Showing papers on "Von Neumann algebra published in 2016"
TL;DR: It is shown that for a finite von Neumann algebra, the states that maximise Segal’s entropy with a given energy level are Gibbs states.
Abstract: We show that for a finite von Neumann algebra, the states that maximise Segal’s entropy with a given energy level are Gibbs states. This is a counterpart of the classical result for the algebra of all bounded linear operators on a Hilbert space and von Neumann entropy.
194 citations
TL;DR: In particular, it is proved the possibility to perform privacy amplification and classical data compression with quantum side information modeled by a von Neumann algebra.
Abstract: We discuss information-theoretic concepts on infinite-dimensional quantum systems. In particular, we lift the smooth entropy formalism as introduced by Renner and collaborators for finite-dimensional systems to von Neumann algebras. For the smooth conditional min- and max-entropy, we recover similar characterizing properties and information-theoretic operational interpretations as in the finite-dimensional case. We generalize the entropic uncertainty relation with quantum side information of Tomamichel and Renner and discuss applications to quantum cryptography. In particular, we prove the possibility to perform privacy amplification and classical data compression with quantum side information modeled by a von Neumann algebra.
46 citations
TL;DR: In this paper, the existence of a Haar trace-preserving embedding of the von Neumann algebra L ∞(G) into an ultrapower of the hyperfinite II_1-factor was studied.
Abstract: For a compact quantum group G of Kac type, we study the existence of a Haar trace-preserving embedding of the von Neumann algebra L^∞(G) into an ultrapower of the hyperfinite II_1-factor (the Connes embedding property for L^∞(G)). We establish a connection between the Connes embedding property for L^∞(G) and the structure of certain quantum subgroups of G, and use this to prove that the II_1-factors L^∞(O_N^+) and L^∞(U_N^+)
associated to the free orthogonal and free unitary quantum groups have the Connes embedding property for all N >= 4. As an application, we deduce that the free entropy dimension of the standard generators of L^∞(O_N^+) equals 1 for all N >= 4. We also mention an application of our work to the problem of classifying the quantum subgroups of O_N^+.
43 citations
TL;DR: In this article, a representation of the Green's function of the classical Neumann problem for the Poisson equation in the unit ball of arbitrary dimension is given, which can be expressed in terms of elementary functions and its explicit form can be written out.
Abstract: Representation of the Green’s function of the classical Neumann problem for the Poisson equation in the unit ball of arbitrary dimension is given. In constructing this function, we use the representation of the fundamental solution of the Laplace equation in the form of a series. It is shown that the Green’s function can be represented in terms of elementary functions and its explicit form can be written out. An explicit form of the Neumann kernel at and .
41 citations
TL;DR: In this article, a new class of non-amenable groups, denoted by ${\bf NC}_1 \cap {\bf Quot}(\mathcal C_{rss})$ which give rise to $\textit{prime}$ von Neumann algebras was introduced.
Abstract: In this paper we introduce a new class of non-amenable groups denoted by ${\bf NC}_1 \cap {\bf Quot}(\mathcal C_{rss})$ which give rise to $\textit{prime}$ von Neumann algebras. This means that for every $\Gamma \in {\bf NC}_1 \cap {\bf Quot}(\mathcal C_{rss})$ its group von Neumann algebra $L(\Gamma)$ cannot be decomposed as a tensor product of diffuse von Neumann algebras. We show ${\bf NC}_1 \cap {\bf Quot}(\mathcal C_{rss})$ is fairly large as it contains many examples of groups intensively studied in various areas of mathematics, notably: all infinite central quotients of pure surface braid groups; all mapping class groups of (punctured) surfaces of genus $0,1,2$; most Torelli groups and Johnson kernels of (punctured) surfaces of genus $0,1,2$; and, all groups hyperbolic relative to finite families of residually finite, exact, infinite, proper subgroups.
32 citations
TL;DR: In this paper, a representation theory of completely positive (CP) instruments on arbitrary von Neumann algebras is discussed, and a condition called the normal extension property (NEP) is introduced to establish a correspondence between CP instruments with the NEP and statistical equivalence classes of measuring processes.
Abstract: In this paper, we aim to establish foundations of measurement theory in local quantum physics. For this purpose, we discuss a representation theory of completely positive (CP) instruments on arbitrary von Neumann algebras. We introduce a condition called the normal extension property (NEP) and establish a one-to-one correspondence between CP instruments with the NEP and statistical equivalence classes of measuring processes. We show that every CP instrument on an atomic von Neumann algebra has the NEP, extending the well-known result for type I factors. Moreover, we show that every CP instrument on an injective von Neumann algebra is approximated by CP instruments with the NEP. The concept of posterior states is also discussed to show that the NEP is equivalent to the existence of a strongly measurable family of posterior states for every normal state. Two examples of CP instruments without the NEP are obtained from this result. It is thus concluded that in local quantum physics not every CP instrument represents a measuring process, but in most of physically relevant cases every CP instrument can be realized by a measuring process within arbitrary error limits, as every approximately finite dimensional von Neumann algebra on a separable Hilbert space is injective. To conclude the paper, the concept of local measurement in algebraic quantum field theory is examined in our framework. In the setting of the Doplicher-Haag-Roberts and Doplicher-Roberts theory describing local excitations, we show that an instrument on a local algebra can be extended to a local instrument on the global algebra if and only if it is a CP instrument with the NEP, provided that the split property holds for the net of local algebras.
