On the norm of an idempotent Schur multiplier on the Schatten class

We show that if the norm of an idempotent Schur multiplier on the Schatten class lies sufficiently close to , then it is necessarily equal to . We also give a simple characterization of those idempotent Schur multipliers on whose norm is .

     

-spaces and finite unions

This article is a continuation of a recent paper by the author and R. Z. Buzyakova. New results are obtained in the direction of the next natural question: how complex can a space be that is the union of two (of a finite family) ``nice" subspaces? Our approach is based on the notion of a -space introduced by E. van Douwen and on a generalization of this notion, the notion of -space. It is proved that if a space is the union of a finite family of subparacompact subspaces, then is an -space. Under , it follows that if a separable normal -space is the union of a finite number of subparacompact subspaces, then is Lindelöf. It is also established that if a regular space is the union of a finite family of subspaces with a point-countable base, then is a -space. Finally, a certain structure theorem for unions of finite families of spaces with a point-countable base is established, and numerous corollaries are derived from it. Also, many new open problems are formulated.

     

Notes on the largest irreducible character degree of a finite group

Let be a finite group and the largest irreducible character degree of . In this note, we show the following results: if , then ; if and, in addition, is -solvable with abelian Sylow -subgroup, then .

     

On embeddings of full amalgamated free product C-algebras

We examine the question of when the -homomorphism of full amalgamated free product C-algebras, arising from compatible inclusions of C-algebras , and , is an embedding. Results giving sufficient conditions for to be injective, as well as classes of examples where fails to be injective, are obtained. As an application, we give necessary and sufficient conditions for the full amalgamated free product of finite-dimensional C-algebras to be residually finite dimensional.

     

Uncountable categoricity for gross models

A model is said to be gross if all infinite definable sets in have the same cardinality (as ). We prove that if for some uncountable , has a unique gross model of cardinality , then for any uncountable , has a unique gross model of cardinality .

     

Weak properties of weighted convolution algebras

Suppose that is a weighted convolution algebra on with the weight normalized so that the corresponding space of measures is the dual space of the space of continuous functions. Suppose that is a continuous nonzero homomorphism, where is also a convolution algebra. If is norm dense in , we show that is (relatively) weak dense in , and we identify the norm closure of with the convergence set for a particular semigroup. When is weak continuous it is enough for to be weak dense in . We also give sufficient conditions and characterizations of weak continuity of . In addition, we show that, for all nonzero in , the sequence converges weak to 0. When is regulated, converges to 0 in norm.

     

Uniqueness of dilation invariant norms

Let be a nontrivial dilation. We show that every complete norm on that makes from into itself continuous is equivalent to . also determines the norm of both and with in a weaker sense. Furthermore, we show that even all the dilations do not determine the norm on .

     

Approximation of holomorphic maps with a lower bound on the rank

Let be a closed polydisc or ball in , and let be a quasi-projective algebraic manifold which is Zariski locally equivalent to , or a complement of an algebraic subvariety of codimension in such a manifold. If is an integer satisfying , then every holomorphic map from a neighborhood of to with rank at every point of can be approximated uniformly on by entire maps with rank at every point of .

     

Symmetric word equations in two positive definite letters

For every symmetric (``palindromic") word in two positive definite letters and for each fixed -by- positive definite and , it is shown that the symmetric word equation has an -by- positive definite solution . Moreover, if and are real, there is a real solution . The notion of symmetric word is generalized to allow non-integer exponents, with certain limitations. In some cases, the solution is unique, but, in general, uniqueness is an open question. Applications and methods for finding solutions are also discussed.

     

Pure Picard-Vessiot extensions with generic properties

Given a connected linear algebraic group over an algebraically closed field of characteristic 0, we construct a pure Picard-Vessiot extension for , namely, a Picard-Vessiot extension , with differential Galois group , such that and are purely differentially transcendental over . The differential field is the quotient field of a -stable proper differential subring with the property that if is any differential field with field of constants and is a Picard-Vessiot extension with differential Galois group a connected subgroup of , then there is a differential homomorphism such that is generated over as a differential field by .

