Neumann Algebra Research Articles

Groupoid Approach to Ergodic Dynamical System of Commutative von Neumann Algebra

Groupoid Approach to Ergodic Dynamical System of Commutative von Neumann Algebra

Unitary connections on Bratteli diagrams

In this paper, we extend Ocneanu’s theory of connections on graphs to define a 2-category whose 0-cells are tracial Bratteli diagrams, and whose 1-cells are generalizations of unitary connections. We show that this 2-category admits an embedding into the 2-category of hyperfinite von Neumann algebras, generalizing fundamental results from subfactor theory to a 2-categorical setting.

On bounded coordinates in bimodules

We provide a comprehensive study on uniformly left [Formula: see text]-bounded (respectively, [Formula: see text]-bounded) orthonormal bases in infinite-dimensional cyclic bimodules associated with c.p. maps between two von Neumann algebras [Formula: see text] and [Formula: see text], where [Formula: see text] and [Formula: see text] are faithful normal states on [Formula: see text] and [Formula: see text], respectively. Separate investigations on cyclic bimodules associated with Markov maps and arbitrary c.p. maps are also provided since the results differ. This generalizes the results of the authors’ previous work on Kadison’s unitary basis problem.

Cartan subalgebras in von Neumann algebras associated with graph product groups

We completely classify the Cartan subalgebras in all von Neumann algebras associated with graph product groups and their free ergodic measure preserving actions on probability spaces.

Bimodule coefficients, Riesz transforms on Coxeter groups and strong solidity

In deformation-rigidity theory, it is often important to know whether certain bimodules are weakly contained in the coarse bimodule. Consider a bimodule H over the group algebra \mathbb{C}[\Gamma] with \Gamma a discrete group. The starting point of this paper is that if a dense set of the so-called coefficients of H is contained in the Schatten \mathcal{S}_{p} class p \in [2, \infty) , then the n -fold tensor power H^{\otimes n}_{\Gamma} for n \geq \frac{p}{2} is quasi-contained in the coarse bimodule. We apply this to gradient bimodules associated with the carré du champ of a symmetric quantum Markov semi-group. For Coxeter groups, we give a number of characterizations of having coefficients in \mathcal{S}_{p} for the gradient bimodule constructed from the word length function. We get equivalence of: (1) the gradient- \mathcal{S}_{p} property introduced by the second named author, (2) smallness at infinity of a natural compactification of the Coxeter group, and for a large class of Coxeter groups, (3) walks in the Coxeter diagram called parity paths. We derive several strong solidity results. In particular, we extend current strong solidity results for right-angled Hecke von Neumann algebras beyond right-angled Coxeter groups that are small at infinity. Our general methods also yield a concise proof of a result by Sinclair for discrete groups admitting a proper cocycle into a p -integrable representation.

Order type Tingley's problem for type I finite von Neumann algebras

Let M and N be von Neumann algebras, S(M)+ be the collection of all positive norm one elements in M, and PM be the projection lattice of M. Let Φ:PM→PN be a metric preserving order isomorphism and Λ:S(M)+→S(N)+ be a bijective isometry. When both M and N are type I finite, we establish that the map Φ extends to a Jordan ⁎-isomorphism from M onto N. On the other hand, if M and N are of the form ⨁n=1k0Mn, where Mn is either zero or a von Neumann algebra of type In, then the map Λ extends to a Jordan ⁎-isomorphism from M onto N. On our way, we also verify that when M and N are general von Neumann algebras, the map Λ extends to a Jordan ⁎-isomorphism if and only if Λ|PM∖{0} is a bi-orthogonality preserving bijection from PM∖{0} onto PN∖{0}.

Notes on the type classification of von Neumann algebras

This paper provides an explanation of the type classification of von Neumann algebras, which has made many appearances in the recent work on entanglement in quantum field theory and quantum gravity. The goal is to bridge a gap in the literature between resources that are too technical for the non-expert reader, and resources that seek to explain the broad intuition of the theory without giving precise definitions. Reading this paper will provide you with: (i) an argument for why “factors” are the fundamental von Neumann algebras that one needs to study; (ii) an intuitive explanation of the type classification of factors in terms of renormalization schemes that turn unnormalizable positive operators into “effective density matrices”; (iii) a mathematical explanation of the different types of renormalization schemes in terms of the allowed traces on a factor; (iv) an intuitive characterization of type I and type II factors in terms of their “standard forms”; and (v) a list of some interesting connections between type classification and modular theory, including the argument for why type III1 factors are believed to be the relevant ones in quantum field theory. None of the material is new, but the pedagogy is different from other sources the author has read; it is most similar in spirit to the recent work on gravity and the crossed product by Chandrasekaran, Longo, Penington, and Witten.

