There are several well-known fundamental theorems in Gabor analysis that are naturally connected to group representation theory and theory of operator algebras.While some of these connections between time-frequency analysis and operator algebras were established by Jon von Neumann in 1930s, they have been extensively investigated more recently mainly due to the developments of wavelet/Gabor theory, or more generally, the theory of frames in the last two decades.In this article, we will discuss some of the main results we obtained in the last few years together with some new results, exposition and open problems.We will be mainly focused on the results that were originated from time-frequency analysis but reflect intrinsic connections with group representation theory.In particular, we give a detailed account on an abstract version of the duality principle in time-frequency analysis for group representations, and its connections with some open problems in the theory of operator algebras.

Let H be a complex Hilbert space and B(H) the algebra of all bounded linear operators on H.Let M⊆B(H) be a von Neumann algebra and " the star partial order in M, that is, for A, B∈M, then we say that AB if A^*A=A^*B and AA^*=BA^*.It is proved that the supremum and infimum of a subset in M with respect to the star partial order are the same as in B(H).Moreover, we give the representation of a star partial order-hereditary subspace in M, that is, a W^* closed nonzero subspace A in M is star partial order-hereditary, which means that for any A∈M and B∈A, A∈A whenever AB, if and only if there is a unique pair of nonzero projections E and F which have the same central carrier in M such that A=EMF.

Let U=Tri(A, M, B) be a triangular algebra.In this paper, under mild assumptions, we prove that if δ:U→U is a linear map satisfying δ([[U, V], W])=[[δ(U), V], W]+[[U, δ(V)], W]+[[U, V], δ(W)], for any U, V, W∈U with UV=UW=0(resp.UV=UW=0), then δ(U)=Φ(U)+h(U) for any U∈U, where Φ:U→U is a derivation, h:U→Z(U) is a linear map vanishing at second commutators with UV=UW=0(resp.UV=UW=0).

Let H be a complex Hilbert space with dim H>2 and A be a selfadjoint standard operator algebra on H.For given positive integer k≥1, the k-skew commutator of operators A and B on H is defined as _*[A, B]_k=_*[A, _*[A, B]_k-1], where _*[A, B]₀=B, _*[A, B]₁=AB-BA^*.Assume k≥4 and Φ is a map on A with range containing all rank one projections.It is shown that, _*[Φ(A), Φ(B)]_k=_*[A, B]_k holds for all A, B∈A if and only if Φ(A)=A for all A∈A, or Φ(A)=-A for all A∈A.The latter case does not occur if k is even.

We show that when the cb-distance dcb(V, W) between two W^*-ternary ring of operators V and W is small, the distance between their linking von Neumann algebras R(V) and R(W) is also small.We show that W^*-ternary ring of operators close to injective W^*-ternary ring of operators is injective again.We also consider similar properties for W^*-ternary ring of operators with property Γ or McDuff property.

We have proved that the algebraic action determined by some elements in the integer group ring of the free group on two generators is ergodic and computed the Fuglede-Kadison determinant of specific element in the Heisenberg group factor.

We introduce some work on Hardy-Sobolev spaces and Fock spaces and their operators and operator algebras, including discussing the boundedness, compactness, Fredholmness, index theory, spectrum and essential spectrum, norm and essential norm and Schatten-p class of some special kinds of operators on these two types of space, and studying the corresponding C^*-algebra generated by them.

We present the recent work on the noncommutative Fourier transform for subfactors and locally compact quantum groups, a Survey.We give a short introduction to subfactors and locally compact quantum groups and their properties;the Hausdorff-Young inequality and its extremal functions;Young's inequality and its extremal pairs;uncertainty principles and its minimizers;a noncommutative sum set theorem.

We investigate the existence of quasi-invariant subspaces with arbitrary index.We first give a general criterion.As applications, we show that both the Focktype spaces F^p(C)={f∈Hol(C):(1)/(π)∫_C|f(z)|^pe^-|z|2dA(z)< +∞, 1≤p< +∞} and the Hilbert space H={f∈Hol(C):f∈Hol(C):(1)/(π)∫_C|f(z)|²e^-|z|dA(z)<+∞} have quasiinvariant subspaces with arbitary index.

We give an equivalent characterization for the α-comparison property of C^*-algebras:any simple unital stably finite C^*-algebra A has the α-comparison property, if and only if, for any <a>, <b>∈W(A), α·d_τ(a) <d_τ(b)(∀τ∈QT(A)) implies that <a>≤<b> holds in W(A).Using this characterization, we prove the following results:C^*-algebras with α-comparison property have weak comparison;C^*-algebras with α-comparison property for α=m+1 have strong tracial m-comparison of positive elements;Z-stability strict comparison α-comparison property for α=m+1 strong tracial m-comparison weak comparison and local weak comparison all agree for the C^*-algebras satisfying the conditions given by Kirchberg-Rørdam;if α:=inf{α'∈(1, ∞)|A has the α-comparison property}<∞, then A has the α-comparison property.

