Free Banach Lattice Research Articles

Banach lattices with upper p-estimates: free and injective objects

AbstractWe study the free Banach lattice $$\textrm{FBL}^{(p,\infty )}[E]$$ FBL ( p , ∞ ) [ E ] with upper p-estimates generated by a Banach space E. Using a classical result of Pisier on factorization through $$L^{p,\infty }(\mu )$$ L p , ∞ ( μ ) together with a finite dimensional reduction, it is shown that the spaces $$\ell ^{p,\infty }(n)$$ ℓ p , ∞ ( n ) witness the universal property of $$\textrm{FBL}^{(p,\infty )}[E]$$ FBL ( p , ∞ ) [ E ] isomorphically. As a consequence, we obtain a functional representation for $$\textrm{FBL}^{(p,\infty )}[E]$$ FBL ( p , ∞ ) [ E ] , answering a question from Oikhberg et al. [Free Banach lattices. J Eur Math Soc (JEMS), 2024]. More generally, our proof allows us to identify the norm of any free Banach lattice over E associated with a rearrangement invariant function space. After obtaining the above functional representation, we take the first steps towards analyzing the fine structure of $$\textrm{FBL}^{(p,\infty )}[E]$$ FBL ( p , ∞ ) [ E ] . Notably, we prove that the norm for $$\textrm{FBL}^{(p,\infty )}[E]$$ FBL ( p , ∞ ) [ E ] cannot be isometrically witnessed by $$L^{p,\infty }(\mu )$$ L p , ∞ ( μ ) and settle the question of characterizing when an embedding between Banach spaces extends to a lattice embedding between the corresponding free Banach lattices with upper p-estimates. To prove this latter result, we introduce a novel push-out argument, which when combined with the injectivity of $$\ell ^p$$ ℓ p allows us to give an alternative proof of the subspace problem for free p-convex Banach lattices. On the other hand, we prove that $$\ell ^{p,\infty }$$ ℓ p , ∞ is not injective in the class of Banach lattices with upper p-estimates, elucidating one of many difficulties arising in the study of $$\textrm{FBL}^{(p,\infty )}[E]$$ FBL ( p , ∞ ) [ E ] .

Local complementation in Banach spaces and its preservation under free constructions

In this work we consider natural generalizations of local complementation in Banach spaces, which include Lipschitz-local complementation. We show that all these notions are indeed equivalent to the classical notion of local complementation of Banach spaces. As an application, we show that local complementation is naturally preserved under certain free constructions in Functional Analysis, including Lipschitz-free spaces and free Banach lattices.

Free dual spaces and free Banach lattices

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 FBL(p)[E⁎⁎], admits a canonical isometric lattice embedding into FBL(p)[E]⁎⁎ and FBL(p)[E⁎⁎] is lattice finitely representable in FBL(p)[E]. Moreover, we also show that for p>1, FBL(p)[E]⁎⁎ can actually be considered as the free dual p-convex Banach lattice generated by E, whereas for p=1 this happens precisely when E does not contain complemented copies of ℓ1.

Free complex Banach lattices

The construction of the free Banach lattice generated by a real Banach space is extended to the complex setting. It is shown that for every complex Banach space E there is a complex Banach lattice FBLC[E] containing a linear isometric copy of E and satisfying the following universal property: for every complex Banach lattice XC, every operator T:E→XC admits a unique lattice homomorphic extension Tˆ:FBLC[E]→XC with ‖Tˆ‖=‖T‖. The free complex Banach lattice FBLC[E] is shown to have analogous properties to those of its real counterpart. However, examples of non-isomorphic complex Banach spaces E and F can be given so that FBLC[E] and FBLC[F] are lattice isometric. The spectral theory of induced lattice homomorphisms on FBLC[E] is also explored.

