Leti ${\Bbb F}$n be the free group of rank n and let $\bigoplus C^{*}({\Bbb F}_{n})$ denote the direct sum of full group C*-algebras $C^{*}({\Bbb F}_{n})$ of ${\Bbb F}_{n} (1\leq n<\infty$). We introduce a new comultiplication Δϕ on $\bigoplus C^{*}({\Bbb F}_{n})$ such that $(\bigoplus C^{*}({\Bbb F}_{n}),\,\Delta_{\varphi})$ is a non-cocommutative C*-bialgebra. With respect to Δϕ, the tensor product π⊗ϕπ′ of any two representations π and π′ of free groups is defined. The operation ×ϕ is associative and non-commutative. We compute its tensor product formulas of several representations.

Some well-known results about the 2-density topology on ${\mathcal R}$ (in particular in the context of the Lusin–Menchoff property) are extended to τbm , i.e. the m-density topology on ${\mathcal R}$ n with m ∈ (n,+∞). Every set of finite perimeter in ${\mathcal R}$ n is equivalent (in measure) to a set in τb m 0 , where m 0=n+1+ ${1\over n-1}$ . There exists a set of finite perimeter in ${\mathcal R}$ n which is not equivalent (in measure) to any member in the a.e.-modification of τbm , whatever m ∈ [n,+∞).

It is known that hypersurfaces in ${\mathbb C}$Pn or ${\mathbb C}$Hn for which the number g of distinct principal curvatures satisfies g ≤ 2, must belong to a standard list of Hopf hypersurfaces with constant principal curvatures, provided that n ≥ 3. In this paper, we construct a two-parameter family of non-Hopf hypersurfaces in ${\mathbb C}$P2 and ${\mathbb C}$H2 with g=2 and show that every non-Hopf hypersurface with g=2 is locally of this form.

For any pair M, N of von Neumann algebras such that the algebraic tensor product M ⊗ N admits more than one C*-norm, the cardinal of the set of C*-norms is at least 2ℵ0. Moreover, there is a family with cardinality 2ℵ0 of injective tensor product functors for C*-algebras in Kirchberg's sense. Let ${\mathbb B}$=∏nMn. We also show that, for any non-nuclear von Neumann algebra M⊂ ${\mathbb B}$(ℓ2), the set of C*-norms on ${\mathbb B}$ ⊗ M has cardinality equal to 22ℵ0.

We construct a subgroup Hd of the iterated wreath product Gd of d copies of the cyclic group of order p with the property that the derived length and the smallest cardinality of a generating set of Hd are equal to d while no proper subgroup of Hd has derived length equal to d. It turns out that the two groups Hd and Gd are the extreme cases of a more general construction that produces a chain Hd=K1<···< Kp−1=Gd of subgroups sharing a common recursive structure. For i ∈ {1,. . .,p−1}, the subgroup Ki has nilpotency class (i+1)d−1.

We study bi-invariant word metrics on groups. We provide an efficient algorithm for computing the bi-invariant word norm on a finitely generated free group and we construct an isometric embedding of a locally compact tree into the bi-invariant Cayley graph of a nonabelian free group. We investigate the geometry of cyclic subgroups. We observe that in many classes of groups, cyclic subgroups are either bounded or detected by homogeneous quasimorphisms. We call this property the bq-dichotomy and we prove it for many classes of groups of geometric origin.

We show that the $\mathbb{Z}$/2-equivariant nth integral Morava K-theory with reality is self-dual with respect to equivariant Anderson duality. In particular, there is a universal coefficients exact sequence in integral Morava K-theory with reality, and we recover the self-duality of the spectrum KO as a corollary. The study of $\mathbb{Z}$/2-equivariant Anderson duality made in this paper gives a nice interpretation of some symmetries of RO($\mathbb{Z}$/2)-graded (i.e. bigraded) equivariant cohomology groups in terms of Mackey functor duality.

In 2006, F. Luca and I. E. Shparlinski (Proc. Indian Acad. Sci. (Math. Sci.)116(1) (2006), 1–8) proved that there are only finitely many pairs (n, m) of positive integers which satisfy the Diophantine equation |τ(n!)|=m!, where τ is the Ramanujan function. In this paper, we follow the same approach of Luca and Shparlinski (Proc. Indian Acad. Sci. (Math. Sci.)116(1) (2006), 1–8) to determine all solutions of the above equation. The proof of our main theorem uses linear forms in two logarithms and arithmetic properties of the Ramanujan function.

We prove uniqueness of positive solutions for the boundary value problem

\begin{equation*} \left\{ \begin{array}{l} -\Delta u=\lambda f(u)\text{ in }\Omega , \\ \ \ \ \ \ \ \ u=0\text{ on }\partial \Omega , \end{array} \right. \end{equation*}

$\mathbb{R}$

The aim of the present paper is the classification of real hypersurfaces M equipped with the condition Al = lA, l = R(., ξ)ξ, restricted in a subspace of the tangent space T p M of M at a point p. This class is large and difficult to classify, therefore a second condition is imposed: (∇ξ l)X = ω(X)ξ + ψ(X)lX, where ω(X), ψ(X) are 1-forms. The last condition is studied for the first time and is much weaker than ∇ξ l = 0 which has been studied so far. The Jacobi Structure Operator satisfying this weaker condition can be called generalized ξ-parallel Jacobi Structure Operator.