We introduce the concept of p-completely bounded frames for p-operator spaces.We prove that a separable p-operator space X has a p-completely bounded frame if and only if it has the p-completely bounded approximation property if and only if it can be p-completely complementedly embedded into a p-operator space with a pcompletely bounded basis.For a non-separable p-operator space with the p-completely bounded approximation property, we prove that its separable subspace always can be p-completely isomorphically embedded into a p-operator space with a p-completely bounded frame.

We introduce compact quantum metric spaces and quantum Gromov-Hausdorff distance defined by Rieffel and quantum Gromov-Hausdorff propinquity recently defined by Latrémolière, and discuss the question of how matrix algebras converge to the sphere in both quantum distances, respectively.

We show that Voiculescu's topological entropy of the canonical automorphism of the C^*-algebra arising from the asymptotic equivalence on every irreducible zero-dimensional Smale space is equal to the topological entropy of the original topological dynamics.For the related C^*-dynamical system, we have the "variational principle" with respect to the CNT-entropy and the topological entropy, and also show that the state defined by the Bowen measure of the Smale space is the unique equilibrium state of the canonical automorphism.

The famous work of Murray and von Neumann about decomposing W^*-algebras into different types(which is known as the classification theory of W^*-algebras) is based on the study of projections in W^*-algebras.Different from W^*-algebras(which are generated by projections), a C^*-algebra may contain no non-zero projection.Therefore, we cannot transport the classification theory of Murray and von Neumann directly to C^*-algebras.In our recent works, we have developed two classifying(or decomposition) schemes of C^*-algebras using the properties of their open projections and properties of their positive elements, respectively.In this note, after a briefing of our two classifying schemes of C^*-algebras, we introduce a more general classification framework that, on top of giving many other possible schemes, can be used to obtain, compare and refine the two classification schemes mentioned above.

Let R be a ring, M be an R-bimodule, m and n be two fixed nonnegative integers with m+n=0.If an additive mapping δ from R into M satisfies(m+n)δ(A²)=2mAδ(A)+2nδ(A)A for every A in R, then δ is called an(m, n)-Jordan derivation.In this paper, we prove that if R is a unital ring and M is a unital Rbimodule with a left(right) separating set generated algebraically by all idempotents in R, then every(m, n)-Jordan derivation from R into M is identical with zero whenever m, n>0 and m=n.We also show that if A and B be two unital rings, M is a faithful unital(A, B)-bimodule(N is a faithful unital(B, A)-bimodule), m, n>0 and m=n, U=[_N^A_B^M] is a |mn(m-n)(m+n)|-torsion-free generalized matrix ring, then every(m, n)-Jordan derivation from U into itself is equal to zero.

The paper studies the structure of restricted pre-Lie algebras. More specifically speaking, we first give the definitions of restricted pre-Lie algebras and restrictable pre-Lie algebras. Second, we obtain some properties of p-mappings and restrictable preLie algebras, and research restricted pre-Lie algebras with semisimple elements. Finally, quasi-toral restricted pre-Lie algebras are discussed and the uniqueness of decomposition for restricted pre-Lie algebras is determined.

We are concerned with the following semilinear elliptic equations with Sobolev-Hardy exponent -Δu-μ(u)/(|x|²)=λu+(|u|^2*(s)-2)/(|x|^s)u+f, in Ω,
u=0, on ∂Ω,
Where 2^*(s)=(2(N-S))/(N-2) is the critical Sobolev-Hardy exponent, N≥3, 0≤s<2, 0≤μ<ū=((N-2)²)/(4). We show that for 0≤λ<λ₁, where λ₁ is the first eigenvalue of the operator -Δ-(μ)/(|x|²) and f∈(H₀¹(Ω)^*), the dual space of H₀¹(Ω), with f(x)0. Under appropriate assumptions on f(x), we show that has at least two solutions. Moreover, if f≥0, the obtained solutions are non-negative.

Let X, Y be complex Banach spaces with dimentions greater than 1. Let A, B be normed closed subalgebras of B(X), B(Y) containing finite rank operators, respectively. For any A, B∈A, we define the quasi-product of A and B as AB=A+B-AB. In this paper, A characterization of additive mappings from A onto B which preserve any one of (left, right) quasi-invertibility and (left, right, semi) quasizero divisors in both directions is given.

