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

Abstract

We study the free Banach lattice $\textrm{FBL}^{(p,\infty )}[E]$ 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, it is shown that the spaces $\ell ^{p,\infty }(n)$ witness the universal property of $\textrm{FBL}^{(p,\infty )}[E]$ isomorphically. As a consequence, we obtain a functional representation for $\textrm{FBL}^{(p,\infty )}[E]$, 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 $\textrm{FBL}^{(p,\infty )}[E]$. Notably, we prove that the norm for $\textrm{FBL}^{(p,\infty )}[E]$ cannot be isometrically witnessed by $L^{p,\infty }(\mu )$ 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 $\ell ^p$ allows us to give an alternative proof of the subspace problem for free p-convex Banach lattices. On the other hand, we prove that $\ell ^{p,\infty }$ is not injective in the class of Banach lattices with upper p-estimates, elucidating one of many difficulties arising in the study of $\textrm{FBL}^{(p,\infty )}[E]$.

1 Introduction

Injective, projective and free objects are a common subject of study in many areas of mathematics. Although this terminology comes from category theory, as pointed out by Semadeni in [37], particular instances of the corresponding notions in different areas usually go under different names. For example, in Banach space theory, injectivity and related properties have been broadly studied in the category of Banach spaces and bounded linear operators: Here, Hahn-Banach theorem can be interpreted as injectivity of the spaces $\ell _\infty (\Gamma )$ (see the monographs [3, 15] for deeper and more recent developments). In the setting of Banach lattices, a notion of injectivity relative to positive operators was considered in the 70s and 80s by a number of authors including Lotz [28], Cartwright [16], Haydon [23], Lindenstrauss and Tzafriri [27], and Mangheni [29]. Interestingly, for the category of Banach lattices and lattice homomorphisms, there are no injective objects (see [19]), although separably injective objects have been constructed recently in [9]. In addition, free objects have also been a fruitful topic of research in functional analysis during the past two decades, due mainly to the introduction of Lipschitz free Banach spaces (see [20] for a survey on the topic) and, more recently, free Banach lattices (cf. [8, 31]). The aim of this paper is to study these notions in restricted classes of Banach lattices such as p-convex Banach lattices or Banach lattices with upper p-estimates.

Given a Banach space E, the free Banach lattice generated by E is a Banach lattice $\textrm{FBL}[E]$ together with a linear isometric embedding $\phi _E: E\rightarrow \textrm{FBL}[E]$ such that for every Banach lattice X and every bounded linear operator $T:E\rightarrow X$ there is a unique lattice homomorphism $\widehat{T}:\textrm{FBL}[E]\rightarrow X$ such that $\widehat{T}\circ \phi _E=T$ and $\Vert \widehat{T}\Vert =\Vert T\Vert .$ In other words, $\textrm{FBL}[E]$ is the (unique) Banach lattice which converts linear operators defined on E into lattice homomorphisms, in the sense of the following diagram:

In categorical terms this can also be paraphrased as follows: If $\mathfrak {Ban}$ denotes the category of Banach spaces and bounded linear operators and $\mathfrak {BL}$ that of Banach lattices and lattice homomorphisms, then $\textrm{FBL}:\mathfrak {Ban}\rightarrow \mathfrak {BL}$ is the left adjoint of the forgetful functor $O:\mathfrak {BL}\rightarrow \mathfrak {Ban}$. Over the past ten years, many novel techniques have been developed to understand the structural properties of $\textrm{FBL}[E]$ as well as the rigidity of the correspondence $E\mapsto \textrm{FBL}[E]$. This has led to a host of new developments in the theory of Banach spaces and Banach lattices, as well as to the resolution of several outstanding open problems. To indicate just a few examples, free Banach lattices are used in [8] to solve a question of J. Diestel on WCG spaces, in [31] to solve a question from [38] regarding basic sequences satisfying maximal inequalities, and in [18] to produce the first example of a lattice homomorphism which does not attain its norm. Free Banach lattices also play a fundamental role in the theory of projective Banach lattices [5, 6, 19] and are used in [9] to construct Banach lattices of universal disposition, separably injective Banach lattices and push-outs. On the other hand, free Banach lattices are known to have a remarkably rigid structure in their own right. Notably, it is proven in [7] that any disjoint collection in $\textrm{FBL}[E]$ must be countable and in [31] that $\textrm{FBL}[E]$ determines E isometrically if $E^*$ is smooth. This latter result has initiated a program to find geometric properties of a Banach space E which can be equivalently characterized via lattice properties of $\textrm{FBL}[E]$—see [31] for several such properties.

Although defined in an abstract manner, free Banach lattices admit a very concrete functional representation. To see this, denote by H[E] the linear subspace of $\mathbb {R}^{E^*}$ consisting of all positively homogeneous functions $f:E^* \rightarrow \mathbb {R}$. For $f\in H[E]$ define

$$\begin{aligned} \Vert f\Vert _{\textrm{FBL}[E]}= \sup \left\{ \sum _{k=1}^n |f(x_k^*)|: \, n\in \mathbb {N}, \, x_1^*,\dots ,x_n^*\in E^*, \, \sup _{x\in B_E} \sum _{k=1}^n |x_k^*(x)|\le 1\right\} . \end{aligned}$$

Given any $x\in E$, let $\delta _x\in H[E]$ be defined by

$$\begin{aligned} \delta _x(x^*):= x^*(x) \quad \text{ for } \text{ all } x^*\in E^*. \end{aligned}$$

