Robust transitivity and topological mixing for -flows

We prove that nontrivial homoclinic classes of -generic flows are topologically mixing. This implies that given , a nontrivial -robustly transitive set of a vector field , there is a -perturbation of such that the continuation of is a topologically mixing set for . In particular, robustly transitive flows become topologically mixing after -perturbations. These results generalize a theorem by Bowen on the basic sets of generic Axiom A flows. We also show that the set of flows whose nontrivial homoclinic classes are topologically mixing is not open and dense, in general.

     

Zero product preserving maps of operator-valued functions

Let be locally compact Hausdorff spaces and , be Banach algebras. Let be a zero product preserving bounded linear map with dense range. We show that is given by a continuous field of algebra homomorphisms from into if is irreducible. As corollaries, such a surjective arises from an algebra homomorphism, provided that is a -algebra and is a semi-simple Banach algebra, or both and are -algebras.

     

Free products in linear groups

Let be a commutative integral domain of characteristic , and let be a finite subgroup of , the projective general linear group of degree over . In this note, we show that if , then also contains the free product , where is the infinite cyclic group generated by the image of a suitable transvection.

     

Exponential nonnegativity

Let be a Banach algebra, , the spectrum of and the spectral abscissa of . If , then we show that there exists an algebra cone such that is exponentially nonnegative with respect to and the spectral radius is increasing on .

     

The Cauchy problem for a class of Kovalevskian pseudo-differential operators

We prove the well-posedness of the forward Cauchy problem for a pseudo-differential operator of order with the Log-Lipschitz continuous symbol in the time variable. The characteristic roots of are distinct and satisfy the necessary Lax-Mizohata condition Im . The Log-Lipschitz regularity has been tested as the optimal one for well-posedness in the case of second-order hyperbolic operators. Our main aim is to present a simple proof which needs only a little of the basic calculus of standard pseudo-differential operators.

     

Self-normalizing Sylow subgroups

Using the classification of finite simple groups we prove the following statement: Let 3$"> be a prime, a group of automorphisms of -power order of a finite group , and a -invariant Sylow -subgroup of . If is trivial, then is solvable. An equivalent formulation is that if has a self-normalizing Sylow -subgroup with 3$"> a prime, then is solvable. We also investigate the possibilities when .

     

Commutants of reflexive algebras and classification of completely distributive subspace lattices

Let be a subspace lattice on a normed space containing a nontrivial comparable element. If commutes with all the operators in , then there exists a scalar such that . Furthermore, we classify the class of completely distributive subspace lattices into subclasses called Type , Type and Type , respectively. It is shown that nontrivial nests or, more generally, completely distributive subspace lattices with a comparable element are Type , and that nontrivial atomic Boolean subspace lattices are Type .

     

Rings with finite Gorenstein injective dimension

In this paper we prove that for any associative ring , and for any left -module with finite projective dimension, the Gorenstein injective dimension equals the usual injective dimension . In particular, if is finite, then also is finite, and thus is Gorenstein (provided that is commutative and Noetherian).

     

A note on Kneser-Haken finiteness

Kneser-Haken finiteness asserts that for each compact 3-manifold there is an integer such that any collection of c(M)$"> closed, essential, 2-sided surfaces in must contain parallel elements. We show here that if is closed, then twice the number of tetrahedra in a (pseudo)-triangulation of suffices for .

     

-extensions in characteristic 3

We present an explicit construction of the complete family of Galois extensions of a field of characteristic 3 with Galois group the central product of a double cover of the symmetric group and the quaternion group , containing a given -extension of the field .

     

The splitting problem for subspaces of tensor products of operator algebras

The main result of this paper is that if is a von Neumann algebra that is a factor and has the weak* operator approximation property (the weak* OAP), and if is a von Neumann algebra, then every -weakly closed subspace of that is an -bimodule (under multiplication) splits, in the sense that there is a -weakly closed subspace of such that . Note that if is a von Neumann subalgebra of , then is an -bimodule if and only if . So this result is a generalization (in the case where has the weak* OAP) of the result of Ge and Kadison that if is a factor, then every von Neumann subalgebra of that contains splits. We also obtain other results concerning the splitting of -weakly closed subspaces of tensor products of von Neumann algebras and the splitting of normed closed subspaces of C*-algebras that generalize results previously obtained for von Neumann subalgebras and C*-subalgebras.

