We prove that certain free products of factors of type ${\rm I}$ and other von Neumann algebras with respect to nontracial, almost periodic states are almost periodic free Araki-Woods factors. In particular, they have the free absorption property and Connes' Sd invariant completely classifies these free products. For example, for $λ, μ\in ]0, 1[$, we show that $$(M_2(\C), ω_λ) \ast (M_2(\C), ω_μ)$$ is isomorphic to the free Araki-Woods factor whose Sd invariant is the subgroup of $\R^*_+$ generated by $λ$ and $μ$. Our proofs are based on algebraic techniques and amalgamated free products. These results give some answers to questions of Dykema and Shlyakhtenko.
Using Speicher central limit Theorem we provide Hiai's q-Araki-Woods von Neumann algebras with nice asymptotic matricial models. Then, we use this model and an elaborated ultraproduct procedure, to show that all q-Araki-Woods von Neumann algebras are QWEP.
We show that Ozawa's recent results on solid von Neumann algebras imply that there are free Araki-Woods factors, which fail to have free absorption. We also show that a free Araki-Woods factors $Γ(μ, n)$ associated to a measure and a multiplicity function $n$ may non-trivially depend on the multiplicity function.
We extend the class of contractions for which the second quantisation on $q$-Araki-Woods algebras can be defined. As a corollary, we prove that all $q$-Araki-Woods algebras possess the Haagerup approximation property.
The generating abelian subalgebras arsing from vectors in the ergodic component of Hiai's construction of the q-deformed Araki-Woods von Neumann algebras are quasi-split.
We show that any amenable von Neumann subalgebra of any free Araki-Woods factor that is globally invariant under the modular automorphism group of the free quasi-free state is necessarily contained in the almost periodic free summand.
We show that Shlyakhtenko's free Araki-Woods factors are strongly solid, meaning that for any diffuse amenable von Neumann subalgebra that is the range of a normal conditional expectation, the normalizer remains amenable. This provides the first class of nonamenable strongly solid type III factors.
The $q$-deformed Araki-Woods von Neumann algebras $Γ_q(\mathcal{H}_\mathbb{R}, U_t)^{\prime \prime}$ are factors for all $q\in (-1,1)$ whenever $dim(\mathcal{H}_\mathbb{R})\geq 3$. When $dim(\mathcal{H}_\mathbb{R})=2$ they are factors as well for all $q$ so long as the parameter defining $(U_t)$ is `small' or $1$ $($trivial$)$ as the case may be.
We show that any free product of finite-dimensional von Neumann algebras equipped with non-tracial states is isomorphic to a free Araki-Woods factor with its free quasi-free state possibly direct sum a finite-dimensional von Neumann algebra. This gives a complete answer to questions posed by Dykema and Shlyakhtenko, which had been partially answered by work of Houdayer and work of Ueda. We also extend this to suitable infinite-dimensional von Neumann algebras with almost periodic states.
By adapting an ultraproduct technique of Junge and Zeng, we prove that radial completely bounded multipliers on $q$-Gaussian algebras transfer to $q$-Araki-Woods algebras. As a consequence, we establish the $w^{\ast}$-complete metric approximation property for all $q$-Araki-Woods algebras. We apply the latter result to show that the canonical ultraweakly dense C$^\ast$-subalgebras of $q$-Araki-Woods algebras are always QWEP.
We introduce a new invariant \mathcal{S}(M) for type III factors M with no almost-periodic weights. We compute this invariant for certain free Araki-Woods factors. We show that Connes' invariant τcannot distinguish all isomorphism classes of free Araki-Woods factors. We show that there exists a continuum of mutually non-isomorphic free Araki-Woods factors, each without almost-periodic weights.
We consider semi-group BMO-spaces associated with arbitrary von Neumann algebras and prove interpolation theorems. This extends results by Junge-Mei for the tracial case. We give examples of multipliers on free Araki-Woods algebras and in particular we find $L_\infty \rightarrow {\rm BMO}$ multipliers. We also provide $L_p$-bounds for a natural generalization of the Hilbert transform.
To any strongly continuous orthogonal representation of $\R$ on a real Hilbert space $\CH_\R$, Hiai constructed $q$-deformed Araki-Woods von Neumann algebras for $-1< q< 1$, which are $W^{\ast}$-algebras arising from non tracial representations of the $q$-commutation relations, the latter yielding an interpolation between the Bosonic and Fermionic statistics. We prove that if the orthogonal representation is not ergodic then these von Neumann algebras are factors whenever $dim(\CH_\R)\geq 2$ and $q\in (-1,1)$. In such case, the centralizer of the $q$-quasi free state has trivial relative commu…
Using Baire category techniques we prove that Araki-Woods factors are not classifiable by countable structures. As a result, we obtain a far reaching strengthening as well as a new proof of the well-known theorem of Woods that the isomorphism problem for ITPFI factors is not smooth. We derive as a consequence that the odometer actions of Z that preserve the measure class of a finite non-atomic product measure are not classifiable up to orbit equivalence by countable structures.