We establish interior and trace embedding results for Sobolev functions on a class of bounded non-smooth domains. Also, we define the corresponding generalized Maz'ya spaces of variable exponent, and obtain embedding results similar as in the constant case. Some relations between the variable exponent Maz'ya spaces and the variable exponent Sobolev spaces are also achieved. At the end, we give an application of the previous results for the well-posedness of a class of quasi-linear equations with variable exponent.

In this paper, we study Hardy spaces associated with non-negative self-adjoint operators and develop their vector-valued theory. The complex interpolation scales of vector-valued tent spaces and Hardy spaces are extended to the endpoint p=1. The holomorphic functional calculus of L is also shown to be bounded on the associated Hardy space H 1 L (X). These results, along with the atomic decomposition for the aforementioned space, rely on boundedness of certain integral operators on the tent space T 1(X).

Consider a simple Lie algebra $\mathfrak{g}$ and $\overline{\mathfrak{g}}$ ⊂ $\mathfrak{g}$ a Levi subalgebra. Two irreducible $\overline{\mathfrak{g}}$-modules yield isomorphic inductions to $\mathfrak{g}$ when their highest weights coincide up to conjugation by an element of the Weyl group W of $\mathfrak{g}$ which is also a Dynkin diagram automorphism of $\overline{\mathfrak{g}}$. In this paper, we study the converse problem: given two irreducible $\overline{\mathfrak{g}}$-modules of highest weight μ and ν whose inductions to $\mathfrak{g}$ are isomorphic, can we conclude that μ and ν are conjugate under the action of an element of W which is also a Dynkin diagram automorphism of $\overline{\mathfrak{g}}$? We conjecture this is true in general. We prove this conjecture in type A and, for the other root systems, in various situations providing μ and ν satisfy additional hypotheses. Our result can be interpreted as an analogue for branching coefficients of the main result of Rajan [6] on tensor product multiplicities.

Suppose that a complex manifold M is locally embedded into a higher-dimensional neighbourhood as a submanifold. We show that, if the local neighbourhood germs are compatible in a suitable sense, then they glue together to give a global neighbourhood of M. As an application, we prove a global version of Hertling–Manin's unfolding theorem for germs of TEP structures; this has applications in the study of quantum cohomology.

We offer a Lebesgue-type decomposition of a representable functional on a *-algebra into absolutely continuous and singular parts with respect to another. Such a result was proved by Zs. Szűcs due to a general Lebesgue decomposition theorem of S. Hassi, H.S.V. de Snoo, and Z. Sebestyén concerning non-negative Hermitian forms. In this paper, we provide a self-contained proof of Szűcs' result and in addition we prove that the corresponding absolutely continuous parts are absolutely continuous with respect to each other.

Recently, a new equivalence relation between weak* closed operator spaces acting on Hilbert spaces has appeared. Two weak* closed operator spaces ${\mathcal{U}}$, ${\mathcal{V}}$ are called weak TRO equivalent if there exist ternary rings of operators ${\mathcal{M}}$i, i = 1, 2 such that ${\mathcal{U}}=[{\mathcal{M}}_2{\mathcal{V}}{\mathcal{M}}_1^*]^{-w^*}, {\mathcal{V}}=[{\mathcal{M}}_2^*{\mathcal{U}}{\mathcal{M}}_1]^{-w^*}.$ Weak TRO equivalent spaces are stably isomorphic, and conversely, stably isomorphic dual operator spaces have normal completely isometric representations with weak TRO equivalent images. In this paper, we prove that if ${\mathcal{U}}$ and ${\mathcal{V}}$ are weak TRO equivalent operator spaces and the space of I × I matrices with entries in ${\mathcal{U}}$, MIw(${\mathcal{U}}$), is hyperreflexive for suitable infinite I, then so is MIw(${\mathcal{V}}$). We describe situations where if ${\mathcal{L}}$1, ${\mathcal{L}}$2 are isomorphic lattices, then the corresponding algebras Alg($\mathcal{L}$1), Alg($\mathcal{L}$2) have the same complete hyperreflexivity constant.

In the paper we describe Kahler QCH surfaces. We prove that any Calabi type and orthotoric Kahler surfaces are QCH Kahler surfaces. We also classify locally homogeneous QCH surfaces.

We construct, for q a root of unity of odd order, an embedding of the projective special linear group PSL(n) into the group of bi-Galois objects over uq(sl(n))*, the coordinate algebra of the Frobenius–Lusztig kernel of SL(n), which is shown to be an isomorphism at n=2.

We characterize those partially ordered sets I for which the canonical maps Mi → colim Mj into colimits of abstract sets are always injective, provided that the transition maps are injective. We also obtain some consequences for colimits of vector spaces.