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We present a numerically exact approach for evaluating vibrationally resolved electronic spectra at finite temperatures using the coherence thermofield dynamics. In this method, which avoids implementing an algorithm for solving the von Neumann equation for coherence, the thermal vibrational ensemble is first mapped to a pure-state wavepacket in an augmented space, and this wavepacket is then propagated by solving the standard, zero-temperature Schrödinger equation with the split-operator Fourier method. We show that the finite-temperature spectra obtained with the coherence thermofield dynamics in a Morse potential agree exactly with those computed by Boltzmann-averaging the spectra of individual vibrational levels. Because the split-operator thermofield dynamics on a full tensor-product grid is restricted to low-dimensional systems, we briefly discuss how the accessible dimensionality can be increased by various techniques developed for the zero-temperature split-operator Fourier method. [ABSTRACT FROM AUTHOR]

We present a robust strategy to numerically sample the Coulomb potential in reciprocal space for periodic Born–von Karman cells of general shape. Our approach tackles two common issues of plane-wave based implementations of Coulomb integrals under periodic boundary conditions: the treatment of the singularity at the Brillouin-zone center and discretization errors, which can cause severe convergence problems in anisotropic cells, necessary for the calculation of low-dimensional systems. We apply our strategy to the Hartree–Fock and coupled cluster (CC) theories and discuss the consequences of different sampling strategies on different theories. We show that sampling the Coulomb potential via the widely used probe-charge Ewald method is unsuitable for CC calculations in anisotropic cells. To demonstrate the applicability of our developed approach, we study two representative, low-dimensional use cases: the infinite carbon chain, for which we report the first periodic CCSD(T) potential energy surface, and a surface slab of lithium hydride, for which we demonstrate the impact of different sampling strategies for calculating surface energies. We find that our Coulomb sampling strategy serves as a vital solution, addressing the critical need for improved accuracy in plane-wave based CC calculations for low-dimensional systems. [ABSTRACT FROM AUTHOR]

We discuss noninjectivity of mixed q-deformed Araki-Woods von Neumann algebras, extending the results in our previous work. These von Neumann algebras generalize those that were constructed by Hiai. [ABSTRACT FROM AUTHOR]

Bimodal probability distributions for planar directions are relevant to many scientific fields. Besides mixtures of two unimodal circular distributions, there are relatively few other circular distributions that allow for bimodality. One of these is the generalized von Mises distribution, which can be symmetric or asymmetric, bimodal or unimodal. Moreover, the GvM model possesses many important theoretical properties that are not available with competing models. This article provides an algebraic analysis of the bimodality of the GvM distribution. The problem of bimodality is reduced to study the nature of the roots of a quartic. Four real roots over the interval [ − 1 , 1 ] of the quartic correspond to bimodality. Situations with complex roots, multiple real roots and real roots outside [ − 1 , 1 ] are analyzed. The set of all parameters of the GvM distribution that yields bimodality is identified. [ABSTRACT FROM AUTHOR]

We consider half-line Dirac operators with operator data of Wigner-von Neumann type. If the data is a finite linear combination of Wigner-von Neumann functions, we show absence of singular continuous spectrum and provide an explicit set containing all embedded pure points that depends only on the L p decay and frequencies of the operator data. For infinite sums of Wigner-von Neumann-like terms, we bound the Hausdorff dimension of the singular part of the spectrum. [ABSTRACT FROM AUTHOR]

We extend theorems of Breuillard–Kalantar–Kennedy–Ozawa on unital reduced crossed products to the non-unital case under mild assumptions. As a result simplicity of C ∗ -algebras is stable under taking reduced crossed products over discrete C ∗ -simple groups, and a similar result for uniqueness of tracial weight. Interestingly, our analysis on tracial weights involves von Neumann algebra theory. Our generalizations have two applications. The first is to locally compact groups. We establish stability results of (non-discrete) C ∗ -simplicity and the unique trace property under discrete group extensions. The second is to the twisted crossed product. Thanks to the Packer–Raeburn theorem, our results lead to (generalizations of) the results of Bryder–Kennedy by a different method. [ABSTRACT FROM AUTHOR]