Extending a work of Carlen and Lieb, Biane has obtained the optimal hypercontractivity of the $q$-Ornstein-Uhlenbeck semigroup on the $q$-deformation of the free group algebra. In this note, we look for an extension of this result to the type III situation, that is for the $q$-Araki-Woods algebras. We show that hypercontractivity from $L^p$ to $L^2$ can occur if and only if the generator of the deformation is bounded.
We define the standard Borel space of free Araki-Woods factors and prove that their isomorphism relation is not classifiable by countable structures. We also prove that equality of $τ$-topologies, arising as invariants of type III factors, as well as coycle and outer conjugacy of actions of abelian groups on free product factors are not classifiable by countable structures.
We show that the operator Hilbert space OH introduced by Pisier embeds into the predual of the hyerfinite III1 factor. The main new tool is a Khintchine type inequality for the generators of the CAR algebra with respect to a quasi-free state. Our approach yields a Khintchine type inequality for the q-gaussian variables for all values q between -1 and 1. These results are closely related to recent results of Pisier and Shlyakhtenko in the free case.
This is a survey paper on the work of D. Shlyakhtenko on free Araki-Woods factors. The paper was written (in French) on the occasion of the Bourbaki seminar. Our original contribution consists in computing the tau invariant for arbitrary free Araki-Woods factors.
Let $(H_{\mathbf{R}}, U_t)$ be any strongly continuous orthogonal representation of $\mathbf{R}$ on a real (separable) Hilbert space $H_{\mathbf{R}}$. For any $q\in (-1,1)$, we denote by $Γ_q(H_{\mathbf{R}},U_t)^{\prime\prime}$ the $q$-deformed Araki-Woods algebra introduced by Shlyakhtenko and Hiai. In this paper, we prove that $Γ_q(H_{\mathbf{R}},U_t)^{\prime\prime}$ has trivial bicentralizer if it is a type $\rm III_1$ factor. In particular, we obtain that $Γ_q(H_{\mathbf{R}},U_t)^{\prime\prime}$ always admits a maximal abelian subalgebra that is the range of a faithful normal conditional e…
We show that for each (0<λ<1), the free Araki-Woods factor of type III(_λ) cannot be written as a tensor product of two diffuse von Neumann algebras (i.e., is prime), and does not contain a Cartan subalgebra.
We prove that the mixed $q$-Gaussian algebra $Γ_{Q}(H_{\mathbb{R}})$ associated to a real Hilbert space $H_{\mathbb{R}}$ and a real symmetric matrix $Q=(q_{ij})$ with $\sup|q_{ij}|<1$, is a factor as soon as $\dim H_{\mathbb{R}}\geq2$. We also discuss the factoriality of $q$-deformed Araki-Woods algebras, in particular showing that the $q$-deformed Araki-Woods algebra $Γ_{q}(H_{\mathbb{R}},U_{t})$ given by a real Hilbert space $H_{\mathbb{R}}$ and a strongly continuous group $U_{t}$ is a factor when $\dim H_{\mathbb{R}}\geq2$ and $U_{t}$ admits an invariant eigenvector.
We prove that the von Neumann algebra generated by q-gaussians is not injective as soon as the dimension of the underlying Hilbert space is greater than 1. Our approach is based on a vector valued Khintchine type inequality for Wick products. The same proof also works for the more general setting of a Yang-Baxter deformation. Our techniques can also be extended to the so called q-Araki-Woods von Neumann algebras recently introduced by Hiai. In this latter case, we obtain the non injectivity under some asssumption on the spectral set of the positive operator associated with the deformation.
We construct interacting quantum fields in 1+1 space-time dimensions, representing charged or neutral scalar bosons at positive temperature and zero chemical potential. Our work is based on prior work by Klein and Landau and Hoegh-Krohn. Generalized path space methods are used to add a spatially cut-off interaction to the free system, which is described in the Araki-Woods representation. It is shown that the interacting KMS state is normal w.r.t. the Araki-Woods representation. The observable algebra and the modular conjugation of the interacting system are shown to be identical to the ones of…
We prove that, for any type III$_1$ free product factor, its continuous core is full if and only if its $τ$-invariant is the usual topology on the real line. This trivially implies, as a particular case, the same result for free Araki--Woods factors. Moreover, our method shows the same result for full (generalized) Bernoulli crossed product factors of type III$_1$.
