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In this paper, we discuss the Schatten-p class (0 < p ≤ ∞) of Toeplitz operators on generalized Fock space with the symbol in positive Borel measure. It makes a great difference from other papers by using the estimates of the kernel and the weight together instead of separately estimating each other. We also get the equivalent conditions when a Toeplitz operator is in the Schatten-p class.

In this paper, we prove the Hyers-Ulam-Rassias stability of isometric homomorphisms in proper CQ*-algebras for the following Cauchy-Jensen additive mapping: $$ 2f\left( {\frac{{x_1 + x_2 }} {2} + y} \right) = f(x_1 ) + f(x_2 ) + 2f(y). $$ The concept of Hyers-Ulam-Rassias stability originated from the Th.M. Rassias’ stability theorem that appeared in the paper: On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc., 72 (1978), 297–300.

In this paper, we introduce the fractional wavelet transformations (FrWT) involving Hankel–Clifford integral transformation (HClIT) on the positive half line and studied some of its basic properties. Also we obtain Parseval’s relation and an inversion formula. Examples of fractional powers of Hankel–Clifford integral transformation (FrHClIT) and FrWT are given. Then, we introduce the concept of fractional wavelet packet transformations FrBWPT and FrWPIT, and investigate their properties.

In this paper, we study positive Toeplitz operators on the harmonic Bergman space via their Berezin transforms. We consider the Toeplitz operators with continuous harmonic symbols on the closed disk and show that the Toeplitz operator is positive if and only if its Berezin transform is nonnegative on the disk. On the other hand, we construct a function such that the Toeplitz operator with this function as the symbol is not positive but its Berezin transform is positive on the disk. We also consider the harmonic Bergman space on the upper half plane and prove that in this case the positive Toeplitz operators with continuous integrable harmonic symbols must be the zero operator.

In this paper, by using the Anick’s resolution and Grobner–Shirshov basis for quantized enveloping algebra of type G2, we compute the minimal projective resolution of the trivial module of Uq+ (G2) and as an application we compute the global dimension of Uq+ (G2).

In this paper, the structure of Jordan higher derivable maps on triangular algebras by commutative zero products is given. As an application, the form of Jordan higher derivable maps of nest algebras by commutative zero products is obtained.

In this paper, we study the structure theory of a class of not-finitely graded Lie algebras related to generalized Heisenberg–Virasoro algebras. In particular, the derivation algebras, the automorphism groups and the second cohomology groups of these Lie algebras are determined.

In this paper, we prove a local odd dimensional equivariant family index theorem which generalizes Freed’s odd dimensional index formula. Then we extend this theorem to the noncommutative geometry framework. As a corollary, we get the odd family Lichnerowicz vanishing theorem and the odd family Atiyah-Hirzebruch vanishing theorem.

In this paper, by using the de Rham model of Chen-Ruan cohomology, we define the relative Chen-Ruan cohomology ring for a pair of almost complex orbifold (G,H) with H being an almost sub-orbifold of G. Then we use the Gromov-Witten invariants of Ĝ, the blow-up of G along H, to give a quantum modification of the relative Chen-Ruan cohomology ring H*CR(G,H) when H is a compact symplectic sub-orbifold of the compact symplectic orbifold G.

In this paper, we give a classification of almost Hermitian metrics with nonpositive holomorphic bisectional curvature on a product of compact almost complex manifolds. This generalizes previous results of Zheng [Ann. of Math. (2), 137(3), 671–673 (1993)] and the author [Proc. Amer. Math. Soc., 139(4), 1469–1472 (2011)].

In this paper, we will estimate an upper bound for the similarity degree of the crossed product of a hyperfinite finite von Neumann algebra by weakly compact action of an infinite discrete group. We will also improve some upper bounds for similarity degrees of some finite von Neumann algebras.

In this paper, we classified the finite p-groups with exactly one A1-subgroup of given structure of order p3.

In this paper, we shall give an elementary proof of a Bishop-Stone-Weierstrass theorem for (M2(ℂ))n with respect to its pure states. To be more precise, we shall show that the pure-state Bishop hull of a unital subalgebra (not necessarily self-adjoint) of (M2(ℂ))n is equal to itself.

