SZEG TYPE FACTORIZATION THEOREM FOR NONCOMMUTATIVE HARDY-LORENTZ SPACES

We introduce noncommutative Hardy-Lorentz spaces and give the Szegō and inner-outer type factorizations of these spaces.

CONVERGENCE OF WEIGHTED AVERAGES OF MARTINGALES IN NONCOMMUTATIVE BANACH FUNCTION SPACES

Let x （xn）≥1 be a martingale on a noncommutative probability space n （M, r） and （wn）n≥1 a sequence of positive numbers such that Wn = ∑ k=1^n wk →∞ as n →∞ We prove that x = （x.）n≥1 converges in E（M） if and only if （σn（x）n≥1 converges in E（.hd）, where E（A//） is a noncommutative rearrangement invariant Banach function space with the Fatou property and σn（x） is given by σn（x） = 1/Wn ∑k=1^n wkxk, n=1, 2, .If in addition, E（Ad） has absolutely continuous norm, then, （an（x））≥1 converges in E（.M） if and only if x = （Xn）n≥1 is uniformly integrable and its limit in measure topology x∞∈ E（M）.

NONCOMMUTATIVE ORLICZ-HARDY SPACES

Let （Φ,Ψ） be a pair of complementary N-functions and HΦ（A） and HΨ（A） be the associated noncommutative Orlicz-Hardy spaces. We extend the Riesz, Szeg＆#168;o and inner-outer type factorization theorems of Hp（A） to this case.

CONDITIONAL EXPECTATIONS AND MARTINGALES IN NONCOMMUTATIVE Lp-SPACES ASSOCIATED WITH CENTER-VALUED TRACES

In this paper we prove the existence of conditional expectations in the noncom- mutative Lp（M, Ф）-spaces associated with center-valued traces. Moreover, their description is also provided. As an application of the obtained results, we establish the norm convergence of weighted averages of martingales in noncommutative Lp（M, Ф）-spaces.

OUTER OPERATORS FOR THE NONCOMMUTATIVE SYMMETRIC HARDY SPACES ASSOCIATED WITH FINITE SUBDIAGONAL ALGEBRA

In this article, we extended main results on outer operators of [6] to the symmetric Hardy spaces, when associated subdiagonal algebra is finite.

COMPLEX INTERPOLATION OF NONCOMMUTATIVE HARDY SPACES ASSOCIATED WITH SEMIFINITE VON NEUMANN ALGEBRAS

We proved a complex interpolation theorem of noncommutative Hardy spaces associated with semi-finite von Neumann algebras and extend the Riesz type factorization to the semi-finite case.

SZEG TYPE FACTORIZATION OF HAAGERUP NONCOMMUTATIVE HARDY SPACES

Let M be a σ-finite von Neumann algebra equipped with a normal faithful state φ, and let A be a maximal subdiagonal algebra of M. We proved a Szeg type factorization theorem for the Haagerup noncommutative H;-spaces.

LITTLEWOOD-PALEY CHARACTERIZATIONS OF ANISOTROPIC HARDY-LORENTZ SPACES

Let p∈（0, 1], q∈（0, ∞] and A be a general expansive matrix on Rn. Let HAp,q （Rn） be the anisotropic Hardy-Lorentz spaces associated with A defined via the non-tangential grand maximal function. In this article, the authors characterize HAp,q（Rn） in terms of the Lusin-area function, the Littlewood-Paley g-function or the Littlewood-Paley gλ＊-function via first establishing an anisotropic Fefferman-Stein vector-valued inequality in the Lorentz space Lp,q（Rn）. All these characterizations are new even for the classical isotropic Hardy-Lorentz spaces on Rn. Moreover, the range of λ in the gλ＊-function characterization of HAp,q （Rn） coincides with the best known one in the classical Hardy space Hp（Rn） or in the anisotropic Hardy space HAp （Rn）.

