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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]

This paper studies a question of Bourin on noncommutative $ L^p $ L p 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 $ L^p $ L p spaces associated with a semifinite von Neumann algebra. We give a partial answer to a question of Bourin on noncommutative $ L^p $ L p spaces. Some related submajorization inequalities are also given. [ABSTRACT FROM AUTHOR]

This paper establishes a necessary and sufficient condition for the coincidence of non-commutative log -algebras constructed from different faithful normal semifinite traces. Consequently, we provide a criterion for the isomorphism of log -algebras built on non-commutative von Neumann algebras with different faithful normal semifinite traces. Additionally, we demonstrate a connection between the isomorphism of non-commutative log -algebras and the isomorphism of the corresponding log -algebras constructed on the center of these von Neumann algebras. Furthermore, we present a necessary and sufficient condition for the isomorphism of log -algebras derived from different von Neumann algebras of type I n . [ABSTRACT FROM AUTHOR]

A new framework for the reflection positivity, based on order-preserving operator inequalities, is proposed. This framework is utilized to investigate one-dimensional fermion–phonon systems, with a particular focus on the detailed examination of ground state properties. Our analysis reveals that the reflection positivity provides a consistent description of the charge-density-wave (CDW) order in such systems. Additionally, we establish the existence of CDW long-range order in the ground state when the Coulomb interaction is long range. [ABSTRACT FROM AUTHOR]

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 (H i l b (G) , C [ G ]) or (R e p (G) , C G ) 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. [ABSTRACT FROM AUTHOR]

We discuss high energy properties of states for (possibly interacting) quantum fields in curved spacetimes. In particular, if the spacetime is real analytic, we show that an analogue of the timelike tube theorem and the Reeh–Schlieder property hold with respect to states satisfying a weak form of microlocal analyticity condition. The former means the von Neumann algebra of observables of a spacelike tube equals the von Neumann algebra of observables of a significantly bigger region that is obtained by deforming the boundary of the tube in a timelike manner. This generalizes theorems by Araki (Helv Phys Acta 36:132–139, 1963) and Borchers (Nuovo Cim (10) 19:787–793, 1961) to curved spacetimes. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Let G be a generalized matrix algebra. We show that under certain conditions, each generalized Lie n -derivation associated with a linear map on G is a sum of a generalized derivation and a central map vanishing on all (n − 1) -th commutators and is also a sum of a generalized inner derivation and a Lie n -derivation. As an application, generalized Lie n -derivations on von Neumann algebras are characterized. [ABSTRACT FROM AUTHOR]

In this paper, we prove a non-commutative version of the Weyl-von Neumann theorem for representations of unital, separable AH algebras into countably decomposable, semifinite, properly infinite, von Neumann factors, where an AH algebra means an approximately homogeneous C*-algebra. We also prove a result for approximate summands of representations of unital, separable AH algebras into finite von Neumann factors. [ABSTRACT FROM AUTHOR]

In this paper, we develop the theory of quasi-invariant (respectively, strongly quasi-invariant) states under the action of a group G of normal ∗-automorphisms of a ∗-algebra (or von Neumann algebra) 풜. We prove that these states are naturally associated to left-G-1-cocycles. If G is compact, the structure of strongly G-quasi-invariant states is determined. For any G-strongly quasi-invariant state φ, we construct a unitary representation associated to the triple (풜,G,φ). 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 풮∞ 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. [ABSTRACT FROM AUTHOR]

In this paper, we prove that the mixed q -deformed Araki-Woods von Neumann algebras are factors for dim (ℋ ℝ) ≥ 3. [ABSTRACT FROM AUTHOR]

We provide an example of two elementarily equivalent countable ICC groups G and H such that G is amenable and H is not inner amenable. As a result, we provide the first example of elementarily equivalent groups whose group von Neumann algebras are not elementarily equivalent, answering a question asked by many researchers. [ABSTRACT FROM AUTHOR]

We fully characterize noncommutative symmetric spaces E(\mathcal {M},\tau) affiliated with a semifinite von Neumann algebra \mathcal {M} equipped with a faithful normal semifinite trace \tau on a (not necessarily separable) Hilbert space having the Gelfand–Phillips property and the WCG-property. The complete list of their relations with other classical structural properties (such as the Dunford–Pettis property, the Schur property and their variations) is given in the general setting of noncommutative symmetric spaces. [ABSTRACT FROM AUTHOR]

