Free Banach lattices over pre-ordered Banach spaces

The notion of the free Banach lattice over a set was introduced by de Pagter and Wickstead in 2015 [de_pagter_wickstead:2015]; it was followed by the definition of the free Banach lattice ${\mathrm{FBL}}[X]$ over a Banach space $X$ by Avilés, Rodríguez, and Tradacete in [aviles_rodriguez_tradacete:2018]. The latter paper was the starting point of an active line of research.

In the present paper, we bring (pre-)ordering into the picture. For a pre-ordered Banach space $X$ with positive wedge $X^{+}$ and a given convexity type ${\mathcalalt C}$ (a notion introduced in the paper), we define the free Banach lattice ${\mathrm{FBL}}^{\mathcalalt C}[(X,X^{+})]$ over $(X,X^{+})$. When taking ${\mathcalalt C}=\emptyset$, our ${\mathrm{FBL}}^{\emptyset}[X,\{0\}]$ is ${\mathrm{FBL}}[X]$ from [aviles_rodriguez_tradacete:2018]; the free Banach lattices ${\mathrm{FBL}}^{\mathcalalt D}[X]$ under convexity conditions from [jardon-sanchez_laustsen_taylor_tradacete_troitsky:2022] all occur as an ${\mathrm{FBL}}^{\mathcalalt C}[(X,\{0\}]$ in the present paper. When $X$ is an ordered Banach space with a (not necessarily generating) closed positive cone $X^{+}$, there is a linear injection from $X$ into ${\mathrm{FBL}}^{\mathcalalt C}[(X,X^{+})]$ that is bipositive. Thus $X$ sits inside ${\mathrm{FBL}}^{\mathcalalt C}[(X,X^{+})]$, and its ordering extends to a lattice ordering on the enveloping space. It is a non-trivial question, left for further research, to find conditions under which the image is closed; this is always the case if $X^{+}=\{0\}$ when the injection is even isometric. When the image is closed, $X$ is embedded (though perhaps not necessarily isometrically) into the Banach lattice ${\mathrm{FBL}}^{\mathcalalt C}[(X,X^{+})]$ as an ordered Banach space. The embedding of a Banach space $X$ into ${\mathrm{FBL}}[X]$ makes a better understanding possible of the relation between Banach spaces and Banach lattices. The same can now be expected for that between ordered Banach spaces and Banach lattices.

Compared to earlier papers on free Banach lattices, our approach to the existence of the free Banach lattice is of a fundamentally different nature. We do not start from a function lattice, but instead from an algebraic free object which we know to exist, and proceed from there. This has not only the advantage that it works in all generality, whereas—as we shall argue—this cannot be expected for the function lattice method, but we also believe that it makes the picture more transparent. As will become clear, one can say quite a bit about the free Banach lattice without having a concrete function lattice realisation of it (which may not exist), or without even knowing that it actually exists.

This paper is organised as follows.

Section 2 contains the necessary preliminaries.

Section 3 contains preparatory material on convexity in Banach lattices. In its first part, using an idea from [jardon-sanchez_laustsen_taylor_tradacete_troitsky:2022], a characterisation of $p$-convex Banach lattices is obtained in terms of vector lattice homomorphisms into $\mathrm{L}_{p}$-spaces or the real numbers. This characterisation, which appears to be new, is instrumental to Section 7. It also allows one to give an alternate proof of the well-known fact that $p$-convex Banach lattices can always be renormed so that the $p$-convexity constant becomes 1. In its second part, the notion of a convexity type is introduced, generalising that of a convexity condition in [jardon-sanchez_laustsen_taylor_tradacete_troitsky:2022]. If, for a Banach lattice, there is a norming family of contractive vector lattice homomorphisms into Banach lattices that are all of the same convexity type, then so is the domain lattice.

Section 4 introduces the free vector lattice ${\mathrm{FVL}}[(V,V^{+})]$ over a pre-ordered vector space $V$ with positive wedge $V^{+}$. It is shown that it exists. One can argue that, once this existence been established (a consequence of a basic principle in universal algebra), it is routine to verify that ${\mathrm{FBL}}^{\mathcalalt C}[(X,X^{+})]$ always exists.