It is easy to see that

$$\begin{aligned} H_1[E]:=\bigl \{f\in H[E]: \Vert f\Vert _{\textrm{FBL}[E]} <\infty \bigr \} \end{aligned}$$

is a sublattice of H[E] and that $\Vert \cdot \Vert _{\textrm{FBL}[E]}$ defines a complete lattice norm on $H_1[E]$. Moreover, $\Vert \delta _x\Vert _{\textrm{FBL}[E]}=\Vert x\Vert$ for every $x\in E$. As shown in [8], $\textrm{FBL}[E]$ coincides with the closed sublattice of $H_1[E]$ generated by $\{\delta _x:x\in E\}$ with respect to $\Vert \cdot \Vert _{\textrm{FBL}[E]}$, together with the map $\phi _E(x)= \delta _x$. In particular, by positive homogeneity, we may view elements of $\textrm{FBL}[E]$ as weak$^*$-continuous functions on the dual ball.

For numerous reasons it is useful to introduce a scale of free Banach lattices, indexed by $p\in [1,\infty ]$. Recall that a Banach lattice X is p-convex if there exists a constant $M\ge 1$ such that for every $x_1,\ldots ,x_m\in X$ we have

$$\begin{aligned} \bigg \Vert \left( \sum _{k=1}^m|x_k|^p\right) ^{\frac{1}{p}} \bigg \Vert \le M \left( \sum _{k=1}^m\Vert x_k\Vert ^p\right) ^{\frac{1}{p}}. \end{aligned}$$

(1.1)

The least constant M satisfying the above inequality is called the p-convexity constant of X and is denoted by $M^{(p)}(X)$. We say that X satisfies an upper p-estimate (or that X is a $(p,\infty )$-convex Banach lattice) if (1.1) holds for all pairwise disjoint vectors $x_1,\ldots ,x_m\in X$; the least constant is then called the upper p-estimate constant of X and is denoted by $M^{(p,\infty )}(X)$. A classical result [26, Proposition 1.f.6] states that a Banach lattice X has an upper p-estimate if and only if

$$\begin{aligned} \bigg \Vert \bigvee _{k=1}^m|x_k|\bigg \Vert \le M\left( \sum _{k=1}^m\Vert x_k\Vert ^p\right) ^{\frac{1}{p}} \end{aligned}$$

(1.2)

for all $x_1,\dots , x_m\in X$. Moreover, the best choice of M in (1.2) is exactly $M^{(p,\infty )}(X)$. By [14, 26, 33], every p-convex Banach lattice (resp. Banach lattice with an upper p-estimate) can be renormed to have p-convexity (resp. upper p-estimate) constant 1.

Given $p\in [1,\infty ]$ and a Banach space E, one defines the free p-convex Banach lattice generated by E analogously to $\textrm{FBL}[E]$: The free p-convex Banach lattice generated by E is a Banach lattice ${\textrm{FBL}}^{(p)}[E]$ with p-convexity constant 1 together with a linear isometric embedding $\phi _E: E\rightarrow {\textrm{FBL}}^{(p)}[E]$ such that for every p-convex Banach lattice X with p-convexity constant 1 and every bounded linear operator $T:E\rightarrow X$ there is a unique lattice homomorphism $\widehat{T}:{\textrm{FBL}}^{(p)}[E]\rightarrow X$ such that $\widehat{T}\circ \phi _E=T$ and $\Vert \widehat{T}\Vert =\Vert T\Vert .$ By replacing p-convexity with upper p-estimates one obtains the definition of the free Banach lattice with upper p-estimates generated by E, which is denoted by $\textrm{FBL}^{(p,\infty )}[E]$.

As shown in [24], ${\textrm{FBL}}^{(p)}[E]$ admits an analogous functional representation to $\textrm{FBL}[E]$, but with $\Vert \cdot \Vert _{\textrm{FBL}[E]}$ replaced by

$$\begin{aligned} & \Vert f\Vert _{{\textrm{FBL}}^{(p)}[E]}=\nonumber \\ & \quad \sup \left\{ \left( \sum _{k=1}^n |f(x_k^*)|^p\right) ^\frac{1}{p}: \, n\in \mathbb {N}, \, x_1^*,\dots ,x_n^*\in E^*, \, \sup _{x\in B_E} \left( \sum _{k=1}^n |x_k^*(x)|^p\right) ^\frac{1}{p}\le 1\right\} \nonumber \\ \end{aligned}$$

(1.3)

and $H_1[E]$ replaced by

$$\begin{aligned} H_p[E]:=\bigl \{f\in H[E]: \Vert f\Vert _{{\textrm{FBL}}^{(p)}[E]} <\infty \bigr \}. \end{aligned}$$

Making use of this explicit representation of ${\textrm{FBL}}^{(p)}[E]$, most of the major results on $\textrm{FBL}[E]$ can be shown to hold for ${\textrm{FBL}}^{(p)}[E]$. On the other hand, [31, Sections 9.6 and 10] identify several interesting properties of ${\textrm{FBL}}^{(p)}[E]$ which depend explicitly on p.

The question of establishing a functional representation for $\textrm{FBL}^{(p,\infty )}[E]$ was left as an open problem in [24, Remark 6.2] and then reiterated in [31, Section 9.6]. The proof of the existence of $\textrm{FBL}^{(p,\infty )}[E]$ in [24] proceeds by equipping the free vector lattice with a “maximal” norm satisfying an upper p-estimate and then completing the resulting space. However, this approach gives no practical information about the norm and cannot even guarantee that the resulting Banach lattice is a space of functions on the dual ball. For this reason, minimal progress has been made on $\textrm{FBL}^{(p,\infty )}[E]$, with the exception of some interesting results being proven in [30].