Let R be a unital ring containing a nontrivial idempotent. In this article, under a mild condition on R , we prove that if a map δ : R → R satisfies δ (P n (A 1 , A 2 , A 3 , ... , A n)) = ∑ i = 1 n P n (A 1 , ... , A i − 1 , δ (A i) , A i + 1 , ... , A n) for any A 1 , A 2 , A 3 , ... , A n ∈ R with A 1 A 2 A 3 = 0 , then δ (A + B) − δ (A) − δ (B) ∈ Z (R) for any A , B ∈ R . In particular, if R is a von Neumann algebra with no central summands of type I1 or a factor, then δ (x) = d (x) + τ (x) for all x ∈ R , where d : R → R is an additive derivation and τ : R → Z (R) is a map vanishing on each (n − 1) -th commutator p n (A 1 , A 2 , A 3 , ... , A n) with A 1 A 2 A 3 = 0 . [ABSTRACT FROM AUTHOR]

Let A be a unital prime *-algebra (over the complex field C ) with a nontrivial projection. For any S 1 , S 2 , ... , S n ∈ A , define q 1 (S 1) = S 1 , q 2 (S 1 , S 2) = [ S 1 , S 2 ] ⋄ = S 1 S 2 − S 2 S 1 ∗ and q n (S 1 , S 2 , ... , S n) = [ q n − 1 (S 1 , S 2 , ... , S n − 1) , S n ] ⋄ for all integers n ≥ 3. In this article, it is shown that if a map ξ : A → A (not necessarily linear) satisfies ξ (q n (S 1 , S 2 , ... , S n)) = ∑ i = 1 n q n (S 1 , ... , S i − 1 , ξ (S i) , S i + 1 , ... , S n) (n ≥ 3) for all S 1 , S 2 , ... , S n ∈ A with S 1 ∗ S 2 ∗ ⋯ S n ∗ = 0 , then ξ is an additive *-derivation. Finally, this result is applied to standard operator algebras and factor von Neumann algebras. [ABSTRACT FROM AUTHOR]

Let U be a von Neumann algebra endowed with the Lie product [A,B] = AB -- BA (A,B ∈ U). In this article, we consider the subsequent condition on an additive mapping ø on the von Neumann algebra U with a suitable projection P ∈ U : ø([[A, B],C]) = [[ø(A), B],C] = [[A,ø(B)],C] for all A,B,C ∈ U with AB = P and we show that ø(A) = WA + ξ(A) for all A ∈ U, where W ∈ Z(U), and ξ: U → Z(U) (Z(U) is the center of U) is an additive map in which ∈ ([[A, B],C]) = 0 for any A, B,C ∈ U with AB = P. We also give some results of the conclusion. [ABSTRACT FROM AUTHOR]

For a semifinite von Neumann algebra M, individual convergence of subsequential, Z (M) (center of M) valued weighted ergodic averages is studied in non commutative Orlicz spaces. In the process, we also derive a maximal ergodic inequality corresponding to such averages in noncommutative Lp (1 ≤ p < ∞) spaces using the weak (1,1) inequality obtained by Yeadon. [ABSTRACT FROM AUTHOR]

Let A be a unital * -algebra. Let pn(A1, A2, …, An) be the polynomial defined by n indeterminates A1, A2, …, An ∈ A and their multiple skew Lie product, and ℕ be the set of non-negative integers. In this paper, under some mild conditions on A, it is shown that if D = {dm}m ∈ ℕ is the family of maps dm : A → A such that d0 = idA, the identity map on A satisfying dm(pn(A1, A2, …, An)) = ∑ /i1 + i2+ …1in=m pn(di1 (A1), di2 (A2), …, din (An)) for all A1, A2, …, An ∈ A and for each m ∈ ℕ, then D = {dm}m ∈ ℕ is an additive * -higher derivation. Moreover, we apply the above result to prime * -algebras, von Neumann algebras with no central summands of type I1, factor von Neumann algebras and standard operator algebras. [ABSTRACT FROM AUTHOR]

Two rigorous mathematical foundations for relativistic quantum field theory are local nets of von Neumann algebras and Wightman field theory. We recall that local nets of von Neumann algebras are local nets of bounded operator algebras, whereas Wightman field theory makes use of unbounded linear operators. In this paper, we introduce local nets of unbounded operator algebras and show how these can be connected with Wightman field theory. This is motivated by the fact that observables in quantum mechanics are unbounded operators in general. [ABSTRACT FROM AUTHOR]

