<Emphasis Type="Italic">K</Emphasis><Subscript>0</Subscript>-group and approximate similarity invariants of strongly irreducible operators

Let H olenote a complex separable Hilbert space and L(H) denote the collection of bounded linear operators on H. An operator T ∈ L(H) is said to be strongly irreducible if T does not commute with any nontrivial idempotent. Herrero and Jiang showed that the norm-closure of the class of all strongly irreducible operators is the class of all operators with connected spectrum. This result can be considered as an approximate inverse of the Riesz decomposition theorem. In the paper, we give a more precise charact...     

Three-dimensional sets with small sumset

We describe the structure of three dimensional sets of lattice points, having a small doubling property. Let be a finite subset of ℤ³ such that dim = 3. If and , then lies on three parallel lines. Moreover, for every three dimensional finite set that lies on three parallel lines, if , then is contained in three arithmetic progressions with the same common difference, having together no more than terms. These best possible results confirm a recent conjecture of Freiman and cannot be sharpened by reducing the quantity υ or by increasing the upper bounds for .     

Baer cotorsion Pairs

LetR be a unital associative ring and two classes of leftR-modules. In [St3] the notion of a ( ) pair was introduced. In analogy to classical cotorsion pairs, a pair (V,W) of subclasses is called a ( ) pair if it is maximal with respect to the classes and the condition Ext _R ¹ (V, W)=0 for all . In this paper we study pairs whereR = ℤ and is the class of all torsion-free abelian groups andT is the class of all torsion abelian groups. A complete characterization is obtained assumingV=L. For example, it is shown that every pair is singly cognerated underV=L. The author was supported by a DFG grant.     

Reality properties of conjugacy classes in algebraic groups

Let be a (not necessarily semi-finite) σ-finite von Neumann algebra. We prove that there exists a finite von Neumann algebra so that for every 1 < p < 2, the Haagerup L ^p -space associated with embeds isomorphically into . We also provide a proof of the following non-commutative generalization of a classical result of Rosenthal: if is a semi-finite von Neumann algebra then every reflexive subspace of embeds isomorphically into L ^r ( ) for some r > 1. Dedicated to Professor H. P. Rosenthal on the occasion of his sixty-fifth birthday Research partially supported by NSF grant DMS-0456781.     

Linear connections of a framed distribution of hyperplane elements in conformal space

We study the geometry of affine and normal connections induced by a complete normalization of mutually orthogonal distributions $ \mathcal{M} We study the geometry of affine and normal connections induced by a complete normalization of mutually orthogonal distributions and in conformal space C _n , where is a distribution of hyperplane elements, and is a distribution of line elements. We consider invariant fields of pencils that are parallel with respect to the normal connection along any curve belonging to the distribution . Original Russian Text ? A.M. Matveeva, 2008, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2008, No. 7, pp. 79–84.     

A note on simultaneous Diophantine approximation on planar curves

Let denote the set of simultaneously - approximable points in and denote the set of multiplicatively ψ-approximable points in . Let be a manifold in . The aim is to develop a metric theory for the sets and analogous to the classical theory in which is simply . In this note, we mainly restrict our attention to the case that is a planar curve . A complete Hausdorff dimension theory is established for the sets and . A divergent Khintchine type result is obtained for ; i.e. if a certain sum diverges then the one-dimensional Lebesgue measure on of is full. Furthermore, in the case that is a rational quadric the convergent Khintchine type result is obtained for both types of approximation. Our results for naturally generalize the dimension and Lebesgue measure statements of Beresnevich et al. (Mem AMS, 179 (846), 1–91 (2006)). Moreover, within the multiplicative framework, our results for constitute the first of their type. The research of Victor V. Beresnevich was supported by an EPSRC Grant R90727/01. Sanju L. Velani is a Royal Society University Research Fellow. For Iona and Ayesha on No. 3.     

Perfect difference sets constructed from Sidon sets

A set of positive integers is a perfect difference set if every nonzero integer has a unique representation as the difference of two elements of . We construct dense perfect difference sets from dense Sidon sets. As a consequence of this new approach we prove that there exists a perfect difference set such that

A class of multipliers for $$\mathcal{W}^ \bot $$

Let denote the class of ergodic probability preserving transformations which are disjoint from every weakly mixing system. Let be the class of multipliers for , i.e. ergodic transformations whose all ergodic joinings with any element of are also in . Fix an ergodic rotationT, a mildly mixing actionS of a locally compact second countable groupG and an ergodic cocycle ϕ forT with values inG. The main result of the paper is a sufficient (and also necessary by [LeP] whenG is countable Abelian andS is Bernoullian) condition for the skew product build fromT, ϕ andS to be an element of . Moreover, the self-joinings of such extensions ofT are described with an application to study semisimple extensions of rotations. Dedicated to Hillel Furstenberg on the occasion of his retirement The first-named author was supported in part by CRDF, grant UM1-2546-KH-03. The second-named author was supported in part by KBN grant 1P03A 03826.     

A question in the theory of saturated formations of finite soluble groups

This paper examines the following question. If and are saturated formations then is defined to be the class of all soluble groups whose belong to . In general is a formation, but need not be a saturated formation. Here the smallest saturated formation containing is studied.     

Parabolic twists for the linear algebras <Emphasis Type="Italic">A</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis>−1</Subscript>

New solutions of twist equations for the universal enveloping algebras U (A_n−1) are found. These solutions can be represented as products of full chains of extended Jordanian twists Abelian factors (“rotations”) , and sets of quasi-Jordanian twists . The latter are generalizations of Jordanian twists (with carrier b²) for special deformed extensions of the Hopf algebra U (b²). The carrier subalgebra for the composition is a nonminimal parabolic subalgebra in A _n−1 such that . The parabolic twisting elements are obtained in an explicit form. Details of the construction are illustrated by considering the examples n = 4 and n = 11. Bibliography: 21 titles. Published in Zapiski Nauchnykh Seminarov POMI, Vol. 347, 2007, pp. 187–213.     