The first aim of this paper is to answer the above question and identify the norm for $\textrm{FBL}^{(p,\infty )}[E]$. As a consequence, we obtain the desired continuous injection $\textrm{FBL}^{(p,\infty )}[E]\hookrightarrow C(B_{E^*})$ and the prospect of ascertaining the fine structure of these spaces. As it turns out, up to equivalence, we have

$$\begin{aligned} & \Vert f\Vert _{\textrm{FBL}^{(p,\infty )}[E]}=\nonumber \\ & \quad \sup \left\{ \Vert (f(x_k^*))_{k=1}^n \Vert _{\ell ^{p,\infty }(n)}: \, n\in \mathbb {N}, \, x_1^*,\dots ,x_n^*\in E^*, \, \sup _{x\in B_E}\Vert (x_k^*(x))_{k=1}^n\Vert _{\ell ^{p,\infty }(n)}\le 1\right\} ,\nonumber \\ \end{aligned}$$

(1.4)

where $\ell ^{p,\infty }(n)$ denotes the canonical n-dimensional weak-$L^p$ space. Moreover, as we will see below, the proof that (1.4) is the correct norm is rather illuminating, in that it yields a formula for the norm for the “free Banach lattice” associated to other classes of lattices.

From the above functional description of $\textrm{FBL}^{(p,\infty )}[E]$, we obtain immediate access to many results. For example, we learn from [7] that $\textrm{FBL}^{(p,\infty )}[E]$ has only countable disjoint collections and from [8] that lattice homomorphic functionals separate the points of $\textrm{FBL}^{(p,\infty )}[E]$. With more work, one can prove analogues of many of the results in [31] which were initially proven for ${\textrm{FBL}}^{(p)}[E]$. However, certain deeper theorems do not immediately generalize to $\textrm{FBL}^{(p,\infty )}[E]$. The purpose of the second half of this paper is to prove results of this type. In particular, we solve the subspace problem for $\textrm{FBL}^{(p,\infty )}$ by characterizing when an embedding $\iota : F\hookrightarrow E$ induces a lattice embedding $\overline{\iota }: \textrm{FBL}^{(p,\infty )}[F]\hookrightarrow \textrm{FBL}^{(p,\infty )}[E]$. Obtaining such a characterization is by no means trivial, as the proof of the subspace problem for ${\textrm{FBL}}^{(p)}$ relied heavily on the $L^p$-type structure of p-convex Banach lattices.

1.1 Outline of the paper

In Sect. 2 we study the free Banach lattice generated by E associated to a non-empty class $\mathcal {C}$ of Banach lattices. We begin in Sect. 2.1 by briefly reviewing the construction of the free vector lattice generated by E. After this, we equip the free vector lattice with the norm

$$\begin{aligned} \rho _{\mathcal {C}}(f):= \sup \bigl \{\Vert \widehat{T}f\Vert _X: X \in {\mathcal {C}},\ T:E\rightarrow X \text { is a linear contraction}\bigr \}, \end{aligned}$$

where $\widehat{T}$ denotes the unique lattice homomorphic extension of T, and define the free Banach lattice generated by E associated to ${\mathcal {C}}$ as the completion of $(\textrm{FVL}[E],\rho _{\mathcal {C}})$. This definition yields a Banach lattice $\textrm{FBL}^\mathcal {C}[E]$ containing an isometric copy of E with the ability to uniquely extend any operator $T:E\rightarrow X\in \mathcal {C}$ to a lattice homomorphism of the same norm. In Sect. 2.2 we “close” the class $\mathcal {C}$ by constructing a class $\overline{\mathcal {C}}\supseteq \mathcal {C}$ such that $\textrm{FBL}^\mathcal {C}[E]=\textrm{FBL}^\mathcal {\overline{C}}[E]\in \overline{\mathcal {C}}$. This then allows us to identify $\textrm{FBL}^\mathcal {C}[E]$ as a universal object in this new class. In Sect. 2.3 we find an explicit formula for the norm of $\textrm{FBL}^\mathcal {C}[E]$ when $\mathcal {C}$ consists of a single r.i. space. This, in particular, allows us to identify $\textrm{FBL}^\mathcal {C}[E]$ as a space of weak$^*$-continuous positively homogeneous functions on $B_{E^*}.$ In Sect. 2.4 we characterize when continuous injective lattice homomorphisms extend injectively to the completion of a normed lattice, elucidating the difficulty in representing $\textrm{FBL}^\mathcal {C}[E]$ inside of $C(B_{E^*})$.

In Sect. 3 we combine the results of Sect. 2 with the Maurey and Pisier factorization theorems to reproduce the norm (1.3) for ${\textrm{FBL}}^{(p)}[E]$ and show that (1.4) is a $(1-\frac{1}{p})^{\frac{1}{p}-1}$-equivalent lattice norm for $\textrm{FBL}^{(p,\infty )}[E]$ with upper p-estimate constant 1. We also characterize in Theorem 3.8 all of the $\mathcal {C}$ for which $\textrm{FBL}^\mathcal {C}[E]={\textrm{FBL}}^{(p)}[E]$ and all of the $\mathcal {C}'$ for which $\textrm{FBL}^{\mathcal {C}'}[E]\approx \textrm{FBL}^{(p,\infty )}[E]$.