We study the maximal abelian von Neumann algebra corresponding to L^\infty (\mathbb {R}) via the Bargmann transform. It is naturally an algebra of operators on the Fock space F^2, but it can also be realized as a function algebra contained in F^2. This provides an interesting example of a C^* algebra whose elements are analytic functions. [ABSTRACT FROM AUTHOR]

We prove that, if a discrete group |$G$| is not inner amenable, then the unit group of the ring of operators affiliated with the group von Neumann algebra of |$G$| is non-amenable with respect to the topology generated by its rank metric. This provides examples of non-discrete irreducible, continuous rings (in von Neumann's sense) whose unit groups are non-amenable with regard to the rank topology. Our argument establishes and uses connections with Eymard–Greenleaf amenability of the action of the unitary group of a |$\textrm{II}_{1}$| factor on the associated space of projections of a fixed trace. [ABSTRACT FROM AUTHOR]

The AdS/CFT correspondence states that certain conformal field theories are equivalent to string theories in a higher-dimensional anti-de Sitter space. One aspect of the correspondence is an equivalence of density matrices or, if one ignores normalizations, of positive operators. On the CFT side of the correspondence, any two positive operators A, B will satisfy the trace inequality Tr(AB) ≤ Tr(A)Tr(B). This relation holds on any Hilbert space H and is deeply associated with the fact that the algebra B(H ) of bounded operators on H is a type I von Neumann factor. Holographic bulk theories must thus satisfy a corresponding condition, which we investigate below. In particular, we argue that the Euclidean gravitational path integral respects this inequality at all orders in the semi-classical expansion and with arbitrary higher-derivative corrections. The argument relies on a conjectured property of the classical gravitational action, which in particular implies a positive action conjecture for quantum gravity wavefunctions. We prove this conjecture for Jackiw-Teitelboim gravity and we also motivate it for more general theories. [ABSTRACT FROM AUTHOR]

We compute the structure constants of topological symmetric orbifold theories up to third order in the large N expansion. The leading order structure constants are dominated by topological metric contractions. The first order interactions are single cycles joining while at second order we can have double joining as well as splitting. At third order, single cycle joining obtains genus one contributions. We also compute illustrative small N structure constants. Our analysis applies to all second quantized Frobenius algebras, a large class of algebras that includes the cohomology ring of the Hilbert scheme of points on K3 among many others. We point out interesting open questions that our results raise. [ABSTRACT FROM AUTHOR]

We will investigate the α-z-Rényi divergence in the general von Neumann algebra setting based on Haagerup non-commutative Lp-spaces. In particular, we establish almost all its expected properties when 0 < α < 1 and some of them when α > 1. In an Appendix we also give an equality condition for generalized Hölder's inequality in Haagerup non-commutative Lp-spaces. [ABSTRACT FROM AUTHOR]

Since the early days of Tomita–Takesaki theory, it is known that a von Neumann algebra 푀 that admits a state 휑 with trivial centralizer M φ must be a type III1 factor, but the converse remained open. We solve this problem and prove that such ergodic states form a dense G δ set among all faithful normal states on any III1 factor with separable predual. Through Connes' Radon–Nikodym cocycle theorem, this problem is related to the existence of ergodic cocycle perturbations for outer group actions, which we consider in the second part of the paper. [ABSTRACT FROM AUTHOR]

We show that the local von Neumann algebra on convex areas of the frustration-free ground state of abelian quantum double models is of type I I ∞ . [ABSTRACT FROM AUTHOR]

We prove that if A is a non-separable abelian tracial von Neuman algebra then its free powers A∗n,2≤n≤∞, are mutually non-isomorphic and with trivial fundamental group, , whenever 2≤n<∞. This settles the non-separable version of the free group factor problem. [ABSTRACT FROM AUTHOR]

Let M be a finite von Neumann algebra with no central summands of type I1. We show that each nonlinear 2-local Lie n-derivation δ : M → M with n ≥ 3 is of the form d + h, where d : M → M is a linear derivation and h is a homogeneous central-valued mapping which annihilates each (n − 1) -th commutator of M. [ABSTRACT FROM AUTHOR]

This paper explores different notions of extreme states on operator systems and on operator spaces. We prove that boundary points of the generalized state space of an operator system in finite dimensions are precisely the restrictions of boundary representations of the generated C ∗ -algebra. We study weak TRO-extreme states on operator spaces in ternary rings of operators as a non self-adjoint version of C ∗ -extreme states on operator systems in C ∗ -algebras. Rectangular boundary point for a rectangular matrix convex set is introduced and its connection with boundary representations of the associated operator space is established in finite dimensional setting. Also, we show that the set of all rectangular boundary points is the minimal spanning set for a compact rectangular matrix convex set in finite dimensions. [ABSTRACT FROM AUTHOR]

