On the Lipschitz classification of normed spaces, unit balls, and spheres

For every normed space , we note its closed unit ball and unit sphere by and , respectively. Let and be normed spaces such that is Lipschitz homeomorphic to , and is Lipschitz homeomorphic to .

We prove that the following are equivalent:

1. is Lipschitz homeomorphic to .

2. is Lipschitz homeomorphic to .

3. is Lipschitz homeomorphic to .

This result holds also in the uniform category, except (2 or 3) 1 which is known to be false.

     

Lifting wreath product extensions

Let and be finite groups and let be a hilbertian field. We show that if has a generic extension over and satisfies the arithmetic lifting property over , then the wreath product of and also satisfies the arithmetic lifting property over . Moreover, if the orders of and are relatively prime and is abelian, then any extension of by (which is necessarily a semidirect product) has the arithmetic lifting property.

     

A bifurcation result for harmonic maps from an annulus to with not symmetric boundary data

We consider the problem of minimizing the energy of the maps from the annulus to such that is equal to for , and to , for , where is a fixed angle.

We prove that the minimum is attained at a unique harmonic map which is a planar map if , while it is not planar in the case \pi^2$">.

Moreover, we show that tends to as , where minimizes the energy of the maps from to , with the boundary condition , .

     

Types in reductive -adic groups: The Hecke algebra of a cover

In this paper, is a non-Archimedean local field and is the group of -points of a connected reductive algebraic group defined over . Also, is an irreducible representation of a compact open subgroup of , the pair being a type in . The pair is assumed to be a cover of a type in a Levi subgroup of . We give conditions, generalizing those of earlier work, under which the Hecke algebra is the tensor product of a canonical image of and a sub-algebra , for a compact open subgroup of containing .

     

Conformal metrics and Ricci tensors in the pseudo-Euclidean space

We consider constant symmetric tensors on , , and we study the problem of finding metrics conformal to the pseudo-Euclidean metric such that . We show that such tensors are determined by the diagonal elements and we obtain explicitly the metrics . As a consequence of these results we get solutions globally defined on for the equation Moreover, we show that for certain unbounded functions defined on , there are metrics conformal to the pseudo-Euclidean metric with scalar curvature .

     

Elements in a commutative Banach algebra determining the norm topology

For an element of a commutative complex Banach algebra we investigate the following property: every complete norm on making the multiplication by from to itself continuous is equivalent to .

     

Single elements of finite CSL algebras

An element of an (abstract) algebra is a single element of if and imply that or . Let be a real or complex reflexive Banach space, and let be a finite atomic Boolean subspace lattice on , with the property that the vector sum is closed, for every . For any subspace lattice the single elements of Alg are characterised in terms of a coordinatisation of involving . (On separable complex Hilbert space the finite distributive subspace lattices which arise in this way are precisely those which are similar to finite commutative subspace lattices. Every distributive subspace lattice on complex, finite-dimensional Hilbert space is of this type.) The result uses a characterisation of the single elements of matrix incidence algebras, recently obtained by the authors.

     

FS-property for -algebras

A -algebra is said to have the FS-property if the set of all self-adjoint elements in has a dense subset of elements with finite spectrum. We shall show that this property is not stable under taking the minimal -tensor products even in case of separable nuclear -algebras.

     

Splitting sets in integral domains

Let be an integral domain. A saturated multiplicatively closed subset of is a splitting set if each nonzero may be written as where and for all . We show that if is a splitting set in , then is a splitting set in , a multiplicatively closed subset of , and that is a splitting set in is an lcm splitting set of , i.e., is a splitting set of with the further property that is principal for all and . Several new characterizations and applications of splitting sets are given.

     

Unbounded quasi-integrals

Let be a locally compact Hausdorff space. We define a quasi-measure in , a quasi-integral on , and a quasi-integral on . We show that all quasi-integrals on are bounded, continuity properties of the quasi-integral on , representation of quasi-integrals on in terms of quasi-measures, and unique extension of quasi-integrals on to .

     

Line bundles for which a projectivized jet bundle is a product

We characterize the triples , consisting of line bundles and on a complex projective manifold , such that for some positive integer , the -th holomorphic jet bundle of , , is isomorphic to a direct sum .

     

An order characterization of commutativity for -algebras

In this paper, we investigate the problem of when a -algebra is commutative through operator-monotonic increasing functions. The principal result is that the function is operator-monotonic increasing on a -algebra if and only if is commutative. Therefore, -algebra is commutative if and only if in for all positive elements in .

     

Irreducible restriction and zeros of characters

Let be a finite group, let be normal in and suppose that is an irreducible complex character of . Then is not irreducible if and only if vanishes on some coset of in .

     

Minimal index of a -crossed product by a finite group

Let be the -crossed product of a simple unital -algebra by a finite group . In this paper we show that the canonical conditional expectation from to has the minimal index if is simple. It is also proved that if is an outer action, then the canonical one is the unique conditional expectation of index-finite type from to , while there are infinitely many conditional expectations when a nontrivial subgroup of acts innerly on .

     

Between the Lindelöf property and countable tightness

We consider a class of compact spaces for which the space of probability Radon measures on has countable tightness in the topology. We show that that class contains those compact zero-dimensional spaces for which is weakly Lindelöf, and, under MA + CH, all compact spaces with having property (C) of Corson.

     

On classification of polarized varieties with non-integral nef values

Let be an -dimensional normal projective variety with only Gorenstein, terminal, -factorial singularities. Let be an ample line bundle on . Let denote the nef value of . The classification of via the nef value morphism is given for the situations when satisfies or .

     

Une preuve courte du principe de Selberg pour un groupe -adique

In 1992, Blanc and Brylinski showed the following property for a -adic group , called the ``abstract Selberg principle': the orbital integrals on conjugacy classes of non-compact elements of the Hattori rank of a finitely generated projective smooth representation of vanish. The proof is by explicit computations of ``low' level ( and cyclic and Hochschild cohomologies. Here we intend to show that this property is actually a direct consequence of two facts: Clozel's integration formula (which leads us to assume the defining characteristic to be zero) and the triviality of the action of unramified characters on the of (which is also proven here, using a standard -theoretic argument due to Grothendieck).

     

Quasitilted extensions of algebras I

Let be a connected finite dimensional -algebra, and let be a nonzero decomposable -module such that the one-point extension is quasitilted. We show here that every nonzero indecomposable direct summand of is directing and is a tilted algebra.

     

Projections of polytopes on the plane and the generalized Baues problem

Given an affine projection of a -polytope onto a polygon , it is proved that the poset of proper polytopal subdivisions of which are induced by has the homotopy type of a sphere of dimension if maps all vertices of into the boundary of . This result, originally conjectured by Reiner, is an analogue of a result of Billera, Kapranov and Sturmfels on cellular strings on polytopes and explains the significance of the interior point of present in the counterexample to their generalized Baues conjecture, constructed by Rambau and Ziegler.