Section 4 is devoted to the subspace problem for $\textrm{FBL}^{(p,\infty )}$; that is, the problem of characterizing those embeddings $\iota : F\hookrightarrow E$ which induce a lattice embedding $\overline{\iota }: \textrm{FBL}^{(p,\infty )}[F]\hookrightarrow \textrm{FBL}^{(p,\infty )}[E]$. This is achieved in Sect. 4.1 by utilizing a novel push-out argument. A benefit of our argument is that it applies equally well to ${\textrm{FBL}}^{(p)}$. However, a priori, it yields a slightly different solution to the subspace problem for ${\textrm{FBL}}^{(p)}$ than the one in [31, Theorem 3.7]. To reconcile this, we prove the injectivity of $\ell ^p$ in the class of p-convex Banach lattices, which immediately yields the equivalence of the two solutions, up to constants. On the other hand, in Sect. 4.3 we show that $\ell ^{p,\infty }$ is not injective in the class of Banach lattices with upper p-estimates, adding an additional novelty to our solution to the subspace problem for $\textrm{FBL}^{(p,\infty )}[E]$.

2 The free Banach lattice associated to a class of Banach lattices

In this section we define and investigate the free Banach lattice $\textrm{FBL}^{\mathcal {C}}[E]$ generated by a Banach space E associated to a class $\mathcal {C}$ of Banach lattices. We begin in Sect. 2.1 with a brief review of the construction of the free vector lattice. Then, in Sect. 2.2, we define $\textrm{FBL}^{\mathcal {C}}[E]$ as the completion of $\textrm{FVL}[E]$ under a certain norm, identify an associated class $\overline{\mathcal {C}}\supseteq \mathcal {C}$ of Banach lattices, and prove the universality of $\textrm{FBL}^{\mathcal {C}}[E]$ in the class $\overline{\mathcal {C}}$. In Sect. 2.3, an explicit formula for the norm of $\textrm{FBL}^{\mathcal {C}}[E]$ is obtained when $\mathcal {C}$ consists of a single rearrangement invariant Banach lattice. This, in particular, yields a continuous injection $\textrm{FBL}^\mathcal {C}[E]\hookrightarrow C(B_{E^*})$ for such $\mathcal {C}$. Finally, in Sect. 2.4 we supplement the above results with a characterization of the normed lattices for which every continuous lattice homomorphic injection extends injectively to the completion.

2.1 Review of the free vector lattice

We start this section by briefly reviewing the main aspects of the construction of the free vector lattice generated by a vector space. These types of arguments have been known since the beginning of the study of such objects [11, 13].

Let E be a Banach space and endow $B_{E^*}$ with the weak$^*$-topology. For any $x\in E$, the function $\delta _x:B_{E^*}\rightarrow {\mathbb {R}}$ given by $\delta _x(x^*) = x^*(x)$ belongs to the space $C(B_{E^*})$ of continuous functions on $B_{E^*}$. Define a sequence of subspaces of $C(B_{E^*})$ as follows:

$$\begin{aligned} E_0= \{\delta _x: x\in E\};\ E_n = {\text {span}}(E_{n-1}\cup |E_{n-1}|) \text { if } n\in {\mathbb {N}}. \end{aligned}$$

Here, for a set A in a vector lattice X, we are denoting $|A|=\{|x|:x\in A\}$. Let $\textrm{FVL}[E]= \bigcup ^\infty _{n=0}E_n$ and note that $E_0$ is a subspace of $C(B_{E^*})$ since $\sum ^n_{k=1}a_k\delta _{x_k} = \delta _{\sum ^n_{k=1}a_kx_k}$.

Proposition 2.1

$\textrm{FVL}[E]$ is the smallest vector sublattice of $C(B_{E^*})$ containing $E_0$.

Proof

Obviously, ${\text {span}}(E_{n-1})\subseteq E_n$ for all $n\in {\mathbb {N}}$. Hence, $\textrm{FVL}[E]$ is a vector subspace of $C(B_{E^*})$ containing $\{\delta _x: x\in E\}$. Assume that $f\in \textrm{FVL}[E]$. By definition, $f\in E_n$ for some $n\in {\mathbb {N}}$. Thus, $|f| \in |E_n| \subseteq E_{n+1} \subseteq \textrm{FVL}[E]$, so that $\textrm{FVL}[E]$ is a vector sublattice of $C(B_{E^*})$.

Assume that G is a vector sublattice of $C(B_{E^*})$ containing $\{\delta _x: x\in E\}=E_0$. If $E_{n-1}\subseteq G$ for some $n\in {\mathbb {N}}$, then $E_{n-1} \cup |E_{n-1}|\subseteq G$. Hence, $E_n \subseteq G$. By induction, $\textrm{FVL}[E] = \bigcup ^\infty _{n=0}E_n \subseteq G$. $\square$

It follows from the expression for $\textrm{FVL}[E]$ that each $f\in \textrm{FVL}[E]$ is positively homogeneous in the following sense: $f(\lambda x^*) = \lambda f(x^*)$ if $x^*\in B_{E^*}$ and $0\le \lambda \le 1$. Define $\phi := \phi _E:E \rightarrow \textrm{FVL}[E]$ by $\phi (x) = \delta _x$. Clearly, $\phi$ is a linear operator.