Free Banach lattices generated by a lattice and projectivity

In this article we deal with the free Banach lattice F B L ⟨ L ⟩ FBL\langle \mathbb {L} \rangle generated by a lattice L \mathbb {L} . We prove that if F B L ⟨ L ⟩ FBL\langle \mathbb {L} \rangle is projective then L \mathbb {L} has a maximum and a minimum. On the other hand, we show that if L \mathbb {L} has maximum and minimum then F B L ⟨ L ⟩ FBL\langle \mathbb {L} \rangle is 2 2 -lattice isomorphic to a C ( K ) C(K) -space. As a consequence, F B L ⟨ L ⟩ FBL\langle \mathbb {L} \rangle is projective if and only if it is lattice isomorphic to a C ( K ) C(K) -space with K K being an absolute neighborhood retract. As an application, we characterize those linearly ordered sets and Boolean algebras for which the corresponding free Banach lattice is projective.

Lattice embeddings in free Banach lattices over lattices

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 $\mathbb{L} \subseteq \mathbb{M}$ induces a Banach lattice homomorphism $\hat \imath\colon FBL \langle \mathbb{L} \rangle \longrightarrow FBL \langle \mathbb{M}\rangle$ between the corresponding free Banach lattices. We show that this mapping $\hat \imath$ might not be an isometric embedding neither an isomorphic embedding. In order to provide sufficient conditions for $\hat \imath$ to be an isometric embedding we define the notion of locally complemented lattices and prove that, if $\mathbb L$ is locally complemented in $\mathbb M$, then $\hat \imath$ yields an isometric lattice embedding from $FBL\langle\mathbb L\rangle$ into $FBL\langle\mathbb M\rangle$. We provide a wide number of examples of locally complemented sublattices and, as an application, we obtain that every free Banach lattice generated by a lattice is lattice isomorphic to an AM-space or, equivalently, to a sublattice of a $C(K)$-space. Furthermore, we prove that $\hat \imath$ is an isomorphic embedding if and only if it is injective, which in turn is equivalent to the fact that every lattice homomorphism $x^*\colon \mathbb{L} \longrightarrow [-1,1]$ extends to a lattice homomorphism $\hat x^*\colon \mathbb{M} \longrightarrow [-1,1]$. Using this characterization we provide an example of lattices $\mathbb{L} \subseteq \mathbb{M}$ for which $\hat \imath$ is an isomorphic not isometric embedding.

A Godefroy—Kalton principle for free Banach lattices

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, a Banach lattice X satisfies the lattice-lifting property if every lattice homomorphism to X having a bounded linear right-inverse must have a lattice homomorphism right-inverse. In terms of free Banach lattices, this can be rephrased into the following question: which Banach lattices embed into the free Banach lattice which they generate as a lattice-complemented sublattice? We will provide necessary conditions for a Banach lattice to have the lattice-lifting property, and show that this property is shared by Banach spaces with a 1-unconditional basis as well as free Banach lattices. The case of C(K) spaces will also be analyzed.

Free Banach lattices under convexity conditions

We prove the existence of free objects in certain subcategories of Banach lattices, including p-convex Banach lattices, Banach lattices with upper p-estimates, and AM-spaces. From this we immediately deduce that projectively universal objects exist in each of these subcategories, extending results of Leung, Li, Oikhberg and Tursi (Israel J. Math. 2019). In the p-convex and AM-space cases, we are able to explicitly identify the norms of the free Banach lattices, and we conclude by investigating the structure of these norms in connection with nonlinear p-summing maps.

Amalgamation and Injectivity in Banach Lattices

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.