AbstractLet A be a unital prime *-algebra (over the complex field C ) with a nontrivial projection. For any S1,S2,…,Sn∈A, define q1(S1)=S1,q2(S1,S2)=[S1,S2]⋄=S1S2−S2S1∗ and qn(S1,S2,…,Sn)=[qn−1(S1,S2,…,Sn−1),Sn]⋄ for all integers n≥3. In this article, it is shown that if a map ξ:A→A (not necessarily linear) satisfies ξ(qn(S1,S2,…,Sn))=∑i=1nqn(S1,…,Si−1,ξ(Si),Si+1,…,Sn) (n≥3) for all S1,S2,…,Sn∈A with S1∗S2∗⋯Sn∗=0, then ξ is an additive *-derivation. Finally, this result is applied to standard operator algebras and factor von Neumann algebras. [ABSTRACT FROM AUTHOR]

We study 3d quantum gravity with two asymptotically anti-de Sitter regions, in particular, using its relation with coupled Alekseev-Shatashvili theories and Liouville theory. Expressions for the Hartle-Hawking state, thermal 2n-point functions, torus wormhole correlators and Wheeler-DeWitt wavefunctions in different bases are obtained using the ZZ boundary states in Liouville theory. Exact results in 2d Jackiw-Teitelboim (JT) gravity are uplifted to 3d gravity, with two copies of Liouville theory in 3d gravity playing a similar role as Schwarzian theory in JT gravity. The connection between 3d gravity and the Liouville ZZ boundary states are manifested by viewing BTZ black holes as Maldacena-Maoz wormholes, with the two wormhole boundaries glued along the ZZ boundaries. In this work, we also study the factorization problem of the Hartle-Hawking state in 3d gravity. With the relevant defect operator that imposes the necessary topological constraint for contractibility, the trace formula in gravity is modified in computing the entanglement entropy. This trace matches with the one from von Neumann algebra considerations, further reproducing the Bekenstein-Hawking area formula from entanglement entropy. Lastly, we propose a calculation for off-shell geometrical quantities that are responsible for the ramp behavior in the late time two-point functions, which follows from the understanding of the Liouville FZZT boundary states in the context of 3d gravity, and the identification between Verlinde loop operators in Liouville theory and "baby universe" operators in 3d gravity. [ABSTRACT FROM AUTHOR]

This letter explores a transition in the type of von Neumann algebra for asymptotically AdS spacetimes from the implementations of the different gravitational constraints. We denote it as the centaur-algebra of observables. In the first part of the letter, we employ a class of flow geometries interpolating between AdS2 and dS2 spaces, the centaur geometries. We study the type II∞ crossed product algebra describing the semiclassical gravitational theory, and we explore the algebra of bounded sub-regions in the bulk theory following T T ¯ deformations of the geometry and study the gravitational constraints with respect to the quasi-local Brown-York energy of the system at a finite cutoff. In the second part, we study arbitrary asymptotically AdS spacetimes, where we implement the boundary protocol of an infalling observer modeled as a probe black hole proposed by [1] to study modifications in the algebra. In both situations, we show how incorporating the constraints requires a type II1 description. [ABSTRACT FROM AUTHOR]

An action of a compact, in particular finite group on a C*-algebra is called properly outer if no automorphism of the group that is distinct from identity is implemented by a unitary element of the algebra of local multipliers of the C*-algebra. In this paper, I define the notion of strictly outer action (similar to the definition for von Neumann factors in [S. Vaes, The unitary implementation of a locally compact group action, J. Funct. Anal. 180 (2001) 426–480]) and prove that for finite groups and prime C*-algebras, it is equivalent to the proper outerness of the action. For finite abelian groups this is equivalent to other relevant properties of the action. [ABSTRACT FROM AUTHOR]

