On Subspaces and Quotients of Banach Spaces C(K, X)

 We study the relation of to the subspaces and quotients of Banach spaces of continuous vector-valued functions , where K is an arbitrary dispersed compact set. More precisely, we prove that every infinite dimensional closed subspace of totally incomparable to X contains a copy of complemented in . This is a natural continuation of results of Cembranos-Freniche and Lotz-Peck-Porta. We also improve our result when K is homeomorphic to an interval of ordinals. Next we show that complemented subspaces (resp., quotients) of which contain no copy of are isomorphic to complemented subspaces (resp., quotients) of some finite sum of X. As a consequence, we prove that every infinite dimensional quotient of which is quotient incomparable to X, contains a complemented copy of . Finally we present some more geometric properties of spaces. Received 8 November 2000; in revised form 7 December 2001     

Every non-normable non-archimedean Köthe space has a quotient without the bounded approximation property

It is proved that any non-archimedean non-normable Fréchet space with a Schauder basis and a continuous norm has a quotient without the bounded approximation property. It follows that any infinite-dimensional non-archimedean Fréchet space, which is not isomorphic to any of the following spaces: , has a quotient without a Schauder basis. Clearly, any quotient of c₀ and has a Schauder basis. It is shown a similar result for and     

SOME ASPECTS OF THE LIFTING PROPERTY OF THE SPACE £∞ (μ, X)

Abstract

Using a lifting of £_∞ (μ, X) ([5],[6]), we construct a lifting ρ x of the seminormed vector space £_∞ (μ, X) of measurable, essentially bounded X-valued functions. We show that in a certain sense such a lifting always exists. If μ is Lebesgue measure on (0, 1) we show that ρ x exists as map from £_∞ ((O, 1), X) → £_∞,((0, l), X) if and only if X is reflexive. In general the lifted function takes its values in X **. Therefore we investigate the question, when f ε £_∞ (μ, X) is strictly liftable in the sense that the lifted function is a map with values even in X.

As an application we introduce the space £_∞ ^strong (μ, L (X, Y**)), a subspace of the space of strongly measurable, essentially bounded L (X, Y, **)-valued functions, and the associated quotient space £_∞ ^strong (μ, L (X,Y**)). We show that this space is a Banach space because there is a kind of a Dunford-Pettis Theorem for a subspace of L (X, £_∞(μ Y**)). Finally we investigate the measurability property of functions in £_∞(μ Y**)) und see that there exists a connection to the Radon-Nikodym property of the space L (X, Y).     

Connected components in the space of composition operators on<Emphasis Type="Italic">H∞</Emphasis> functions of many variables

LetE be a complex Banach space with open unit ballB _e. The structure of the space of composition operators on the Banach algebra H∞, of bounded analytic functions onB _e with the uniform topology, is studied. We prove that the composition operators arising from mappings whose range lies strictly insideB _e form a path connected component. WhenE is a Hilbert space or aC _o(X)- space, the path connected components are shown to be the open balls of radius 2. The research of this author was supported by grant number SAB1999-0214 from the Ministerio de Educación, Cultura y Deporte during his stay at the Universidad de Valencia. The research of this author was partially supported DGES(Spain) pr. 96-0758. The research of this author was partially supported by Magnus Ehrnrooths stiftelse.     

The weak metric approximation property

We introduce and investigate the weak metric approximation property of Banach spaces which is strictly stronger than the approximation property and at least formally weaker than the metric approximation property. Among others, we show that if a Banach space has the approximation property and is 1-complemented in its bidual, then it has the weak metric approximation property. We also study the lifting of the weak metric approximation property from Banach spaces to their dual spaces. This enables us, in particular, to show that the subspace of c₀, constructed by Johnson and Schechtman, does not have the weak metric approximation property. The research of the second-named author was partially supported by Estonian Science Foundation Grant 5704 and the Norwegian Academy of Science and Letters.     

