The Non-commutative Flow of Weights on a Von Neumann Algebra

Falcone, Tony; Takesaki, Masamichi

doi:10.1006/jfan.2000.3718

Cited by 33 publications

(54 citation statements)

References 27 publications

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“…A more functorial construction of the continuous core, known as the noncommutative flow of weights can be given, see [Co72,CT76,FT99]. If P ⊂ 1 P M 1 P is a von Neumann subalgebra with expectation, we have a canonical trace preserving inclusion c(P ) ⊂ 1 P c(M )1 P .…”

Section: By Lemma 23 and Theorem 25 We Getmentioning

confidence: 99%

Amalgamated free product type III factors with at most one Cartan subalgebra

Boutonnet¹,

Houdayer²,

Raum³

2013

Compositio Math.

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International audienceWe investigate Cartan subalgebras in nontracial amalgamated free product von Neumann algebras M1 * B M2 over an amenable von Neumann subalgebra B. First, we settle the problem of the absence of Cartan subalgebra in arbitrary free product von Neumann algebras. Namely, we show that any nonamenable free product von Neumann algebra (M1, ϕ1) * (M2, ϕ2) with respect to faithful normal states has no Cartan subalgebra. This generalizes the tracial case that was established in [Io12a]. Next, we prove that any countable nonsingular ergodic equivalence relation R defined on a standard measure space and which splits as the free product R = R1 * R2 of recurrent subequivalence relations gives rise to a nonamenable factor L(R) with a unique Cartan subalgebra, up to unitary conjugacy. Finally, we prove unique Cartan decomposition for a class of group measure space factors L ∞ (X) ⋊ Γ arising from nonsingular free ergodic actions Γ (X, µ) on standard measure spaces of amalgamated groups Γ = Γ1 * Σ Γ2 over a finite subgroup Σ

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Section: By Lemma 23 and Theorem 25 We Getmentioning

confidence: 99%

Amalgamated free product type III factors with at most one Cartan subalgebra

Boutonnet¹,

Houdayer²,

Raum³

2013

Compositio Math.

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“…Take ω ∈ Z 2 ( G) c with β = ω * ω 21 from Theorem 3. 16. In fact, we may and do assume that ω ∈ R β ⊗ R β because β is a skew symmetric bicharacter on R β .…”

