Transversals for strongly almost disjoint families

For a family of sets , and a set , is said to be a transversal of if and for each . is said to be a Bernstein set for if for each . Erdos and Hajnal first studied when an almost disjoint family admits a set such as a transversal or Bernstein set. In this note we introduce the following notion: a family of sets is said to admit a -transversal if can be written as such that each admits a transversal. We study the question of when an almost disjoint family admits a -transversal and related questions.

     

The universal central extension of the three-point loop algebra

We consider the three-point loop algebra,

where denotes a field of characteristic 0 and is an indeterminate. The universal central extension of was determined by Bremner. In this note, we give a presentation for via generators and relations, which highlights a certain symmetry over the alternating group . To obtain our presentation of , we use the realization of as the tetrahedron Lie algebra.

     

Some Hopf Galois structures arising from elementary abelian -groups

Let be an odd prime, , the elementary abelian -group of rank , and let be the group of principal units of the ring . If is a Galois extension with Galois group , then we show that for , the number of Hopf Galois structures on afforded by -Hopf algebras with associated group is greater than , where .

     

Order-weakly compact operators from vector-valued function spaces to Banach spaces

Let be an ideal of over a  -finite measure space , and let stand for the order dual of . For a real Banach space let be a subspace of the space of -equivalence classes of strongly -measurable functions and consisting of all those for which the scalar function belongs to . For a real Banach space a linear operator is said to be order-weakly compact whenever for each the set is relatively weakly compact in . In this paper we examine order-weakly compact operators . We give a characterization of an order-weakly compact operator in terms of the continuity of the conjugate operator of with respect to some weak topologies. It is shown that if is an order continuous Banach function space, is a Banach space containing no isomorphic copy of and is a weakly sequentially complete Banach space, then every continuous linear operator is order-weakly compact. Moreover, it is proved that if is a Banach function space, then for every Banach space any continuous linear operator is order-weakly compact iff the norm is order continuous and is reflexive. In particular, for every Banach space any continuous linear operator is order-weakly compact iff is reflexive.

     

A structure theorem for quasi-Hopf comodule algebras

If is a quasi-Hopf algebra and is a right -comodule algebra such that there exists a morphism of right -comodule algebras, we prove that there exists a left -module algebra such that . The main difference when comparing to the Hopf case is that, from the multiplication of , which is associative, we have to obtain the multiplication of , which in general is not; for this we use a canonical projection arising from the fact that becomes a quasi-Hopf -bimodule.

     

Noncoherence of some rings of functions

Let , denote the unit disc and unit circle, respectively, in , with center 0. If , then let denote the set of complex-valued functions defined on that are analytic in , and continuous and bounded on . Then is a ring with pointwise addition and multiplication. We prove that if the intersection of with the set of limit points of is not empty, then the ring is not coherent.

     

Semiprime smash products and -stable prime radicals for PI-algebras

Assume that is a finite-dimensional Hopf algebra over a field and that is an -module algebra satisfying a polynomial identity (PI). We prove that if is semisimple and is -semiprime, then is semiprime. If is cosemisimple, we show that the prime radical of is -stable.

     

A strong comparison principle for the -Laplacian

We consider weak solutions of the differential inequality of p-Laplacian type

such that on a smooth bounded domain in and either or is a weak solution of the corresponding Dirichlet problem with zero boundary condition. Assuming that on the boundary of the domain we prove that , and assuming that on the boundary of the domain we prove unless . The novelty is that the nonlinearity is allowed to change sign. In particular, the result holds for the model nonlinearity with .

     

Level algebras with bad properties

This paper can be seen as a continuation of the works contained in the recent article (J. Alg., 305 (2006), 949-956) of the second author, and those of Juan Migliore (math. AC/0508067). Our results are:

1). There exist codimension three artinian level algebras of type two which do not enjoy the Weak Lefschetz Property (WLP). In fact, for , we will construct a codimension three, type two -vector of socle degree such that all the level algebras with that -vector do not have the WLP. We will also describe the family of those algebras and compute its dimension, for each .

2). There exist reduced level sets of points in of type two whose artinian reductions all fail to have the WLP. Indeed, the examples constructed here have the same -vectors we mentioned in 1).

3). For any integer , there exist non-unimodal monomial artinian level algebras of codimension . As an immediate consequence of this result, we obtain another proof of the fact (first shown by Migliore in the above-mentioned preprint, Theorem 4.3) that, for any , there exist reduced level sets of points in whose artinian reductions are non-unimodal.

     

Dimension des familles de courbes lisses sur une surface quartique normale de

Dans cette note, on montre que les courbes, lisses connexes, de degré et genre , tracées sur une surface quartique normale variable de , et n'y étant pas intersection complète, forment des familles de dimensions . Cette majoration est la meilleure possible. Comme application on prouve que le schéma de Hilbert des courbes lisses connexes de de degré et genre est irréductible.