Norm-attaining lattice homomorphisms

In this paper we study the structure of the set ${\\rm Hom}(X,\\mathbb{R})$ of all lattice homomorphisms from a Banach lattice $X$ into $\\mathbb{R}$. Using the relation among lattice homomorphisms and disjoint families, we prove that the topological dual of the free Banach lattice ${\\rm FBL}(A)$ generated by a set $A$ contains a disjoint family of cardinality $2^{|A|}$, answering a question of B. de Pagter and A. W. Wickstead. We also deal with norm-attaining lattice homomorphisms. For classical Banach lattices, as $c_0$, $L_p$- and $C(K)$-spaces, every lattice homomorphism on it attains its norm, which shows, in particular, that there is no James theorem for this class of functions. We prove that, indeed, every lattice homomorphism on $X$ and $C(K,X)$ attains its norm whenever $X$ has order continuous norm. On the other hand, we provide what seems to be the first example in the literature of a lattice homomorphism which does not attain its norm. In general, we study the existence and characterization of lattice homomorphisms not attaining their norm in free Banach lattices. As a consequence, it is shown that no Bishop–Phelps type theorem holds true in the Banach lattice setting, i.e., not every lattice homomorphism can be approximated by norm-attaining lattice homomorphisms.

Octahedral norms in free Banach lattices

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 M-embedded Banach space, the disc algebra or the projective tensor product under some hypothesis, then the norm of FBL[E] is octahedral. We get the analogous result when the topological dual $$E^*$$ of E is almost square. We finish the paper by proving that the norm of the free Banach lattice generated by a Banach space of dimension $$ \ge 2$$ is nowhere Frechet differentiable. Moreover, we discuss some open problems on this topic.

On the Banach lattice $$c_0$$

We show that $c_0$ is not a projective Banach lattice, answering a question of B. de Pagter and A. Wickstead. On the other hand, we show that $c_0$ is complemented in the free Banach lattice generated by itself (seen as a Banach space). As a consequence, the free Banach lattice generated by $c_0$ is not projective.

Projectivity of the free Banach lattice generated by a lattice

We study the projectivity of the free Banach lattice generated by a lattice $\mathbb{L}$ in two cases: when the lattice is finite, and when the lattice is an infinite linearly ordered set. We prove that in the first case it is projective while in the second case, if the linear order contains either an increasing sequence without upper bounds or a decreasing sequence without lower bounds, then it is not projective.

Simple constructions of $$\mathrm {FBL}(A)$$ FBL ( A ) and $$\mathrm {FBL}[E]$$ FBL [ E

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 $$\nu (a)\leqslant 1$$ for all $$a\in A$$ . We present a similar construction for the free Banach lattice $$\mathrm {FBL}[E]$$ generated by a Banach space E.

The free Banach lattices generated by $$\ell _p$$ ℓ p and $$c_0$$ c 0

We prove that, when $$2<p<\infty $$ , in the free Banach lattice generated by $$\ell _p$$ (respectively by $$c_0$$ ), the absolute values of the canonical basis form an $$\ell _r$$ -sequence, where $$\frac{1}{r} = \frac{1}{2} + \frac{1}{p}$$ (respectively an $$\ell _2$$ -sequence). In particular, in any Banach lattice, the absolute values of any $$\ell _p$$ sequence always have an upper $$\ell _r$$ -estimate. Quite surprisingly, this implies that the free Banach lattices generated by the nonseparable $$\ell _p(\Gamma )$$ for $$2<p<\infty $$ , as well as $$c_0(\Gamma )$$ , are weakly compactly generated whereas this is not the case for $$1\le p\le 2$$ .

The free Banach lattice generated by a lattice

We introduce the free Banach lattice generated by a lattice L. We give an explicit description of it and we study some of its properties for the case when $\mathbb{L}$ is a linear order, like the countable chain condition.

Chain conditions in free Banach lattices

We show that for an arbitrary Banach space E, the free Banach lattice FBL[E] generated by E satisfies the σ-bounded chain condition.

The free Banach lattice generated by a Banach space

The free Banach lattice over a Banach space is introduced and analyzed. This generalizes the concept of free Banach lattice over a set of generators, and allows us to study the Nakano property and the density character of non-degenerate intervals on these spaces, answering some recent questions of B. de Pagter and A.W. Wickstead. Moreover, an example of a Banach lattice which is weakly compactly generated as a lattice but not as a Banach space is exhibited, thus answering a question of J. Diestel.

Free and projective Banach lattices

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 &gt; 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.