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 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 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 Solution of the Word Problem for Certain Types of Groups I

1956 ◽  
Vol 3 (1) ◽  
pp. 45-54 ◽  
Author(s):  
J. L. Britton
Keyword(s):  
Finite Number ◽  
Word Problem ◽  
Infinite Number ◽  
Finite System ◽  
Defining Relations ◽  
Number Of Generators ◽  
Infinite Word ◽  
Formed Part ◽  
Free Product Of Groups ◽  
Product Of Groups

The main result of this series of papers is a theorem on the free product of groups (Theorem 1) which formed part of a doctoral thesis. This theorem has an immediate application to the word problem (Theorem 2). Usually the word problem refers to a finite system of generators and a finite number of defining relations, but in this context it is more natural to allow an infinite number of generators and defining relations. This (infinite) word problem is not solvable in general (Example 2).

scholarly journals WHEN LATTICE HOMOMORPHISMS OF ARCHIMEDEAN VECTOR LATTICES ARE RIESZ HOMOMORPHISMS

2009 ◽  
Vol 87 (2) ◽  
pp. 263-273 ◽  
Author(s):  
MOHAMED ALI TOUMI
Keyword(s):  
London Math ◽  
Lattice Homomorphism ◽  
Vector Lattices ◽  
Lattice Homomorphisms ◽  
Orthogonal Systems

AbstractLet A, B be Archimedean vector lattices and let (ui)i∈I, (vi)i∈I be maximal orthogonal systems of A and B, respectively. In this paper, we prove that if T is a lattice homomorphism from A into B such that $T\left ( \lambda u_{i}\right ) =\lambda v_{i}$ for each λ∈ℝ+ and i∈I, then T is linear. This generalizes earlier results of Ercan and Wickstead (Math. Nachr279 (9–10) (2006), 1024–1027), Lochan and Strauss (J. London Math. Soc. (2) 25 (1982), 379–384), Mena and Roth (Proc. Amer. Math. Soc.71 (1978), 11–12) and Thanh (Ann. Univ. Sci. Budapest. Eotvos Sect. Math.34 (1992), 167–171).

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 On Semiprime Finitely Generated Right Alternative Weakly M-Rings

2013 ◽  
Vol 6 ◽  
pp. 9-12
Author(s):  
K. Jayalakshmi ◽  
C. Manjula
Keyword(s):  
Finite Number ◽  
Minimum Condition ◽  
Finitely Generated ◽  
Number Of Generators ◽  
Additional Assumptions

Right alternative rings, satisfying the weakly M-ring identity (w,xy,z) = x(w,y,z) are studied. Any one of the following additional assumptions imply associativity: semi prime and a finite number of generators: prime with char. ≠ 2 and minimum condition on either right ideals or on trivial left ideals, or simple and char. ≠ 2.

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.

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