Given a semifinite von Neumann algebra \mathcal M equipped with a faithful normal semifinite trace \tau, we prove that the spaces L^0(\mathcal M,\tau) and \mathcal R_\tau are complete with respect to pointwise—almost uniform and bilaterally almost uniform—convergences in L^0(\mathcal M,\tau). Then we show that the pointwise Cauchy property for a special class of nets of linear operators in the space L^1(\mathcal M,\tau) can be extended to pointwise convergence of such nets in any fully symmetric space E\subset \mathcal R_\tau, in particular, in any space L^p(\mathcal M,\tau), 1\leq p<\infty. Some applications of these results in the noncommutative ergodic theory are discussed. [ABSTRACT FROM AUTHOR]

In this article, we work with a unital twisted partial action of a group G on a ring R. We study when the twisted partial skew generalized power series rings are Von Newman regular. Moreover, we study when the partial skew Malcev-Newman ring is simple, we provide some examples where the assumption of our principal results about Von Newman regularity can not be dropped and we discuss a conjecture given by Tuganbaev in the case of partial actions. [ABSTRACT FROM AUTHOR]

Let M$\mathcal {M}$ be a semifinite von Neumann algebra equipped with an increasing filtration (Mn)n⩾1$(\mathcal {M}_n)_{n\geqslant 1}$ of (semifinite) von Neumann subalgebras of M$\mathcal {M}$. For 1⩽p⩽∞$1\leqslant p \leqslant \infty$, let Hpc(M)$\mathcal {H}_p^c(\mathcal {M})$ denote the noncommutative column martingale Hardy space constructed from column square functions associated with the filtration (Mn)n⩾1$(\mathcal {M}_n)_{n\geqslant 1}$ and the index p$p$. We prove the following real interpolation identity: If 0<θ<1$0<\theta <1$ and 1/p=1−θ$1/p=1-\theta$, then H1c(M),H∞c(M)θ,p=Hpc(M).$$\begin{equation*} \hspace*{48pt}{\left(\mathcal {H}_1^c(\mathcal {M}), \mathcal {H}_\infty ^c(\mathcal {M})\right)}_{\theta,p}=\mathcal {H}_p^c(\mathcal {M}).\hspace*{-48pt} \end{equation*}$$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. [ABSTRACT FROM AUTHOR]

Let A be a unital A-algebra containing a non-trivial projection. In this paper, we prove that if a map Ω: A → A such that Ω([K, F]A ⊙ D) = [Ω(K), F]A ⊙ D + [K, Ω(F)]A ⊙ D + [K, F]A ⊙ Ω(D), where [K, F]A = KF - FKA and K ⊙ F = KAF + FKA for all K, F, D ∈ A, then Ω is an additive A-derivation. Furthermore, we extend its results on factor von Neumann algebras, standard operator algebras and prime A-algebras. Additionally, we provide an example illustrating the existence of such maps. [ABSTRACT FROM AUTHOR]

Peculiarities of multiqubit measurement are for the most part similar to peculiarities of measurement for qudit — quantum object with finite-dimensional Hilbert space. Three different interpretations of measurement concept are analyzed. One of those is purely quantum and is in collection, for a given state of the object to be measured, of incompatible observable measurement results in amount enough for reconstruction of the state. Two others make evident the difference between the reduced density matrix and the density matrices of physical objects involved in the measurement. It is shown that the von Neumann projectors, in combination with the concept of qudit phase space, produce an idea of a phase portrait of qudit state as a set of mathematical expectations for projectors on the possible pure states. The phase portrait is not a probability distribution since the projectors on nonorthogonal states are incompatible observables. Along with that, the phase portrait includes probability distributions for all the resolutions of identity of the qudit observable algebra. Additional peculiarities of measurement of the qudit degenerated observables, caused by the possibility of independent measurement of the observables for the particles, make possible to distinguish the local reduction of the qudit particle states from the entanglement of the local measurement results. The phase portrait of a composite system comprised by a qudit pair generates local and conditional phase portraits of particles. The entanglement is represented by the dependence of the shape of conditional phase portrait on the properties of the observable used in the measurement for the other particle. Analysis of the properties of a conditional phase portrait of a multiqubit qubits shows that absence of the entanglement is possible only in the case of substantial restrictions imposed on the method of multiqubit decomposition into qubits. Such a special method for determination of particles exists for each multiqubit state. [ABSTRACT FROM AUTHOR]