Proposition 2.2

Let $f\in \textrm{FVL}[E]$. There is a finite subset A of E so that f belongs to the sublattice of $\textrm{FVL}[E]$ generated by $\{\delta _x: x\in A\}$.

Proof

The assertion is trivial if $f\in E_0$. Assume that it holds whenever $f\in E_{n-1}$. Let $f\in E_n$ and represent $f = \sum ^r_{k=1}(a_kf_k + b_k|f_k|)$ where $a_k, b_k\in {\mathbb {R}}$ and $f_k\in E_{n-1}$. By the inductive hypothesis, there is a finite set A so that $f_k, 1\le k\le r$, belong to the sublattice generated by $\{\delta _x:x\in A\}$. By construction, f belongs to the same sublattice.$\square$

An alternative way to prove Propositions 2.1 and 2.2 is to utilize [2, Exercise 8, p. 204]. The following statement is implicit in [31, Theorem 2.1], but we include the details for the convenience of the reader.

Proposition 2.3

$(\textrm{FVL}[E],\phi )$ is the free vector lattice generated by E in the following sense: If X is an Archimedean vector lattice and $T:E\rightarrow X$ is a linear operator then there is a unique linear vector lattice homomorphism $\widehat{T}: \textrm{FVL}[E] \rightarrow X$ such that $\widehat{T}\phi = T$.

Proof

Let A be a finite set in E and let $L_A$ be the sublattice of $C(B_{E^*})$ generated by $\{\delta _x: x\in A\}$. Let $I=I_A$ be the principal lattice ideal in X generated by $\sum _{x\in A}|Tx|$. By standard theory, there is a compact Hausdorff space K and an injective vector lattice homomorphism $i:I \rightarrow C(K)$. For each $t\in K$, define $\ell _t: \text {span}\, A \rightarrow {\mathbb {R}}$ by $\ell _t(\sum _{x\in A}a_xx) = \sum _{x\in A}a_x(iTx)(t)$. Note that if $\sum _{x\in A}a_xx=0$ then $\sum _{x\in A}a_xTx = 0$. This implies that $\sum _{x\in A}a_xiTx=0$ and thus $\ell _t(\sum _{x\in A}a_xx) = 0$. Hence, $\ell _t$ is a well-defined linear functional on $\text {span}\, A$. Since $\text {span}\, A$ is a finite dimensional subspace of E, $\ell _t$ is bounded. Let $x_t^*\in E^*$ be a Hahn-Banach extension of $\ell _t$.

Suppose that $f\in \textrm{FVL}[E]$. By Proposition 2.2, there exists a finite set $A = \{x_1,\dots , x_n\}$ in E so that $f\in L_A$. We can therefore write $f = G(\delta _{x_1},\dots , \delta _{x_n})$ where G is a finite sequence of operations of taking linear combinations and $|\cdot |$. Define $\widehat{T}f = G(Tx_1,\dots ,Tx_n)\in X$. To see that $\widehat{T}$ is well-defined, we have to show that $\widehat{T}(\sum a_k f_k)=0$ if $\sum a_kf_k=0$, for any finite sum. We may assume that $f_k\in L_A$ for all k. In particular, we have $\widehat{T}f_k\in I_A$. Write $f_k = G_k(\delta _{x_1},\dots , \delta _{x_n})$ for some lattice-linear $G_k$ and note that for any $t\in K$,

$$\begin{aligned} \left[ i\widehat{T}\left( \sum a_kf_k\right) \right] (t)&= \sum a_k\, iG_k(Tx_1,\dots , Tx_n)(t)\\&=\sum a_kG_k(iTx_1(t),\dots , iTx_n(t))\\&= \sum a_k G_k(x^*_t(x_1),\dots , x^*_t(x_n))\\&= \sum a_k G_k(\delta _{x_1}(x^*_t),\dots , \delta _{x_n}(x^*_t))\\&= \sum a_kf_k(x^*_t). \end{aligned}$$

Thus, $\sum a_kf_k =0$ implies $i\widehat{T}(\sum a_kf_k) =0$. Since $i:I_A\rightarrow C(K)$ is injective, $\widehat{T}(\sum a_kf_k) =0$. This proves that $\widehat{T}$ is well-defined. It follows from its definition that $\widehat{T}$ is a linear vector lattice homomorphism.

It is easy to see that for any $x\in E$, $\widehat{T}\phi x = \widehat{T}(\delta _x) = Tx$. Let $S:\textrm{FVL}[E]\rightarrow X$ be a vector lattice homomorphism so that $S\phi x = Tx$. Then for any $f = G(\delta _{x_1},\dots , \delta _{x_n})$ we must have

$$\begin{aligned} Sf = G(S\delta _{x_1}, \dots , S\delta _{x_n}) = G(Tx_1,\dots , Tx_n) = \widehat{T}f. \end{aligned}$$

This proves the uniqueness of $\widehat{T}$.$\square$

Remark 2.4

In this paper we are identifying $\textrm{FVL}[E]$ as a space of functions on $B_{E^*}$. This is in contrast to previous works where $\textrm{FVL}[E]$ was identified as a space of functions on $E^*$. By positive homogeneity, these interpretations are clearly equivalent.