Let x:Mⁿ→Sⁿ⁺¹ be a hypersurface in the (n+1)-dimensional unit sphere Sⁿ⁺¹ without umbilics. Four basic invariants of x under the Moebius transformation group in Sⁿ⁺¹ are Moebius metric g, Moebius second fundamental form B; Moebius form Φ and Blaschke tensor A. We classify the Moebius isoparametric hypersurfaces in Sⁿ⁺¹ with four distinct principal curvatures which multiplies are 1, 1, 1, m(m≥2).

By using a generalized Grunsky inequality, we obtain some estimates of the essential norm of the Grunsky operator for a univalent function in terms of the boundary distortion of the quasiconformal extension. As a corollary, we deduce the compactness criterion of the Grunsky operator.

According to the denseness and closedness of range of an operator, its point spectrum and residual spectrum are split into 1, 2-point-spectrum and 1, 2-residualspectrum, respectively. For 3×3 upper triangular operator matrices, the possible point spectra and possible residual spectra, ∪_{D, E, F}σ_*, i(M_{D, E, F})(*=p, r; i=1, 2), are given by means of the analysis method and block operator technique.

The automorphism group of an extraspecial Z-group is determined. Let G be an extraspecial Z-group, where G=|α_j∈Z, j=1, 2, ..., 2n+1,
let Aut_cG be the normal subgroup of AutG consisting of all elements of AutG which act trivially on ζG. Then AutG=Aut_cGZ₂, and there is an exact sequence 1→→Aut_cG→Sp(2n, Z)→1.

Let W be a self-orthogonal class of left R-modules. This paper concerns the class of n-strongly W-Gorenstein modules, which is a common generalization of strongly W-Gorenstein modules, strongly Gorenstein projective modules and n-strongly Gorenstein projective modules. Special attention is given to n-strongly W_P-Gorenstein mod-ules and n-strongly W_I-Gorenstein modules. The stability of n-strongly W-Gorenstein category is considered, some concrete characterizations of W_P-Gorenstein modules in B_C(R) are given and new versions of Foxby equivalence with respect to n-strongly W_P-Gorenstein (resp., n-strongly W_I-Gorenstein) modules are established. The properties of n-strongly W_F-Gorenstein modules are also investigated.

We establish the first and the second-order asymptotics of distributions of normalized maxima of independent and non-identically distributed bivariate Gaussian triangular arrays, where each vector of the n-th row follows from a bivariate Gaussian distribution with correlation coefficient being a monotone continuous positive function of i/n. Furthermore, parametric inference for this unknown function is studied. Some simulation study and real data sets analysis are also presented.

Longitudinal data are usually analyzed using mixed effects models under the assumption of normal distributions. A departure from normality may result in invalid inference. Compared with the traditional mean regression, quantile regression can characterize a complete scan of the conditional distribution of the response variable and provide more robust inferences for nonnormal error distributions. In this paper, we focus on the quantile estimation and variable selection of censored mixed effects models. Firstly, the inverse censoring probability weighted (ICPW) method is utilized to obtain parameters estimation. Furthermore, the LASSO penalties are incorporated into the ICPW method to implement variable selection. Monte Carlo simulations demonstrate that the proposed method performs superior to the "naive" method which ignores censored data. Finally, an AIDS data set is analyzed to illustrate the proposed method.

We studied the derivations and the second cohomology group of the classical N=2 Lie conformal superalgebra. Furthermore, we investigated the universal central extension of this Lie conformal superalgebra by applying the result on the second cohomology group.

The paper is about the asymptotic properties of Degasperis-Procesi equation. That is, using the method of asymptotic density, under the assumption that it is unique, the paper proves that the positive momentum density of the Degasperis-Procesi equation is a combination of Dirac measures supported on the positive axis. This means that as time goes to infinity, the momentum density concentrates in small intervals moving right with different constant speeds.

We study the weighted Poincaré inequality and log-Sobolev inequality for one dimensional Cauchy distribution. We offer the optimal weighted functions, prove the sharpness, and estimate the orders of the constants in the inequalities.

Let μ_M,D be the self-affine measure uniquely determined by the iterated function system (IFS){φ_d(x)=M^-1(x+d)}d∈D with equal weight, where M ∈ M_n(R) is an expanding matrix and D⊂Rⁿ is a finite digit set. The spectrality or nonspectrality of μ_M,D is directly connected with the existence of orthogonal exponential basis (Fourier basis) in the Hilbert space L²(μ_M,D), and has been received much attention in recent years. In this paper, we provide several sufficient conditions for self-affine measures to be non-spectral. The results here extend the corresponding non-spectral criterions of Dutkay, Jorgensen and others in a simple manner.