A non-trivial Fréchet quotient of the space of real analytic functions

We prove that the space of real analytic functions ${\cal A}(\Omega)$ on an arbitrary open set $\Omega \subseteq \mathbb{R}^d$ has a Fréchet infinite dimensional quotient space with a continuous norm. Received: 4 February 2002     

Integrality of L2L^2-Betti numbers

The Atiyah conjecture predicts that the -Betti numbers of a finite CW-complex with torsion-free fundamental group are integers. We establish the Atiyah conjecture, under the condition that it holds for G and that is a normal subgroup, for amalgamated free products . Here F is a free group and is an arbitrary semi-direct product. This includes free products G*F and semi-direct products . We also show that the Atiyah conjecture holds (with an additional technical condition) for direct and inverse limits of groups for which it is true. As a corollary it holds for positive 1-relator groups with torsion free abelianization. Putting everything together we establish a new (bigger) class of groups for which the Atiyah conjecture holds, which contains all free groups and in particular is closed under taking subgroups, direct sums, free products, extensions with torsion-free elementary amenable quotient or with free quotient, and under certain direct and inverse limits. Received: 22 August 1998/ Revised: 10 Jannary 2000 / Published online: 28 June 2000     

Weakly compact operators onH ∞

We prove that a weakly compact operator fromH or any of its even duals into an arbitrary Banach space is uniformly convexifying. By using this, we establish three dicothomies: (1) every operator defined onH or any of its even duals either fixes a copy ofl or factors through a Banach space having the Banach-Saks property; (2) every quotient ofH or any of its even duals either contains a copy ofl or is super-reflexive; (3) every subspace ofL ¹/H ₀ ¹ or any of its even duals either contains a complemented copy ofl ¹ or is super-reflexive.     

The density condition in subspaces and quotients of Frechet spaces

The aim of this note is to investigate the topological structure (in particular the density condition) of subspaces and separated quotients of Fréchet spaces. Our main result is the following one: LetE be a Fréchet space which is neither Montel nor isomorphic to a closed subspace ofX × , withX a Banach space, also assume thatE can be written asFG withF andG infinite dimensional closed subspaces ofE not isomorphic to , thenE contains a closed subspace with basis and not satisfying the density condition. We also prove that every Köthe echelon space of orderp, 1<p<, which is not quasinormable has a separated quotient with basis which does not satisfy the density condition.     

Essentially normal Hilbert modules and <Emphasis Type="Italic">K</Emphasis>-homology

This paper mainly concerns the essential normality of graded submodules. Essentially all of the basic Hilbert modules that have received attention over the years are p-essentially normal—including the d-shift Hilbert module, the Hardy and Bergman modules of the unit ball. Arveson conjectured graded submodules over the unit ball inherit this property and provided motivations to seek an affirmative answer. Some positive results have been obtained by Arveson and Douglas. However, the problem has been resistant. In dimensions d = 2, 3, this paper shows that the Arveson’s conjecture is true. In any dimension, the paper also gives an affirmative answer in the case of the graded principal submodule. Finally, the paper is associated with K-homology invariants arising from graded quotient modules, by which geometry of the quotient modules and geometry of algebraic varieties are connected. In dimensions d = 2, 3, it is shown that K-homology invariants determined by graded quotients are nontrivial. The paper also establishes results on p-smoothness of K-homology elements, and gives an explicit expression for K-homology invariant in dimension d = 2.     

Isometries between the unit spheres of lβγ -sum of strictly convex normed spaces

Abstract

In this paper, the author introduces a new F-space the l ^βγ-sum of strictly convex normed spaces, and obtains the representation of onto isometry between the unit spheres, then concludes that such mappings can be extended to the whole space as real linear isometries.     

Commutative Sequences of Integrable Functions and Best Approximation With Respect to the Weighted Vector Measure Distance

Let λ be a countably additive vector measure with values in a separable real Hilbert space H. We define and study a pseudo metric on a Banach lattice of integrable functions related to λ that we call a λ-weighted distance. We compute the best approximation with respect to this distance to elements of the function space by the use of sequences with special geometric properties. The requirements on the sequence of functions are given in terms of a commutation relation between these functions that involves integration with respect to λ. We also compare the approximation that is obtained in this way with the corresponding projection on a particular Hilbert space.     

Locally convex quotient lattice cones

Walter Roth has investigated certain equivalence relations on locally convex cones in [W. Roth, Locally convex quotient cones, J. Convex Anal. 18, No. 4, 903–913 (2011)] which give rise to the definition of a locally convex quotient cone. In this paper, we investigate some special equivalence relations on a locally convex lattice cone by which the locally convex quotient cone becomes a lattice. In the case of a locally convex solid Riesz space, this reduces to the known concept of locally convex solid quotient Riesz space. We prove that the strict inductive limit of locally convex lattice cones is a locally convex lattice cone. We also study the concept of locally convex complete quotient lattice cones.     

