Bogomolov multipliers for some <Emphasis Type="Italic">p</Emphasis>-groups of nilpotency class 2

The Bogomolov multiplier B ₀(G) of a finite group G is defined as the subgroup of the Schur multiplier consisting of the cohomology classes vanishing after restriction to all abelian subgroups of G. The triviality of the Bogomolov multiplier is an obstruction to Noether’s problem. We show that if G is a central product of G ₁ and G ₂, regarding K _i ≤ Z(G _i ), i = 1, 2, and θ: G ₁ → G ₂ is a group homomorphism such that its restriction \(\theta {|_{{K_1}}}:{K_1} \to {K_2}\) is an isomorphism, then the triviality of B ₀(G ₁/K ₁),B ₀(G ₁) and B ₀(G ₂) implies the triviality of B ₀(G). We give a positive answer to Noether’s problem for all 2-generator p-groups of nilpotency class 2, and for one series of 4-generator p-groups of nilpotency class 2 (with the usual requirement for the roots of unity).     

Unitary multiplier and dilation of projective isometric representation

In this paper we study unitary operator-valued multiplier σ on a normal subsemigroup S of a group G with its extension to G. A dilation of a projective isometric σ-representations of S to a projective unitary Φ(σ)-representation of G is established for a suitable unitary operator-valued multiplier Φ(σ) associated with the multiplier σ which is explicitly constructed during the study.     

Weakly compact multipliers on group algebras

We study left multipliers on the second dual spaces L¹(G)″ and M(G)″. We answer a question of Ghahramani and Lau, showing that for non-compact G a non-zero left multiplier on these spaces cannot be weakly compact.     

Hörmander-type multipliers on locally compact Vilenkin groups: theL 1(G)-case

In [3] and [4]Kitada presented Hörmander-type multiplier theorems for Lebesgue and Hardy spaces defined over a locally compact Vilenkin groupG. Like in the classical case, multipliers for the spaceL ¹(G) were not included in these results. In the present paper we discuss this particular case and we show how we need to modify the usual Hörmander multiplier condition to obtainL ¹ (G)-multipliers.     

Groups whose vanishing class sizes are not divisible by a given prime

Let G be a finite group. An element ${g\in G}Let G be a finite group. An element g ? G{g\in G} is a vanishing element of G if there exists an irreducible complex character χ of G such that χ(g) = 0: if this is the case, we say that the conjugacy class of g in G is a vanishing conjugacy class of G. In this paper we show that, if the size of every vanishing conjugacy class of G is not divisible by a given prime number p, then G has a normal p-complement and abelian Sylow p-subgroups.     

Simplicity of stable principal sheaves

Let M be a compact connected Kähler manifold, and let Gbe a connected complex reductive linear algebraic group. Weprove that a principal G-sheaf on M admits an admissible Einstein–Hermitianconnection if and only if the principal G-sheaf is polystable.Using this it is shown that the holomorphic sections of theadjoint vector bundle of a stable principal G-sheaf on M aregiven by the center of the Lie algebra of G. The Bogomolov inequalityis shown to be valid for polystable principal G-sheaves.     

Asymptotic purity for very general hypersurfaces of ℙ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript> × ℙ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript> of bidegree (<Emphasis Type="Italic">k,k</Emphasis>)

For a complex irreducible projective variety, the volume function and the higher asymptotic cohomological functions have proven to be useful in understanding the positivity of divisors as well as other geometric properties of the variety. In this paper, we study the vanishing properties of these functions on hypersurfaces of ℙ ⁿ × ℙ ⁿ . In particular, we show that very general hypersurfaces of bidegree (k, k) obey a very strong vanishing property, which we define as asymptotic purity: at most one asymptotic cohomological function is nonzero for each divisor. This provides evidence for the truth of a conjecture of Bogomolov and also suggests some general conditions for asymptotic purity.     

Note on a Result of Kerman and Weit II

Let G be a locally compact topological group, equipped with a fixed left Haar measure μ. We show that if f is a compactly supported real valued continuous function on G which has a unique maximum or a unique minimum at a point in G, then the space generated by the span of left translations of {f ⁿ ∣n=1,2,3,…} is dense in L ^p (G,μ), 1≤p<∞, in the space of continuous functions, continuous compactly supported functions and in the space of continuous functions vanishing at ∞. Similar results are true when the group G is substituted by G-spaces with compact isotropy group.     

Module homomorphisms and multipliers on locally compact quantum groups

For a Banach algebra A with a bounded approximate identity, we investigate the A-module homomorphisms of certain introverted subspaces of A^∗, and show that all A-module homomorphisms of A^∗ are normal if and only if A is an ideal of A∗∗. We obtain some characterizations of compactness and discreteness for a locally compact quantum group G. Furthermore, in the co-amenable case we prove that the multiplier algebra of L¹(G) can be identified with M(G). As a consequence, we prove that G is compact if and only if LUC(G)=WAP(G) and M(G)≅Z(LUC^∗(G)); which partially answer a problem raised by Volker Runde.     

Maximization of lower semi-continuous convex functional on bounded subsets of locally convex spaces. II: Quasi-Lagrangian duality theorems

We prove, for a proper lower semi-continuous convex functional ? on a locally convex space E and a bounded subset G of E, a formula for sup ?(G) which is symmetric to the Lagrange multiplier theorem for convex minimization, obtained in [7], with the difference that for sup ?(G) Lagrange multiplier functionals need not exist. When ? is also continuous we give some necessary conditions for g₀ ∈ G to satisfy ?(g₀) = sup ?(G). Also, we give some applications to deviations and farthest points. Finally, we show the connections with the “hyperplane theorems” of our previous paper [8].     