     

Simple algebras of Weyl type, II

Over a field of any characteristic, for a commutative associative algebra , and for a commutative subalgebra of , the vector space which consists of polynomials of elements in with coefficients in and which is regarded as operators on forms naturally an associative algebra. It is proved that, as an associative algebra, is simple if and only if is -simple. Suppose is -simple. Then, (a) is a free left -module; (b) as a Lie algebra, the subquotient is simple (except for one case), where is the center of . The structure of this subquotient is explicitly described. This extends the results obtained by Su and Zhao.

     

Hereditary and maximal crossed product orders

We first deal with classical crossed products , where is a finite group acting on a Dedekind domain and (the -invariant elements in ) a DVR, admitting a separable residue fields extension. Here is a 2-cocycle. We prove that is hereditary if and only if is semi-simple. In particular, the heredity property may hold even when is not tamely ramified (contradicting standard textbook references). For an arbitrary Krull domain , we use the above to prove that under the same separability assumption, is a maximal order if and only if its height one prime ideals are extended from . We generalize these results by dropping the residual separability assumptions. An application to non-commutative unique factorization rings is also presented.

     

Embeddings of -dimensional separable metric spaces into the product of Sierpinski curves

We give a short proof of the following fact: the set of embeddings of any -dimensional separable metric space into a certain -dimensional subset of the -product of Sierpinski curves is residual in .

     

Polynomial maps and even dimensional spheres

We construct, for every even dimensional sphere , , and every odd integer , a homogeneous polynomial map of Brouwer degree and algebraic degree .

     

Operator theory on noncommutative varieties, II

An -tuple of operators on a Hilbert space is called a -constrained row contraction if and

where is a WOT-closed two-sided ideal of the noncommutative analytic Toeplitz algebra and is defined using the -functional calculus for row contractions.

We show that the constrained characteristic function associated with and is a complete unitary invariant for -constrained completely non-coisometric (c.n.c.) row contractions. We also provide a model for this class of row contractions in terms of the constrained characteristic functions. In particular, we obtain a model theory for -commuting c.n.c. row contractions.

     

On the poles of the resolvent in Calkin algebra

In the present note, we study the problem of lifting poles in Calkin algebra on a separable infinite-dimensional complex Hilbert space . We show by an example that such lifting is not possible in general, and we prove that if zero is a pole of the resolvent of the image of an operator in the Calkin algebra, then there exists a compact operator for which zero is a pole of if and only if the index of is zero on a punctured neighbourhood of zero. Further, a useful characterization of poles in Calkin algebra in terms of essential ascent and descent is provided.

     

The Banach algebra induced by a double centralizer

Given a Banach algebra , R. Larsen defined, in his book ``An introduction to the theory of multipliers", a Banach algebra by means of a multiplier on , and essentially used it in the case of a commutative semisimple Banach algebra to prove a result on multiplications which preserve regular maximal ideals. Here, we consider the analogue Banach algebra induced by a bounded double centralizer of a Banach algebra . Then, our main concern is devoted to the relationships between , , and the algebras of bounded double centralizers and of and , respectively. By removing the assumption of semisimplicity, we generalize some results proven by Larsen.

     

On local irreducibility of the spectrum

Let be the algebra of complex matrices, and for denote by and the spectrum and spectral radius of respectively. Let be a domain in containing 0, and let be a holomorphic map. We prove: (1) if for , then for ; (2) if for , then again for . Both results are special cases of theorems expressing the irreducibility of the spectrum near .

     

A new estimate in optimal mass transport

Let be a bounded Lipschitz regular open subset of and let be two probablity measures on . It is well known that if is absolutely continuous, then there exists, for every , a unique transport map pushing forward on and which realizes the Monge-Kantorovich distance . In this paper, we establish an bound for the displacement map which depends only on , on the shape of and on the essential infimum of the density .

     

Some exact sequences for Toeplitz algebras of spherical isometries

A family of commuting bounded operators on a Hilbert space is said to be a spherical isometry if in the weak operator topology. We show that every commuting family of spherical isometries is jointly subnormal, which means that it has a commuting normal extension on some Hilbert space Suppose now that the normal extension is minimal. Then we show that every bounded operator in the commutant of has a unique norm preserving extension to an operator in the commutant of Moreover, if is the commutator ideal in then is *-isomorphic to We also show that the commutant of the minimal normal extension is completely isometric, via the compression mapping, to the space of Toeplitz-type operators associated to We apply these results to construct exact sequences for Toeplitz algebras on generalized Hardy spaces associated to strictly pseudoconvex domains.