Subspaces of Maximal Operator Spaces

We explore subspaces of maximal operator spaces ( submaximal spaces) and give a new characterization of such spaces. We show that the set of n-dimensional submaximal spaces is closed in the topology of c.b. distance, but not compact. We also investigate subspaces of MAX(L ) and prove that any homogeneous Hilbertian subspace of MAX(L ₁) is completely isomorphic to R + C.     

Remainder terms in a higher order Sobolev inequality

For higher order Hilbertian Sobolev spaces, we improve the embedding inequality for the critical L ^p -space by adding a remainder term with a suitable weak norm.     

Restriction varieties and geometric branching rules

This paper develops a new method for studying the cohomology of orthogonal flag varieties. Restriction varieties are subvarieties of orthogonal flag varieties defined by rank conditions with respect to (not necessarily isotropic) flags. They interpolate between Schubert varieties in orthogonal flag varieties and the restrictions of general Schubert varieties in ordinary flag varieties. We give a positive, geometric rule for calculating their cohomology classes, obtaining a branching rule for Schubert calculus for the inclusion of the orthogonal flag varieties in Type A flag varieties. Our rule, in addition to being an essential step in finding a Littlewood–Richardson rule, has applications to computing the moment polytopes of the inclusion of SO(n) in SU(n), the asymptotic of the restrictions of representations of SL(n) to SO(n) and the classes of the moduli spaces of rank two vector bundles with fixed odd determinant on hyperelliptic curves. Furthermore, for odd orthogonal flag varieties, we obtain an algorithm for expressing a Schubert cycle in terms of restrictions of Schubert cycles of Type A flag varieties, thereby giving a geometric (though not positive) algorithm for multiplying any two Schubert cycles.     

Clifford structures on Riemannian manifolds

We introduce the notion of even Clifford structures on Riemannian manifolds, which for rank r=2 and r=3 reduce to almost Hermitian and quaternion-Hermitian structures respectively. We give the complete classification of manifolds carrying parallel rank r even Clifford structures: Kähler, quaternion-Kähler and Riemannian products of quaternion-Kähler manifolds for r=2,3 and 4 respectively, several classes of 8-dimensional manifolds (for 5?r?8), families of real, complex and quaternionic Grassmannians (for r=8,6 and 5 respectively), and Rosenfeld?s elliptic projective planes OP², (C⊗O)P², (H⊗O)P² and (O⊗O)P², which are symmetric spaces associated to the exceptional simple Lie groups F₄, E₆, E₇ and E₈ (for r=9,10,12 and 16 respectively). As an application, we classify all Riemannian manifolds whose metric is bundle-like along the curvature constancy distribution, generalizing well-known results in Sasakian and 3-Sasakian geometry.     

On alternating and symmetric groups as Galois groups

Fix an integern≧3. We show that the alternating groupA _n appears as Galois group over any Hilbertian field of characteristic different from 2. In characteristic 2, we prove the same whenn is odd. We show that any quadratic extension of Hilbertian fields of characteristic different from 2 can be embedded in anS _n-extension (i.e. a Galois extension with the symmetric groupS _n as Galois group). Forn≠6, it will follow thatA _n has the so-called GAR-property over any field of characteristic different from 2. Finally, we show that any polynomialf=X ⁿ+…+a₁X+a₀ with coefficients in a Hilbertian fieldK whose characteristic doesn’t dividen(n-1) can be changed into anS _n-polynomialf ^* (i.e the Galois group off ^* overK Gal(f ^*, K), isS _n) by a suitable replacement of the last two coefficienta ₀ anda ₁. These results are all shown using the Newton polygon. The author acknowledges the financial support provided through the European Community’s Human Potential Programme under contract HPRN-CT-2000-00114, GTEM.     

On metrizability of compactoid sets in non-archimedean locally convex spaces

In 2003, N. De Grande-De Kimpe, J. Kąkol and C. Perez-Garcia using t-frames and some machinery concerning tensor products proved that compactoid sets in non-archimedean (LM)-spaces (i.e. the inductive limits of a sequence of non-archimedean metrizable locally convex spaces) are metrizable. In this paper we show a similar result for a large class of non-archimedean locally convex space with a £-base, i.e. a decreasing base (U_α)_αN^N of neighbourhoods of zero. This extends the first mentioned result since every non-archimedean (LM)-space has a £-base. We also prove that compactoid sets in non-archimedean (DF)-spaces are metrizable.     

