On the Converse of Aliprantis and Burkinshaw's Theorem

We prove a complete converse of Aliprantis and Burkinshaw’s Theorem [2]. Also we obtain a generalization of Wickstead’s Theorem [9] about this converse, and we give some interesting consequences. revised 4 April, 18 October, and 26 December 2005     

Representations of Archimedean Riesz spaces by continuous functions

A brief survey of representations of Archimedean Riesz spaces in spaces of continuous extended real-valued functions, together with an example of their use in proving results about Riesz spaces     

Pointfree prime representation of real Riesz maps

In classical topology it is proved, nonconstructively, that for a topological space X, every bounded Riesz map ϕ in C(X) is of the form for a point x ∈ X. In this paper our main objective is to give the pointfree version of this result. In fact, we constructively represent each real Riesz map on a compact frame M by prime elements. Received March 23, 2004; accepted in final form May 14, 2005.     

Denting points in K?the-Bochner spaces

A characterization of denting points in the unit ball of Köthe-Bochner spaceE(X) is given. This characterization is a generalization of some analogous theorems for Bochner and Musielak Orlicz spaces and it is compared to similar known results concerning strongly extreme points inE(X).Scholar of Ministere de la Recherche et de la Technologie of France, while on leave from Technical University at Pozna, Poland.     

Krivine's theorem and the indices of a Banach lattice

In this paper we shall present an exposition of a fundamental result due to J.L. Krivine about the local structure of a Banach lattice. In [3] Krivine proved that _p (1p) is finitely lattice representable in any infinite dimensional Banach lattice. At the end of the introduction of [3] it is then stated that a value of p for which this holds is given by, what we will call below, the upper index of the Banach lattice. He states that this follows from the methods of his paper and of the paper [5] of Maurey and Pisier. One can ask whether the theorem also holds for p equal to the lower index of the Banach lattice. At first glance this is not obvious from [3], since many theorems in [3] have as a hypothesis that the upper index of the Banach lattice is finite. This can e.g. also be seen from the book [6] of H.U. Schwarz, where only the result for the upper index is stated, while both indices are discussed. One purpose of this paper is clarify this point and to present an exposition of all the ingredients of a proof of Krivine's theorem for both the upper and lower index of a Banach lattice. We first gather some definitions and state some properties of the indices of a Banach lattice. For a discussion of these indices we refer to the book of Zaanen[7].     

On the Levi and Lebesgue properties

The paper is devoted to a question if the Levi property is preserved by direct sums and quotients. The three-space problem for the Levi and Lebesgue properties in topological Riesz spaces is also investigated.     

Some fundamental properties for duals of Orlicz spaces

In this paper some basic properties of Orlicz spaces are extended to their dual spaces, and finally, criteria for extreme points in these spaces are given.     

Some properties of the injective tensor product of L[0,1] and a Banach space

Let X be a (real or complex) Banach space and 1<p,p′<∞ such that 1/p+1/p′=1. Then , the injective tensor product of L^p[0,1] and X, has the Radon-Nikodym property (resp. the analytic Radon-Nikodym property, the near Radon-Nikodym property, contains no copy of c₀, is weakly sequentially complete) if and only if X has the same property and each continuous linear operator from L^p′[0,1] to X is compact.     

EQUIMEASURABLE SETS IN RIESZ SPACES

Abstract

We discuss the notion of equimeasurability in the general setting of Riesz spaces and obtain a characterization for (Carleman) abstract kernel operators in terms of equimea=surable sets.     

Characterization of locally convex solid topologies of lebesgue on the vector latticeC(X)

We give some sufficient and necessary conditions such that the vector lattice of continuous functions on a compact topological space, not necessary metrizable, admits a locally convex and solid topology which is Lebesgue.

     

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     

The duality problem for the Class of b-weakly compact operators

We establish necessary and sufficient conditions under which b-weakly compact operators between Banach lattices have b-weakly compact adjoints or operators with b-weakly compact adjoints are themselves b-weakly compact. Also, we give some consequences.      

Operator-valued Fourier multiplier theorems on Lp(X) and geometry of Banach spaces

This paper gives the optimal order l of smoothness in the Mihlin and Hörmander conditions for operator-valued Fourier multiplier theorems. This optimal order l is determined by the geometry of the underlying Banach spaces (e.g. Fourier type). This requires a new approach to such multiplier theorems, which in turn leads to rather weak assumptions formulated in terms of Besov norms.     

Rearrangement Invariant Continuous Linear Functionals on Weak <Emphasis Type="Italic">L</Emphasis><Superscript>1</Superscript>

We show that in the dual of Weak L¹ the subspace of all rearrangement invariant continuous linear functionals is lattice isometric to a space L¹(μ) and is the linear hull of the maximal elements of the dual unit ball. We also show that the dual of Weak L¹ contains a norm closed weak* dense ideal which is lattice isometric to an ℓ¹-sum of spaces of type C(K). Helmut H. Schaefer in memoriam     

Some remarks on disjointness preserving operators

In this note we present a simple proof of the following results: if T: E E is a lattice homomorphism on a Banach lattice E, then: i) (T)={1} implies T=I; and ii) r(T–I)<1 implies TZ(E), the center of E.     

Limit laws in vector lattices

Abstract

We prove a law of large numbers and a central limit theorem with respect to the order convergence topology in vector lattices.     

An isomorphic characterization of L-spaces

We show that a sequentially (τ)-complete topological vector lattice X_τ is isomorphic to some L¹(μ), if and only if the positive cone can be written as X₊ = ₊B for some convex, (τ)-bounded, and (τ)-closed set B X₊ {0}. The same result holds under weaker hypotheses, namely the Riesz decomposition property for X (not assumed to be a vector lattice) and the monotonic σ-completeness (monotonic Cauchy sequences converge). The isometric part of the main result implies the well-known representation theorem of Kakutani for (AL)-spaces. As an application we show that on a normed space Y of infinite dimension, the “ball-generated” ordering induced by the cone Y₊ = ₊ (for u >) cannot have the Riesz decomposition property. A second application deals with a pointwise ordering on a space of multivariate polynomials.