<Emphasis Type="Italic">C</Emphasis>*-Homomorphisms and duality of <Emphasis Type="Italic">C</Emphasis>*-discrete quantum groups

Let

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be a C*-discrete quantum group and let

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be the discrete quantum group associated with

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. Suppose that there exists a continuous action of

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on a unital C*-algebra

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so that

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becomes a

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-algebra. If there is a faithful irreducible vacuum representation π of

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on a Hilbert space H =

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with a vacuum vector Ω, which gives rise to a

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-invariant state, then there is a unique C*-representation (θ, H) of

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supplemented by the action. The fixed point subspace of

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under the action of

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is exactly the commutant of θ(

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).

     

C-spaces and simplicial complexes

Given a class

of simplicial complexes G, we introduce the notion of a

-C-space. In the definition of a C-space, open disjoint families v _i refine coverings u _i . The nerves of these families are zero-dimensional complexes. In our definition, the nerve of a family vi must embed in the complex G _i of the class

. We give a complete characterization of bicompact

-C-spaces.

     

On Jordan ideals and submodules: Algebraic and analytic aspects

Let

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be an algebra, and let X be an arbitrary

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-bimodule. A linear space Y ? X is called a Jordan

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-submodule if Ay + yA ∈ Y for all A ∈

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and y ∈ Y. (For X =

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, this coincides with the notion of a Jordan ideal.) We study conditions under which Jordan submodules are subbimodules. General criteria are given in the purely algebraic situation as well as for the case of Banach bimodules over Banach algebras. We also consider symmetrically normed Jordan submodules over C*-algebras. It turns out that there exist C*-algebras in which not all Jordan ideals are ideals.

     

Boundedness and compactness of an integral operator between <Emphasis Type="Italic">H</Emphasis><Superscript>∞</Superscript> and a mixed norm space on the polydisk

This addendum to [1] completely characterizes the boundedness and compactness of a recently introduced integral type operator from the space of bounded holomorphic functions H ^∞(\(\mathbb{D}^n \)) on the unit polydisk \(\mathbb{D}^n \) to the mixed norm space

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with p, q ∈ [1,∞) and α = (α₁, ..., α _n ) such that α _j > ?1 for every j = 1, ..., n. We show that the operator is bounded if and only if it is compact and if and only if g ∈

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, where \(\vec q\) = (q, ..., q).

     

Sur les fonctions arithmétiques non entières

A theorem of Baker says that a function F entire on ? ^d such that F(? ^d ) ? ? and increasing slower (in a precise sense) than \(2^{z_1 + \cdots + z_d } \) is necessarily a polynomial. This is a multivariate generalisation of the celebrated theorem of Pólya (case d = 1). Using the theory of analytic functionals with non-compact carrier, Yoshino proved a general theorem dealing with the growth of arithmetic analytic functions, which implies that the conclusion of Baker’s theorem holds if F is only assumed to be holomorphic on the domain

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, j = 1,..., d}.

The case d = 1 was also treated in a different way by Gel’fond and Pólya by means of the characteristic function of Carlson-Nörlund. This function was introduced to bound in a nearly optimal way the growth of holomorphic functions of one variable that can be expanded in a Newton interpolation series in the half-plane

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.

In this article, we show how this characteristic function can also be used to bound the growth of multivariate functions defined on

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that can be expanded in multiple Newton series. These considerations enable us to improve Gel’fond-Pólya’s and Yoshino’s theorems, in particular, to remove or to weaken certain of their technical conditions.

     

Wavelet transform and radon transform on the quaternion Heisenberg group

Let Q be the quaternion Heisenberg group,and let P be the affine automorphism group of Q.We develop the theory of continuous wavelet transform on the quaternion Heisenberg group via the unitary representations of P on L2(Q).A class of radial wavelets is constructed.The inverse wavelet transform is simplified by using radial wavelets.Then we investigate the Radon transform on Q.A Semyanistyi–Lizorkin space is introduced,on which the Radon transform is a bijection.We deal with the Radon transform on Q both by the Euclidean Fourier transform and the group Fourier transform.These two treatments are essentially equivalent.We also give an inversion formula by using wavelets,which does not require the smoothness of functions if the wavelet is smooth.In addition,we obtain an inversion formula of the Radon transform associated with the sub-Laplacian on Q.     

Commuting variety of Witt algebra

Let \(\mathfrak{g} = W_1 \) be the Witt algebra over an algebraically closed field k of characteristic p > 3; and let Open image in new window be the commuting variety of g. In contrast with the case of classical Lie algebras of P. Levy [J. Algebra, 2002, 250: 473–484], we show that the variety Open image in new window is reducible, and not equidimensional. Irreducible components of Open image in new window and their dimensions are precisely given. As a consequence, the variety Open image in new window is not normal.     

Geodesic mapping onto Kählerian spaces of the first kind

In the present paper a generalized Kählerian space Open image in new window of the first kind is considered as a generalized Riemannian space \(\mathbb{G}\mathbb{R}_N \) with almost complex structure F _i ^h that is covariantly constant with respect to the first kind of covariant derivative.Using a non-symmetric metric tensor we find necessary and sufficient conditions for geodesic mappings f: Open image in new window with respect to the four kinds of covariant derivatives. These conditions have the form of a closed system of partial differential equations in covariant derivatives with respect to unknown components of the metric tensor and the complex structure of the Kählerian space Open image in new window .     