In this paper, we give discreteness criteria of subgroups of the special linear group on ℚp or ℂp in two and higher dimensions. Jorgensen’s inequality gives a necessary condition for a non-elementary group of Mobius transformations to be discrete. We give a version of Jorgensen’s inequality for SL(m,ℂp).

In this paper, we first establish an abstract inequality for lower order eigenvalues of a self-adjoint operator on a Hilbert space which generalizes and extends the recent results of Cheng et al. (Calc. Var. Partial Differential Equations, 38, 409–416 (2010)). Then, making use of it, we obtain some universal inequalities for lower order eigenvalues of the biharmonic operator on manifolds admitting some special functions. Moreover, we derive a universal inequality for lower order eigenvalues of the poly-Laplacian with any order on the Euclidean space.

In this paper, the concepts of approximate character amenability (contractibility), uniform approximate character amenability (contractibility) and w*-approximate character amenability are introduced. We are concerned with the relations among the generalized concepts of character amenability for Banach algebras. We show that approximate character amenability, w*-approximate character amenability and approximate character contractibility are the same properties, as uniform approximate character amenability and character amenability, as uniform approximate character contractibility and character contractibility. The general theory for these concepts is also developed. Moreover, approximate character amenability of several concrete classes of Banach algebras related to locally compact groups and also some discrete semigroups is considered.

The loop-Witt algebra is the Lie algebra of the tensor product of the Witt algebra and the Laurent polynomial algebra. In this paper we study the universal central extension, derivations and automorphism group for the loop-Witt algebra.

Let q be a sufficiently large integer and χ be a Dirichlet character modulo q. In this paper, we extend the product Πχ(−1)=−1L(1, χ) with prime q, arising from the Kummer conjecture, to the products of some general Dirichlet series, and give some meaningful estimates for them.

In this paper, the concepts of probabilistic normed Riesz space and probabilistic Banach lattice are introduced, and their basic properties are studied. In this context, some continuity and convergence theorems are proved.

In this paper, continuous homogeneous selections for the set-valued metric generalized inverses T∂ of linear operators T in Banach spaces are investigated by means of the methods of geometry of Banach spaces. Necessary and sufficient conditions for bounded linear operators T to have continuous homogeneous selections for the set-valued metric generalized inverses T∂ are given. The results are an answer to the problem posed by Nashed and Votruba.

This paper firstly discusses the existence of strongly irreducible operators on Banach spaces. It shows that there exist strongly irreducible operators on Banach spaces with w*-separable dual. It also gives some properties of strongly irreducible operators on Banach spaces. In particular, if T is a strongly irreducible operator on an infinite-dimensional Banach space, then T is not of finite rank and T is not an algebraic operator. On Banach spaces with subsymmetric bases, including infinite-dimensional separable Hilbert spaces, it shows that quasisimilarity does not preserve strong irreducibility. In addition, we show that the strong irreducibility of an operator does not imply the strong irreducibility of its conjugate operator, which is not the same as the property in Hilbert spaces.

In this paper, we give a class of C*-algebras with non-stable K1-group property which include the example non-simple tracial topological rank zero and stable rank two C*-algebra given by Lin and Osaka.

In this paper, we introduce the concept of VT-congruence triples on a regular semigroup S and show how such triples can be constructed by using the equivalences on S/ℒ, S/R and the special congruences on S. Also, such congruence triples are characterized so that an associated congruence can be uniquely determined by a given congruence triple. Moreover, we also consider the VH-congruence pairs on an orthocryptogroup.

Let (X, µ) be a measure space. In this paper, using some ideas from Grafakos and Kalton, the authors establish an off-diagonal Marcinkiewicz interpolation theorem for a quasilinear operator T in Lorentz spaces L p,q (X) with p, q ∈ (0,∞], which is a corrected version of Theorem 1.4.19 in [Grafakos, L.: Classical Fourier Analysis, Second Edition, Graduate Texts in Math., No. 249, Springer, New York, 2008] and which, in the case that T is linear or nonnegative sublinear, p ∈ [1,∞) and q ∈ [1,∞), was obtained by Stein and Weiss [Introduction to Fourier Analysis on Euclidean Spaces, Princeton University Press, Princeton, N.J., 1971].

