Predual Space Research Articles

Free Banach spaces and extension of Lipschitz maps

Let X be a metric space. We study the free Banach space B ( X ) over X , that is a predual space of the Banach space of all Lipschitz functions on X which preserve a marked point θ ∈ X . Some applications to the extension theory of Lipschitz and two-Lipschitz functions are obtained.

Predual spaces for Q spaces

To find the predual spaces Pα( R n ) of Qα( R n ) is an important motivation in the study of Q spaces. In this article, wavelet methods are used to solve this problem in a constructive way. First, an wavelet tent atomic characterization of Pα( R n ) is given, then its usual atomic characterization and Poisson extension characterization are given. Finally, the continuity on Pα of Calderón-Zygmund operators is studied, and the result can be also applied to give the Morrey characterization of Pα( R n ).

The duality between Morrey spaces and its pre-dual spaces characterized by heat kernel

Let L be an elliptic operator, we give arguments about the duality estimate between Morrey spaces and its pre-dual spaces characterized by the heat kernel associated to L that is given in [6],[8].

Construction and properties of quasi-linear functionals

Quasi-linear functionals are shown to be uniformly continuous and decomposable into a difference of two quasi-integrals. A predual space for the quasi-linear functionals inducing the weak-topology is given. General constructions of quasi-linear functionals by solid set-functions and q-functions are given.

Compact operators on the weighted Bergman space A1(ψ)

We show that a bounded linear operator $S$ on the weighted Bergman space $A^1(\psi )$ is compact and the predual space $A_0(\varphi )$ of $A^1(\psi )$ is invariant under $S^\ast $ if and only if $Sk_z \rightarrow 0$ as $z\rightarrow \partial D$, where $k_

Approximation of Holomorphic Mappings on Infinite Dimensional Spaces

In this article we examine necessary and sufficient conditions for the predual of the space of holomorphic mappings of bounded type, Gb(U), to have the approximation property and the compact approximation property and we consider when the predual of the space of holomorphic mappings, G(U), has the compact approximation property. We obtain also similar results for the preduals of spaces of m-homogeneous polynomials, Q(mE).

Preduals of Spaces of Vector-Valued Holomorphic Functions

For U a balanced open subset of a Frechet space E and F a dual-Banach space we introduce the topology τγ on the space \(H\left( {U,F} \right)\) of holomorphic functions from U into F. This topology allows us to construct a predual for \(\left( {H\left( {U,F} \right),\tau \delta } \right)\) which in turn allows us to investigate the topological structure of spaces of vector-valued holomorphic functions. In particular, we are able to give necessary and sufficient conditions for the equivalence and compatibility of various topologies on spaces of vector-valued holomorphic functions.

Chebyshev centres and centrable sets

In this paper we characterize real Banach spaces whose duals are isometric to L 1 (μ) spaces (the so-called L 1 -predual spaces) as those spaces in which every finite set is centrable. For a locally compact, non-compact set X and for an L 1 -predual E, we give a complete description of the extreme points and denting points of the dual unit ball of C 0 (X, E), equipped with the diameter norm.

Locally Asplund Preduals of Spaces of Holomorphic Functions

Defant [5] introduced the local Radon–Nikodým property for duals of locally convex spaces. This is a generalization of Asplund spaces as defined in Banach space theory. In this paper we generalise Dunford"s Theorem [7] to Banach spaces with Schauder decompositions and apply this result to spaces of holomorphic functions on balanced domains in a Banach space.

A dual characterisation of the Radon-Nikodym property

We prove that a Banach space X has the Radon-Nikodym property if, and only if, every weak*-lower semicontinuous convex continuous function f of X* is Gâteaux differentiable at some point of its domain with derivative in the predual space x.

The predual of an order-unit Banach space

We give a shortened and simplified proof of theknown theorem that a Banach space which is a predual ofan order-unit Banach space is a base-normedspace.

The Daugavet Equation for Operators on Function Spaces

We prove the norm identity ‖Id+T‖=1+‖T‖, which is known as the Daugavet equation, for weakly compact operatorsTon natural function spaces such as function algebras andL1-predual spaces, provided a non-discreteness assumption is met. We also considerc0-factorable operators and operators onCΛ-spaces.

Amenability of Banach algebras of compact operators

In this paper we study conditions on a Banach spaceX that ensure that the Banach algebraК(X) of compact operators is amenable. We give a symmetrized approximation property ofX which is proved to be such a condition. This property is satisfied by a wide range of Banach spaces including all the classical spaces. We then investigate which constructions of new Banach spaces from old ones preserve the property of carrying amenable algebras of compact operators. Roughly speaking, dual spaces, predual spaces and certain tensor products do inherit this property and direct sums do not. For direct sums this question is closely related to factorization of linear operators. In the final section we discuss some open questions, in particular, the converse problem of what properties ofX are implied by the amenability ofК(X).

Distinguished Preduals of Spaces of Holomorphic Functions

Distinguished Preduals of Spaces of Holomorphic Functions

Geometry of the Dual Ball of the Spin Factor

The main result of this paper is a geometric characterization of the unit ball of the dual of a complex spin factor. THEOREM. A strongly facially symmetric space of type I 2 in which every proper norm closed face in the unit ball is norm exposed, and which satisfies symmetry of transition probabilities, is linearly isometric to the dual of a complex spin factor. This result is an important step in the authors' program of showing that the class of all strongly facially symmetric spaces satisfying certain natural and physically significant axioms is equivalent isometricaIly to the class of all predual spaces JBW.-triples

Facial Structure in Operator Algebra Theory

We give a complete description of the closed faces in twelve kinds of convex sets that appear in operator algebra theory. These consist of positive parts of unit balls for C*-algebras and their dual spaces, and for von Neumann algebras and their pre-duals; of self-adjoint parts of unit balls in the same four classes and finally of general unit balls in the four classes. All these faces are shown to be semi-exposed and naturally paired with a polar face in the dual (or pre-dual) space

Preduals of Spaces of Holomorphic Functions

Preduals of Spaces of Holomorphic Functions

On the facial structure of the unit balls in a GL-space and its dual

In the early sixties Effros[9] and Prosser[14] studied, in independent work, the duality of the faces of the positive cones in a von Neumann algebra and its predual space. In an implicit way, this work was generalized to certain ordered Banach spaces in papers of Alfsen and Shultz [3] in the seventies, the duality being given in terms of faces of the base of the cone in a base norm space and the faces of the positive cone of the dual space. The present paper is concerned with the facial structure of the unit balls in an ordered Banach space and its dual as well as the duality that reigns between these structures. Specifically, the main results concern the sets of norm-exposed and norm-semi-exposed faces of the unit ball V1 in a GL-space or complete base norm space V and the sets of weak*-exposed and weak*-semi-exposed faces of the unit ball in its dual space V* which forms a unital GM-space or a complete order unit space.

INTEGRATION ON JORDAN ALGEBRAS

In this paper the author gives a nonassociative generalization of abstract integration on JBW-algebras - Jordan Banach algebras having a predual space. Using a faithful normal finite trace on a JBW-algebra , a topology of convergence in measure is introduced and the Jordan algebra of all measurable elements with respect to is constructed as the completion of in this topology. The spaces and are introduced for and it is shown that they can be considered as the spaces of all integrable and square-integrable elements, respectively, of . As in the case of von Neumann algebras it is proved that is isometrically isomorphic to the Banach space predual to .Bibliography: 32 titles.

SeparableL 1 preduals are quotients ofC(Δ)

It is proved that every separable predual space of anL 1 space is a quotient space ofC(Δ).