A finite-dimensional normed space with two non-equivalent symmetric bases

A sequence of finite-dimensional normed spaces is constructed, each with two symmetric bases, such that the sequence of equivalence constants between these bases is unbounded. An essential tool in the proof is the edge-isoperimetric inequality in the discrete cube.     

Embedding Banach spaces with unconditional bases into spaces with symmetric bases

Every reflexive Banach space with unconditional basis is isomorphic to a complemented subspace of a reflexive Banach space with symmetric basis.     

A Banach space block finitely universal for monotone bases

A reflexive Banach space with a basis is constructed having the property that every monotone basis is block finitely representable in each block basis of .

     

On the uniqueness of symmetric bases in finite dimensional Banach spaces

Suppose{e _i} _i=1 ⁿ and{f _i} _i=1 ⁿ are symmetric bases of the Banach spacesE andF. Letd(E,F)≦C andd(E,l _n ² )≧n' for somer>0. Then there is a constantC _r=C_r(C)>0 such that for alla _i∈Ri=1,...,n $$C_r^{ - 1} \left\| {\sum\limits_{i = 1}^n {a_i e_i } } \right\| \leqq \left\| {\sum\limits_{i = 1}^n {a_i f_i } } \right\| \leqq C_r \left\| {\sum\limits_{i = 1}^n {a_i e_i } } \right\|$$ We also give a partial uniqueness of unconditional bases under more restrictive conditions.     

A Banach space with few operators

Assuming the axiom (of set theory)V=L (explained below), we construct a Banach space with density character ℵ₁ such that every (linear bounded) operatorT fromB toB has the forma I+T ₁, whereI is the identity, andT ₁ has a separable range. The axiomV=L means that all the sets in the universe are in the classL of sets constructible from ordinals; in a sense this is the minimal universe. In fact, we make use of just one consequence of this axiom, ℵ₁ proved by Jensen, which is widely used by mathematical logicians.     

A remark on symmetric bases

Every separable Banach space with an unconditional basis is isomorphic to a complemented subspace of a space with a symmetric basis.     

On Banach spaces with unconditional bases

LetX be a Banach space with a sequence of linear, bounded finite rank operatorsR _n:X→X such thatR _nR_m=R_min(_n,m) ifn≠m and lim _n→∞ R _n x=x for allx∈X. We prove that, ifR _n−R_n −1 factors uniformly through somel _p and satisfies a certain additional symmetry condition, thenX has an unconditional basis. As an application, we study conditions on Λ ⊂ ℤ such thatL _Λ=closed span , where , has an unconditional basis. Examples include the Hardy space .     

Polynomials on Banach spaces with unconditional bases

We study the classes of homogeneous polynomials on a Banach space with unconditional Schauder basis that have unconditionally convergent monomial expansions relative to this basis. We extend some results of Matos, and we show that the homogeneous polynomials with unconditionally convergent expansions coincide with the polynomials that are regular with respect to the Banach lattices structure of the domain.

     

Banach space properties forcing a reflexive, amenable Banach algebra to be trivial

It is an open problem whether an infinite-dimensional amenable Banach algebra exists whose underlying Banach space is reflexive. We give sufficient conditions for a reflexive, amenable Banach algebra to be finite-dimensional (and thus a finite direct sum of full matrix algebras). If \frak A {\frak A} is a reflexive, amenable Banach algebra such that for each maximal left ideal L of \frak A {\frak A} (i) the quotient \frak A/L {\frak A}/L has the approximation property and (ii) the canonical map from \frak A \check? L^{^} {\frak A} \check{\otimes} L^\perp to (\frak A / L) \check? L^{^} ({\frak A} / L) \check{\otimes} L^\perp is open, then \frak A {\frak A} is finite-dimensional. As an application, we show that, if \frak A {\frak A} is an amenable Banach algebra whose underlying Banach space is an \scr L^p {\scr L}^p -space with p ? (1,￥) p\in (1,\infty) such that for each maximal left ideal L the quotient \frak A/L {\frak A}/L has the approximation property, then \frak A {\frak A} is finite-dimensional.     

The Boden-Hu conjecture holds precisely up to rank 8

We consider moduli schemes of vector bundles over a smooth projective curve endowed with parabolic structures over a marked point. Boden and Hu observed that a slight variation of the weights leads to a desingularisation of the moduli scheme, and they conjectured that one can always obtain a small resolution this way. The present text proves this conjecture in some cases (including all bundles of rank up to eight) and gives counterexamples in all other cases (in particular in every rank beyond eight). The main tool is a generalisation of Ext-groups involving more than two quasiparabolic bundles.Mathematics Subject Classification (2000): 14H60, 14D20     

A projectively symmetric Finsler space

Symmetric (Riemannian) spaces were introduced and developed by Cartan [1, 2] which led to the discovery of projectively symmetric (Riemannian) spaces by Soós [9]. Recently the theory of symmetric spaces has been extended to Finsler geometry by the present author [5]. The current paper deals with that class of Finsler spaces throughout which their projective curvature tensors possess vanishing covariant derivatives. Following Soós' terminology such spaces are calledprojectively symmetric Finsler spaces. Examples, conditions for a symmetric Finsler space to be projectively symmetric, reduction of various identities, and the discussion of a decomposed projectively symmetric Finsler space form the skeleton of the paper.     

Primitive symmetric designs with up to 2500 points

By this article we conclude the construction of all primitive ( v, k,λ ) symmetric designs with v < 2500 , up to a few unsolved cases. Complementary to the designs with prime power number of points published previously, here we give 55 primitive symmetric designs with v ≠ p ^m , p prime and m positive integer, together with the analysis of their full automorphism groups. The research involves programming and wide‐range computations. We make use of the software package GAP and the library of primitive groups which it contains. © 2011 Wiley Periodicals, Inc. J Combin Designs 19:463‐474, 2011     

A non-reflexive Banach space with all contractions mean ergodic

We construct on any quasi-reflexive of order 1 separable real Banach space an equivalent norm, such that all contractions on the space and all contractions on its dual are mean ergodic, thus answering negatively a question of Louis Sucheston.     

A new Banach space with Valdivia dual unit ball

We give an example of a Banach space which admits no projectional resolution of the identity but whose dual unit ball in weak* topology is a Valdivia compact. This answers a question asked by M. Fabian, G. Godefroy and V. Zizler. Partially supported by Research grants GAUK 277/2001, GAUK 160/1999 and MSM 113200007.