Amalgamation and Injectivity in Banach Lattices

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.

A quantitative approach to disjointly non-singular operators

We introduce and study some operational quantities which characterize the disjointly non-singular operators from a Banach lattice E to a Banach space Y when E is order continuous, and some other quantities which characterize the disjointly strictly singular operators for arbitrary E.

Geometric Characterization of Injective Banach Lattices

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.

Domination and Kwapień’s factorization theorems for positive Cohen p–nuclear m–linear operators

In this paper, we introduce and study the concept of positive Cohen p-nuclear multilinear operators between Banach lattice spaces. We prove a natural analog to the Pietsch domination theorem for this class. Moreover, we give like the Kwapień’s factorization theorem. Finally, we investigate some relations with another known classes.

Bishop-Phelps-Bollobás property for positive operators when the domain is L∞

We prove that the class of positive operators from [Formula: see text] to [Formula: see text] has the Bishop-Phelps-Bollobás property for any positive measure [Formula: see text], whenever [Formula: see text] is a uniformly monotone Banach lattice with a weak unit. The same result also holds for the pair [Formula: see text] for any uniformly monotone Banach lattice [Formula: see text] Further we show that these results are optimal in case that [Formula: see text] is strictly monotone.

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

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 .