31 citations
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10 Feb 2016
TL;DR: In this article, it was shown that the Dyadic group can also be approximated by an increasing sequence of finite dimensional subalgebras with a free, dense orbit.
Abstract: Let S be the Stone space of a complete, non-atomic, Boolean algebra Let G be a countably infinite group of homeomorphisms of S Let the action of G on S have a free dense orbit Then we prove that, on a generic subset of S, the orbit equivalence relation coming from this action can also be obtained by an action of the Dyadic Group, L Z2 As an application, we show that if M is the monotone cross product C�-algebra, arising from the natural action of G on C(S), and if the projection lattice in C(S) is countably generated then M can be approximated by an increasing sequence of finite dimensional subalgebras On each S, in a class considered earlier, we construct a natural action of L Z2 with a free, dense orbit Using this we exhibit a huge family of small monotone complete C � -algebras, (B�,� ∈ �) with the following properties: (i) Each Bis a Type III factor which is not a von Neumann algebra (ii) Each Bis a quotient of the Pedersen-Borel envelope of the Fermion algebra and hence is strongly hyperfinite The cardinality of � is 2 c , where c = 2 @0 When � 6 µ then Band Bµ take different values in the classification semi-group; in particular, they cannot be isomorphic
26 citations
TL;DR: In this paper, the *-Jordan derivation on a factor von Neumann algebra has been studied, where the derivation for every factor is additive *-derivation, and the derivations on the factor are shown to be additive.
Abstract: Let be a factor von Neumann algebra and be the *-Jordan derivation on , that is, for every , where , then is additive *-derivation.
26 citations
TL;DR: In this paper, a generalized framework for private quantum codes using von Neumann algebras and the structure of commutants is introduced, which leads naturally to a more general notion of complementary channel, which is used to establish a generalized complementarity theorem between private and correctable subalgebra that applies to both the finite and infinite dimensional settings.
Abstract: We introduce a generalized framework for private quantum codes using von Neumann algebras and the structure of commutants. This leads naturally to a more general notion of complementary channel, which we use to establish a generalized complementarity theorem between private and correctable subalgebras that applies to both the finite and infinite-dimensional settings. Linear bosonic channels are considered and specific examples of Gaussian quantum channels are given to illustrate the new framework together with the complementarity theorem.
24 citations
TL;DR: In this paper, the authors consider the closed algebra A d generated by polynomial multipliers on the Drury-Arveson space and identify A d ⁎ as a direct sum of the preduals of the full multiplier algebra and of a commutative von Neumann algebra.
24 citations
TL;DR: In this paper, the authors studied analogues of classical Hilbert transforms as fourier multipliers on free groups and proved their complete boundedness on non commutative spaces associated with the free group von Neumann algebras for all $1
Abstract: We study analogues of classical Hilbert transforms as fourier multipliers on free groups. We prove their complete boundedness on non commutative $L^p$ spaces associated with the free group von Neumann algebras for all $1
TL;DR: In this article, a bijective map Φ between two von Neumann algebras, one of which has no central abelian projections, was investigated, and it was shown that Φ(I)Φ is a sum of a linear ∗-isomorphism and a conjugate linear ∆-ISomorphism, where I is a self-adjoint central element in the range with Φ (I)2=I.
Abstract: This article investigates a bijective map Φ between two von Neumann algebras, one of which has no central abelian projections, satisfying Φ([[A,B]∗,C]∗)=[[Φ(A),Φ(B)]∗,Φ(C)]∗ for all A,B,C in the domain, where [A,B]∗=AB−BA* is the skew Lie product of A and B. We show that the map Φ(I)Φ is a sum of a linear ∗-isomorphism and a conjugate linear ∗-isomorphism, where Φ(I) is a self-adjoint central element in the range with Φ(I)2=I.
TL;DR: In this article, the authors give a list of axioms that completely determine the sequential product on all von Neumann algebras simultaneously (Theorem 4) and give a proof of the correctness of the results.