On Quantum Mechanics on Noncommutative Quantum Phase Space

In this work,we develop a general framework in which Noncommutative Quantum Mechanics (NCQM), characterized by a space noncommutativity matrix parameter θ=∈_(ji)~kθ_k and a momentum noncommutativity matrix parameter β_(ij)=∈_(ij)~kβ_k,is shown to be equivalent to Quantum Mechanics (QM) on a suitable transformed Quantum Phase Space (QPS).Imposing some constraints on this particular transformation,we firstly find that the product of the two parameters θ and β possesses a lower bound in direct relation with Heisenberg incertitude relations,and secondly that the two parameters are equivalent but with opposite sign,up to a dimension factor depending on the physical system under study.This means that noncommutativity is represented by a unique parameter which may play the role of a fundamental constant characterizing the whole NCQPS.Within our framework,we treat some physical systems on NCQPS:free particle,harmonic oscillator,system of two-charged particles,Hydrogen atom.Among the obtained results, we discover a new phenomenon which consists of a free particle on NCQPS viewed as equivalent to a harmonic oscillator with Larmor frequency depending on β,representing the same particle in presence of a magnetic field=q~(-1).For the other examples,additional correction terms depending on β appear in the expression of the energy spectrum.Finally,in the two-particle system case,we emphasize the fact that for two opposite charges noncornmutativity is effectively feeled with opposite sign.

WEAK ATOMIC DECOMPOSITIONS OF MARTINGALE HARDY-LORENTZ SPACES AND APPLICATIONS

In this article, we establish some atomic decomposition theorems for martin- gale Hardy-Lorentz spaces. As applications, with the help of weak atomic decompositions, some interpolation theorems for martingale Hardy-Lorentz spaces are proved.

Solution of the spin-one DKP oscillator under an external magnetic field in noncommutative space with minimal length

The spin-one Duffin-Kemmer-Petiau （DKP） oscillator under a magnetic field in the presence of Ihe minimal length in the noncommutative coordinate space is studied by using the momentum space representation. The explicit form of energy eigenvalues is found, and the eigenfunctions are obtained in terms of the Jacobi polynomials. It shows that for the same azimuthal quantum number, the energy E increases monotonically with respect to the noncomnmtative parameter and the minimal length parameter. Additionally, we also report some special cases aiming to discuss the effect of the noncommutative coordinate space and the minimal length in the energy spectrum.

A Relation of the Noncommutative Parameters in Generalized Noncommutative Phase Space

We introduce the deformed boson operators which satisfy a deformed boson algebra in some special types of generalized noncommutative phase space. Based on the deformed boson algebra, we construct coherent state representations. We calculate the variances of the coordinate operators on the coherent states and investigate the corresponding Heisenberg uncertainty relations. It is found that there are some restriction relations of the noncommutative parameters in these special types of noncommutative phase space.

State-Vector Space and Canonical Coherent States in Noncommutative Plane

The structure of the state-vector space of identical bosons in noncommutative spaces is investigated. To maintain Bose-Einstein statistics the commutation relations of phase space variables should simultaneously include coordinate-coordinate non-commutativity and momentum-momentum non-commutativity, which leads to a kind of deformed Heisenberg-Weyl algebra. Although there is no ordinary number representation in this state-vector space, several set of orthogonal and complete state-vectors can be derived which are common eigenvectors of corresponding pairs of commuting Hermitian operators. As a simple application of this state-vector space, an explicit form of two-dimensional canonical coherent state is constructed and its properties are discussed.

Relativistic Oscillators in a Noncommutative Space and in a Magnetic Field

In this work, we study the relativistic oscillators in a noncommutative space and in a magnetic field. It is shown that the effect of the magnetic field may compete with that of the noncommutative space and that is able to vanish the effect of the noncommutative space.

Minimal Length Quantum Mechanics of Dirac Particles in Noncommutative Space

We study the two-dimensional harmonic oscillator in commutative and noncommutative space within the framework of minimal length quantum mechanics for spin-l^2 particles. The energy spectra and the eigenfunction are obtained in both cases. Special cases are also deduced.

