scholarly journals Amalgamation and Injectivity in Banach Lattices

2021 ◽  
Author(s):  
Antonio Avilés ◽  
Pedro Tradacete
Keyword(s):  
Banach Lattice ◽  
Amalgamation Property ◽  
Banach Lattices ◽  
Lattice Construction ◽  
Lattice Homomorphisms

Abstract We study distinguished objects in the category $\mathcal{B}\mathcal{L}$ of Banach lattices and lattice homomorphisms. The free Banach lattice construction introduced by de Pagter and Wickstead [ 8] generates push-outs, and combining this with an old result of Kellerer [ 17] on marginal measures, the amalgamation property of Banach lattices is established. This will be the key tool to prove that $L_1([0,1]^{\mathfrak{c}})$ is separably $\mathcal{B}\mathcal{L}$-injective, as well as to give more abstract examples of Banach lattices of universal disposition for separable sublattices. Finally, an analysis of the ideals on $C(\Delta ,L_1)$, which is a separably universal Banach lattice as shown by Leung et al. [ 21], allows us to conclude that separably $\mathcal{B}\mathcal{L}$-injective Banach lattices are necessarily non-separable.

scholarly journals Free and projective Banach lattices

2015 ◽  
Vol 145 (1) ◽  
pp. 105-143 ◽  
Author(s):  
Ben de Pagter ◽  
Anthony W. Wickstead
Keyword(s):  
Finite Number ◽  
Banach Lattice ◽  
Banach Lattices ◽  
Closed Ideal ◽  
Lattice Homomorphism ◽  
Linear Lattice ◽  
Fundamental Properties ◽  
Number Of Generators ◽  
Lattice Homomorphisms

We define and prove the existence of free Banach lattices in the category of Banach lattices and contractive lattice homomorphisms, and establish some of their fundamental properties. We give much more detailed results about their structure in the case when there are only a finite number of generators, and give several Banach lattice characterizations of the number of generators being, respectively, one, finite or countable. We define a Banach lattice P to be projective if, whenever X is a Banach lattice, J is a closed ideal in X, Q : X → X/J is the quotient map, T : P → X/J is a linear lattice homomorphism and ε > 0, there exists a linear lattice homomorphism : P → X such that T = Q º and ∥∥ ≤ (1 + ε)∥T∥. We establish the connection between projective Banach lattices and free Banach lattices, describe several families of Banach lattices that are projective and prove that some are not.

scholarly journals Banach spaces with property (w)

1993 ◽  
Vol 35 (2) ◽  
pp. 207-217 ◽  
Author(s):  
Denny H. Leung
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Banach Lattice ◽  
Banach Lattices ◽  
Weakly Compact ◽  
Type Spaces

A Banach space E is said to have Property (w) if every operator from E into E' is weakly compact. This property was introduced by E. and P. Saab in [9]. They observe that for Banach lattices, Property (w) is equivalent to Property (V*), which in turn is equivalent to the Banach lattice having a weakly sequentially complete dual. Thus the following question was raised in [9].Does every Banach space with Property (w) have a weakly sequentially complete dual, or even Property (V*)?In this paper, we give two examples, both of which answer the question in the negative. Both examples are James type spaces considered in [1]. They both possess properties stronger than Property (w). The first example has the property that every operator from the space into the dual is compact. In the second example, both the space and its dual have Property (w). In the last section we establish some partial results concerning the problem (also raised in [9]) of whether (w) passes from a Banach space E to C(K, E).

scholarly journals Weak and unbounded order convergence in Banach lattices

1977 ◽  
Vol 24 (3) ◽  
pp. 312-319 ◽  
Author(s):  
A. W. Wickstead
Keyword(s):  
Weak Convergence ◽  
Banach Lattice ◽  
Vector Lattice ◽  
Banach Lattices ◽  
Order Convergence ◽  
The Relationship

AbstractA net (xy) in a vector lattice is unbounded order convergent (uo-convergent) to 0 if u ∧ |xv| order converges to 0 for all u ≧ 0. We consider, in a Banach lattice, the relationship between weak and uo-convergence. We characterise those Banach lattices in which weak convergence implies uo-convergence and those in which uo-convergence of a bounded net implies weak convergence. Finally we combine the results to characterise those Banach lattices in which weak and uo-convergence coincide for bounded nets.