Abstract: We study the sequential product the operation p∗q=pqp on the set of effects, [0, 1]𝒜, of a von Neumann algebra 𝒜 that represents sequential measurement of first p and then q. In their work [J. Math. Phys. 49(5), 052106 (2008)], Gudder and Latremoliere give a list of axioms based on physical grounds that completely determines the sequential product on a von Neumann algebra of type I, that is, a von Neumann algebra ℬ(ℋ) of all bounded operators on some Hilbert space ℋ. In this paper we give a list of axioms that completely determines the sequential product on all von Neumann algebras simultaneously (Theorem 4).
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TL;DR: A model of Selinger and Valiron's quantum lambda calculus based on von Neumann algebras is presented, and it is shown that the model is adequate with respect to the operational semantics.
Abstract: We present a model of Selinger and Valiron's quantum lambda calculus based on von Neumann algebras, and show that the model is adequate with respect to the operational semantics.
TL;DR: In particular, if the von Neumann algebra is infinite type I factors, a concrete characterization of is given as discussed by the authors, where a nonlinear skew Jordan derivable map is an additive -derivation.
Abstract: Let be a factor von Neumann algebra. Suppose that is a nonlinear skew Jordan derivable map. Then, is an additive -derivation. In particular, if the von Neumann algebra is infinite type I factors, a concrete characterization of is given.
TL;DR: In this paper, the authors show that the Hecke algebra C q [W ] of a Coxeter group W, associated to parameter q, can be completed to a von Neumann algebra N q (W ) in the case where W is right-angled.
TL;DR: In this article, the theory of direct integral decompositions of both bounded and unbounded operators is further developed; in particular, results about spectral projections, functional calculus and affiliation to von Neumann algebras are proved.
TL;DR: In this article, the authors provide criteria for self-adjointness and −homness of first and second order differential operators acting on sections of infinite dimensional bundles, whose fibers are modules of finite type over a von Neumann algebra endowed with a trace.
Abstract: We provide criteria for self-adjointness and {\tau}-Fredhomness of first and second order differential operators acting on sections of infinite dimensional bundles, whose fibers are modules of finite type over a von Neumann algebra A endowed with a trace {\tau}. We extend the Callias-type index to operators acting on sections of such bundles and show that this index is stable under compact perturbations.
TL;DR: In this paper, it was shown that the predual of any von Neumann algebra is 1-Plichko, i.e., it has a countably 1-norming Markushevich basis.
Abstract: We prove that the predual of any von Neumann algebra is 1-Plichko, i.e., it has a countably 1-norming Markushevich basis. This answers a question of the third author who proved the same for preduals of semifinite von Neumann algebras. As a corollary we obtain an easier proof of a result of U. Haagerup that the predual of any von Neumann algebra enjoys the separable complementation property. We further prove that the selfadjoint part of the predual is 1-Plichko as well.
TL;DR: In this article, the authors give a sufficient and necessary condition for a self-adjoint contraction to be a difference of a pair of orthogonal projections, and characterize the von Neumann algebra generated by such pairs.
TL;DR: In this article, the dual of a von Neumann-Schatten p-frame in separable Banach spaces is discussed, and a classical perturbation result to von-Neumann p-frames is presented.
Abstract: In this paper, we discuss the dual of a von Neumann–Schatten p-frames in separable Banach spaces and obtain some of their characterizations. Moreover, we present a classical perturbation result to von Neumann–Schatten p-frames.
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TL;DR: In this paper, a spectral gap characterization of fullness for the continuous core of a type of von Neumann algebra is given, and it is shown that the core is full if and only if its topology is the usual topology on the plane.
Abstract: We give a spectral gap characterization of fullness for type $\mathrm{III}$ factors which is the analog of a theorem of Connes in the tracial case. Using this criterion, we generalize a theorem of Jones by proving that if $M$ is a full factor and $\sigma : G \rightarrow \mathrm{Aut}(M)$ is an outer action of a discrete group $G$ whose image in $\mathrm{Out}(M)$ is discrete then the crossed product von Neumann algebra $M \rtimes_\sigma G$ is also a full factor. We apply this result to prove the following conjecture of Tomatsu-Ueda: the continuous core of a type $\mathrm{III}_1$ factor $M$ is full if and only if $M$ is full and its $\tau$ invariant is the usual topology on $\mathbb{R}$.
TL;DR: In this paper, a τ is a faithful normal semianninite trace on von Neumann algebra M, where τ is defined as a τ-compact operator and B is a τ compact operator such that |A| log+|A|, ep|B| ∈ L1(M, τ) then AB,BA ∈ M, τ.
Abstract: Let τ be a faithful normal semifinite trace on von Neumann algebra M, 0 0. If A is a τ-compact operator and \(B \in \widetilde M\) is such that |A| log+|A|, ep|B| ∈ L1(M, τ) then AB,BA ∈ L1(M, τ).