2.2 Universality of the free Banach lattice associated to a class of Banach lattices

Let ${\mathcal {C}}$ be a (non-empty) class of Banach lattices. For instance, ${\mathcal {C}}$ could be the class of weak-$L^p$ spaces or the class of Banach lattices satisfying an upper p-estimate with constant 1. Given a Banach space E, define a norm $\rho _{\mathcal {C}}$ on $\textrm{FVL}[E]$ by

$$\begin{aligned} \rho _{\mathcal {C}}(f) = \sup \bigl \{\Vert \widehat{T}f\Vert _X: X \in {\mathcal {C}},\ T:E\rightarrow X \text { is a linear contraction}\bigr \}. \end{aligned}$$

Clearly, $\rho _\mathcal {C}$ is a lattice norm on $\textrm{FVL}[E]$. The free Banach lattice generated by E associated to ${\mathcal {C}}$ is the completion of $(\textrm{FVL}[E],\rho _{\mathcal {C}})$ and is denoted by $\textrm{FBL}^{\mathcal {C}}[E]$. It is easy to see that $\phi _E: E\rightarrow \textrm{FBL}^{\mathcal {C}}[E]$ is a linear isometric embedding. Define an enlarged class of Banach lattices $\overline{{\mathcal {C}}}$ so that $Y\in \overline{{\mathcal {C}}}$ if and only if Y is lattice isometric to a closed sublattice of a Banach lattice of the form $(\bigoplus _{\gamma \in \Gamma }X_\gamma )_\infty$, where $X_\gamma \in {\mathcal {C}}$ for all $\gamma$. Note that we do not require that the $X_\gamma$’s be distinct.

Proposition 2.5

Suppose that $Y\in \overline{{\mathcal {C}}}$ and $T:E\rightarrow Y$ is a bounded linear operator. Then $\widehat{T}:(\textrm{FVL}[E],\rho _{\mathcal {C}})\rightarrow Y$ is a vector lattice homomorphism such that $\Vert \widehat{T}\Vert = \Vert T\Vert$.

Proof

Since Y is an Archimedean vector lattice, $\widehat{T}:\textrm{FVL}[E]\rightarrow Y$ is a vector lattice homomorphism. Note that $\phi _E$ is an isometric embedding and $\widehat{T}\phi _E = T$. Hence, $\Vert T\Vert \le \Vert \widehat{T}\Vert$. On the other hand, suppose that Y is a closed sublattice of X, where $X = (\bigoplus _{\gamma \in \Gamma }X_\gamma )_\infty$, with $X_\gamma \in {\mathcal {C}}$ for all $\gamma$. Let $\pi _\gamma$ be the projection from X onto $X_\gamma$. Then $\pi _\gamma T:E\rightarrow X_\gamma$ is a bounded linear operator with $\Vert \pi _\gamma T\Vert \le \Vert T\Vert$. By definition of $\rho _{\mathcal {C}}$, $\Vert \widehat{\pi _\gamma T} f\Vert \le \Vert T\Vert \, \rho _{\mathcal {C}}(f)$ for any $f\in \textrm{FVL}[E]$. As $\pi _\gamma$ is a lattice homomorphism, by the uniqueness of $\widehat{\pi _\gamma T}$, we have $\widehat{\pi _\gamma T} = \pi _\gamma \widehat{T}$. Thus, $\Vert \widehat{T}f\Vert = \sup _\gamma \Vert \pi _\gamma \widehat{T}f\Vert \le \Vert T\Vert \, \rho _{\mathcal {C}}(f)$ for any $f\in \textrm{FVL}[E]$. This shows that $\Vert \widehat{T}\Vert \le \Vert T\Vert$.$\square$

Obviously, the operator $\widehat{T}$ in Proposition 2.5 extends to a lattice homomorphism from $\textrm{FBL}^{\mathcal {C}}[E]$ to Y, still denoted by $\widehat{T}$.

Corollary 2.6

Let E be a Banach space. For any non-empty class ${\mathcal {C}}$ of Banach lattices, the norms $\rho _{\mathcal {C}}$ and $\rho _{\overline{{\mathcal {C}}}}$ agree on $\textrm{FVL}[E]$. In particular, $\textrm{FBL}^{\mathcal {C}}[E] = \textrm{FBL}^{\overline{{\mathcal {C}}}}[E]$ as Banach lattices.

Proof

Since ${\mathcal {C}}\subseteq \overline{{\mathcal {C}}}$ it is clear that $\rho _{\mathcal {C}}\le \rho _{\overline{{\mathcal {C}}}}$. On the other hand, suppose that $f\in \textrm{FVL}[E]$ and $\varepsilon >0$ are given. There exists $Y\in \overline{{\mathcal {C}}}$ and a linear contraction $T: E\rightarrow Y$ such that $\rho _{\overline{{\mathcal {C}}}}(f) \le (1+\varepsilon ) \Vert \widehat{T}f\Vert$. By Proposition 2.5, $\Vert \widehat{T}f\Vert \le \Vert T\Vert \, \rho _{\mathcal {C}}(f)= \rho _{\mathcal {C}}(f)$. It follows that $\rho _{\overline{{\mathcal {C}}}} \le \rho _{\mathcal {C}}$. $\square$

Proposition 2.7

The Banach lattice $\textrm{FBL}^{\mathcal {C}}[E]$ belongs to the class $\overline{{\mathcal {C}}}$.