Banach-stone theorems and separating maps

LetC(X,E) andC(Y,F) denote the spaces of continuous functions on the Tihonov spacesX andY, taking values in the Banach spacesE andF, respectively. A linear mapH:C(X,E)→C(Y,F) isseparating iff(x)g(x)=0 for allx inX impliesHf(y)Hg(y)=0 for ally inY. Some automatic continuity properties and Banach-Stone type theorems (i.e., asserting that isometries must be of a certain form) for separating mapsH between spaces of real- and complex-valued functions have already been developed. The extension of such results to spaces of vector-valued functions is the general subject of this paper. We prove in Theorem 4.1, for example, for compactX andY, that a linear isometryH betweenC(X,E) andC(Y,F) is a “Banach-Stone” map if and only ifH is “biseparating (i.e,H andH ⁻¹ are separating). The Banach-Stone theorems of Jerison and Lau for vector-valued functions are then deduced in Corollaries 4.3 and 4.4 for the cases whenE andF or their topological duals, respectively, are strictly convex. Research supported by the Fundació Caixa Castelló, MI/25.043/92     

Dimensions of very ample pluricanonical subsystems for compact quotients of bounded symmetric domains

A compact complex manifold X obtained by taking quotient of a bounded symmetric domain has an ample canonical line bundle. We prove that the dimension of very ample pluricanonical subsystem is strictly bigger than 2n, where n is the dimension of X. Received: 23 June 2000 / Revised version: 30 March 2001     

On Peano's theorem in Banach spaces

We show that if X is an infinite-dimensional separable Banach space (or more generally a Banach space with an infinite-dimensional separable quotient) then there is a continuous mapping f:X→X such that the autonomous differential equation x^′=f(x) has no solution at any point.     

Locally trivial 𝔾a−actions on ℂ5 with singular algebraic quotients

Simple examples are given of proper algebraic actions of the additive group of complex numbers on ?⁵ whose geometric quotients are, respectively, a?ne, strictly quasia?ne, and algebraic spaces which are not schemes. Moreover, a Zariski locally trivial action is given whose ring of invariant regular functions defines a singular factorial a?ne fourfold embedded in ?¹². The geometric quotient for the action embeds as a strictly quasia?ne variety in the smooth locus of the algebraic quotient with complement isomorphic to the normal a?ne surface with the A₂?singularity at the origin.     

Quotient rings of algebras of functions and operators

Abstract. Various quotient rings of rings B of Banach algebra A-valued continuous functions on a completely regular Hausdorff Space X are constructed in terms of continuous functions defined on dense open subsets of X taking values in the maximal quotient ring of the Banach algebra A. This extends the results proved by N. J. Fine, L., Gillman and J. Lambek (1965) for the case of A, the field of real numbers. The pattern is similar and utilizes as well as generalises the results proved for algebras of multipliers of B by C. A. Akemann, G. K. Pedersen and J. Tomiyama (1973). The techniques combine those from algebra, analysis and topology. The details of the cases when A is the normed division algebra of real quaternions or the operator algebra B(H) of a Hilbert space H are given to illustrate our results. Received February 1, 1999; in final form June 18, 1999 / Published online May 8, 2000     

A class of Banach spaces with few non-strictly singular operators

We construct a family (X_γ) of reflexive Banach spaces with long (countable as well as uncountable) transfinite bases but with no unconditional basic sequences. The method we introduce to achieve this allows us to considerably control the structure of subspaces of the resulting spaces as well as to precisely describe the corresponding spaces on non-strictly singular operators. For example, for every pair of countable ordinals γ,β, we are able to decompose every bounded linear operator from X_γ to X_β as the sum of a diagonal operator and an strictly singular operator. We also show that every finite-dimensional subspace of any member X_γ of our class can be moved by and (4+?)-isomorphism to essentially any region of any other member X_δ or our class. Finally, we find subspaces X of X_γ such that the operator space L(X,X_γ) is quite rich but any bounded operator T from X into X is a strictly singular pertubation of a scalar multiple of the identity.     

Unconditional families in Banach spaces

It is shown that for every separable Banach space X with non-separable dual, the space contains an unconditional family of size . The proof is based on Ramsey Theory for trees and finite products of perfect sets of reals. Among its consequences, it is proved that every dual Banach space has a separable quotient.