A CHARACTERIZATION OF COMPLEX LATTICE HOMOMORPHISMS ON BANACH LATTICE ALGEBRAS

1995 ◽  
Vol 18 (1-3) ◽  
pp. 131-140 ◽  
Author(s):  
C. B. Huijsmans
Keyword(s):  
Banach Lattice ◽  
Complex Lattice ◽  
Lattice Homomorphisms

A characterization of complex lattice homomorphisms

Positivity ◽  
2008 ◽  
Vol 13 (1) ◽  
pp. 31-37 ◽  
Author(s):  
Youssef Azouzi ◽  
Karim Boulabiar
Keyword(s):  
Complex Lattice ◽  
Lattice Homomorphisms

scholarly journals Strongly exposed points and a characterization of l1 (Γ) by the Schur property

1987 ◽  
Vol 30 (3) ◽  
pp. 397-400 ◽  
Author(s):  
Ioannis A. Polyrakis
Keyword(s):  
Banach Spaces ◽  
Banach Lattice ◽  
Positive Cone ◽  
Schur Property ◽  
Positive Elements ◽  
Strongly Exposed Points ◽  
Ordered Banach Spaces ◽  
Convex Subsets ◽  
Quasi Interior

In this paper we study the existence of strongly exposed points in unbounded closed and convex subsets of the positive cone of ordered Banach spaces and we prove the following characterization for the space l1(Γ): A Banach lattice X is order-isomorphic to l1(Γ) iff X has the Schur property and X* has quasi-interior positive elements.

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 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 Bounded homomorphisms and finitely generated fiber products of lattices

2020 ◽  
Vol 30 (04) ◽  
pp. 693-710
Author(s):  
William DeMeo ◽  
Peter Mayr ◽  
Nik Ruškuc
Keyword(s):  
Word Problem ◽  
Time Algorithm ◽  
Finitely Generated ◽  
Bounded Lattice ◽  
Exponential Time ◽  
Unpublished Result ◽  
Lattice Homomorphisms ◽  
Finitely Presented ◽  
Exponential Time Algorithm

We investigate when fiber products of lattices are finitely generated and obtain a new characterization of bounded lattice homomorphisms onto lattices satisfying a property we call Dean’s condition (D) which arises from Dean’s solution to the word problem for finitely presented lattices. In particular, all finitely presented lattices and those satisfying Whitman’s condition satisfy (D). For lattice epimorphisms [Formula: see text], [Formula: see text], where [Formula: see text], [Formula: see text] are finitely generated and [Formula: see text] satisfies (D), we show the following: If [Formula: see text] and [Formula: see text] are bounded, then their fiber product (pullback) [Formula: see text] is finitely generated. While the converse is not true in general, it does hold when [Formula: see text] and [Formula: see text] are free. As a consequence, we obtain an (exponential time) algorithm to decide boundedness for finitely presented lattices and their finitely generated sublattices satisfying (D). This generalizes an unpublished result of Freese and Nation.

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 An extension of the Kakutani–Bohnenblust characterization of $$L^p$$-spaces to $$p\in (0,\infty )$$

Positivity ◽  
2020 ◽  
Vol 24 (5) ◽  
pp. 1461-1477
Author(s):  
S. Teerenstra ◽  
A. C. M. van Rooij
Keyword(s):  
Banach Lattice ◽  
Lattice Equation

Abstract For $$p\in [1,\infty )$$ p ∈ [ 1 , ∞ ) , S. Kakutani and H.F. Bohnenblust have given characterizations of $$L^p$$ L p as a Banach lattice. We generalize that result to $$p\in (0,\infty )$$ p ∈ ( 0 , ∞ ) . In particular, we show that a quasi-Banach lattice "Equation missing" that satisfies $$\rfloor \negthickspace \rfloor u+v\lfloor \negthickspace \lfloor ^p=\rfloor \negthickspace \rfloor u\lfloor \negthickspace \lfloor ^p +\rfloor \negthickspace \rfloor v\lfloor \negthickspace \lfloor ^p$$ ⌋ ⌋ u + v ⌊ ⌊ p = ⌋ ⌋ u ⌊ ⌊ p + ⌋ ⌋ v ⌊ ⌊ p if $$u\wedge v =0$$ u ∧ v = 0 , is isometrically Riesz isomorphic to $$L^p$$ L p .

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 A survey of weighted substitution operators and generalizations of Banach-stone theorem

2005 ◽  
Vol 2005 (6) ◽  
pp. 937-948 ◽  
Author(s):  
R. K. Singh
Keyword(s):  
Function Spaces ◽  
Open Problems ◽  
Disjointness Preserving ◽  
Lattice Homomorphisms ◽  
Stone Theorem

The classical Banach-Stone theorem characterizes linear surjective isometries betweenC(K)-spaces. The main aim of this paper is to present a survey of Banach-Stone-theorem-type results between some function spaces. The weighted substitution operators play an important role in characterization of isometries, disjointness preserving operators, and lattice homomorphisms. Some open problems are given for further investigation.

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