We show that Araki and Masuda's weighted non-commutative vector valued $L_p$-spaces [Araki \& Masuda, Publ. Res. Inst. Math. Sci., 18:339 (1982)] correspond to an algebraic generalization of the sandwiched Rényi divergences with parameter $α= \frac{p}{2}$. Using complex interpolation theory, we prove various fundamental properties of these divergences in the setup of von Neumann algebras, including a data-processing inequality and monotonicity in $α$. We thereby also give new proofs for the corresponding finite-dimensional properties. We discuss the limiting cases $α\to \{\frac{1}{2},1,\infty\…
We present some results concerning the relative modular operator in semifinite von Neumann algebras. These results allow one to prove some basic formula for trace, to obtain equivalence between Araki's relative entropy and Umegaki's information as well as to derive some formulae for quasi-entropies, and Rényi's relative entropy known in finite dimension.
We make a systematic study of standard $f$-divergences in general von Neumann algebras. An important ingredient of our study is to extend Kosaki's variational expression of the relative entropy to an arbitary standard $f$-divergence, from which most of the important properties of standard $f$-divergences follow immediately. In a similar manner we give a comprehensive exposition on the Rényi divergence in von Neumann algebra. Some results on relative hamiltonians formerly studied by Araki and Donald are improved as a by-product.
We introduce the notion of the $α$-Haagerup approximation property for $α\in[0,1/2]$ using a one-parameter family of positive cones studied by Araki and show that the $α$-Haagerup approximation property actually does not depend on a choice of $α$. This gives us a direct proof of the fact that two characterizations of the Haagerup approximation property are equivalent, one in terms of the standard form and the other in terms of completely positive maps. We also discuss the $L^p$-Haagerup approximation property for a non-commutative $L^p$-spaces associated with a von Neumann algebra ($1<p<\infty…
A breakthrough took place in the von Neumann algebra theory when the Tomita-Takesaki theory was established around 1970. Since then, many important issues in the theory were developed through 1970's by Araki, Connes, Haagerup, Takesaki and others, which are already very classics of the von Neumann algebra theory. Nevertheless, it seems still difficult for beginners to access them, though a few big volumes on the theory are available. These lecture notes are delivered as an intensive course in 2019, April at Department of Mathematical Analysis, Budapest University of Technology and Economics. T…
Given a finite, directed, connected graph $Γ$ equipped with a weighting $μ$ on its edges, we provide a construction of a von Neumann algebra equipped with a faithful, normal, positive linear functional $(\mathcal{M}(Γ,μ),φ)$. When the weighting $μ$ is instead on the vertices of $Γ$, the first author showed the isomorphism class of $(\mathcal{M}(Γ,μ),φ)$ depends only on the data $(Γ,μ)$ and is an interpolated free group factor equipped with a scaling of its unique trace (possibly direct sum copies of $\mathbb{C}$). Moreover, the free dimension of the interpolated free group factor is easily com…
In the eighties, A. Connes and E. J. Woods made a connection between hyperfinite von Neumann algebras and Poisson boundaries of time dependent random walks. The present paper explains this connection and gives a detailed proof of two theorems quoted there: the construction of a large class of states on a hyperfinite von Neumann algebra (due to A. Connes) and the ergodic decomposition of a Markov measure via harmonic functions (a classical result in probability theory). The crux of the first theorem is a model for conditional expectations on finite dimensional C*-algebras. The proof of the seco…
We study the modular Hamiltonian associated with a Gaussian state on the Weyl algebra. We obtain necessary/sufficient criteria for the local equivalence of Gaussian states, independently of the classical results by Araki and Yamagami, Van Daele, Holevo. We then present a criterion for a Bogoliubov automorphism to be weakly inner in the GNS representation. We also describe the vacuum modular Hamiltonian associated with a time-zero interval in the scalar, massless, free QFT in two spacetime dimensions, thus complementing the recent results in higher space dimensions. In particular, we have the f…
The two-sided Bogoliubov inequality in von Neumann algebras conceptualizes the free energy--quantum correlations link
Zuse Institute Berlin（柏林齐谢研究所）；Institute of Mathematics, Freie Universität Berlin（柏林自由大学数学研究所）；Institute of Theoretical Physics, Universität Leipzig（莱比锡大学理论物理研究所）
Trace inequalities are general techniques with many applications in quantum information theory, often replacing classical functional calculus in noncommutative settings. The physics of quantum field theory and holography, however, motivate entropy inequalities in type III von Neumann algebras that lack a semifinite trace. The Haagerup and Kosaki $L_p$ spaces enable re-expressing trace inequalities in non-tracial von Neumann algebras. In particular, we show this for the generalized Araki-Lieb-Thirring and Golden-Thompson inequalities from (Sutter, Berta \& Tomamichel 2017). Then, using the Haag…