The sharp weighted maximal inequalities for noncommutative martingales

The purpose of the paper is to establish weighted maximal Lp-inequalities in the context of operator-valued martingales on semifinite von Neumann algebras. The main emphasis is put on the optimal dependence of the Lp constants on the characteristic of the weight involved. As applications, we establish weighted estimates for the noncommutative version of Hardy-Littlewood maximal operator and weighted bounds for noncommutative maximal truncations of a wide class of singular integrals.

Some inequalities about Bourin's question in noncommutative L p -spaces

This paper studies a question of Bourin on noncommutative spaces. Some inequalities due to Hayajneh et al. [Norm inequalities for positive semidefinite matrices and a question of Bourin II. Int J Math. 2021;32:2150043] and Hayajneh et al. [Norm inequalities for positive semidefinite matrices and a question of Bourin III. Positivity. 2022;26:23] are extended to the case of operators in noncommutative spaces associated with a semifinite von Neumann algebra. We give a partial answer to a question of Bourin on noncommutative spaces. Some related submajorization inequalities are also given.

Characterization of Lie-Type Higher Derivations of von Neumann Algebras with Local Actions

Let m and n be fixed positive integers. Suppose that A is a von Neumann algebra with no central summands of type I1, and Lm:A→A is a Lie-type higher derivation. In continuation of the rigorous and versatile framework for investigating the structure and properties of operators on Hilbert spaces, more facts are needed to characterize Lie-type higher derivations of von Neumann algebras with local actions. In the present paper, our main aim is to characterize Lie-type higher derivations on von Neumann algebras and prove that in cases of zero products, there exists an additive higher derivation ϕm:A→A and an additive higher map ζm:A→Z(A), which annihilates every (n−1)th commutator pn(S1,S2,⋯,Sn) with S1S2=0 such that Lm(S)=ϕm(S)+ζm(S)forallS∈A. We also demonstrate that the result holds true for the case of the projection product. Further, we discuss some more related results.

Maximal von Neumann subalgebras in type I von Neumann algebras

In this paper we investigate maximal von Neumann subalgebras and obtain complete description results of such subalgebras in all type I von Neumann algebras M with separable predual. This is done by first studying the cases M being L(Z) and B(H), and then lifting to the direct sums and tensor products of such algebras.

Trace- and Improved Data-Processing Inequalities for von Neumann Algebras

We prove a version of the data-processing inequality for the relative entropy for general von Neumann algebras with an explicit lower bound involving the measured relative entropy. The inequality, which generalizes previous work by Sutter et al. on finite-dimensional density matrices, yields a bound for how well a quantum state can be recovered after it has been passed through a channel. Some natural applications of our results are in quantum field theory where the von Neumann algebras are known to be of type III. Along the way we generalize various multi-trace inequalities to general von Neumann algebras.

Von Neumann algebra description of inflationary cosmology

We study the von Neumann algebra description of the inflationary quasi-de Sitter (dS) space. Unlike perfect dS space, quasi-dS space allows the nonzero energy flux across the horizon, which can be identified with the expectation value of the static time translation generator. Moreover, as a dS isometry associated with the static time translation is spontaneously broken, the fluctuation in time is accumulated, which induces the fluctuation in the energy flux. When the inflationary period is given by (ϵHH)-1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(\\epsilon _H H)^{-1}$$\\end{document} where ϵH\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\epsilon _H$$\\end{document} is the slow-roll parameter measuring the increasing rate of the Hubble radius, both the energy flux and its fluctuation diverge in the G→0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$G \\rightarrow 0$$\\end{document} limit. Taking the fluctuation in the energy flux and that in the observer’s energy into account, we argue that the inflationary quasi-dS space is described by Type II∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$_\\infty $$\\end{document} algebra. As the entropy is not bounded from above, this is different from Type II1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$_1$$\\end{document} description of perfect dS space in which the entropy is maximized by the maximal entanglement. We also show that our result is consistent with the observation that the von Neumann entropy for the density matrix reflecting the fluctuations above is interpreted as the generalized entropy.