In Section 5, the free Banach lattice ${\mathrm{FBL}}^{\mathcalalt C}[(X,X^{+})]$ of a given convexity type over a pre-ordered Banach space $(X,X^{+})$ is defined. The definition is purely categorical, in terms of positive contractions that factor over contractive vector lattice homomorphisms, without any conditions on maps being injective or isometric, or on norms of operators being equal. The remainder of the section is a study of its properties, should it exist. It is not disappointing what can already be said a priori. Our definition of ${\mathrm{FVL}}^{\emptyset}[(X,\{0\})]$, for example, can be shown to be equivalent to the seemingly stronger one for ${\mathrm{FBL}}[X]$ in [aviles_rodriguez_tradacete:2018]. A formula for the norm can be derived, and it can be determined when ${\mathrm{FBL}}^{\mathcalalt C}[(X,X^{+})]$ contains ${\mathrm{FVL}}[(V,V^{+})]$. This is always the case when $X^{+}=\{0\}$. Consequently, the free Banach lattices in [aviles_rodriguez_tradacete:2018, jardon-sanchez_laustsen_taylor_tradacete_troitsky:2022] always contain a copy of the free vector lattice over the Banach space $X$. In the notation of these papers, the vector lattices of functions on the dual space that is generated by the $\delta_{x}$ is the free vector lattice over $X$. This seems to have gone unnoticed in the papers on free Banach lattices thus far, and one can regard it is an explanation why the approach in [aviles_rodriguez_tradacete:2018, jardon-sanchez_laustsen_taylor_tradacete_troitsky:2022], where one starts with a function lattice on the dual unit ball, actually works. We also give arguments why this will in all likelihood not work when $X^{+}\neq\{0\}$. It can also be determined a priori when precisely there is a canonical realisation of ${\mathrm{FBL}}^{\mathcalalt C}[(X,X^{+})]$ as a function lattice on the positive part of the dual unit ball.

In Section 6, the existence of ${\mathrm{FBL}}^{\mathcalalt C}[(X,X^{+})]$ is established in a manner as was already done in [de_jeu:2021] for ${\mathrm{FBL}}[X]$, and as is systematically exploited in [van_amstel_THESIS:2023]. The idea, which can already be found in the books by Dixmier and presumably also elsewhere, is to start with a free object in an algebraic context and proceed routinely along a standard path. There is no role for function lattices on the dual space, or even for the dual space itself.

In Section 7, it is shown that the free $p$-convex Banach lattices with $p$-convexity constant 1 over $(X,X^{+})$ can be realised as lattices of homogeneous continuous functions on the positive part of the dual unit ball. When $X^{+}=\{0\}$, one retrieves the realisations in [jardon-sanchez_laustsen_taylor_tradacete_troitsky:2022]. This section is clearly indebted to the ingenious techniques in [jardon-sanchez_laustsen_taylor_tradacete_troitsky:2022], but in spite of the similarity the road to these realisations is very different. Whereas in [jardon-sanchez_laustsen_taylor_tradacete_troitsky:2022] the proof of the existence of the free Banach lattice is essentially redone by showing that the proposed Banach function lattice has the correct universal property, we start from the free object itself, which we know to exist, and analyse it to the extent that it becomes a triviality that it has a realisation as a function lattice on the positive part of the dual unit ball. One could say that in the present paper these realisations are derived, and that in [aviles_rodriguez_tradacete:2018, jardon-sanchez_laustsen_taylor_tradacete_troitsky:2022] they are found.

A words on notation. Firstly, compared to many papers on free Banach lattices, our canonical choice of letters is reversed: $X$ is a Banach space and $E$ and $F$ are Banach lattices or vector lattices. We believe that this reflects the notation in most of the existing textbooks on Banach lattices and vector lattices, and apologise for any inconvenience this may cause. Secondly, we have consistently used Greek letters for positive contractions and contractive vector lattice homomorphisms, rather than the usual operator letters such as $T$. This fits better into our categorical approach, where these are the morphisms we work with.

All vector spaces are over the real numbers. A vector lattice need not be Archimedean.

If $V$ is a vector space, then a wedge in $V$ is a non-empty subset $V^{+}$ such that $\alpha_{1}x_{1}+\alpha_{2}x_{2}\in V^{+}$ for all $\alpha_{1},\alpha_{2}\geq 0$ and $x_{1},x_{2}\in V^{+}$. A wedge $V^{+}$ is a cone when $V^{+}\cap(-V^{+})=\{0\}$. There is a well-known correspondence between pre-orderings on $V$ and wedges in $V$, where the partial orderings correspond to the cones. A pre-ordered vector space is a pair $(V,V^{+})$, where $V^{+}$ is a wedge in the vector space $V$. If $(V,V^{+})$ and $(W,W^{+})$ are pre-ordered vector spaces, then a linear map $\varphi\colon V\to W$ is positive when $\varphi(V^{+})\subseteq W^{+}$. When $V^{+}=\{0\}$, these are just the linear maps.

A pre-ordered Banach space is a pair $(X,X^{+})$, where $X^{+}$ is a wedge in the Banach space $X$. It need not be closed.

If $X$ is a Banach space, then $X^{*}$ denotes its dual. When $(X,X^{+})$ is a pre-ordered Banach space, we write ${\mathrm{B}}_{X}$ for its unit ball, ${\mathrm{B}}_{X^{\ast}}$ for the unit ball of its dual, and ${\mathrm{B}}_{X^{\ast}_{+}}$ for the positive elements of ${\mathrm{B}}_{X^{\ast}}$. If $X^{+}=\{0\}$, then ${\mathrm{B}}_{X^{\ast}_{+}}={\mathrm{B}}_{X^{\ast}}$.