Hom-Leibniz algebra is a natural generalization of Leibniz algebras and Hom-Lie algebras. In this paper, we develop some structure theory (such as (co)homology groups, universal central extensions) of Hom-Leibniz algebras based on some works of Loday and Pirashvili.

We quantize the W-algebra W(2, 2), whose Verma modules, Harish-Chandra modules, irreducible weight modules and Lie bialgebra structures have been investigated and determined in a series of papers recently.

In this paper we establish two theta function identities with four parameters by the theory of theta functions. Using these identities we introduce common generalizations of Hirschhorn-Garvan-Borwein cubic theta functions, and also re-derive the quintuple product identity, one of Ramanujan’s identities, Winquist’s identity and many other interesting identities.

This paper is primarily concerned with the proof of the Lp boundedness of parabolic Littlewood-Paley operators with kernels belonging to certain block spaces. Some known results are essentially extended and improved.

Block introduced certain analogues of the Zassenhaus algebras over a field of characteristic 0. The nongraded infinite-dimensional simple Lie algebras of Block type constructed by Xu can be viewed as generalizations of the Block algebras. In this paper, we construct a family of irreducible modules in terms of multiplication and differentiation operators on “polynomials” for four-devivation nongraded Lie algebras of Block type based on the finite-dimensional irreducible weight modules with multiplicity one of general linear Lie algebras. We also find a new series of submodules from which some irreducible quotient modules are obtained.

The purpose of this paper is to study derivations d, d′ defined on a Banach algebra A such that the spectrum σ([dx,d′x]) is finite for all x ∈ A. In particular we show that if the algebra is semisimple, then there exists an element a in the socle of A such that [d,d′] is the inner derivation implemented by a.

In this paper, we give a necessary and sufficient condition for a comodule algebra over a weak Hopf algebra to have a total integral, thus extending the classical theory developed by Doi in the Hopf algebra setting. Also, from these results, we deduce a version of Maschke’s Theorem for (H,B)-Hopf modules associated with a weak Hopf algebra H and a right H-comodule algebra B.

The noncommutative Singer-Wermer conjecture states that every linear (possibly unbounded) derivation on a (possibly noncommutative) Banach algebra maps into its Jacobson radical. This conjecture is still an open question for more than thirty years. In this paper we approach this question via linear left ϑ-derivations.

In this paper we discuss the K-groups of Wiener algebra . For the 1-shift space X GM2, we obtain a characterization of Fredholm operators on X 2 for all n ∈ ℕ. We also calculate the K-groups of operator algebra on the 1-shift space X GM2.

In the present paper we describe a specialization of prinjective Ringel-Hall algebra to 1, for prinjective modules over incidence algebras of posets of finite prinjective type, by generators and relations. This gives us a generalisation of Serre relations for semisimple Lie algebras. Connections of prinjective Ringel-Hall algebras with classical Lie algebras are also discussed.

Toroidal Leibniz algebras are the universal central extensions of the iterated loop algebras ˙ g⊗C(t ±1 1 ,...,t ±1 ν ) in the category of Leibniz algebras. In this paper, some properties and representations of toroidal Leibniz algebras are studied. Some general theories of central extensions of Leibniz algebras are also obtained.

In this paper, we consider the principal eigenvalue problem for Hormander’s Laplacian on R n , and we find a comparison principle for such principal eigenvalues. We also study a related semi-linear sub-elliptic equation in the whole R n and prove that, under a suitable condition, we have infinitely many positive solutions of the problem.

In this paper, we prove several results concerning meromorphic functions f and g of hyperorder less than one such that f(j) and g(j) have the same zeros and poles for j = 0, 1, 2. We provide some examples to show that our results are sharp.

In the present paper, a problem of Ioana Mihaila is negatively answered on the invertibility of composition operators on Riemann surfaces, and it is proved that the composition operator Cp is Fredholm if and only if it is invertible if and only if p is invertible for some special cases. In addition, the Toeplitz operators on \( \Lambda _{2,a}^1 (M) \)for Riemann surface M are defined and some properties of these operators are discussed.