Strong homology and dimension

We show that strong homology groups H̄_p(X; G) of a space X vanish if p is greater than the shape dimension sd X. For p=sd X, H̄_p(X; G) coincides with the Čech homology groups Ȟ_p(X; G). We also show that there exist 1-dimensional spaces, which do not admit 1-dimensional ANR-resolutions. Therefore, the vanishing of H̄_p(X; G) for p>dim X is a nontrivial fact.     

Certain quotient spaces are countably separated,III

Let G be a locally compact group, N a closed normal subgroup, and α a multiplier for G such that every α-representation of G is Type I and such that every α-representation of N is Type I. Let $\text{N}\text{?}{}^{\mathrm{<mtext>\alpha </mtext>}}$ be the α-dual space of N. Necessary and sufficient conditions are given such that $\text{N}\text{?}{}^{\mathrm{<mtext>\alpha </mtext>}}\text{G}$ is countably separated. Generalizations of many known sufficient conditions for countable separability are easy corollaries.     

Zeros of Brauer characters

The authors obtain a sufficient condition to determine whether an element is a vanishing regular element of some Brauer character. More precisely, let G be a finite group and p be a fixed prime, and H = G′ O^p′ (G); if g ∈ G⁰ - H⁰ with o(gH) coprime to the number of irreducible p-Brauer characters of G, then there always exists a nonlinear irreducible p-Brauer character which vanishes on g. The authors also showin this note that the sums of certain irreducible p-Brauer characters take the value zero on every element of G⁰ - H⁰.     

Extensions for finite Chevalley groups III: Rational and generic cohomology

Let G be a connected reductive algebraic group and B   be a Borel subgroup defined over an algebraically closed field of characteristic p>0$p>0$. In this paper, the authors study the existence of generic G-cohomology and its stability with rational G-cohomology groups via the use of methods from the authors' earlier work. New results on the vanishing of G and B  -cohomology groups are presented. Furthermore, vanishing ranges for the associated finite group cohomology of G(_Fq)$G({\mathbb{F}}_q)$ are established which generalize earlier work of Hiller, in addition to stability ranges for generic cohomology which improve on seminal work of Cline, Parshall, Scott and van der Kallen.     

Recognition of some families of finite simple groups by order and set of orders of vanishing elements

Let G be a finite group. An element g ∈ G is called a vanishing element if there exists an irreducible complex character χ of G such that χ(g)= 0. Denote by Vo(G) the set of orders of vanishing elements of G. Ghasemabadi, Iranmanesh, Mavadatpour (2015), in their paper presented the following conjecture: Let G be a finite group and M a finite nonabelian simple group such that Vo(G) = Vo(M) and |G| = |M|. Then G ≌ M. We answer in affirmative this conjecture for M = Sz(q), where q = 2²ⁿ⁺¹ and either q ? 1, \(q - \sqrt {2q} + 1\) or q + \(\sqrt {2q} + 1\) is a prime number, and M = F⁴(q), where q = 2 ⁿ and either q⁴ + 1 or q⁴ ? q² + 1 is a prime number.     

The Drinfel’d Double for Group-cograded Multiplier Hopf Algebras

Let G be any group and let K(G) denote the multiplier Hopf algebra of complex functions with finite support in G. The product in K(G) is pointwise. The comultiplication on K(G) is defined with values in the multiplier algebra M(K(G) ⊗K(G )) by the formula for all and . In this paper we consider multiplier Hopf algebras B (over ) such that there is an embedding I: K(G) →M(B). This embedding is a non-degenerate algebra homomorphism which respects the comultiplication and maps K(G) into the center of M(B). These multiplier Hopf algebras are called G-cograded multiplier Hopf algebras. They are a generalization of the Hopf group-coalgebras as studied by Turaev and Virelizier. In this paper, we also consider an admissible action π of the group G on a G-cograded multiplier Hopf algebra B. When B is paired with a multiplier Hopf algebra A, we construct the Drinfel’d double D ^π where the coproduct and the product depend on the action π. We also treat the ^*-algebra case. If π is the trivial action, we recover the usual Drinfel’d double associated with the pair . On the other hand, also the Drinfel’d double, as constructed by Zunino for a finite-type Hopf group-coalgebra, is an example of the construction above. In this case, the action is non-trivial but related with the adjoint action of the group on itself. Now, the double is again a G-cograded multiplier Hopf algebra. Presented by K. Goodearl.     

Upper and lower bounds for the energy of bipartite graphs

Using Lagrange's multiplier rule, we find upper and lower bounds of the energy of a bipartite graph G, in terms of the number of vertices, edges and the spectral moment of fourth order. Moreover, the upper bound is attained in a graph G if and only if G is the graph of a symmetric balanced incomplete block design (BIBD). Also, we determine the graphs for which the lower bound is sharp.     

On p-approximation properties for p-operator spaces

This paper has a two-fold purpose. Let 1<p<∞. We first introduce the p-operator space injective tensor product and study various properties related to this tensor product, including the p-operator space approximation property, for p-operator spaces on Lp-spaces. We then apply these properties to the study of the pseudofunction algebra PFp(G), the pseudomeasure algebra PMp(G), and the Figà-Talamanca-Herz algebra Ap(G) of a locally compact group G. We show that if G is a discrete group, then most of approximation properties for the reduced group C^∗-algebra , the group von Neumann algebra VN(G), and the Fourier algebra A(G) (related to amenability, weak amenability, and approximation property of G) have the natural p-analogues for PFp(G), PMp(G), and Ap(G), respectively. The p-completely bounded multiplier algebra McbAp(G) plays an important role in this work.