     

The duality theory of a finite dimensional discrete quantum group

Suppose that is a finite dimensional discrete quantum group and is a Hilbert space. This paper shows that if there exists an action of on so that is a modular algebra and the inner product on is -invariant, then there is a unique C*-representation of on supplemented by the The commutant of in is exactly the -invariant subalgebra of . As an application, a new proof of the classical Schur-Weyl duality theory of type A is given.

     

The number of Hall -subgroups of a -separable group

We observe a simple formula to compute the number of Hall -subgroups of a -separable finite group in terms of only the action of a fixed Hall -subgroup of on a set of normal -sections of . As a consequence, we obtain that divides whenever is a subgroup of a finite -separable group . This generalizes a recent result of Navarro. In addition, our method gives an alternative proof of Navarro's result.

     

Spectral subspaces of subscalar and related operators

For a bounded linear operator on a complex Banach space and a closed subset of the complex plane this note deals with algebraic representations of the corresponding analytic spectral subspace from local spectral theory. If is the restriction of a generalized scalar operator to a closed invariant subspace, then it is shown that for all sufficiently large integers where denotes the largest linear subspace of for which for all Moreover, for a wide class of operators that satisfy growth conditions of polynomial or Beurling type, it is shown that is closed and equal to

     

Spectrally bounded operators on simple -algebras

A linear mapping from a subspace of a Banach algebra into another Banach algebra is called spectrally bounded if there is a constant such that for all , where denotes the spectral radius. We prove that every spectrally bounded unital operator from a unital purely infinite simple -algebra onto a unital semisimple Banach algebra is a Jordan epimorphism.

     

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.

     

Bundles with periodic maps and mod Chern polynomial

Suppose that is a vector bundle with a linear periodic map of period ; the map is assumed free on the outside of the -section. A polynomial , called a mod Chern polynomial of , is defined. It is analogous to the Stiefel-Whitney polynomial defined by Dold for real vector bundles with the antipodal involution. The mod Chern polynomial can be used to measure the size of the periodic coincidence set for fibre preserving maps of the unit sphere bundle of into another vector bundle.

     

-adic formal series and primitive polynomials over finite fields

In this paper, we investigate the Hansen-Mullen conjecture with the help of some formal series similar to the Artin-Hasse exponential series over -adic number fields and the estimates of character sums over Galois rings. Given we prove, for large enough , the Hansen-Mullen conjecture that there exists a primitive polynomial over of degree with the -th ( coefficient fixed in advance except when if is odd and when if is even.

     

Contractible Fréchet algebras

A unital Fréchet algebra is called contractible if there exists an element such that and for all where is the canonical Fréchet -bimodule morphism. We give a sufficient condition for an infinite-dimensional contractible Fréchet algebra to be a direct sum of a finite-dimensional semisimple algebra and a contractible Fréchet algebra without any nonzero finite-dimensional two-sided ideal (see Theorem 1). As a consequence, a commutative lmc Fréchet -algebra is contractible if, and only if, it is algebraically and topologically isomorphic to for some . On the other hand, we show that a Fréchet algebra, that is, a locally -algebra, is contractible if, and only if, it is topologically isomorphic to the topological Cartesian product of a certain countable family of full matrix algebras.

     

Approximate fixed point sequences and convergence theorems for Lipschitz pseudocontractive maps

Let be a nonempty closed convex subset of a real Banach space and be a Lipschitz pseudocontractive self-map of with . An iterative sequence is constructed for which as . If, in addition, is assumed to be bounded, this conclusion still holds without the requirement that Moreover, if, in addition, has a uniformly Gâteaux differentiable norm and is such that every closed bounded convex subset of has the fixed point property for nonexpansive self-mappings, then the sequence converges strongly to a fixed point of . Our iteration method is of independent interest.