On amenability of group algebras, I

We study amenability of algebras and modules (based on the notion of almost-invariant finite-dimensional subspace), and apply it to algebras associated with finitely generated groups. We show that a group G is amenable if and only if its group ring G is amenable for some (and therefore for any) field . Similarly, a G-set X is amenable if and only if its span X is amenable as a G-module for some (and therefore for any) field .     

Hall chains in normal subgroups of finite p-groups

Some suffcient conditions for existence of k-admissible Hall chains (= -chains) in normal subgroups of finite p-groups are established (for irregular p-groups we consider only the case k = p). In Propositions 13–15 we study p-groups without -chains, and metacyclic 2-groups with the above property are classified. Abelian p-groups with exactly one -chain are characterized in Proposition 12.     

Denseness of holomorphic functions attaining their numerical radii

For two complex Banach spaces X and Y, (B _X; Y) will denote the space of bounded and continuous functions from B _X to Y that are holomorphic on the open unit ball. The numerical radius of an element h in (B _X; X) is the supremum of the set

Classes of strictly singular operators and their products

V. D. Milman proved in [20] that the product of two strictly singular operators on L _p [0, 1] (1 ⩽ p < 1) or on C[0, 1] is compact. In this note we utilize Schreier families in order to define the class of -strictly singular operators, and then we refine the technique of Milman to show that certain products of operators from this class are compact, under the assumption that the underlying Banach space has finitely many equivalence classes of Schreier-spreading sequences. Finally we define the class of -hereditarily indecomposable Banach spaces and we examine the operators on them. This project originated at SUMIRFAS conference in 2005 in Texas A&M University. The authors wish to thank the organizers of the SUMIRFAS conference for their hospitality.     

Reduced Minimum Modulus Preserving in Banach Space

Let be the algebra of all bounded linear operators on a complex Banach space X and γ(T) be the reduced minimum modulus of operator . In this work, we prove that if , is a surjective linear map such that is an invertible operator, then , for every , if and only if, either there exist two bijective isometries and such that for every , or there exist two bijective isometries and such that for every . This generalizes for a Banach space the Mbekhta’s theorem [12].      

Completion of restricted Lie algebras

We study pro-‘finite dimensional finite exponent’ completions of restricted Lie algebras over finite fields of characteristicp. These compact Hausdorff topological restricted Lie algebras, called pro- restricted Lie algebras, are the restricted Lie-theoretic analogues of pro-p groups. A structure theory for pro- restricted Lie algebras with finite rank is developed. In particular, the centre of such a Lie algebra is shown to be open. As an application we examinep-adic analytic pro-p groups in terms of their associated pro- restricted Lie algebras. Supported by NSERC of Canada.     

Blaschke-decomposable convex bodies

For a given centred convex bodyK of ℝ,n≥3, let be the class of all convex bodies with the same projection body asK. The question whetherK can be expressed as a Blaschke average of two non-homothetic bodies from is considered. Necessary and sufficient conditions onK to be Blaschke decomposable in are given. The paper provides also a characterization of the bodiesK such that the Blaschke indecomposable bodies in are dense in itself.     

Remarks on Thurston’s Construction of Pseudo-Anosov Maps of Riemann Surfaces

It is well known that certain isotopy classes of oseudo-Anosov maos on a Riemann surface S of non-excluded type can be defined through Dehn twists tα and tβ along simple closed geodesics α and β on S,respectively. Let G be the corresponding Fuchsian group acting on the hyperbolic plane H so that H/G≌S.For any point α∈S,define S = S／{α}.In this article, the author gives explicit parabolic elements of G from which he constructs pseudo-Anosov classes on S that can be projected to a given pseudo-Anosov class on S obtained from Thurston＇s construction.     

The hardy and bellman operators in spaces connected with H( \mathbb{T} ) and BMO( \mathbb{T} )

Assume that 1 ≤ p < ∞ and a function f ∈ L ^p [0, π] has the Fourier series $ \sum\limits_{n = 1}^\infty {a_n } Assume that 1 ≤ p < ∞ and a function f ∈ L ^p [0, π] has the Fourier series cos nx. According to one result of G.H. Hardy, the series cos nx is the Fourier series for a certain function (f) ∈ L ^p [0, π]. But if 1 < p ≤ ∞ and f ∈ L ^p [0, π], then the series cos nx is the Fourier series for a certain function (f) ∈ L ^p [0, π]. Similar assertions are true for sine series. This allows one to define the Hardy operator on L ^p (), 1 ≤ p < ∞, and to define the Bellman operator on L ^p (), 1 < p ≤ ∞. In this paper we prove that the Bellman operator boundedly acts in VMO(), and the Hardy operator also maps a certain subspace C() onto VMO(). We also prove the invariance of certain classes of functions with given majorants of modules of continuity or best approximations in the spaces H(), L(), VMO() with respect to the Hardy and Bellman operators. Original Russian Text ? S.S. Volosivets and B.I. Golubov, 2008, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2008, No. 5, pp. 4–13.     

On ideals and quotients of A \mathcal{T} -algebras

Some results on A -algebras are given. We study the problem when ideals, quotients and hereditary subalgebras of A -algebras are A -algebras or A -algebras, and give a necessary and sufficient condition of a hereditary subalgebra of an A -algebra being an A -algebra.