Dynamics of Two-Level Trapped Ion in a Standing Wave Laser in Noncommutative Space

We study the dynamics of a two-level trapped ion in a standing wave electromagnetic field in two-dimensional （2D） noncommutative spaces in the Lamb-Dicke regime under the rotating wave approximation. We obtain the explicit analytical expressions for the energy spectra, energy eigenstates, unitary time evolution operator, atomic inversion, and phonon number operators. The Rabi oscillations, the collapse, and revivals in the average atomic inversion and the average phonon number are explicitly shown to contain the information of the parameter of the space noncommutativity, which sheds light on proposing new schemes based on the dynamics of trapped ion to test the noncommutativity.

Noncommutative Phase Space and the Two Dimensional Quantum Dipole in Background Electric and Magnetic Fields

The two dimensional quantum dipole springs in background uniform electric and magnetic fields are first studied in the conventional commutative coordinate space, leading to rigorous results. Then, the model is studied in the framework of the noncommutative (NC) phase space. The NC Hamiltonian and angular momentum do not commute any more in this space. By the means of the su(1,1) symmetry and the similarity transformation, exact solutions are obtained for both the NC angular momentum and the NC Hamiltonian.

Decoherence of a Damped Anisotropic Harmonic Oscillator under Magnetic Field Effects in a Two-Dimensional Noncommutative Phase-Space

In this paper, decoherence of a damped anisotropic harmonic oscillator in the presence of a magnetic field is studied in the framework of the Lindblad theory of open quantum systems in noncommutative phase-space. General fundamental conditions that should follow our quantum mechanical diffusion coefficients appearing in the master equation are kindly derived. From the master equation, the expressions of density operator, the Wigner distribution function, the expectation and variance with respect to coordinates and momenta are obtained. Based on these quantities, the total energy of the system is evaluated and simulations show its dependency to phase-space structure and its improvement due to noncommutativity effects and the environmental temperature as well. In addition, we also evaluate the decoherence time scale and show that it increases with noncommutativity phase-space effects as compared to the commutative case. It turns out from simulations that this time scale is significantly improved under magnetic field effects.

LENTICULAR NONCOMMUTATIVE TORI

All C*-algebras of sections of locally trivial C* -algebra bundles over ∏i=1sLki(ni) with fibres Aw Mc(C) are constructed, under the assumption that every completely irrational noncommutative torus Aw is realized as an inductive limit of circle algebras, where Lki (ni) are lens spaces. Let Lcd be a cd-homogeneous C*-algebra over whose cd-homogeneous C*-subalgebra restricted to the subspace Tr × T2 is realized as C(Tr) A1/d Mc(C), and of which no non-trivial matrix algebra can be factored out.The lenticular noncommutative torus Lpcd is defined by twisting in by a totally skew multiplier p on Tr+2 × Zm-2. It is shown that is isomorphic to if and only if the set of prime factors of cd is a subset of the set of prime factors of p, and that Lpcd is not stablyisomorphic to if the cd-homogeneous C*-subalgebra of Lpcd restricted to some subspace LkiLki (ni) is realized as the crossed product by the obvious non-trivial action of Zki on a cd/ki-homogeneous C*-algebra over S2ni+1 for ki an integer greater than 1.

Noncommutative martingale Hardy-Orlicz spaces:Dualities and inequalities

We investigate dualities and inequalities related to noncommutative martingale Hardy-Orlicz spaces.More precisely,for a concave Orlicz functionΦ,we characterize the dual spaces of noncommutative martingale Hardy-Orlicz spaces HΦc(R)and hΦc(M),where R denotes a hyperfinite finite von Neumann algebra and M is a finite von Neumann algebra.The first duality is new even for classical martingales,which partially answers the problem raised by Conde-Alonso and Parcet(2016).We establish as well asymmetric martingale inequalities associated with Orlicz functions that are p-convex and q-concave for 0<p≤q<2.