We generalize Koopman–von Neumann classical mechanics to poly symplectic fields and recover De Donder–Weyl's theory. Compared with Dirac's Hamiltonian density, it inspires a new Hamiltonian formulation with a canonical momentum field that is Lorentz-covariant with symplectic geometry. We provide commutation relations for the classical and quantum fields that generalize the Koopman–von Neumann and Heisenberg algebras. The classical algebra requires four fields that generalize spacetime, energy–momentum, frequency–wavenumber, and the Fourier conjugate of energy–momentum. We clarify how first and second quantization can be found by simply mapping between operators in classical and quantum commutator algebras. [ABSTRACT FROM AUTHOR]

We construct a two-tensor model with order-3 and present its W-representation. Moreover we derive the compact expressions of correlators from the W-representation and analyze the free energy in large N limit. In addition, we establish the correspondence between two colored Dyck walks in the Fredkin spin chain and tree operators in the ring. Based on the classification Dyck walks, we give the number of tree operators with the given level. Furthermore, we show the entanglement scaling of Fredkin spin chain beyond logarithmic scaling in the ordinary critical systems from the viewpoint of tensor model. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Let \mathcal {N}=L_{\infty }(G_{\mathbf {m}})\bar {\otimes }\mathcal {M} where \mathcal {M} is a semifinite von Neumann algebra and G_{\mathbf {m}} is a bounded Vilenkin group. Considering the partial sums S_n(f) of the Vilenkin-Fourier series of f\in L_1(\mathcal {N}), we prove that there is a universal constant c independent of f, n and \mathcal {M} such that \begin{equation*} \|S_n(f)\|_{L_{1,\infty }(\mathcal {N})}\leq c\|f\|_{L_1(\mathcal {N})}. \end{equation*} Consequently, by transference argument, we obtain Scheckter and Sukochev's result in the noncommutative bounded Vilenkin system setting. We also show the H_1-L_1 boundedness for modified partial sums. [ABSTRACT FROM AUTHOR]

A subalgebra \mathfrak A in a \sigma-finite von Neumann algebra \mathcal M has the left (resp. right) partial factorization property if for any invertible operator S\in \mathcal M, there exists an isometry (resp. a co-isometry) U\in \mathcal M such that U^*S, S^{-1}U\in \mathfrak A. We show that the relative invariant subspace lattice {Lat}_{\mathcal M}\mathfrak A consisting of those projections in \mathcal M whose ranges are invariant under \mathfrak A is commutative. Moreover if \mathcal M has a separable predual, then {Lat}_{\mathcal M}\mathfrak A is generated by a nest together with the lattice of projections in the center of \mathcal M. In particular, if \mathfrak A is a von Neumann subalgebra, then \mathfrak A=\mathcal M. [ABSTRACT FROM AUTHOR]

In this correspondence, we prove the von Bahr–Esseen moment inequality for pairwise independent random vectors in Hilbert spaces. Our constant in the von Bahr–Esseen moment inequality is better than that obtained for the real-valued random variables by Chen et al. [The von Bahr–Esseen moment inequality for pairwise independent random variables and applications, J. Math. Anal. Appl. 419 (2014), 1290–1302], and Chen and Sung [Generalized Marcinkiewicz–Zygmund type inequalities for random variables and applications, J. Math. Inequal. 10(3) (2016), 837–848]. The result is then applied to obtain mean convergence theorems for triangular arrays of rowwise and pairwise independent random vectors in Hilbert spaces. Some results in the literature are extended. [ABSTRACT FROM AUTHOR]

We provide a comprehensive study on uniformly left ψ -bounded (respectively, (φ , ψ) -bounded) orthonormal bases in infinite-dimensional cyclic bimodules associated with c.p. maps between two von Neumann algebras M and N , where φ and ψ are faithful normal states on M and N , 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. [ABSTRACT FROM AUTHOR]

The problem of how measurement in quantum mechanics takes place has existed since its formulation. Von Neumann proposed a scheme where he treated measurement as a two-part process— a unitary evolution in the full system-ancilla space and then a projection onto one of the pointer states of the ancilla (representing the 'collapse' of the wavefunction). The Lindblad master equation, which has been extensively used to explain dissipative quantum phenomena in the presence of an environment, can effectively describe the first part of the von Neumann measurement scheme when the jump operators in the master equation are Hermitian. We have proposed a non-Hermitian Hamiltonian formalism to emulate the first part of the von Neumann measurement scheme. We have used the embedding protocol to dilate a non-Hermitian Hamiltonian that governs the dynamics in the system subspace into a higher-dimensional Hermitian Hamiltonian that evolves the full space unitarily. We have obtained the various constraints and the required dimensionality of the ancilla Hilbert space in order to achieve the required embedding. Using this particular embedding and a specific projection operator, one obtains non-Hermitian dynamics in the system subspace that closely follow the Lindblad master equation. This work lends a new perspective to the measurement problem by employing non-Hermitian Hamiltonians. [ABSTRACT FROM AUTHOR]

