer actions of a countable discrete abelian group A can be viewed as outer actions of G by pulling back the outer action via the quotient map G → A. Thanks to all hard analytic work on the cocycle conjugacy classification in the past, as cited in the references, our work is very algebraic and indeed done by cohomological computations.We will begin by relating the discrete core of R λ and the...

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Cohomology and Extensions of von Neumann Algebras, II

Cited by 44 publications

References 13 publications

Conformal Orbifold Theories and Braided Crossed G-Categories

Conformal Orbifold Theories and Braided Crossed G-Categories

Locally trivial W∗-bundles

Outer actions of a discrete amenable group on approximately finite-dimensional factors, III: The type III_λcase, 0<λ <1, asymmetrization and examples

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Cohomology and Extensions of von Neumann Algebras, II

Cited by 44 publications

References 13 publications

Conformal Orbifold Theories and Braided Crossed G-Categories

Conformal Orbifold Theories and Braided Crossed G-Categories

Locally trivial W∗-bundles

Outer actions of a discrete amenable group on approximately finite-dimensional factors, III: The type IIIλcase, 0<λ <1, asymmetrization and examples

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Product

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Outer actions of a discrete amenable group on approximately finite-dimensional factors, III: The type III_λcase, 0<λ <1, asymmetrization and examples