Neumann Algebra Research Articles

On the nonequilibrium dynamics of gravitational algebras

Abstract We explore nonequilibrium features of certain operator algebras which appear in quantum gravity. The algebra of observables in a black hole background is a Type II ∞ von Neumann algebra. We discuss how this algebra can be coupled to the algebra of observable of an infinite reservoir within the canonical ensemble, aiming to induce nonequilibrium dynamics. The resulting dynamics can lead the system towards a nonequilibrium steady state which can be characterized through modular theory. Within this framework we address the definition of entropy production and its relationship to relative entropy, alongside exploring other applications.

Nonlinear Mixed $\lambda$-Jordan Triple Derivation on $\ast$-algebras

Let $\mathcal{A}$ be a $\ast$-algebra with unit $I$ and $P_1$ and $P_2 = I - P_1$ includes a non-trivial projections, and let $\lambda \in \mathbb{C}\setminus\{0,-1\}$. In this paper, We aim to study the characterization of nonlinear mixed $\lambda$-Jordan triple derivation on $\ast$-algebras. As an application, we can also apply our results on prime $\ast$-algebras, factor von-Neumann algebras and standard operator algebras.

Nonlinear mixed ∗-Jordan n-derivations on ∗-algebras

Let [Formula: see text] be a ∗-algebra with unity [Formula: see text] and a nontrivial projection [Formula: see text]. In this paper, it is shown that if a map [Formula: see text] satisfies [Formula: see text] [Formula: see text] for all [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] for all [Formula: see text], where [Formula: see text], then [Formula: see text] is an additive ∗-derivation. Moreover, we shall call such a map [Formula: see text], a nonlinear mixed ∗-Jordan [Formula: see text]-derivation. As applications, we will characterize nonlinear mixed ∗-Jordan [Formula: see text]-derivation in standard operator algebras and von Neumann algebras.

Lattices of Logmodular Algebras

A subalgebra \mathcal{A} of a C^{*} -algebra \mathcal{M} is logmodular (resp. has factorization) if the set \{a^{*}a;\ \text{$a\in\mathcal{M}$ is invertible with $a,a^{-1}\in\mathcal{A}$}\} is dense in (resp. equal to) the set of all positive and invertible elements of \mathcal{M} . In this paper, we show that the lattice of projections in a (separable) von Neumann algebra \mathcal{M} whose ranges are invariant under a logmodular algebra in \mathcal{M} , is a commutative subspace lattice. Further, if \mathcal{M} is a factor then this lattice is a nest. As a special case, it follows that all reflexive (in particular, completely distributive CSL) logmodular subalgebras of type I factors are nest algebras, thus answering in the affirmative a question by Paulsen and Raghupathi (Trans. Amer. Math. Soc. 363 (2011) 2627–2640). We also give a complete characterization of logmodular subalgebras in finite-dimensional von Neumann algebras.

On relative commutants of subalgebras in group and tracial crossed product von Neumann algebras

On relative commutants of subalgebras in group and tracial crossed product von Neumann algebras

Noncommutative Nest Hardy Spaces and Martingales

Abstract Let $(\mathcal{M},\tau )$ be a finite von Neumann algebra equipped with a normalized faithful trace and let $\mathbb{A}$ be a nest of order type $\mathbb{N}$. The nest algebra is defined by $H_{\infty }^{r}(\mathbb{A})=\{x\in \mathcal{M}:ex=exe,e\in \mathbb{A}\}$, and the related noncommutative Hardy space $H_{p}^{r}(\mathbb{A})$ is the completion of $H_{\infty }^{r}(\mathbb{A})$ in $L_{p}(\mathcal{M})$. In the present paper, we make a connection between $H_{p}^{r}(\mathbb{A})$ and certain noncommutative martingale Hardy space via showing that triangular truncation is a concrete martingale transform. We include several related results for noncommutative nest Hardy spaces in the spirit of recent development of noncommutative martingale theory.

On the amenable subalgebras of group von Neumann algebras

We approach the study of sub-von Neumann algebras of the group von Neumann algebra L(Γ) for countable groups Γ from a dynamical perspective. It is shown that L(Γ) admits a maximal invariant amenable subalgebra. The notion of invariant probability measures (IRAs) on the space of subalgebras is introduced, analogous to the concept of Invariant Random Subgroups. And it is shown that amenable IRAs are supported on the maximal amenable invariant subalgebra.