For multiplicative perturbations of unitary operators, it is presented a version of Weyl- von Neumann theorem and a sufficient conditions for generic (in the intensity parameter) singular continuous spectrum under unitary rank one perturbations. [ABSTRACT FROM AUTHOR]

This paper is concerned with the noncommutative analog of the normal subgroup theorem for certain groups. Inspired by Kalantar and Panagopoulos (arXiv:2108.02928, 2021, 16), we show that all Γ$\Gamma$‐invariant subalgebras of LΓ$L\Gamma$ and Cr∗(Γ)$C^*_r(\Gamma)$ are (Γ$\Gamma$‐)coamenable. The groups we work with satisfy a singularity phenomenon described by Bader et al. (Invent. Math. 229 (2022), 929–985). The setup of singularity allows us to obtain a description of Γ$\Gamma$‐invariant intermediate von Neumann subalgebras L∞(X,ξ)⊂M⊂L∞(X,ξ)⋊Γ$L^{\infty }(X,\xi)\subset \mathcal {M}\subset L^{\infty }(X,\xi)\rtimes \Gamma$ in terms of the normal subgroups of Γ$\Gamma$. [ABSTRACT FROM AUTHOR]

Quantum measurements can be interpreted as a generalization of probability vectors, in which non-negative real numbers are replaced by positive semi-definite operators. We extrapolate this analogy to define a generalization of doubly stochastic matrices that we call doubly normalized tensors (DNTs), and investigate a corresponding version of Birkhoff–von Neumann's theorem, which states that permutations are the extremal points of the set of doubly stochastic matrices. We prove that joint measurability appears naturally as a mathematical feature of DNTs in this context and that this feature is necessary and sufficient for a characterization similar to Birkhoff–von Neumann's. Conversely, we also show that DNTs arise from a particular instance of a joint measurability problem, remarking the relevance of this quantum-theoretical property in general operator theory. [ABSTRACT FROM AUTHOR]

We consider quantum algebras of observables associated with subregions in theories of Einstein gravity coupled to matter in the GN → 0 limit. When the subregion is spatially compact or encompasses an asymptotic boundary, we argue that the algebra is a type II von Neumann factor. To do so in the former case we introduce a model of an observer living in the region; in the latter, the ADM Hamiltonian effectively serves as an observer. In both cases the entropy of states on which this algebra acts is UV finite, and we find that it agrees, up to a state-independent constant, with the generalized entropy. For spatially compact regions the algebra is type II1, implying the existence of an entropy maximizing state, which realizes a version of Jacobson's entanglement equilibrium hypothesis. The construction relies on the existence of well-motivated but conjectural states whose modular flow is geometric at an instant in time. Our results generalize the recent work of Chandrasekaran, Longo, Penington, and Witten on an algebra of operators for the static patch of de Sitter space. [ABSTRACT FROM AUTHOR]

Let m and n be fixed positive integers. Suppose that A is a von Neumann algebra with no central summands of type I 1 , and L m : 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 ϕ m : A → A and an additive higher map ζ m : A → Z (A) , which annihilates every (n − 1) t h commutator p n (S 1 , S 2 , ⋯ , S n) with S 1 S 2 = 0 such that L m (S) = ϕ m (S) + ζ m (S) f o r a l l S ∈ A. We also demonstrate that the result holds true for the case of the projection product. Further, we discuss some more related results. [ABSTRACT FROM AUTHOR]

We prove a concentration inequality for invariant means on topological groups, namely for such adapted to a chain of amenable topological subgroups. The result is based on an application of Azuma's martingale inequality and provides a method for establishing extreme amenability. Building on this technique, we exhibit new examples of extremely amenable groups arising from von Neumann's continuous geometries. Along the way, we also answer a question by Pestov on dynamical concentration in direct products of amenable topological groups. [ABSTRACT FROM AUTHOR]