scholarly journals Cone characterization of reflexive Banach lattices

1995 ◽  
Vol 37 (1) ◽  
pp. 65-67 ◽  
Author(s):  
Ioannis A. Polyrakis
Keyword(s):  
Banach Lattice ◽  
Normal Cone ◽  
Banach Lattices

AbstractWe prove that a Banach lattice X is reflexive if and only if X+ does not contain a closed normal cone with an unbounded closed dentable base.

scholarly journals The Köthe Dual of an Abstract Banach Lattice

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
E. Jiménez Fernández ◽  
M. A. Juan ◽  
E. A. Sánchez-Pérez
Keyword(s):  
Integral Representation ◽  
Banach Lattice ◽  
Broad Class ◽  
Banach Lattices ◽  
Vector Measures ◽  
Integrable Functions ◽  
Suitable Definition ◽  
Integral Structures ◽  
Definition Of ◽  
Köthe Dual

We analyze a suitable definition of Köthe dual for spaces of integrable functions with respect to vector measures defined onδ-rings. This family represents a broad class of Banach lattices, and nowadays it seems to be the biggest class of spaces supported by integral structures, that is, the largest class in which an integral representation of some elements of the dual makes sense. In order to check the appropriateness of our definition, we analyze how far the coincidence of the Köthe dual with the topological dual is preserved.

scholarly journals Operators on Injective Banach Lattices

2016 ◽  
Author(s):  
A.G. Kusraev
Keyword(s):  
Banach Lattice ◽  
Linear Operators ◽  
Banach Lattices ◽  
Transfer Principle ◽  
Bounded Linear Operators ◽  
Bounded Linear

The paper deals with some properties of bounded linear operators on injective Banach lattice using a Boolean-valued transfer principle from AL-spaces to injectives stated in author's previous work.

The order completeness of some spaces of vector-valued functions

1974 ◽  
Vol 11 (1) ◽  
pp. 57-61 ◽  
Author(s):  
Donald I. Cartwright
Keyword(s):  
Banach Lattice ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Banach Lattices ◽  
Nuclear Operators ◽  
Order Completeness ◽  
Necessary And Sufficient ◽  
Vector Valued Functions ◽  
Order Intervals ◽  
Vector Valued

Let E be a Banach lattice. Necessary and sufficient conditions are given for the order completeness of the Banach lattices C(X, E) and L1(μ, E) in terms of the compactness of the order intervals in E. The results have interpretations in terms of spaces of compact and nuclear operators.

scholarly journals Geometric Characterization of Injective Banach Lattices

Mathematics ◽  
2021 ◽  
Vol 9 (3) ◽  
pp. 250
Author(s):  
Anatoly Kusraev ◽  
Semën Kutateladze
Keyword(s):  
Banach Space ◽  
Unit Ball ◽  
Banach Spaces ◽  
Set Theory ◽  
Banach Lattice ◽  
Dual Space ◽  
Banach Lattices ◽  
Transfer Principle ◽  
Ordered Banach Spaces

This is a continuation of the authors’ previous study of the geometric characterizations of the preduals of injective Banach lattices. We seek the properties of the unit ball of a Banach space which make the space isometric or isomorphic to an injective Banach lattice. The study bases on the Boolean valued transfer principle for injective Banach lattices. The latter states that each such lattice serves as an interpretation of an AL-space in an appropriate Boolean valued model of set theory. External identification of the internal Boolean valued properties of the corresponding AL-spaces yields a characterization of injective Banach lattices among Banach spaces and ordered Banach spaces. We also describe the structure of the dual space and present some dual characterization of injective Banach lattices.

scholarly journals An absolute continuity for positive operators on Banach lattices

1991 ◽  
Vol 14 (1) ◽  
pp. 83-92
Author(s):  
W. Feldman ◽  
C. Piston ◽  
Calvin E. Piston
Keyword(s):  
Banach Lattice ◽  
Absolute Continuity ◽  
Compact Operators ◽  
Positive Operators ◽  
Banach Lattices ◽  
Absolutely Continuous ◽  
Special Cases

For positive operators on a Banach lattice, absolute contnuity conditions are considered. An operator absolutely continuous with respect toTis compared to sums of compositions ofTtogether with orthomorphisms or in special cases projections. Consequences For compact operators on functions spacesC(X)are considered.

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