     

Distinguished representations and poles of twisted tensor -functions

Let be a quadratic extension of -adic fields. If is an admissible representation of that is parabolically induced from discrete series representations, then we prove that the space of -invariant linear functionals on has dimension one, where is the mirabolic subgroup. As a corollary, it is deduced that if is distinguished by , then the twisted tensor -function associated to has a pole at . It follows that if is a discrete series representation, then at most one of the representations and is distinguished, where is an extension of the local class field theory character associated to . This is in agreement with a conjecture of Flicker and Rallis that relates the set of distinguished representations with the image of base change from a suitable unitary group.

     

Characterizing Cohen-Macaulay local rings by Frobenius maps

Let be a commutative noetherian local ring of prime characteristic. Denote by the ring regarded as an -algebra through -times composition of the Frobenius map. Suppose that is F-finite, i.e., is a finitely generated -module. We prove that is Cohen-Macaulay if and only if the -modules have finite Cohen-Macaulay dimensions for infinitely many integers .

     

Subspaces of

The relationship of the Hardy space and the space of integrable functions is examined in terms of intermediate spaces of functions that are described as sums of atoms. It is proved that these spaces have dual spaces that lie between the space of functions of bounded mean oscillation, , and . Furthermore, the spaces intermediate to and are shown to be dual to spaces similar to the space of functions of vanishing mean oscillation. The proofs are extensions of classical proofs.

     

A criterion for satellite 1-genus 1-bridge knots

Let be a knot in a closed orientable irreducible 3-manifold . Suppose admits a genus 1 Heegaard splitting and we denote by the splitting torus. We say is a -genus -bridge splitting of if intersects transversely in two points, and divides into two pairs of a solid torus and a boundary parallel arc in it. It is known that a -genus -bridge splitting of a satellite knot admits a satellite diagram disjoint from an essential loop on the splitting torus. If and the slope of the loop is longitudinal in one of the solid tori, then is obtained by twisting a component of a -bridge link along the other component. We give a criterion for determining whether a given -genus -bridge splitting of a knot admits a satellite diagram of a given slope or not. As an application, we show there exist counter examples for a conjecture of Ait Nouh and Yasuhara.

     

Simple corona -algebras

Let be a non-unital and -unital simple -algebra. We show that if is simple, then is purely infinite. We also show that is simple if and only if has a continuous scale provided that is not isomorphic to the compact operators.

     

A note on commutativity up to a factor of bounded operators

In this note, we explore commutativity up to a factor for bounded operators and in a complex Hilbert space. Conditions on possible values of the factor are formulated and shown to depend on spectral properties of the operators. Commutativity up to a unitary factor is considered. In some cases, we obtain some properties of the solution space of the operator equation and explore the structures of and that satisfy for some A quantum effect is an operator on a complex Hilbert space that satisfies The sequential product of quantum effects and is defined by We also obtain properties of the sequential product.

     

Convergence of zeta functions of graphs

The -zeta function of an infinite graph (defined previously in a ball around zero) has an analytic extension. For a tower of finite graphs covered by , the normalized zeta functions of the finite graphs converge to the -zeta function of .

     

The dual spectral set conjecture

Suppose that where are real numbers such that and The union is not assumed to be disjoint. It is shown that the translates , , tile the real line for some bounded measurable set if and only if the exponentials , , form an orthogonal basis for some bounded measurable set

     

Gorenstein injective modules and local cohomology

In this paper we assume that is a Gorenstein Noetherian ring. We show that if is also a local ring with Krull dimension that is less than or equal to 2, then for any nonzero ideal of , is Gorenstein injective. We establish a relation between Gorenstein injective modules and local cohomology. In fact, we will show that if is a Gorenstein ring, then for any -module its local cohomology modules can be calculated by means of a resolution of by Gorenstein injective modules. Also we prove that if is -Gorenstein, is a Gorenstein injective and is a nonzero ideal of , then is Gorenstein injective.

     

On -supplemented maximal and minimal subgroups of Sylow subgroups of finite groups

This paper proves: Let be a saturated formation containing . Suppose that is a group with a normal subgroup such that .

(1) If all maximal subgroups of any Sylow subgroup of are -supple- mented in , then ;

(2) If all minimal subgroups and all cyclic subgroups with order 4 of are -supplemented in , then .