Let 풜 and ℬ be two von Neumann algebras. In this paper, we show that a bijective unital map Φ : 풜 → ℬ which preserves the bi-skew Jordan product is an additive ∗ -isomorphism. Moreover, we show that Φ is a linear ∗ -isomorphism or a conjugate linear ∗ -isomorphism. [ABSTRACT FROM AUTHOR]

Let G be a semi-simple Lie group in the Harish-Chandra class with maximal compact subgroup K. Let \Omega _K be minus the radial Casimir operator. Let \frac {1}{4} \dim (G/K) < S_G < \frac {1}{2} \dim (G/K), s \in (0, S_G] and p \in (1,\infty) be such that \[ \left | \frac {1}{p} - \frac {1}{2} \right | < \frac {s}{2 S_G}. \] Then, there exists a constant C_{G,s,p} >0 such that for every m \in L^\infty (G) \cap L^2(G) bi-K-invariant with m \in Dom(\Omega _K^s) and \Omega _K^s(m) \in L^{2S_G/s}(G) we have, \begin{equation} \Vert T_m: L^p(\widehat {G}) \rightarrow L^p(\widehat {G}) \Vert \leq C_{G, s,p} \Vert \Omega _K^s(m) \Vert _{L^{2S_G/s}(G)}, \end{equation} where T_m is the Fourier multiplier with symbol m acting on the non- commutative L^p-space of the group von Neumann algebra of G. This gives new examples of L^p-Fourier multipliers with decay rates becoming slower when p approximates 2. [ABSTRACT FROM AUTHOR]

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 (Jones Index) . [ABSTRACT FROM AUTHOR]

Assume that R be a unital prime ring with whose characteristic is not 2 and containing a nontrivial idempotent P. In this paper, it is shown that under certain conditions, ϕ ([ A , B ]) = [ ϕ (A) , B ] where ϕ an additive map on R and AB = P if and only if it has the form ϕ (A) = λ A + h (A) , where h is an additive map into its center vanishing at commutators [ A , B ] with A B = P and λ ∈ Z (R) . An application of our results we characterize generalized Lie derivations on R. Using main result we apply several classical examples of unital prime rings with nontrivial idempotents such as Banach space standard operator algebras and factor von Neumann algebras. [ABSTRACT FROM AUTHOR]

This article explores adapted mathematical methods to solve the coupled nonlinear Schrödinger (C-NLS) equation through analytical and numerical methods. To obtain exact solutions for the (C-NLS) equation, we utilize the improved modified, extended tanh-function method. By separating the Schrödinger equation into real and imaginary parts, we can obtain four coupled equations, which we then analyze using the generalized tanh method to extract exact solutions. This system of equations is essential for understanding the behavior of quantum systems and has various applications in quantum mechanics. We obtain an analytical solution and demonstrate numerical solutions using implicit finite difference. Studies have shown that this scheme is second-order in space and time, and the von Neumann stability analysis confirms its unconditional stability. We introduce the comparison between numerical and exact solutions. [ABSTRACT FROM AUTHOR]

Building on Simpson's original definition over the complex numbers, we introduce the notion of restricted sheaf Λ of rings of differential operators on a variety defined over a field of positive characteristic. We define the notion of p -curvature for Λ-modules and the analogue of the Hitchin map on the moduli space of Λ-modules. We show that under certain conditions this Hitchin map descends under the Frobenius map of the underlying variety and we give examples. [ABSTRACT FROM AUTHOR]

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 I 2 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. [ABSTRACT FROM AUTHOR]

The studies were carried out in a test facility based on the original biaxial testing machine created at the Institute of Engineering Science of the Ural Branch of the Russian Academy of Sciences (IES UB RAS) and designed to determine the physical properties of materials during elastic-plastic deformation independently along two axes. Finite element modeling of the stress-strain state of a cruciform nickel specimen has shown that under symmetric biaxial tension, equivalent von Mises stresses are uniformly distributed over the central part. The results of microstructural studies indicate that dynamic recrystallization occurs in the material during biaxial deformation at room temperature. The results of studying the behavior of hysteretic magnetic characteristics, magnetostriction, Barkhausen magnetic noise parameters, and electromagnetic parameters of nickel under biaxial symmetrical tension are presented. A unique relationship between coercive force and equivalent stresses in the domain of plastic deformation has been revealed. It has been shown that such magnetic parameters as the maximum of differential magnetic permeability and maximum magnetic permeability, as well as the amplitude and frequency of the eddy current signal determined at a frequency of 10 kHz vary monotonically with increasing equivalent stresses under both elastic and plastic deformations. [ABSTRACT FROM AUTHOR]