We supply ${\mathrm{B}}_{X^{\ast}_{+}}$ with the relative weak*-topology, making it into a compact Hausdorff space, and let $\mathrm{C}^{\mathrm{h}}({\mathrm{B}}_{X^{\ast}_{+}})$ denote the continuous, homogeneous, real-valued functions on it. For $x\in X$, we define $\delta_{x}\in\mathrm{C}^{\mathrm{h}}({\mathrm{B}}_{X^{\ast}_{+}})$ by setting

for $x^{*}\in{\mathrm{B}}_{X^{\ast}_{+}}$. If $X^{+}=\{0\}$, then $\delta_{x}$ is defined on ${\mathrm{B}}_{X^{\ast}}$.

Contractions between Banach spaces are linear.

To conclude, we recall the following definition of free objects from [adamek_herrlich_strecker_ABSTRACT_AND_CONCRETE_CATEGORIES_THE_JOY_OF_CATS:2006, Definition 8.22].

A free object $(j,\textup{F}_{{\textup{{\tiny{Cat}}}}_{1}}^{{\textup{{\tiny{Cat}}}}_{2}}\!\left[O_{1}\right])$ as in Definition 2.1 need not exist. However, if it exists, and if $(j^{\prime},\textup{F}_{{\textup{{\tiny{Cat}}}}_{1}}^{{\textup{{\tiny{Cat}}}}_{2}}\!\left[O_{1}\right]^{\prime})$ is another such pair, then a standard argument shows that the unique morphism $\overline{j^{\prime}}:\textup{F}_{{\textup{{\tiny{Cat}}}}_{1}}^{{\textup{{\tiny{Cat}}}}_{2}}\!\left[O_{1}\right]\to\textup{F}_{{\textup{{\tiny{Cat}}}}_{1}}^{{\textup{{\tiny{Cat}}}}_{2}}\!\left[O_{1}\right]^{\prime}$ of $\textup{{\footnotesize Cat}}_{2}$ such that $U(\overline{j^{\prime}})\circ j=j^{\prime}$ is, in fact, an isomorphism with as its inverse the unique morphism $\overline{j}:\textup{F}_{{\textup{{\tiny{Cat}}}}_{1}}^{{\textup{{\tiny{Cat}}}}_{2}}\!\left[O_{1}\right]^{\prime}\to\textup{F}_{{\textup{{\tiny{Cat}}}}_{1}}^{{\textup{{\tiny{Cat}}}}_{2}}\!\left[O_{1}\right]$ of $\textup{{\footnotesize Cat}}_{2}$ such that $U(\overline{j})\circ j^{\prime}=j$ . In particular, a free object over $O_{1}$ of $\textup{{\footnotesize Cat}}_{2}$ with respect to $U$, if it exists, is determined up to an isomorphism of $\textup{{\footnotesize Cat}}_{2}$. In the case of existence we shall, therefore, sometimes simply speak of ‘the’ free object $\textup{F}_{{\textup{{\tiny{Cat}}}}_{1}}^{{\textup{{\tiny{Cat}}}}_{2}}\!\left[O_{1}\right]$ over $O_{1}$ of $\textup{{\footnotesize Cat}}_{2}$, the accompanying morphism $j$ often being tacitly understood from the context.

In the current paper, we are concerned with two contexts for Definition 2.1, an algebraic and an analytic one. The results in the algebraic context are the stepping stone for those in the analytic context.

We apply Definition 2.1 in the following context. For $\textup{{\footnotesize Cat}}_{1}$, we take the pre-ordered vector spaces as objects and the positive linear maps as morphisms. For $\textup{{\footnotesize Cat}}_{2}$, we take the vector lattices as objects and the vector lattice homomorphisms as morphisms. We define the functor $U$ from $\textup{{\footnotesize Cat}}_{1}$ to $\textup{{\footnotesize Cat}}_{2}$ by sending a vector lattice $E$ to the pre-ordered vector space $(E,E^{+})$ and viewing a vector lattice homomorphism as a positive linear map. Simplifying the notation and terminology a little, this leads to the following definition.

The following separation result is elementary, and in analogous forms obviously valid in many other contexts. We still include its proof to put later results in a similar vein into perspective.

Fix a convexity type ${\mathcalalt C}$. In the general context of Definition 2.1, we take for $\textup{{\footnotesize Cat}}_{1}$ the pre-ordered Banach spaces as objects and the positive contractions as morphisms. Note that, under our standing assumption that ${\mathbbm{R}}$ be ${\mathcalalt C}$-convex, ${\mathrm{B}}_{X^{\ast}_{+}}$ consists of morphisms in $\textup{{\footnotesize Cat}}_{1}$ for every pre-ordered Banach space $(X,{X^{+}})$. For $\textup{{\footnotesize Cat}}_{2}$, we take the ${\mathcalalt C}$-convex Banach lattices as objects and the contractive vector lattice homomorphisms as morphisms. The functor $U$ is defined by sending a Banach lattice $E$ to the pre-ordered Banach space $(E,E^{+})$ and viewing a contractive vector lattice homomorphism as a positive contraction. This leads to the following definition.