The Torsion Theory Generated by <Emphasis Type="Italic">M</Emphasis>-Small Modules

Let M be a right R-module, the class of all M-small modules, and P a projective cover of M in [M]. We consider the torsion theories = ( ), = ( ), and = ( ) in [M], where is the torsion theory generated by is the torsion theory cogenerated by , and is the dual Lambek torsion theory. We study some conditions for to be cohereditary, stable, or split, and prove that Rej(M, ) = M = (= = ) = Gen_M(P) .2000 Mathematics Subject Classification: 16S90     

Elements of Prescribed Order,Prescribed Traces and Systems of Rational Functions Over Finite Fields

Let k 1 and be a system of rational functions forming a strongly linearly independent set over a finite field . Let be arbitrarily prescribed elements. We prove that for all sufficiently large extensions , there is an element of prescribed order such that is the relative trace map from onto We give some applications to BCH codes, finite field arithmetic and ordered orthogonal arrays. We also solve a question of Helleseth et~al. (Hypercubic 4 and 5-designs from Double-Error-Correcting codes, Des. Codes. Cryptgr. 28(2003). pp. 265–282) completely.classification 11T30, 11G20, 05B15     

A unified approach to the topological centre problem for certain Banach algebras arising in abstract harmonic analysis

Let be a locally compact group. Consider the Banach algebra , equipped with the first Arens multiplication, as well as the algebra LUC , the dual of the space of bounded left uniformly continuous functions on , whose product extends the convolution in the measure algebra M . We present (for the most interesting case of a non-compact group) completely different - in particular, direct - proofs and even obtain sharpened versions of the results, first proved by Lau-Losert in [9] and Lau in [8], that the topological centres of the latter algebras precisely are and M , respectively. The special interest of our new approach lies in the fact that it shows a fairly general pattern of solving the topological centre problem for various kinds of Banach algebras; in particular, it avoids the use of any measure theoretical techniques. At the same time, deriving both results in perfect parallelity, our method reveals the nature of their close relation.Received: 1 January 2002     

Syndetically Hypercyclic Operators

Given a continuous linear operator T L(x) defined on a separable -space X, we will show that T satisfies the Hypercyclicity Criterion if and only if for any strictly increasing sequence of positive integers such that the sequence is hypercyclic. In contrast we will also prove that, for any hypercyclic vector x X of T, there exists a strictly increasing sequence such that and is somewhere dense, but not dense in X. That is, T and do not share the same hypercyclic vectors.     

Spectra of Compact Composition Operators Over Bounded Symmetric Domains

Let H²(D) denote the Hardy space of a bounded symmetric domain in its standard Harish-Chandra realization, and let be the weighted Bergman space with and where is a critical value depending on D. Suppose that is holomorphic. We show that if the composition operator defined by is compact (or, more generally, power-compact) on H²(D) or then has a unique fixed point z₀ in D. We then prove that the spectrum of as an operator on these function spaces is precisely the set consisting of 0, 1, and all possible products of eigenvalues of These results extend previous work by Caughran/Schwartz and MacCluer. As a corollary, we now have that MacCluers previous spectrum results on the unit ball B_n extend to H^p(n) (not only for p = 2 but for all p > 1) and (for p 1), where ⁿ is the polydisk in      

Metric Geometry in Homogeneous Spaces of the Unitary Group of a C*-Algebra. Part II. Geodesics Joining Fixed Endpoints

This article focuses on the study of the metric geometry of homogeneous spaces (the unitary group of a C*-algebra modulo the unitary group of a C*-subalgebra ) where the invariant Finsler metric in is induced by the quotient norm of Under the assumption that is of compact type, i.e. when the unitary group is relatively compact in the strong operator topology, this work presents local and global versions of Hopf-Rinow-like theorems: given points there exists a minimal uniparametric group curve joining ρ₀ and ρ₁.     

Complete Positivity of Elementary Operators on C*-Algebras

Let be a C*-algebra. We obtain some conditions that are equivalent to the statement that every n-positive elementary operator on is completely positive.