Relative to any nonrecursive set

There is a countable first order structure such that for any set of integers , is not recursive if and only if there is a presentation of which is recursive in .

     

A note on norm attaining functionals

We are concerned in this paper with the density of functionals which do not attain their norms in Banach spaces. Some earlier results given for separable spaces are extended to the nonseparable case. We obtain that a Banach space is reflexive if and only if it satisfies any of the following properties: (i) admits a norm with the Mazur Intersection Property and the set of all norm attaining functionals of contains an open set, (ii) the set of all norm one elements of contains a (relative) weak* open set of the unit sphere, (iii) has and contains a (relative) weak open set of the unit sphere, (iv) is , has and contains a (relative) weak open set of the unit sphere. Finally, if is separable, then is reflexive if and only if contains a (relative) weak open set of the unit sphere.

     

Common operator properties of the linear operators

Let and be bounded linear operators defined on Banach spaces, , . When , then the operators and have many basic operator properties in common. This situation is studied in this paper.

     

Enveloping algebras of lie color algebras: primeness versus graded-primeness

Let be a finite abelian group and let be a, possibly restricted, -graded Lie color algebra. Then the enveloping algebra is also -graded, and we consider the question of whether being graded-prime implies that it is prime. The first section of this paper is devoted to the special case of Lie superalgebras over a field of characteristic . Specifically, we show that if and if has a unique minimal graded-prime ideal, then this ideal is necessarily prime. As will be apparent, the latter result follows quickly from the existence of an anti-automorphism of whose square is the automorphism of the enveloping algebra associated with its -grading. The second section, which is independent of the first, studies more general Lie color algebras and shows that if is graded-prime and if most homogeneous components of are infinite dimensional over , then is prime. Here we use -methods to study the grading on the extended centroid of . In particular, if is generated by the infinite support of , then we prove that is homogeneous.

     

Every complete doubling metric space carries a doubling measure

We prove that a complete metric space carries a doubling measure if and only if is doubling and that more precisely the infima of the homogeneity exponents of the doubling measures on and of the homogeneity exponents of are equal. We also show that a closed subset of carries a measure of homogeneity exponent . These results are based on the case of compact due to Volberg and Konyagin.

     

Products of similar matrices

Let and be matrices of determinant over a field , with or . We show that if is not a scalar matrix, then is a product of matrices similar to . Analogously, we conjecture that if and are elements of a semisimple algebraic group over a field of characteristic zero, and if there is no normal subgroup of containing but not , then is a product of conjugates of . The conjecture is verified for orthogonal groups and symplectic groups, and for all semisimple groups over local fields. Thus, in a connected, semisimple Lie group with finite center, the only conjugation-invariant subsemigroups are the normal subgroups.

     

Vector bundles on log terminal varieties

Let be an -dimensional variety and an ample vector bundle on of rank . We give a complete classification of pairs , with log terminal and such that is not ample. The results we obtain were conjectured by Fujita, and recently by Zhang.

     

Neither first countable nor Cech-complete spaces are maximal Tychonoff connected

A connected Tychonoff space is called maximal Tychonoff connected if there is no strictly finer Tychonoff connected topology on . We show that if is a connected Tychonoff space and locally separable spaces, locally \v{C}ech-complete spaces, first countable spaces, then is not maximal Tychonoff connected. This result is new even in the cases where is compact or metrizable.

     

A remark on normal derivations

Given a Hilbert space , let be operators on . Anderson has proved that if is normal and , then for all operators . Using this inequality, Du Hong-Ke has recently shown that if (instead) , then for all operators . In this note we improve the Du Hong-Ke inequality to for all operators . Indeed, we prove the equivalence of Du Hong-Ke and Anderson inequalities, and show that the Du Hong-Ke inequality holds for unitarily invariant norms.

     

A remark on Gelfand-Kirillov dimension

Let be a finitely generated non-PI Ore domain and the quotient division algebra of . If is the center of , then .