mentioning

confidence: 99%

Classification of actions of discrete Kac algebras on injective factors

Masuda

Tomatsu

2017

Memoirs of the AMS

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We will study two kinds of actions of a discrete amenable Kac algebra. The first one is an action whose modular part is normal. We will construct a new invariant which generalizes a characteristic invariant for a discrete group action, and we will present a complete classification. The second is a centrally free action. By constructing a Rohlin tower in an asymptotic centralizer, we will show that the Connes-Takesaki module is a complete invariant.2000 Mathematics Subject Classification. Primary 46L65; Secondary 46L55. 1 4 a ∈ M, the functionals ϕa and aϕ in M * are defined by ϕa(x) := ϕ(ax) and aϕ(x) := ϕ(xa) for all x ∈ M. We denote by U(M), Z(M) and W (M) the set of all unitary elements in M, the center of M and the set of faithful normal semifinite weights on M, respectively. For a positive φ ∈ M * , we set |x| φ := φ(|x|),Denote by Aut(M) the set of automorphisms on M. For α ∈ Aut(M) and ϕ ∈ M * , we let α(ϕ) := ϕ • α −1 . Note that Ad u(ϕ) = uϕu * for a unitary u, where Ad u(x) := uxu * for x ∈ M.Let H ⊂ M be a subspace. We say that H is a Hilbert space in M if H ⊂ M is σ-weakly closed and η * ξ ∈ C for all ξ, η ∈ H [52]. Then the coupling ξ, η := η * ξ gives a complete inner product on H . The smallest projection e ∈ M such that eH = H is called the support of H . If the support of H equals 1, then we have the endomorphism ρ H defined by ρ H (x) = i v i xv * i , where {v i } i is an orthonormal basis. We say that an endomorphism σ is inner when σ = ρ H for some Hilbert space H with support 1.Next we recall theory of endomorphisms on a factor. Our standard references are [22,23,40,41]. We denote by End(M) and Sect(M) the set of normal endomorphisms and sectors on M, that is, Sect(M) is the set of unitary equivalence classes of endomorphisms on M. For two endomorphisms ρ, σ ∈ End(M), we letFor a factor M, we denote by End(M) 0 the set of endomorphisms with finite Jones-Kosaki index. For ρ ∈ End(M) 0 , E ρ denotes the minimal expectation from M onto ρ(M) in the sense of [20]. We define the standard left inverse of ρ by1.2. Canonical extensions. In [23], Izumi introduced the canonical extension of an endomorphism, which plays an important role in this paper.Let M be a factor and M the core of M as defined in [16, Definition 2.5]. The core M is the von Neumann algebra generated by M and a one-parameter unitary group {λ ϕ (t)} t∈R , ϕ ∈ W (M), satisfying the following relations:The action θ is called the dual action, and the flow of weights of M means the restriction of θ on the center Z( M).Definition 1.1 (Izumi). Let ρ be an endomorphism on a factor M with finite index. Then the canonical extension ρ of ρ is the endomorphism on M defined byfor all x ∈ M, t ∈ R and ϕ ∈ W (M).Note that ρ commutes with the dual action θ. Conversely, if β ∈ End( M ) is commuting with θ, then the restriction β| M gives an endomorphism on M because of M = M θ , the fixed point algebra by θ. In the following lemma, we study a relationship between β and β| M . On a probability index for an inclusion of von Neumann algebras, readers ar...

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“…In particular, if we restrict the tensor product isomorphism from C ℜ≥0 to I and allow µ ∈ W sf (M ) in the definition of smooth sections, we obtain a convolution product on the space of distributional sections of L restricted to I with bounded Fourier transform, which turns it into an I-graded von Neumann algebra (the grading is the composition of the isomorphism R → Hom(I, U) and the multiplication action), which is canonically isomorphic toM . This resembles the approach used by Falcone and Takesaki [38,Theorem 2.4] to constructM . The case ofM is similar if we do not restrict to I.…”

Section: 33mentioning

confidence: 93%

“…[34] reformulated the original results of Haagerup [15] in a more convenient algebraic setting of modular algebras and defined L a (M ) for an arbitrary a ∈ C ℜ≥0 . Sherman [39] along with Falcone and Takesaki [38] give alternative expositions. The principal idea of the modular algebra approach is to construct a C-graded *-algebra (the modular algebra of M ) whose component in grading a ∈ C happens to be the space L a (M ) (zero for ℜa < 0).…”

Section: Another Series Of Approaches Usesmentioning

confidence: 99%

Algebraic tensor products and internal homs of noncommutative L -spaces

Pavlov

2017

Journal of Mathematical Analysis and Applications

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We prove that the multiplication map L a (M ) ⊗ M L b (M ) → L a+b (M ) is an isometric isomorphism of (quasi)Banach M -M -bimodules. Here L a (M ) = L 1/a (M ) is the noncommutative L p -space of an arbitrary von Neumann algebra M and ⊗ M denotes the algebraic tensor product over M equipped with the (quasi)projective tensor norm, but without any kind of completion. Similarly, the left multiplication map L a (M ) → Hom M (L b (M ), L a+b (M )) is an isometric isomorphism of (quasi)Banach M -M -bimodules, where Hom M denotes the algebraic internal hom without any continuity assumptions. In a forthcoming paper these results will be applied to L p -modules introduced by Junge and Sherman, establishing explicit algebraic equivalences between the categories of right L p (M )-modules for all p ≥ 0.

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The Non-commutative Flow of Weights on a Von Neumann Algebra

Cited by 33 publications

References 27 publications

Amalgamated free product type III factors with at most one Cartan subalgebra

Amalgamated free product type III factors with at most one Cartan subalgebra

Classification of actions of discrete Kac algebras on injective factors

Algebraic tensor products and internal homs of noncommutative L -spaces

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