A subalgebra A of a C*-algebra M is logmodular (resp. has factorization) if the set {a*a; a ∈M is invertible with a, a--1 ∈ A} is dense in (resp. equal to) the set of all positive and invertible elements of M. In this paper, we show that the lattice of projections in a (separable) von Neumann algebra M whose ranges are invariant under a logmodular algebra in M, is a commutative subspace lattice. Further, if 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. [ABSTRACT FROM AUTHOR]

In this paper, we describe the minimal reducing subspaces of Toeplitz operators induced by non-analytic monomials on the weighted Bergman spaces and Dirichlet spaces over the unit ball B 2 . It is proved that each minimal reducing subspace M is finite dimensional, and if dim M ≥ 3, then M is induced by a monomial. Furthermore, the structure of commutant algebra ν (T w ¯ N z N ) := { M w N ∗ M z N , M z N ∗ M w N } ′ is determined by N and the two dimensional minimal reducing subspaces of T w ¯ N z N . We also give some interesting examples. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the concept of a “von Neumann regular C∞-ring”, which is a model for a specific equational theory. We delve into the characteristics of these rings and demonstrate that each Boolean space can be effectively represented as the image of a von Neumann regular C∞-ring through a specific functor. Additionally, we establish that every homomorphism between Boolean algebras can be expressed through a C∞- ring homomorphism between von Neumann regular C∞-rings. [ABSTRACT FROM AUTHOR]

We introduced the quasicentral modulus to study normed ideal perturbations of operators. It is a limit of condenser quasicentral moduli in view of a recently noticed analogy with capacity in nonlinear potential theory. We prove here some basic properties of the condenser quasicentral modulus and compute a simple example. We also discuss some associated noncommutative variational problems. Part of the results are in the more general setting of a semifinite von Neumann algebra. [ABSTRACT FROM AUTHOR]

In this paper, the notion of proper proximality (introduced by Boutonnet, Ioana, and Peterson [Ann. Sci. Éc. Norm. Supér. (4) 54 (2021), 445-482]) is studied and classified in various families of groups. We show that if a group acts non-elementarily by isometries on a tree such that, for any two edges, the intersection of their edge stabilizers is finite, then G is properly proximal. We show that the wreath product G ... H is properly proximal if and only if H is non-amenable. We then completely classify proper proximality among graph products of non-trivial groups. Our results generalize the recent work of Duchesne, Tucker-Drob, and Wesolek classifying inner amenability for these families of groups. Our results also recover some rigidity results associated to the group von Neumann algebras by virtue of being properly proximal. A key idea in the proofs of our theorems is a technique to upgrade from relative proper proximality using computations in the double dual of the small at infinity boundary. [ABSTRACT FROM AUTHOR]

An improvised collocation scheme is applied for the numerical treatment of the nonlinear generalized Burgers-Fisher's (gBF) equation using splines of degree three. In the proposed methodology, some subsequent rectifications are done in the spline interpolant, which resulted in the magnification of the order of convergence along the space direction. A finite difference approach is followed to integrate the time direction. Von Neumann methodology is opted to discuss the stability of the method. The error bounds and convergence study show that the technique has (s4 + Δt2) order of convergence. The correspondence between the approximate and analytical solutions is shown by graphs, plotted using MATLAB and by evaluating absolute error. [ABSTRACT FROM AUTHOR]

In the present work, we present an investigation of Collins Conjecture, a hypothesis made by D. M. Collins in 1993 relating correlation energy and entanglement entropy, by calculating the ground state and singly‐excited triplet‐spin 1s2s 3S and 1s3s 3S state energies of the helium iso‐electronic sequence, with Z = 2–15. By using extensive orthonormal configuration interaction (CI) type wave functions with B‐Spline basis up to about 6000 terms, linear entropy and von Neumann entropy for the abovementioned atomic systems are determined. Together with the Hartree‐Fock energies obtained following a self‐consistent field theory, we have found that there exist linearly proportionalities between the renormalized correlation energies and entanglement entropies of both linear and von Neumann, showing a support for Collins Conjecture applicable to the ground and singly‐excited triplet‐spin states in the helium sequence, for a range of finite Z‐values. [ABSTRACT FROM AUTHOR]