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 (ϵ H H) - 1 where ϵ H 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 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 ∞ algebra. As the entropy is not bounded from above, this is different from Type II 1 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. [ABSTRACT FROM AUTHOR]

We give a characterization of all the unitarily invariant norms on a finite von Neumann algebra acting on a separable Hilbert space. The characterization is analogous to von Neumann's characterization for the n × n complex matrices and the characterization in Fang et al. (J Funct Anal 255(1):142–183, 2008) for I I 1 factors. [ABSTRACT FROM AUTHOR]

We give a free probabilistic proposal to compute the fine-grained radiation entropy for an arbitrary bulk radiation state, in the context of the Penington-Shenker-Stanford-Yang (PSSY) model where the gravitational path integral can be implemented with full control. We observe that the replica trick gravitational path integral is combinatorially matching the free multiplicative convolution between the spectra of the gravitational sector and the matter sector respectively. The convolution formula computes the radiation entropy accurately even in cases when the island formula fails to apply. It also helps to justify this gravitational replica trick as a soluble Hausdorff moment problem. We then work out how the free convolution formula can be evaluated using free harmonic analysis, which also gives a new free probabilistic treatment of resolving the separable sample covariance matrix spectrum. The free convolution formula suggests that the quantum information encoded in competing quantum extremal surfaces can be modelled as free random variables in a finite von Neumann algebra. Using the close tie between free probability and random matrix theory, we show that the PSSY model can be described as a random matrix model that is essentially a generalization of Page's model. It is then manifest that the island formula is only applicable when the convolution factorizes in regimes characterized by the one-shot entropies. We further show that the convolution formula can be reorganized to a generalized entropy formula in terms of the relative entropy. [ABSTRACT FROM AUTHOR]

We show how geometric phases may be used to fully describe quantum systems, with or without gravity, by providing knowledge about the geometry and topology of its Hilbert space. We find a direct relation between geometric phases and von Neumann algebras. In particular, we show that a vanishing geometric phase implies the existence of a well-defined trace functional on the algebra. We discuss how this is realised within the AdS/CFT correspondence for the eternal black hole. On the other hand, a non-vanishing geometric phase indicates missing information for a local observer, associated to reference frames covering only parts of the quantum system considered. We illustrate this with several examples, ranging from a single spin in a magnetic field to Virasoro Berry phases and the geometric phase associated to the eternal black hole in AdS spacetime. For the latter, a non-vanishing geometric phase is tied to the presence of a centre in the associated von Neumann algebra. [ABSTRACT FROM AUTHOR]

We propose a conceptual frame to interpret the prolate differential operator, which appears in Communication Theory, as an entropy operator; indeed, we write its expectation values as a sum of terms, each subject to an entropy reading by an embedding suggested by Quantum Field Theory. This adds meaning to the classical work by Slepian et al. on the problem of simultaneously concentrating a function and its Fourier transform, in particular to the "lucky accident" that the truncated Fourier transform commutes with the prolate operator. The key is the notion of entropy of a vector of a complex Hilbert space with respect to a real linear subspace, recently introduced by the author by means of the Tomita-Takesaki modular theory of von Neumann algebras. We consider a generalization of the prolate operator to the higher dimensional case and show that it admits a natural extension commuting with the truncated Fourier transform; this partly generalizes the one-dimensional result by Connes to the effect that there exists a natural selfadjoint extension to the full line commuting with the truncated Fourier transform. [ABSTRACT FROM AUTHOR]

We show that every ring isomorphism between the algebras of locally measurable operators for type II∞$_\infty$ von Neumann algebras is similar to a real ∗${}^*$‐isomorphism. This together with previous results by the author and Ayupov–Kudaybergenov completely describes ring isomorphisms between the algebras of locally measurable operators as well as lattice isomorphisms between the projection lattices for a general pair of von Neumann algebras without finite type I direct summands. [ABSTRACT FROM AUTHOR]