Algebras Generated by the Bergman and Anti-Bergman Projections and by Multiplications by Piecewise Continuous Functions

The C*-algebra generated by the Bergman and anti-Bergman projections and by the operators of multiplication by piecewise continuous functions on the Lebesgue space L²(Π) over the upper half-plane is studied. Making use of a local principle, limit operators techniques, and the Plamenevsky results on two-dimensional singular integral operators with coefficients admitting homogeneous discontinuities we reduce the study to simpler C*-algebras associated with points and pairs We construct a symbol calculus for unital C*-algebras generated by n orthogonal projections sum of which equals the unit and by m one-dimensional orthogonal projections. Such algebras are models of local algebras at points z ∈∂Π being the discontinuity points of coefficients. A symbol calculus for the C*- algebra and a Fredholm criterion for the operators are obtained. Finally, a C*-algebra isomorphism between the quotient algebra where is the ideal of compact operators, and its analogue for the unit disk is constructed.     

On the structure of level sets of uniform and Lipschitz quotient mappings from $$ \mathbb{R}^n $$ to $$ \mathbb{R} $$

We study two questions posed by Johnson, Lindenstrauss, Preiss, and Schechtman, concerning the structure of level sets of uniform and Lipschitz quotient mappings from . We show that if , is a uniform quotient mapping then for every has a bounded number of components, each component of separates and the upper bound of the number of components depends only on and the moduli of co-uniform and uniform continuity of .Next we prove that all level sets of any co-Lipschitz uniformly continuous mapping from to are locally connected, and we show that for every pair of a constant and a function with , there exists a natural number , so that for every co-Lipschitz uniformly continuous map with a co-Lipschitz constant and a modulus of uniform continuity , there exists a natural number and a finite set with card so that for all has exactly components, has exactly components and each component of is homeomorphic with the real line and separates the plane into exactly 2 components. The number and form of components of for are also described - they have a finite tree structure.     

The Ideal Structure of Toeplitz Algebras

We investigate the ideal structure of the Toeplitz algebra of a totally ordered abelian group . We show that the primitive ideals of are parametrised by the disjoint union of the duals of the order ideals of , and identify the hull-kernel topology on when the chain of orderideals in is isomorphic to a subset of      

Beckman-Quarles type theorems for mappings from $$ \mathbb{R}^n $$ to $$ \mathbb{C}^n $$

Summary. Let We say that preserves the distance d 0 if for each implies Let A _n denote the set of all positive numbers d such that any map that preserves unit distance preserves also distance d. Let D _n denote the set of all positive numbers d with the property: if and then there exists a finite set S _xy with such that any map that preserves unit distance preserves also the distance between x and y. Obviously, We prove: (1) (2) for n 2 D _n is a dense subset of (2) implies that each mapping f from to (n 2) preserving unit distance preserves all distances, if f is continuous with respect to the product topologies on and      

On Shadows Of Intersecting Families

The shadow minimization problem for t-intersecting systems of finite sets is considered. Let be a family of k-subsets of . The -shadow of is the set of all (k-)-subsets contained in the members of . Let be a t-intersecting family (any two members have at least t elements in common) with . Given k,t,m the problem is to minimize (over all choices of ). In this paper we solve this problem when m is big enough.     

Random generators of given orders and the smallest simple Moufang loop

The probability that m randomly chosen elements of a finite power associative loop have prescribed orders and generate is calculated in terms of certain constants related to the action of Aut( ) on the subloop lattice of . As an illustration, all meaningful probabilities of random generation by elements of given orders are found for the smallest nonassociative simple Moufang loop.     

Cancellative residuated lattices

Cancellative residuated lattices are natural generalizations of lattice-ordered groups ( -groups). Although cancellative monoids are defined by quasi-equations, the class of cancellative residuated lattices is a variety. We prove that there are only two commutative subvarieties of that cover the trivial variety, namely the varieties generated by the integers and the negative integers (with zero). We also construct examples showing that in contrast to -groups, the lattice reducts of cancellative residuated lattices need not be distributive. In fact we prove that every lattice can be embedded in the lattice reduct of a cancellative residuated lattice. Moreover, we show that there exists an order-preserving injection of the lattice of all lattice varieties into the subvariety lattice of .We define generalized MV-algebras and generalized BL-algebras and prove that the cancellative integral members of these varieties are precisely the negative cones of -groups, hence the latter form a variety, denoted by . Furthermore we prove that the map that sends a subvariety of -groups to the corresponding class of negative cones is a lattice isomorphism from the lattice of subvarieties of to the lattice of subvarieties of . Finally, we show how to translate equational bases between corresponding subvarieties, and briefly discuss these results in the context of R. McKenzies characterization of categorically equivalent varieties.     