In this paper, we prove directly that α-times integrated groups define algebra homomorphisms. We also give a theorem of equivalence between smooth distribution groups and α-times integrated groups.

In this paper, the authors study some properties of Littlewood–Paley g–functions g ψ (f), Lusin area functions S ψ,α (f) and Littlewood–Paley $$ g^{ * }_{{\psi ,\lambda }} {\left( f \right)} $$ functions defined on ℍ n , where α, λ > 0 and ψ, f are suitable functions. They are the generalization of the corresponding operators on ℝ n .

This paper gives a sufficient and necessary condition for a bialgebra to be a Long bialgebra, and proves a braided product to be a Long bialgebra under some conditions. It also gives a direct sum decomposition of quantum Yang–Baxter modules over Long bialgebras.

In this paper, we interpret Massey products in terms of realizations (twitsting cochains) of certain differential graded coalgebras with values in differential graded algebras. In the case where the target algebra is the cobar construction of a differential graded commutative Hopf algebra, we construct the tensor product of realizations and show that the tensor product is strictly associative, and commutative up to homotopy.

In this paper, we present some modules over the rank–three quantized Weyl algebra, which are closely related to modules over some vertex algebras. The isomorphism classes among these modules are also determined.

Suppose that G is a finite group and D(G) the double algebra of G. For a given subgroup H of G, there is a sub–Hopf algebra D(G;H) of D(G). This paper gives the concrete construction of a D(G;H)–invariant subspace \({\fancyscript A}\)H in field algebra of G–spin model and proves that if H is a normal subgroup of G, then \({\fancyscript A}\)H is Galois closed.

The aim of this paper is to investigate higher level orderings on modules over commutative rings. On the basis of the theory of higher level orderings on fields and commutative rings, some results involving existence of higher level orderings are generalized to the category of modules over commutative rings. Moreover, a strict intersection theorem for higher level orderings on modules is established.

The notion of generic extensions of representations of a Dynkin quiver plays a big role in the study of the structure of the corresponding quantum group. In this paper, we describe the generic extensions of a simple representation by any representation and that of any representation by a simple representation of a Dynkin quiver Q of type D.

In this paper, we generalize the Takesaki–Takai duality theorem in Hilbert C*-modules; that is to say, if (H, V, U) is a Kac-system, where H is a Hilbert space, V is a multiplicative unitary operator on H ⊗ H and U is a unitary operator on H, and if E is an \( {\fancyscript S} \)-compatible Hilbert \( {\fancyscript A} \)-module, then \( E \times {\hat{\fancyscript S}} \times {\fancyscript S} \cong E \otimes K(H), \)where K(H) is the set of all compact operators on H, and \( {\fancyscript S} \)and \( {\hat{\fancyscript S}} \)are Hopf C*-algebras corresponding to the Kac-system (H, V, U).

The aim of this paper is to study quasi-bicrossed products and a especially quantum quasi-doubles. Firstly, we construct one new kind of quasi-bicrossed products by weak Hopf algebras and then devote a brief discussion to this matter. And, we discuss the conditions for quasi-bicrossed products to possess the structure of almost weak Hopf algebras, containing the case of a special smash product. At the end, we give some properties on the quantum quasi-double, respectively on the quasi- R-isomorphism, the representation-theoretic interpretation and the regularity of the quasi-R-matrix.

In this paper we define a very simple invariant η\(\eta \left( {\overrightarrow V } \right)\) for a k-tuple \(\eta \left( {\overrightarrow V } \right)\) of unitaries in a finite factor von Neumann algebra, and we show how this invariant can replace free entropy in many of the important applications. We also introduce a notion of metric free entropy and some related concepts. We include proofs, using η, of the theorems of Liming Ge and of D. Voiculescu, respectively, on the primeness of and on the absence of Cartan subalgebras in the noncommutative free group factors.

During the last ten to fifteen years, a lot of progress has been achieved in the study of complex operator spaces. In this paper, we show that a corresponding theory can be developed for real operator spaces. With some appropriate modifications, many complex results still hold for real operator spaces.