Boundary algebras of the Kitaev quantum double model

The recent article by Jones et al. [arXiv:2307.12552 (2023)] gave local topological order (LTO) axioms for a quantum spin system, showed they held in Kitaev’s Toric Code and in Levin-Wen string net models, and gave a bulk boundary correspondence to describe bulk excitations in terms of the boundary net of algebras. In this article, we prove the LTO axioms for Kitaev’s Quantum Double model for a finite group G. We identify the boundary nets of algebras with fusion categorical nets associated to (Hilb(G),C[G]) or (Rep(G),CG) depending on whether the boundary cut is rough or smooth, respectively. This allows us to make connections to the work of Ogata [Ann. Henri Poincaré 25, 2353–2387 (2024)] on the type of the cone von Neumann algebras in the algebraic quantum field theory approach to topological superselection sectors. We show that the boundary algebras can also be calculated from a trivial G-symmetry protected topological phase (G-SPT), and that the gauging map preserves the boundary algebras. Finally, we compute the boundary algebras for the (3 + 1)D Quantum Double model associated to an Abelian group.

Pointwise Convergence of Noncommutative Fourier Series

This paper is devoted to the study of pointwise convergence of Fourier series for group von Neumann algebras and quantum groups. It is well-known that a number of approximation properties of groups can be interpreted as summation methods and mean convergence of the associated noncommutative Fourier series. Based on this framework, this paper studies the refined counterpart of pointwise convergence of these Fourier series. As a key ingredient, we develop a noncommutative bootstrap method and establish a general criterion of maximal inequalities for approximate identities of noncommutative Fourier multipliers. Based on this criterion, we prove that for any countable discrete amenable group, there exists a sequence of finitely supported positive definite functions tending to 1 1 pointwise, so that the associated Fourier multipliers on noncommutative L p L_p -spaces satisfy the pointwise convergence for all p > 1 p>1 . In a similar fashion, we also obtain results for a large subclass of groups (as well as quantum groups) with the Haagerup property and the weak amenability. We also consider the analogues of Fejér and Bochner-Riesz means in the noncommutative setting. Our approach heavily relies on the noncommutative ergodic theory in conjunction with abstract constructions of Markov semigroups, inspired by quantum probability and geometric group theory. Finally, we also obtain as a byproduct the dimension free bounds of the noncommutative Hardy-Littlewood maximal inequalities associated with convex bodies.

Hyponormal measurable operators, affiliated to a semifinite von Neumann algebra

Hyponormal measurable operators, affiliated to a semifinite von Neumann algebra

Algebraic Aspects and Functoriality of the Set of Affiliated Operators

Abstract In this article, we aim to provide a satisfactory algebraic description of the set of affiliated operators for von Neumann algebras. Let $\mathscr{M}$ be a von Neumann algebra acting on a Hilbert space $\mathcal{H}$, and let $\mathscr{M}_{\text{aff}}$ denote the set of unbounded operators of the form $T = AB^{\dagger }$ for $A, B \in \mathscr{M}$ with $\text{null} (B) \subseteq \text{null} (A)$, where $(\cdot )^{\dagger }$ denotes the Kaufman inverse. We show that $\mathscr{M}_{\text{aff}}$ is closed under sum, product, Kaufman-inverse, and adjoint, and has the structure of a (right) near-semiring. Moreover, the above quotient representation of an operator in $\mathscr{M}_{\text{aff}}$ is essentially unique. Thus, we may view $\mathscr{M}_{\text{aff}}$ as the multiplicative monoid of unbounded operators on $\mathcal{H}$ generated by $\mathscr{M}$ and $\mathscr{M}^{\dagger }$. We further show that our definition of affiliation, as reflected in $\mathscr{M}_{\text{aff}}$, subsumes the traditional one. Let $\Phi $ be a unital normal *-homomorphism between represented von Neumann algebras $(\mathscr{M}; \mathcal{H})$ and $(\mathscr{N}; \mathcal{K})$. Using the quotient representation, we obtain a canonical extension of $\Phi $ to a mapping $\Phi _{\text{aff}}: \mathscr{M}_{\text{aff}} \to \mathscr{N}_{\text{aff}}$ which is a near-semiring homomorphism that respects Kaufman-inverse and adjoint; in addition, $\Phi _{\text{aff}}$ respects Murray–von Neumann affiliation of operators and also respects strong sum and strong product. Thus, $\mathscr{M}_{\text{aff}}$ is intrinsically associated with $\mathscr{M}$ and transforms functorially as we change representations of $\mathscr{M}$. Furthermore, $\Phi _{\text{aff}}$ preserves operator properties such as being symmetric, or positive, or accretive, or sectorial, or self-adjoint, or normal, and also preserves the Friedrichs and Krein–von Neumann extensions of densely defined closed positive operators. As a proof of concept, we transfer some well-known results about closed unbounded operators to the setting of closed affiliated operators for properly infinite von Neumann algebras, via “abstract nonsense”.