Small codimension subvarieties in homogeneous spaces

We prove Bertini type theorems for the inverse image, under a proper morphism, of any Schubert variety in an homogeneous space. Using generalisations of Deligne's trick, we deduce connectedness results for the inverse image of the diagonal in X² where X is any isotropic grassmannian. We also deduce simple connectedness properties for subvarieties of X. Finally we prove transplanting theorems à la Barth-Larsen for the Picard group of any isotropic grassmannian of lines and for the Neron-Severi group of some adjoint and coadjoint homogeneous spaces.     

Orthogonal polynomials on the disc

We consider the space P_n of orthogonal polynomials of degree n on the unit disc for a general radially symmetric weight function. We show that there exists a single orthogonal polynomial whose rotations through the angles , j=0,1,…,n forms an orthonormal basis for P_n, and compute all such polynomials explicitly. This generalises the orthonormal basis of Logan and Shepp for the Legendre polynomials on the disc.Furthermore, such a polynomial reflects the rotational symmetry of the weight in a deeper way: its rotations under other subgroups of the group of rotations forms a tight frame for P_n, with a continuous version also holding. Along the way, we show that other frame decompositions with natural symmetries exist, and consider a number of structural properties of P_n including the form of the monomial orthogonal polynomials, and whether or not P_n contains ridge functions.     

Orthogonal exponentials on the generalized plane Sierpinski gasket

The self-affine measure μ_Mp,D corresponding tois supported on the the generalized plane Sierpinski gasket T(M_p,D). In the present paper we show that there exist at most 3 mutually orthogonal exponential functions in L²(μ_Mp,D), and the number 3 is the best. This generalizes several known results on the non-spectral self-affine measure problem.     

The Baum–Connes conjecture for free orthogonal quantum groups

We prove an analogue of the Baum–Connes conjecture for free orthogonal quantum groups. More precisely, we show that these quantum groups have a γ-element and that γ=1. It follows that free orthogonal quantum groups are K-amenable. We compute explicitly their K-theory and deduce in the unimodular case that the corresponding reduced C^?-algebras do not contain nontrivial idempotents.Our approach is based on the reformulation of the Baum–Connes conjecture by Meyer and Nest using the language of triangulated categories. An important ingredient is the theory of monoidal equivalence of compact quantum groups developed by Bichon, De Rijdt and Vaes. This allows us to study the problem in terms of the quantum group SUq(2). The crucial part of the argument is a detailed analysis of the equivariant Kasparov theory of the standard Podle? sphere.     

Nonvanishing of Kronecker coefficients for rectangular shapes

We prove that for any partition (λ₁,…,λd²) of size ?d there exists k?1 such that the tensor square of the irreducible representation of the symmetric group Sk?d with respect to the rectangular partition (k?,…,k?) contains the irreducible representation corresponding to the stretched partition (kλ₁,…,kλd²). We also prove a related approximate version of this statement in which the stretching factor k is effectively bounded in terms of d. We further discuss the consequences for geometric complexity theory which provided the motivation for this work.     

Finiteness obstructions and Euler characteristics of categories

We introduce notions of finiteness obstruction, Euler characteristic, L²-Euler characteristic, and Möbius inversion for wide classes of categories. The finiteness obstruction of a category Γ of type (FPR) is a class in the projective class group K₀(RΓ); the functorial Euler characteristic and functorial L²-Euler characteristic are respectively its RΓ-rank and L²-rank. We also extend the second author's K-theoretic Möbius inversion from finite categories to quasi-finite categories. Our main example is the proper orbit category, for which these invariants are established notions in the geometry and topology of classifying spaces for proper group actions. Baez and Dolan's groupoid cardinality and Leinster's Euler characteristic are special cases of the L²-Euler characteristic. Some of Leinster's results on Möbius–Rota inversion are special cases of the K-theoretic Möbius inversion.     

Stochastic evolution equations driven by nuclear-space-valued martingales

This paper presents a theory of stochastic evolution equations for nuclear-space-valued processes and provides a unified treatment of several examples from the field of applications. (C ₀ , 1) reversed evolution systems on countably Hilbertian nuclear spaces are also investigated.Research supported by the Air Force Office of Scientific Research Grant No. F49620 85 C 0144. The research of V. Perez-Abreu was also supported by CONACYT Grants PCEXCNA-040651 and PCMTCNA-750220 (Mexico).     