TL;DR: In this paper, a matrix trace inequality for von Neumann algebras was extended to semi-final von Nomenclature alges, and a generalization of the Lieb-Thirring-type inequality in von Numann algesbras due to Kosaki was given.
Abstract: This paper extends a recent matrix trace inequality of Bourin–Lee to semifinite von Neumann algebras. This provides a generalization of the Lieb–Thirring-type inequality in von Neumann algebras due to Kosaki. Some new inequalities, even in the matrix case, are also given for the Heinz means.
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TL;DR: For a noncommutative Orlicz space associated with a semifnite von Neumann algebra, a faithful normal semidefinite trace and an OrlicZ function satisfying the $(\delta_2,\Delta_2)-condition, an individual ergodic theorem is proved as mentioned in this paper.
Abstract: For a noncommutative Orlicz space associated with a semifnite von Neumann algebra, a faithful normal semifnite trace and an Orlicz function satisfying $(\delta_2,\Delta_2)-$condition, an individual ergodic theorem is proved.
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23 Oct 2016TL;DR: In this paper, the Rohlin property for flows on von Neumann algebras and classify them up to strong cocycle conjugacy was introduced. But this result was not applicable to flows on the injective type II$_1$ factor, the classification of injective Type III factors due to Connes, Krieger and Haagerup and the non-fullness of type III$_0$ factors.
Abstract: We will introduce the Rohlin property for flows on von Neumann algebras and classify them up to strong cocycle conjugacy. This result provides alternative approaches to some preceding results such as Kawahigashi's classification of flows on the injective type II$_1$ factor, the classification of injective type III factors due to Connes, Krieger and Haagerup and the non-fullness of type III$_0$ factors. Several concrete examples are also studied.
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TL;DR: In this paper, it was shown that a synaptic algebra is a vector lattice if and only if the elements of the identity element and the carrier of the element are pairwise connected.
Abstract: A synaptic algebra $A$ is a generalization of the self-adjoint part of a von Neumann algebra. We study a linear subspace $V$ of $A$ in regard to the question of when $V$ is a vector lattice. Our main theorem states that if $V$ contains the identity element of $A$ and is closed under the formation of both the absolute value and the carrier of its elements, then $V$ is a vector lattice if and only if the elements of $V$ commute pairwise.
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TL;DR: In this paper, the authors present a survey of derivations, local derivations and 2-local derivations on various algebras of measurable operators affiliated with von Neumann algesbras.
Abstract: The present paper presents a survey of some recent results devoted to derivations, local derivations and 2-local derivations on various algebras of measurable operators affiliated with von Neumann algebras. We give a complete description of derivation on these algebras, except the case where the von Neumann algebra is of type II$_1$. In the latter case the result is obtained under an extra condition of measure continuity of derivations. Local and 2-local derivations on the above algebras are also considered. We give sufficient conditions on a von Neumann algebra $M$, under which every local or 2-local derivation on the algebra of measurable operators affiliated with $M$ is automatically becomes a derivation. We also give examples of commutative algebras of measurable operators admitting local and 2-local derivations which are not derivations.
01 Feb 2016
TL;DR: In this paper, a new classification in terms of extensions of inverse semigroups was proposed, which is more algebraic in character and less point-based than that of Feldman and Moore.
Abstract: In the 1970s, Feldman and Moore classified separably acting von Neumann algebras containing Cartan maximal abelian self-adjoint subalgebras (MASAs) using measured equivalence relations and 2-cocycles on such equivalence relations. In this paper we give a new classification in terms of extensions of inverse semigroups. Our approach is more algebraic in character and less point-based than that of Feldman and Moore. As an application, we give a restatement of the spectral theorem for bimodules in terms of subsets of inverse semigroups. We also show how our viewpoint leads naturally to a description of maximal subdiagonal algebras.
TL;DR: In this article, Jozsa et al. studied the fidelity of a pair of mixed quantum states in the context of two types of operator algebras: algebraic and operator theoretic.
Abstract: Josza’s definition of fidelity [R. Jozsa, J. Mod. Opt. 41(12), 2315–2323 (1994)] for a pair of (mixed) quantum states is studied in the context of two types of operator algebras. The first setting is mainly algebraic in that it involves unital C∗-algebras A that possess a faithful trace functional τ. In this context, the role of quantum states (that is, density operators) in the classical quantum-mechanical framework is assumed by positive elements ρ ∈ A for which τ(ρ) = 1. The second setting is more operator theoretic: by fixing a faithful normal semifinite trace τ on a semifinite von Neumann algebra M, we define and consider the fidelity of pairs of positive operators in M of unit trace. The main results of this paper address monotonicity and preservation of fidelity under the action of certain trace-preserving positive linear maps of A or of the predual M∗. Our results in the von Neumann algebra setting are novel in that we focus on the Schrodinger picture rather than the Heisenberg picture, and they a...