Local automorphisms and derivations on

The aim of this note is to give a short proof that 2-local derivations on , the matrix algebra over the complex numbers are derivations and to give a shorter proof that 2-local *-automorphisms on are *-automorphisms.

     

Polynomial approximation on real-analytic varieties in

We prove: Let be a compact real-analytic variety in . Assume (i) is polynomially convex and (ii) every point of is a peak point for . Then . This generalizes a previous result of the authors on polynomial approximation on three-dimensional real-analytic submanifolds of .

     

interval maps not Borel conjugate to any map

We show that there exist interval maps that are not Borel conjugate to any map. These examples can be chosen to be topologically mixing and , for any finite, arbitrarily large .

     

The Lusternik-Schnirelmann category of

We show that the Lusternik-Schnirelmann category of the symplectic group is . This L-S category coincides with the cone length and the stable weak category.

     

Pseudocompact spaces and -spaces

Let be a completely regular Hausdorff space, and let be the space of continuous real-valued functions on endowed with the compact-open topology. We find various equivalent conditions for to be a -space, resolving an old question of Jarchow and consolidating work by Jarchow, Mazon, McCoy and Todd. Included are analytic characterizations of pseudocompactness and an example that shows that, for , Grothendieck's -spaces do not coincide with Jarchow's -spaces. Any such example necessarily answers a thirty-year-old question on weak barrelledness properties for , our original motivation.

     

Minimal degrees which are but not

We give a short proof of the existence of minimal Turing degrees which are but not .

     

Banach spaces embedding isometrically into when

For we give examples of Banach spaces isometrically embedding into but not into any with

     

A concrete description of -spaces as -spaces and its applications

We prove that for a compact Hausdorff space without isolated points, and are isometrically Riesz isomorphic spaces under a certain topology on . Moreover, is a closed subspace of . This provides concrete examples of compact Hausdorff spaces such that the Dedekind completion of is (= the set of all bounded real-valued functions on ) since the Dedekind completion of is ( and spaces as Banach lattices).

     

Morava -theory of extraspecial -groups

We compute the Morava -theory of some extraspecial 2-groups and associated compact groups.

     

Real -flats tangent to quadrics in

Let and denote the dimension and the degree of the Grassmannian , respectively. For each there are (a priori complex) -planes in tangent to general quadratic hypersurfaces in . We show that this class of enumerative problems is fully real, i.e., for there exists a configuration of real quadrics in (affine) real space so that all the mutually tangent -flats are real.

     

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.

     

Lifts of - and -morphisms to -morphisms

Let be the Hochschild complex of cochains on and let be the space of multivector fields on . In this paper we prove that given any -structure (i.e. Gerstenhaber algebra up to homotopy structure) on , and any -morphism (i.e. morphism of a commutative, associative algebra up to homotopy) between and , there exists a -morphism between and that restricts to . We also show that any -morphism (i.e. morphism of a Lie algebra up to homotopy), in particular the one constructed by Kontsevich, can be deformed into a -morphism, using Tamarkin's method for any -structure on . We also show that any two of such -morphisms are homotopic.

     

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.

     

Triangular actions on

Every locally trivial action of the additive group of complex numbers on four-dimensional complex affine space that is given by a triangular derivation is conjugate to a translation. A criterion for a proper action on complex affine -space to be locally trivial is given, along with an example showing that the hypotheses of the criterion are sharp.

     

On the elliptic equation on

In this paper we consider the existence problem for the elliptic equation on , which arises in the study of conformal deformation of the hyperbolic disc. We prove an existence result for the above equation.

     

Stable minimal surfaces in with degenerate Gauss map

A complete oriented stable minimal surface in is a plane, but in , there are many non-flat examples such as holomorphic curves. The Gauss map plays an important role in the theory of minimal surfaces. In this paper, we prove that a complete oriented stable minimal surface in with -degenerate Gauss map (for 1/4$">) is a plane.

     

On -extensions in characteristic

We study the relationship between generic polynomials and generic extensions over a finite ground field, using dihedral extensions as an example.

     

Spectral pictures of and

The spectral pictures of products and of Banach space operators are compared; in particular when one of them is `of index zero'.

     

-bounding and -induction

Working in the base theory of , we show that for all , the bounding principle for -formulas ( ) is equivalent to the induction principle for -formulas ( ). This partially answers a question of J. Paris.

     

Biliaison classes of curves in

We characterize the curves in that are minimal in their biliaison class. Such curves are exactly the curves that do not admit an elementary descending biliaison. As a consequence we have that every curve in can be obtained from a minimal one by means of a finite sequence of ascending elementary biliaisons.