We give the Karhunen–Loève expansion of the covariance function of a family of discrete weighted Brownian bridges, appearing as discrete analogues of continuous Gaussian processes related to Cramér –von Mises and Anderson–Darling statistics. This analogy enables us to introduce a discrete Cramér–von Mises statistic and show that this statistic satisfies a property of local asymptotic Bahadur optimality for a statistical test involving the classical hypergeometric distributions. Our statistic and the goodness-of-fit problem we deal with are based on basic properties of Hahn polynomials and are, therefore, subject to some extension to all families of classical orthogonal polynomials, as well as their q-analogues. Due probably to computational difficulties, the family of discrete Cramér–von Mises statistics has received less attention than its continuous counterpart—the aim of this article is to bridge part of this gap. [ABSTRACT FROM AUTHOR]

Let be a -algebra acting on a separable complex Hilbert space . We show that the inclusion of into factors through an -space only if is a finite type algebra [ABSTRACT FROM AUTHOR]

We prove a generalization of the strong subadditivity of the von Neumann entropy for bosonic quantum Gaussian systems. Such generalization determines the minimum values of linear combinations of the entropies of subsystems associated to arbitrary linear functions of the quadratures, and holds for arbitrary quantum states including the scenario where the entropies are conditioned on a memory quantum system. We apply our result to prove new entropic uncertainty relations with quantum memory, a generalization of the quantum Entropy Power Inequality, and the linear time scaling of the entanglement entropy produced by quadratic Hamiltonians. [ABSTRACT FROM AUTHOR]

The interaction between Unruh-De Witt spin 1/2 detectors and a real scalar field is scrutinized by making use of the Tomita-Takesaki modular theory as applied to the Von Neumann algebra of the Weyl operators. The use of the modular theory enables to evaluate in an exact way the trace over the quantum field degrees of freedom. The resulting density matrix is employed to the study of the Bell-CHSH correlator. It turns out that, as a consequence of the interaction with the quantum field, the violation of the Bell-CHSH inequality exhibits a decreasing as compared to the case in which the scalar field is absent. [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]

We develop a fast and accurate approach to approximate posterior distributions in the Bayesian empirical likelihood framework. Bayesian empirical likelihood allows for the use of Bayesian shrinkage without specification of a full likelihood but is notorious for leading to several computational difficulties. By coupling the stochastic variational Bayes procedure with an adjusted empirical likelihood framework, the proposed method overcomes the intractability of both the exact posterior and the arising evidence lower bound objective, and the mismatch between the exact posterior support and the variational posterior support. The optimization algorithm achieves fast algorithmic convergence by using the variational expected gradient of the log adjusted empirical likelihood function. We prove the consistency of the proposed approximate posterior distribution and an empirical likelihood analogue of the variational Bernstein-von-Mises theorem. Through several numerical examples, we confirm the accuracy and quick algorithmic convergence of our proposed method. for this article are available online. [ABSTRACT FROM AUTHOR]

The description of (commutative and noncommutative) Lp$L_p$‐isometries has been studied thoroughly since the seminal work of Banach. In the present paper, we provide a complete description for the limiting case, isometries on noncommutative L0$L_0$‐spaces, which extends the Banach–Stone theorem and Kadison's theorem for isometries of von Neumann algebras. The result is new even in the commutative setting. [ABSTRACT FROM AUTHOR]

We construct the noncommutative Poisson boundaries of tracial von Neumann algebras through the ultraproducts of von Neumann algebras. As an application of this result, we complete the proof of Kaimanovich-Vershik's fundamental theorems regarding noncommutative entropy. We also prove the Amenability-Trivial Boundary equivalence and Choquet-Deny-Type I equivalence for tracial von Neumann algebras. [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]

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 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]

Assume that is a faithful normal semifinite trace on a von Neumann algebra , is the unit of , is the -algebra of -measurable operators, and is the Banach space of -integrable operators. We present a new proof of the following generalization of Putnam's theorem (1951): No positive self-commutator with is invertible in . If is infinite then no positive self-commutator with can be of the form , where is a nonzero complex number and is a -compact operator. Given with we seek for the conditions that . If and with then . If and then . If a partial isometry lies in and for some integer then is a commutator and implies that . [ABSTRACT FROM AUTHOR]

This paper concentrates on examining the characterization of nonlinear mixed bi-skew Lie triple ∗- derivations within an ∗-algebra denoted by A which contains a nontrivial projection with a unit I. Additionally, we expand this investigation to applications by describing these derivations within prime ∗-algebras, von Neumann algebras, and standard operator algebras. [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]