Proof

For any $f\in \textrm{FVL}[E]$ and $n\in {\mathbb {N}}$, there exists $X(f,n)\in {\mathcal {C}}$ and a linear contraction $T = T_{f,n}:E\rightarrow X(f,n)$ so that $\rho _{\mathcal {C}}(f)\le (1+ \frac{1}{n})\Vert \widehat{T}f\Vert$. Let $X = (\bigoplus _{f\in \textrm{FVL}[E], n\in {\mathbb {N}}}X(f,n))_\infty$. Clearly, $X\in \overline{{\mathcal {C}}}$. Define $j:\textrm{FBL}^{\mathcal {C}}[E]\rightarrow X$ by $jg = (\widehat{T_{f,n}}g)_{f,n}$. Obviously, j is a vector lattice homomorphism. Let $g\in \textrm{FVL}[E]$. By definition, $\rho _{\mathcal {C}}(g) \ge \Vert \widehat{T_{f,n}}g\Vert$ for all (f, n). Hence, j is a linear contraction. On the other hand, $\Vert jg\Vert \ge \Vert \widehat{T_{g,n}}g\Vert \ge \frac{n}{n+1}\,\rho _{\mathcal {C}}(g)$ for all n, implying that $\Vert jg\Vert \ge \rho _{\mathcal {C}}(g)$. This shows that $j:\textrm{FVL}[E]\rightarrow X$ is a lattice isometry. Consequently, $j:\textrm{FBL}^{\mathcal {C}}[E]\rightarrow X$ is as well. $\square$

The above discussion motivates the question of finding necessary and sufficient conditions on classes of Banach lattices ${\mathcal {C}}$ and ${\mathcal {D}}$ so that $\rho _{\mathcal {C}}$ and $\rho _{\mathcal {D}}$ coincide, or at least, are equivalent. In Sect. 3.1, after some tools have been developed, we will be able to provide a complete solution to this question when ${\mathcal {C}}$ is either the class of p-convex Banach lattices with constant 1, or the class of Banach lattices satisfying an upper p-estimate with constant 1.

2.3 Rearrangement invariant spaces and the norm for $\textrm{FBL}^{\mathcal {C}}[E]$

In this subsection, let $(\Omega , \Sigma ,\mu )$ be a non-atomic $\sigma$-finite measure space and let X be a rearrangement invariant (r.i.) Banach function space on $\Omega$ in the sense of [10, Definition II.4.1]. To be specific, X is a vector lattice ideal of $L^0(\Omega ,\Sigma ,\mu )$, equipped with a complete lattice norm $\Vert \cdot \Vert$, so that $f \in X$, $g \in L^0(\Omega ,\Sigma ,\mu )$ and $\mu \{|f|> t\} = \mu \{|g| > t\}$ for all $t \ge 0$ imply that $g\in X$ and $\Vert f\Vert = \Vert g\Vert$. We further assume that X has the Fatou property: Whenever $f_n\in X$, $0\le f_n \uparrow f$ a.e. and $\sup \Vert f_n\Vert <\infty$ then $f\in X$ and $\Vert f\Vert = \sup _n\Vert f_n\Vert$.

Let ${\mathcal {C}}$ be the class consisting of X only. Given a Banach space E, the aim of this subsection will be to produce a formula for the norm $\rho _{\mathcal {C}}$ on $\textrm{FVL}[E]$ and show that $\textrm{FBL}^{\mathcal {C}}[E] \subseteq C(B_{E^*})$.

Given a sequence ${\mathcal {U}}= (U_n)$ of disjoint measurable subsets of $\Omega$ so that $0<\mu (U_n) < \infty$ for all n, we define for each $f\in L^0(\Omega ,\Sigma ,\mu )$ the conditional expectation over ${\mathcal {U}}$ by

$$\begin{aligned} P_{\mathcal {U}}f = \sum _n \frac{\int _{U_n}f\,d\mu }{\mu (U_n)}\chi _{U_n}. \end{aligned}$$

Note that the above sum has at most one non-zero term at any given point of $\Omega$, so is well-defined as long as $\int _{U_n}f\,d\mu$ is finite for every $n\in {\mathbb {N}}$. This will be the case when $f\in X$.

Proposition 2.8

Let ${\mathcal {U}}= (U_n)$ be a sequence of disjoint measurable subsets of $\Omega$ so that $\mu (U_n) < \infty$ for all n. Then $P_{\mathcal {U}}f\in X$ if $f\in X$ and $P_{\mathcal {U}}$ is a positive contraction on X.

Proof

First note that, since X is a Banach function space, $\int _{U_n}f\,d\mu$ is finite for every $n\in {\mathbb {N}}$ (see property 5 in [10, Definition I.1.1]), so $P_{\mathcal {U}}f$ is well-defined. It is straightforward to check that $P_{\mathcal {U}}$ is a positive contraction from $L^1(\Omega ,\Sigma ,\mu )$ to $L^1(\Omega ,\Sigma ,\mu )$ and from $L^\infty (\Omega ,\Sigma ,\mu )$ to $L^\infty (\Omega ,\Sigma ,\mu )$. Thus, $P_{\mathcal {U}}$ is an admissible operator for the compatible pair $(L^1(\Omega ,\Sigma ,\mu ), L^\infty (\Omega ,\Sigma ,\mu ))$. Moreover, since X is a r.i. space over a non-atomic $\sigma$-finite (in particular, resonant) measure space, it follows from [10, Theorem II.2.2] that X is an exact interpolation space, so $P_{\mathcal {U}}:X\rightarrow X$ is a positive contraction.$\square$

Lemma 2.9

Let $h_1,\dots , h_n\in X$ and set $h = \sum ^n_{k=1}|h_k|$. Given $\varepsilon > 0$, there is a sequence ${\mathcal {U}}= (U_n)$ of disjoint measurable subsets of $\Omega$ so that $0<\mu (U_n) < \infty$ for all n and $|h_k - {\mathcal {P}}_{\mathcal {U}}h_k| \le \varepsilon h$, $1\le k\le n$.