     

Probability measures in -algebras in Hilbert spaces with conjugation

Let be a real -algebra of -real bounded operators containing no central summand of type in a complex Hilbert space with conjugation . Denote by the quantum logic of all -orthogonal projections in the von Neumann algebra . Let be a probability measure. It is shown that contains a finite central summand and there exists a normal finite trace on such that , .

     

Sufficient conditions for one domain to contain another in a space of constant curvature

As an application of the analogue of C-S. Chen's kinematic formula in the 3-dimensional space of constant curvature , that is, Euclidean space , -sphere , hyperbolic space (, respectively), we obtain sufficient conditions for one domain to contain another domain in either an Euclidean space , or a -sphere or a hyperbolic space .

     

A characterization for spaces of sections

The space of smooth sections of a bundle over a compact smooth manifold can be equipped with a manifold structure, called an -manifold, where represents the Fréchet algebra of real valued smooth functions on . We prove that the -manifold structure characterizes the spaces of sections of bundles over and its open subspaces. We also describe the -maps between -manifolds.

     

Certain invariant subspace structure of

In this note, we study certain structure of an invariant subspace of . Considering the largest -invariant (resp. -invariant) subspace in the wandering subspace of with respect to the shift operator , we give an alternative characterization of Beurling-type invariant subspaces. Furthermore, we consider a certain class of invariant subspaces.

     

Ensembles sur lesquels les polynômes sont déterminés par leur image

Let be a non-empty subset of the complex plane , and , two complex polynomials. If and having the same image on implies , we say that is a generalized unicity set (for polynomials). We construct in this paper a subset of such that and are generalized unicity sets, and we give an example of a generalized unicity set which is open, connected and unbounded.

RÉSUMÉ. Soit un sous-ensemble non vide du plan complexe , et , deux fonctions polynômes à coefficients complexes. Si l'égalité entraîne , on dira que est un ensemble d'unicité généralisée (pour les polynômes). On construit dans cet article un sous-ensemble de tel que et sont d'unicité généralisée, et on donne aussi l'exemple d'un ensemble d'unicité généralisée qui est ouvert, connexe et non borné.

     

Continuous multiplicative mappings on

Let and be compact Hausdorff topological spaces, and let and be real Banach algebras of all real-valued continuous functions on and , respectively. The general form of continuous multiplicative mappings is given.

     

-convexity of Orlicz-Bochner spaces

A characterization of -convexity of arbitrary Banach space is given. Moreover, it is proved that the Orlicz-Bochner function space is P-convex if and only if both spaces and are -convex. In particular, the Lebesgue-Bochner space with is -convex iff is -convex.

     

A twistor correspondence and Penrose transform for odd-dimensional hyperbolic space

For odd-dimensional hyperbolic space , we construct transforms between the cohomology of certain line bundles on (a twistor space for ) and eigenspaces of the Laplacian and of the Dirac operator on . The transforms are isomorphisms. As a corollary we obtain that every eigenfunction of or on extends as a holomorphic eigenfunction of the corresponding holomorphic operator on a certain region of the complexification of . We also obtain vanishing theorems for the cohomology of a class of line bundles on .

     

The exposed points of the set of invariant means on an ideal

Let be a -compact locally compact nondiscrete group and let be a -invariant ideal of . We denote the set of left invariant means on that are zero on (i.e. for all ) by . We show that, when is amenable as a discrete group and the closed -invariant subset of the spectrum of corresponding to is a -set, is very large in the sense that every nonempty -subset of contains a norm discrete copy of , where is the Stone- compactification of the set of positive integers with the discrete topology. In particular, we prove that has no exposed points in this case and every nonempty -subset of the set of left invariant means on contains a norm discrete copy of .

     

Remarks on the non-Cohen-Macaulay locus of Noetherian schemes

In this paper we give a notion of polynomial type of a Noetherian scheme and define the function by for all Then we show that if admits a dualizing complex and is equidimensional, is (lower) semicontinuous; moreover, in that case, the non-Cohen-Macaulay locus nCM is not Cohen-Macaulay} is biequidimensional iff is constant on nCM