Coloured Permutations Containing and Avoiding Certain Patterns

Let be the set of all coloured permutations on the symbols 1, 2, . . . , n with colours 1, 2, . . . , r, which is the analogous of the symmetric group when r = 1, and the hyperoctahedral group when r = 2. Let be a subset of d colours; we define to be the set of all coloured permutations . We prove that the number of -avoiding coloured permutations in . We then prove that for any , the number of coloured permutations in which avoid all patterns in except for and contain exactly once equals . Finally, for any , this number equals . These results generalize recent results due to Mansour, Mansour and West, and Simion.AMS Subject Classification: 05A05, 05A15.     

Transformations of Grassmannians preserving the class of base subsets

Let be a finite-dimensional projective space and be the Grassmannian consisting of all k-dimensional subspaces of . In the paper we show that transformations of sending base subsets to base subsets are induced by collineations of to itself or to the dual projective space . This statement generalizes the main result of the authors paper [19].     

Commutators on Half-Spaces

We study the boundedness and compactness of commutators on , where and are defined by and respectively. If satisfies some upper and lower estimates, then we obtain a necessary and sufficient condition for to be bounded or compact on for . The reproducing kernel of the harmonic Bergman space of can be shown to satisfy all the required estimates. Our result is the real variable analogue of the complex variable one for commutators associated with an analytic reproducing kernel.     

Generalized Harish-Chandra Modules: A New Direction in the Structure Theory of Representations

Let be a reductive Lie algebra over C. We say that a -module M is a generalized Harish-Chandra module if, for some subalgebra , M is locally -finite and has finite -multiplicities. We believe that the problem of classifying all irreducible generalized Harish-Chandra modules could be tractable. In this paper, we review the recent success with the case when is a Cartan subalgebra. We also review the recent determination of which reductive in subalgebras are essential to a classification. Finally, we present in detail the emerging picture for the case when is a principal 3-dimensional subalgebra.     

Cyclicity of zeros of families of analytic functions

Let be a family of holomorphic functions in the unit disk , which are also holomorphic in a parameter . We express cyclicity (=generalized multiplicity) of a zero of at via some algebraic characteristics of the ideal generated by the Taylor coefficients of . As an example we estimate the cyclicity of the family of generalized exponential polynomials.     

On the Schwarz reflection principle for monogenic functions

Let ∑ be either an oriented hyperplane or the unit sphere in , let be open and connected and let be an open and connected domain in such that . If in is a null solution of the Dirac operator (also called a monogenic function in ) which is continuously extendable to , then conditions upon are given enabling the monogenic extension of across . In such a way Schwarz reflection type principles for monogenic functions are established in the Spin (1) and Spin cases. The Spin (1) case includes the classical Schwarz reflection principle for holomorphic functions in the plane. The Spin case deals with so-called “half boundary value problems” for the Dirac operator. Received: 2 February 2006     

Set of involutions arising from an egglike inversive plane

To every egglike inversive plane there is associated a family of involutions of the point set of such that circles of are the fixed point sets of the involutions in . Korchmaros and Olanda characterized a family of involutions on a set of size n² + 1to be for an egglike inversive plane of order n by four conditions. In this paper, we give an alternative proof where the Galois space PG(3,n) in which is embedded is built up directly by using concepts and results on finite linear spaces.     

On the Singularities of the Magnetic Spectral Shift Function at the Landau Levels

We consider the three-dimensional Schrödinger operators and where , A is a magnetic potential generating a constant magnetic field of strength , and where decays fast enough at infinity. Then, A. Pushnitskis representation of the spectral shift function (SSF) for the pair of operators is well defined for energies We study the behaviour of the associated representative of the equivalence class determined by the SSF, in a neighbourhood of the Landau levels Reducing our analysis to the study of the eigenvalue asymptotics for a family of compact operators of Toeplitz type, we establish a relation between the type of the singularities of the SSF at the Landau levels and the decay rate of V at infinity. Communicated by Bernard HelfferSubmitted 23/09/03, accepted 15/01/04     

Uniform Convergence of Trigonometric Series with Rarely Changing Coefficients

We consider the series and whose coefficients satisfy the condition for , where the sequence can be expressed as the union of a finite number of lacunary sequences. The following results are obtained. If as , then the series is uniformly convergent. If for all , then the sequence of partial sums of this series is uniformly bounded. If the series is convergent for and as , then this series is uniformly convergent. If the sequence of partial sums of the series for is bounded and for all , then the sequence of partial sums of this series is uniformly bounded. In these assertions, conditions on the rates of decrease of the coefficients of the series are also necessary if the sequence is lacunary. In the general case, they are not necessary.