A basic norm equivalence for the theory of multilevel methods

Summary Subspace decompositions of finite element spaces based onL ₂-like orthogonal projections play an important role for the construction and analysis of multigrid like iterative methods. Recently several authors have proved the equivalence of the associated discrete norms with theH ¹-norm. The present paper gives an elementary, self-contained derivation of this result which is based on the use ofK-functionals known from the theory of interpolation spaces.     

Paranorm <Emphasis Type="Italic">I</Emphasis>-convergent sequence spaces

In this article we introduced the sequence spaces c ^I (p), c ₀ ^I (p), m ^I (p) and m ₀ ^I (p) for p = (p _k ), a sequence of positive real numbers. We study some algebraic and topological properties of these spaces. We prove the decomposition theorem and obtain some inclusion relations.      

GENERALIZED WEIGHTED SOBOLEV SPACES AND APPLICATIONS TO SOBOLEV ORTHOGONAL POLYNOMIALS Ⅱ

We present a definition of general Sobolev spaces with respect to arbitrary measures ,W^k,p(Ω,μ) for 1≤p≤∞,In[RARP] we proved that these spaces are complete under very light conditions.Now we prove that if we consider certain general types of measures,then Cc^∞(R) is dense in these spaces,As an application to Sobolev orthogonal polynomials,we study the boundedness of the multiplication poerator,THis gives an estimation of the zeroes of Sobolev orthogonal polynomials.     

Estimates for the L 2-Projection onto Continuous Finite Element Spaces in a Weighted L p -Norm

We consider the orthogonal L₂-projection P onto continuous finite element spaces. We prove estimates for P in a weighted L_p-norm and use these to prove corresponding approximation properties. AMS subject classification (2000) 65N60, 35K85     

Hilbert spaces of distributions having an orthogonal basis of exponentials

We characterize the Hilbert spaces H whose elements are distributions supported on the interval [0, 1] and which have the property that the system of exponentials {e^2πinx}_n∈Z forms a complete orthogonal system for H, generalizing in this way the classical situation where H=L²([0, 1]) and the system is actually orthonormal. This characterization is extended to the more general setting of spectral pairs and is used to obtain sampling results in various related spaces of functions, that generalize the classical Shannon sampling theorem.     

The isometric extension of the into mapping from the unit sphere S 1( E ) to S 1( l ∞(Γ))

This paper considers the isometric extension problem concerning the mapping from the unit sphere S ₁(E) of the normed space E into the unit sphere S ₁(l ^∞(Γ)). We find a condition under which an isometry from S ₁(E) into S ₁(l ^∞(Γ)) can be linearly and isometrically extended to the whole space. Since l ^∞(Γ) is universal with respect to isometry for normed spaces, isometric extension problems on a class of normed spaces are solved. More precisely, if E and F are two normed spaces, and if V ₀: S ₁(E) → S ₁(F) is a surjective isometry, where c ₀₀(Γ) ⊆ F ⊆ l ^∞(Γ), then V ₀ can be extended to be an isometric operator defined on the whole space. This work is supported by Natural Science Foundation of Guangdong Province, China (Grant No. 7300614)     

Classification of contractively complemented Hilbertian operator spaces

We construct some separable infinite-dimensional homogeneous Hilbertian operator spaces and , which generalize the row and column spaces R and C (the case m=0). We show that a separable infinite-dimensional Hilbertian JC^∗-triple is completely isometric to one of , , , or the space Φ spanned by creation operators on the full anti-symmetric Fock space. In fact, we show that (respectively ) is completely isometric to the space of creation (respectively annihilation) operators on the m (respectively m+1) anti-symmetric tensors of the Hilbert space. Together with the finite-dimensional case studied in [M. Neal, B. Russo, Representation of contractively complemented Hilbertian operator spaces on the Fock space, Proc. Amer. Math. Soc. 134 (2006) 475-485], this gives a full operator space classification of all rank-one JC^∗-triples in terms of creation and annihilation operator spaces.We use the above structural result for Hilbertian JC^∗-triples to show that all contractive projections on a C^∗-algebra A with infinite-dimensional Hilbertian range are “expansions” (which we define precisely) of normal contractive projections from A** onto a Hilbertian space which is completely isometric to R, C, R∩C, or Φ. This generalizes the well-known result, first proved for B(H) by Robertson in [A.G. Robertson, Injective matricial Hilbert spaces, Math. Proc. Cambridge Philos. Soc. 110 (1991) 183-190], that all Hilbertian operator spaces that are completely contractively complemented in a C^∗-algebra are completely isometric to R or C. We use the above representation on the Fock space to compute various completely bounded Banach-Mazur distances between these spaces, or Φ.