On Haagerup noncommutative quasi [formula omitted] spaces

Let M be a σ-finite von Neumann algebra, equipped with a normal faithful state φ, and let A be a maximal subdiagonal subalgebra of M. We have proved that for 0<p<1, Hp(A) is independent of φ. Furthermore, in the case that A is a type 1 subdiagonal subalgebra, we have obtained an interpolation theorem for Hp(A) in the case where 0<θ<1, 0<p0,p1≤∞ and p=1−θp0+θp1≥1.

Quasi-invariant states

In this paper, we develop the theory of quasi-invariant (respectively, strongly quasi-invariant) states under the action of a group [Formula: see text] of normal ∗-automorphisms of a ∗-algebra (or von Neumann algebra) [Formula: see text]. We prove that these states are naturally associated to left-[Formula: see text]-[Formula: see text]-cocycles. If [Formula: see text] is compact, the structure of strongly [Formula: see text]-quasi-invariant states is determined. For any [Formula: see text]-strongly quasi-invariant state [Formula: see text], we construct a unitary representation associated to the triple [Formula: see text]. We prove, under some conditions, that any quantum Markov chain with commuting, invertible and Hermitian conditional density amplitudes on a countable tensor product of type I factors is strongly quasi-invariant with respect to the natural action of the group [Formula: see text] of local permutations and we give the explicit form of the associated cocycle. This provides a family of nontrivial examples of strongly quasi-invariant states for locally compact groups obtained as inductive limit of an increasing sequence of compact groups.

On 2-local derivations of von Neumann algebras

Let A be a semi-finite factor von Neumann algebra. We prove that if a map δ : A → A satisfies that for any A , B ∈ A there is a linear derivation δ A , B : A → A such that δ ( A ) B + Aδ ( B ) = δ A , B ( AB ) , then δ is a linear derivation. It is a positive answer to the problem on semi-finite factor von Neumann algebra posed by Molnár in the paper [A new look at local maps on algebraic structures of matrices and operators, New York J Math 28 (2022)].

The space of tracial states on a [formula omitted]-algebra

We give a simple and elementary proof that the tracial state space of a unital C∗-algebra is a Choquet simplex, using the center-valued trace on a finite von Neumann algebra.

SAT actions of discrete quantum groups and minimal injective extensions of their von Neumann algebras

We introduce a natural generalization of the notion of strongly approximately transitive (SAT) states for actions of locally compact quantum groups. In the case of discrete quantum groups of Kac type, we show that the existence of unique stationary SAT states entails rigidity results concerning injective extensions of quantum group von Neumann algebras.