Using the Littlewood-Paley decomposition technique, Fourier transform and inverse Fourier transform, we study the boundedness of bilinear Fourier multiplier operators on the scales of inhomogeneous Triebel-Lizorkin and Besov spaces with positive smoothness.

The relationship between James constant J(X) and the Jordan-von Neumann type constant C_-∞(X) is given to show there exists a example such that C_-∞(X)< C_Z(X). Moreover, some sufficient conditions for normal structure in terms of the constant C_-∞(X) and the Benavides constant R(a, X) are presented. These results improve some known results.

We studied the best restriction approximation problems using entire functions of exponential type as the approximation tools on some generalized Sobolev classes of smooth functions defined by the differential operator induced by an algebraic polynomial with only real zeros. By the methods of Fourier transform and periodization, etc, we obtained the exact constants of the average relative widths and the best restriction approximation on the generalized Sobolev classes in the L²(R) norm, and obtained the asymptotic results of the best restriction approximation on the generalized Sobolev classes in the L₁(R) norm and the uniform norm for the case that the polynomial has a zero of multiplicity at most 2 at the point 0.

In a series of papers Mauduit and Sárközy introduced and studied the measures of finite sequences of k symbols. In this paper we construct large family of pseudorandom sequences of k symbols with length pq using the residue class ring modulo pq and the methods of discrete logarithm, and study the pseudorandom properties.

The existence of the iterative solution for the operator equation A(x, y) = L_x is studied by the techniques of partially order and the theorem of cone in Banach space, where neither A nor L need to be continues or compactness. Besides, we construct some new iterative sequences and study their approximation, then we get some new theorems. Finally, we apply the new results presented in this paper to study the solvability of a class of integral-differential equation.

The matrix representations of a class of fourth order boundary value problems with eigenparameter-dependent boundary conditions which have a finite spectrum are investigated. We first prove that for any positive integer m, the considered problem has at most 2m + 6 eigenvalues. Next, we show that this fourth order boundary value problem with eigenparameter-dependent boundary condition is equivalent to a class of matrix eigenvalue problem in the sense that they have exactly the same eigenvalues.

Full subsemigroups are defined as subsemigroups containing all idempotents. A semigroup is said to be a ▽_fs-semigroup if its full subsemigroups form a chain under inclusion. The aim of this paper is to investigate abundant ▽_fs-semigroups. Some characterizations of such semigroups are obtained. In particular, the structures of completely 0-simple ▽_fs-semigroups and primitive abundant ▽_fs-semigroups satisfying the regularity condition are established.

We investigate the asymptotic properties of the estimators of quantile difference based on left truncated and right censored data. The TJW product-limit estimator of the distribution function with the left truncated and right censored data is used to provide the empirical estimator of the quantile difference. Meanwhile, another smoothed kernel estimator for the quantile difference is established. Using the theory of empirical process, the expressions of the asymptotic bias and variance of the two estimators are derived. The large sample properties, such as consistency and asymptotic normality, for the estimators are obtained. A small simulation study shows that in the sense of mean squared loss, the smoothed estimator is more efficient than the non-smoothed estimator.

The structure of the finitely generated nilpotent groups with infinite cyclic Frattini subgroups are completely determined. More exactly, the following theorem is proved. Let G be a finitely generated nilpotent group. Then the Frattini subgroup of G is infinite cyclic if and only if G has a decomposition G = S×F×T, where F is a free abelian group of rank s, T = Z_m1 ⊕ Z_m2 ⊕ … ⊕ Z_mu, m₁, m₂, …, m_u are square free integers greater than 1, m₁|m₂|… |m_u,
where d₁, d₂, …, dr are integers and d₁|d₂|… |d_r. Moreover, (d₁, d₂, …, d_r; s; m₁, m₂, …, m_u) is an isomorphic invariant of G. That is to say, if H is also a finitely generated nilpotent group with infinite cyclic Frattini subgroup, then G is isomorphic to H if and only if they have the same invariants.

It is proved in this paper that a connected cochain DG algebra A is a Koszul Calabi-Yau DG algebra if either its cohomology graded algebra H(A) or its underlying graded algebra A^# is the algebra k<x,y>/(xy+yx) generated by degree 1 elements x,y.

A finite group G is called a PNN-group if every minimal subgroup X of G is either normal in G or self-normalizing. In this paper the authors investigate properties of a PNN-group in general and then classify the non-PNN-groups whose proper subgroups are all PNN-groups.