Wreath-like products of groups and their von Neumann algebras I: W^∗-superrigidity

Wreath-like products of groups and their von Neumann algebras I: W^∗-superrigidity

Domination of semigroups on standard forms of von Neumann algebras

Domination of semigroups on standard forms of von Neumann algebras

A block projection operator in the algebra of measurable operators

Let τ be a faithful normal semifinite trace on a von Neumann algebra M. We investigate the block projection operator Pn (n ≥ 2) in the ∗-algebra S(M, τ ) of all τ -measurable operators. We show that A ≤ nPn(A) for any operator A ∈ S(M, τ )+. If an operator A ∈ S(M, τ )+ is invertible in S(M, τ ) then Pn(A) is invertible in S(M, τ ). Consider A = A∗ ∈ S(M, τ ). Then (i) if Pn(A) ≤ A (or if Pn(A) ≥ A) then Pn(A) = A; (ii) Pn(A) = A if and only if PkA = APk for all k = 1, . . . , n; (iii) if A, n(A) are projections then n(A) = A. We obtain 4 corollaries. We also refined and reinforced one example from the paper “A. Bikchentaev, F. Sukochev, Inequalities for the block projection operators, J. Funct. Anal. 280 (7), article 108851, 18 p. (2021)”.

Vanishing first cohomology and strong 1-boundedness for von Neumann algebras

We obtain a new proof of Shlyakhtenko's result which states that if G is a sofic, finitely presented group with vanishing first \ell^2 -Betti number, then L(G) is strongly 1-bounded. Our proof of this result adapts and simplifies Jung's technical arguments which showed strong 1-boundedness under certain conditions on the Fuglede–Kadison determinant of the matrix capturing the relations. Our proof also features a key idea due to Jung which involves an iterative estimate for the covering numbers of microstate spaces. We also use the works of Shlyakhtenko and Shalom to give a short proof that the von Neumann algebras of sofic groups with Property (T) are strongly 1 bounded, which is a special case of another result by the authors.

Channel Divergences and Complexity in Algebraic QFT

We consider a notion of divergence between quantum channels in relativistic continuum quantum field theory (QFT) that is derived from the Belavkin–Staszewski relative entropy and the concept of bimodules for general von Neumann algebras. Key concepts of the divergence that we shall prove based on a new variational formulation of that relative entropy are the subadditivity under composition and additivity under the tensor product between channels. Based on these properties, we propose to use the channel divergence relative to the trivial (identity-) channel as a novel measure of complexity. Using the properties of our channel divergence, we prove in the prerequisite generality necessary for the algebras in QFT that the corresponding complexity has several reasonable properties: (i) the complexity of a composite channel is not larger than the sum of its parts, (ii) it is additive for channels localized in spacelike separated regions, (iii) it is convex, (iv) for an N-ary measurement channel it is log N, (v) for a conditional expectation associated with an inclusion of QFTs with finite Jones index it is given by log (text {Jones Index}).

Tight inclusions of $C^*$-dynamical systems

We study a notion of tight inclusions of C^* - and W^* -dynamical systems, which is meant to capture a tension between topological and measurable rigidity of boundary actions. An important case of such inclusions is C(X)\subset L^\infty(X, \nu) for measurable boundaries with unique stationary compact models. We discuss the implications of this phenomenon in the description of Zimmer amenable intermediate factors. Furthermore, we prove applications in the problem of maximal injectivity of von Neumann algebras.

Gleason’s theorem for composite systems

Gleason’s theorem (Gleason 1957 J. Math. Mech. 6 885) is an important result in the foundations of quantum mechanics, where it justifies the Born rule as a mathematical consequence of the quantum formalism. Formally, it presents a key insight into the projective geometry of Hilbert spaces, showing that finitely additive measures on the projection lattice extend to positive linear functionals on the algebra of bounded operators . Over many years, and by the effort of various authors, the theorem has been broadened in its scope from type I to arbitrary von Neumann algebras (without type factors). Here, we prove a generalisation of Gleason’s theorem to composite systems. To this end, we strengthen the original result in two ways: first, we extend its scope to dilations in the sense of Naimark (1943 Dokl. Akad. Sci. SSSR 41 359) and Stinespring (1955 Proc. Am. Math. Soc. 6 211) and second, we require consistency with respect to dynamical correspondences on the respective (local) algebras in the composition (Alfsen and Shultz 1998 Commun. Math. Phys. 194 87). We show that neither of these conditions changes the result in the single system case, yet both are necessary to obtain a generalisation to bipartite systems.