Convolution functions of several variables with generalized bounded variation

Let L ¹ be the class of all complex-valued functions, with period 2π in each variable, in the space , where $\mathbb{T} = [0,2\pi )$ is the one-dimensional torus. Here, it is observed that L ¹ * E ? E for E = Lip(p; α ₁, α ₂, ..., α _N ) over , for , for , and for in the sense of Vitali as well as Hardy.     

Strongly Gauduchon Spaces

The authors define strongly Gauduchon spaces and the class■■, which are generalization of strongly Gauduchon manifolds in complex spaces. Comparing with the case of Kahlerian, the strongly Gauduchon space and the class■are similar to the Kahler space and the Fujiki class■■ respectively. Some properties about these complex spaces are obtained. Moreover, the relations between the strongly Gauduchon spaces and the class■■ are studied.     

Continuous embeddings of Besov-Morrey function spaces

We study embeddings of spaces of Besov-Morrey type, M Bp1,q1s1,r1(Rd ) → M Bp2 ,q2s2 ,r2 (R d ), and obtain necessary and sufficient conditions for this. Moreover, we can also characterise the special weighted situation Bp1 ,r1s1 (R d , w) → M Bp2 ,q2s2 ,r2 (Rd ) for a Muckenhoupt A ∞ weight w, with wα(x) = |x|α , α -d1, as a typical example.     

Some remarks on projective generators and injective cogenerators

In this paper,we give a relationship between projective generators(resp.,injective cogenerators) in the category of R-modules and the counterparts in the category of complexes of R-modules.As a consequence,we get that complexes of W-Gorenstein modules are actually W-Gorenstein complexes whenever W is a subcategory of R-modules satisfying W⊥W,where W is the subcategory of exact complexes with all cycles in W.We also study when all cycles of a W-Gorenstein complexes are W-Gorenstein modules.     

On minimal non- -groups

A finite group G is called an J N J-group if every proper subgroup H of G is either subnormal in G or self-normalizing. We determinate the structure of non-J N J-groups in which all proper subgroups are J N J- groups.     

Double Hilbert transform on

We introduce a space , where is the testing function space whose functions are infinitely differentiable and have bounded support, and is the space the double Hilbert transform acting on the testing function space. We prove that the double Hilbert transform is a homeomorphism from onto itself.     

Weakly infinite-dimensional spaces modulo simplicial complexes

The classes of spaces

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-wid and ?-wid are introduced for the class

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of finite simplicial complexes and the class ? of compact polyhedra. If

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, then

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-wid = wid, ?-wid = S-wid. It is proved that S-wid ? ?-wid and ?-wid = S-? _τ -wid for any triangulation τ of the class ?.

     

Optimization of the boundary control of string vibrations in the excitation problem with a fixed endpoint in the class <Emphasis Type="Italic">W</Emphasis><Stack><Subscript>2</Subscript><Superscript>2</Superscript></Stack>

We consider the problem of optimal boundary control of string vibrations by a displacement at one endpoint with the other endpoint being fixed; the problem is considered in the space

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and then, more generally, in the space

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, p ≥ 1. The control brings the vibration process from the quiescent state to an arbitrarily prescribed state in a time that is a multiple of the double string length.

     

BGP-reflection functors and Lusztig’s symmetries of modified quantized enveloping algebras

Let U be a quantized enveloping algebra and  its modified form.Lusztig gives some symmetries on U and.In view of the realization of U by the reduced Drinfeld double of the Ringel-Hall algebra,one can apply the BGP-reflection functors to the double Ringel-Hall algebra to obtain Lusztig's symmetries on U and their important properties,for instance,the braid relations.In this paper,we define a modified form ■ of the Ringel-Hall algebra and realize the Lusztig's symmetries on  by applying the BGP-reflection functors to ■.     

Weighted Solyanik Estimates for the Hardy–Littlewood Maximal Operator and Embedding of into

Let Open image in new window denote a weight in Open image in new window which belongs to the Muckenhoupt class Open image in new window and let Open image in new window denote the uncentered Hardy–Littlewood maximal operator defined with respect to the measure Open image in new window . The sharp Tauberian constant of Open image in new window with respect to Open image in new window , denoted by Open image in new window , is defined by

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In this paper, we show that the Solyanik estimate

$$\begin{aligned} \lim _{\alpha \rightarrow 1^-}\mathsf{C}_{w}(\alpha ) = 1 \end{aligned}$$

holds. Following the classical theme of weighted norm inequalities we also consider the sharp Tauberian constants defined with respect to the usual uncentered Hardy–Littlewood maximal operator Open image in new window and a weight Open image in new window :

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We show that we have Open image in new window if and only if Open image in new window . As a corollary of our methods we obtain a quantitative embedding of Open image in new window into Open image in new window .

     

Positive definite and Gram tensor complementarity problems

Given a continuous function Open image in new window and Open image in new window , the non-linear complementarity problem \(\text{ NCP }(g,q)\) is to find a vector Open image in new window such that

$$\begin{aligned} x \ge 0,~~y:=g(x) +q\ge 0~~\text{ and }~~x^Ty=0. \end{aligned}$$

We say that g has the Globally Uniquely Solvable (\(\text{ GUS }\))-property if \(\text{ NCP }(g,q)\) has a unique solution for all Open image in new window and C-property if \(\mathrm{NCP}(g,q)\) has a convex solution set for all Open image in new window . In this paper, we find a class of non-linear functions that have the \(\text{ GUS }\)-property and C-property. These functions are constructed by some special tensors which are positive semidefinite. We call these tensors as Gram tensors.