When left and right disagree: entropy and von Neumann algebras in quantum gravity with general AlAdS boundary conditions

Euclidean path integrals for UV-completions of d-dimensional bulk quantum gravity were recently studied in [1] by assuming that they satisfy axioms of finiteness, reality, continuity, reflection-positivity, and factorization. Sectors HB\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\\mathcal{H}}_{\\mathcal{B}} $$\\end{document} of the resulting Hilbert space were then defined for any (d − 2)-dimensional surface B\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{B} $$\\end{document}, where B\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{B} $$\\end{document} may be thought of as the boundary ∂Σ of a bulk Cauchy surface in a corresponding Lorentzian description, and where B\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{B} $$\\end{document} includes the specification of appropriate boundary conditions for bulk fields. Cases where B\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{B} $$\\end{document} was the disjoint union B ⊔ B of two identical (d − 2)-dimensional surfaces B were studied in detail and, after the inclusion of finite-dimensional ‘hidden sectors,’ were shown to provide a Hilbert space interpretation of the associated Ryu-Takayanagi entropy. The analysis was performed by constructing type-I von Neumann algebras ALB\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\\mathcal{A}}_L^B $$\\end{document}, ARB\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\\mathcal{A}}_R^B $$\\end{document} that act respectively at the left and right copy of B in B ⊔ B.Below, we consider the case of general B\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{B} $$\\end{document}, and in particular for B\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{B} $$\\end{document} = BL ⊔ BR with BL, BR distinct. For any BR, we find that the von Neumann algebra at BL acting on the off-diagonal Hilbert space sector HBL⊔BR\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\\mathcal{H}}_{B_L\\bigsqcup {B}_R} $$\\end{document} is a central projection of the corresponding type-I von Neumann algebra on the ‘diagonal’ Hilbert space HBL⊔BL\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\\mathcal{H}}_{B_L\\bigsqcup {B}_L} $$\\end{document}. As a result, the von Neumann algebras ALBL\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\\mathcal{A}}_L^{B_L} $$\\end{document}, ARBL\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\\mathcal{A}}_R^{B_L} $$\\end{document} defined in [1] using the diagonal Hilbert space HBL⊔BL\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\\mathcal{H}}_{B_L\\bigsqcup {B}_L} $$\\end{document} turn out to coincide precisely with the analogous algebras defined using the full Hilbert space of the theory (including all sectors HB\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\\mathcal{H}}_{\\mathcal{B}} $$\\end{document}). A second implication is that, for any HBL⊔BR\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\\mathcal{H}}_{B_L\\bigsqcup {B}_R} $$\\end{document}, including the same hidden sectors as in the diagonal case again provides a Hilbert space interpretation of the Ryu-Takayanagi entropy. We also show the above central projections to satisfy consistency conditions that lead to a universal central algebra relevant to all choices of BL and BR.

Mini-Workshop: Standard Subspaces in Quantum Field Theory and Representation Theory

Real standard subspaces of complex Hilbert spaces are long known to provide the right language for Tomita–Takesaki modular theory of von Neumann algebras. In recent years they have also become an object of prominent interest in mathematical quantum field theory (QFT) and unitary representation theory of Lie groups. This workshop brought together mathematicians and physicists working with standard subspaces, particularly in QFT (construction of QFT models, characterization of entropy, information-theoretic aspects), nets of standard subspaces on causal homogeneous spaces and aspects of reflection positivity and euclidean models related to standard subspaces and modular theory.

Optimal transport for types and convex analysis for definable predicates in tracial W⁎-algebras

We investigate the connections between continuous model theory, free probability, and optimal transport/convex analysis in the context of tracial von Neumann algebras. In particular, we give an analog of Monge-Kantorovich duality for optimal couplings where the role of probability distributions on Cn is played by model-theoretic types, the role of real-valued continuous functions is played by definable predicates, and the role of continuous function Cn→Cn is played by definable functions. In the process, we also advance the understanding of definable predicates and definable functions by showing that all definable predicates can be approximated by “C1 definable predicates” whose gradients are definable functions. As a consequence, we show that every element in the definable closure of W⁎(x1,…,xn) can be expressed as a definable function of (x1,…,xn). We give several classes of examples showing that the definable closure can be much larger than W⁎(x1,…,xn) in general.

P. Jones' interpolation theorem for noncommutative martingale Hardy spaces II

AbstractLet be a semifinite von Neumann algebra equipped with an increasing filtration of (semifinite) von Neumann subalgebras of . For , let denote the noncommutative column martingale Hardy space constructed from column square functions associated with the filtration and the index . We prove the following real interpolation identity: If and , then This is new even for classical martingale Hardy spaces as it is previously known only under the assumption that the filtration is regular. We also obtain an analogous result for noncommutative column martingale Orlicz–Hardy spaces.