Related papers: Free complex Banach lattices
We investigate the structure of the free $p$-convex Banach lattice $FBL^{(p)}[E]$ over a Banach space $E$. After recalling why such a free lattice exists, and giving a convenient functional representation of it, we focus our study on how…
The relation between the free Banach lattice generated by a Banach space and free dual spaces is clarified. In particular, it is shown that for every Banach space $E$ the free $p$-convex Banach lattice generated by $E^{**}$, denoted…
In this article we deal with the free Banach lattice generated by a lattice and its behavior with respect to subspaces. In general, any lattice embedding $i\colon \mathbb{L} \longrightarrow \mathbb{M}$ between two lattices $\mathbb{L}…
Motivated by the construction of the free Banach lattice generated by a Banach space, we introduce and study several vector and Banach lattices of positively homogeneous functions defined on the dual of a Banach space $E$. The relations…
We show that the free Banach lattice $\mathrm{FBL}(A)$ may be constructed as the completion of $\mathrm{FVL}(A)$ with respect to the maximal lattice seminorm $\nu$ on $\mathrm{FVL}(A)$ with $\nu(a)\le 1$ for all $a\in A$. We present a…
In this article we deal with the free Banach lattice $FBL\langle \mathbb{L} \rangle$ generated by a lattice $\mathbb{L}$. We prove that if $FBL\langle \mathbb{L} \rangle$ is projective then $\mathbb{L}$ has a maximum and a minimum. On the…
We study structural properties of the free Banach lattice $FBL\langle L\rangle$ generated by a distributive lattice $L$. We characterize when $FBL\langle L\rangle$ has a strong unit, compute its density character, analyze the density…
We study the strong Nakano property in Banach lattices with a special focus on free Banach lattices. We show that for every finite dimensional Banach space $E$, the free Banach lattice $FBL[E]$ has the strong Nakano property with a constant…
Motivated by the Lipschitz-lifting property of Banach spaces introduced by Godefroy and Kalton, we consider the lattice-lifting property, which is an analogous notion within the category of Banach lattices and lattice homomorphisms. Namely,…
We begin by describing the unit ball of the free $p$-convex Banach lattice over a Banach space $E$ (denoted by ${\mathrm{FBL}}^{(p)}[E]$) as a closed solid convex hull of an appropriate set. Based on it, we show that, if a Banach space $E$…
We prove the existence of a separable approximately ultra-homogeneous Banach lattice $\mathfrak{BL}$ that is isometrically universal for separable Banach lattices. This is done by showing that the class of Banach lattices has the…
We study different versions of \emph{free objects} in the setting of quasi-Banach spaces and quasi-Banach lattices. Special attention is devoted to the free $p$-convex $p$-Banach lattice $\operatorname{FpBL}^{(p)}[E]$ generated by a…
We show that a free vector lattice over a real vector space $V$ can be realised canonically as a vector lattice of real-valued positively homogeneous functions on any linear subspace of its dual space that separates the points of $V$. This…
We show that for an arbitrary Banach space $E$, the free Banach lattice $FBL[E]$ generated by $E$ satisfies the $\sigma$-bounded chain condition.
We study the free Banach lattice $FBL^{(p,\infty)}[E]$ with upper $p$-estimates generated by a Banach space $E$. Using a classical result of Pisier on factorization through $L^{p,\infty}(\mu)$ together with a finite dimensional reduction,…
We construct and analyze the free Banach $f\!$-algebra $\operatorname{FB{\it f}A}[E]$ generated by a Banach space $E$, extending recent developments on free Banach lattices to the setting of Banach $f\!$-algebras, where multiplication…
We survey recent developments on the structure of complemented subspaces of Banach lattices, including in particular the construction of a complemented subspace of a $C(K)$-space which is not linearly isomorphic to any Banach lattice.…
In this paper, we study octahedral norms in free Banach lattices $FBL[E]$ generated by a Banach space $E$. We prove that if $E$ is an $L_1(\mu)$-space, a predual of von Neumann algebra, a predual of a JBW$^*$-triple, the dual of an…
We study free products, that is, coproducts, in the category of Banach lattices and contractive lattice homomorphisms. We give a concrete construction of the free product of an arbitrary family of Banach lattices as a quotient of a free…
First we develop a technique to construct Banach lattices of homogeneous polynomials. We obtain, in particular, conditions for the linear spans of all positive compact and weakly compact $n$-homogeneous polynomials between the Banach…