Proof

We may assume that $\varepsilon <1$. Let $V_i = \{\varepsilon ^{i} \le h < \varepsilon ^{i-1}\}$. Then $(V_i)_{i\in {\mathbb {Z}}}$ is a sequence of disjoint measurable subsets of $\Omega$ so that $h = h_k = 0$ on $(\bigcup _i V_i)^c$ for any k. Since the measure space is $\sigma$-finite, for each i there is an at most countable measurable partition $(U_{ij})_{j\in J_i}$ of $V_i$ with $0<\mu (U_{ij})<\infty$ and $b_{kij} \in {\mathbb {R}}$ so that

$$\begin{aligned} \left| h_k\chi _{V_i} - \sum _{j\in J_i} b_{kij}\chi _{U_{ij}}\right| \le \frac{\varepsilon ^{i+1}}{2},\ i\in {\mathbb {Z}},\ 1\le k\le n. \end{aligned}$$

Thus, for each $1\le k\le n$ we have

$$\begin{aligned} \left| h_k - \sum _{i,j} b_{kij}\chi _{U_{ij}}\right| \le \sum _i \frac{\varepsilon ^{i+1}}{2}\chi _{V_i}\le \frac{\varepsilon h}{2} \end{aligned}$$

(2.1)

and

$$\begin{aligned} \left| \frac{1}{\mu (U_{ij})}\int _{U_{ij}}h_k \,d\mu - b_{kij} \right| = \left| \frac{1}{\mu (U_{ij})}\int _{U_{ij}}h_k - b_{kij}\chi _{U_{ij}}\,d\mu \right| \le \frac{\varepsilon ^{i+1}}{2}. \end{aligned}$$

Take ${\mathcal {U}}= (U_{ij})_{i\in {\mathbb {Z}}, j\in J_i}$. Note that ${\mathcal {P}}_{\mathcal {U}}h_k = 0$ on the set $(\bigcup _{ij}U_{ij})^c$. For $1\le k\le n$,

$$\begin{aligned} \left| {\mathcal {P}}_{\mathcal {U}}h_k - \sum _{i,j}b_{kij}\chi _{U_{ij}}\right| \le \sum _{i,j}\frac{\varepsilon ^{i+1}}{2}\chi _{U_{ij}} = \sum _i \frac{\varepsilon ^{i+1}}{2}\chi _{V_i}\le \frac{\varepsilon h}{2}. \end{aligned}$$

(2.2)

Summing (2.1) and (2.2) gives the desired result.$\square$

Let $I = \{(m,r) \in {\mathbb {N}}^2: \frac{m}{r} \le \mu (\Omega )\}$. If $(m,r)\in I$, take any sequence $(V_i)^m_{i=1}$ of disjoint measurable subsets of $\Omega$ such that $\mu (V_i) = r^{-1}$, $1\le i\le m$. Define a norm $\rho _{mr}$ on ${\mathbb {R}}^m$ by

$$\begin{aligned} \rho _{mr}(a_1,\dots ,a_m) = \left\| \sum ^m_{i=1}a_i\chi _{V_i}\right\| _X. \end{aligned}$$

Let E be a Banach space and define $\rho :C(B_{E^*})\rightarrow [0,\infty ]$ by

$$\begin{aligned} \rho (f) = \sup \rho _{mr}(f(x^*_1),\dots , f(x^*_m)), \end{aligned}$$

where the supremum is taken over all $(m,r)\in I$ and $x^*_1,\dots , x^*_m\in E^*$ which satisfy the constraint $\sup _{x\in B_E}\rho _{mr}(x_1^*(x),\dots , x_m^*(x)) \le 1$. By definition, $\rho (\delta _x) = \Vert x\Vert$ for any $x\in E$. Moreover, it is easy to check that $\rho$ is a complete lattice norm on the vector lattice $\{f\in C(B_{E^*}): \rho (f) < \infty \}$ which contains $\textrm{FVL}[E]$. For our next result, recall that we are considering the case ${\mathcal {C}}=\{X\}$.

Theorem 2.10

The norms $\rho$ and $\rho _{\mathcal {C}}$ are equal on $\textrm{FVL}[E]$. In particular, $\textrm{FBL}^{\mathcal {C}}[E] \subseteq \{f\in C(B_{E^*}): \rho (f) < \infty \}$.

Proof

Suppose that $f\in \textrm{FVL}[E]$. There are $x_1,\dots , x_n\in E$ and a lattice-linear function G so that $f = G(\delta _{x_1},\dots , \delta _{x_n})$. Let $\varepsilon >0$ be given. There are $(m,r)\in I$ and $x^*_1,\dots , x^*_m\in E^*$ so that $\sup _{x\in B_E}\rho _{mr}(x_1^*(x),\dots , x_m^*(x)) \le 1$ and

$$\begin{aligned} \rho (f) \le \rho _{mr}(f(x^*_1),\dots , f(x^*_m)) + \varepsilon = \left\| \sum ^m_{i=1}f(x^*_i)\chi _{V_i}\right\| _X + \varepsilon , \end{aligned}$$

where $(V_i)^m_{i=1}$ is the sequence of disjoint measurable subsets of $\Omega$ chosen above, so that $\mu (V_i) = r^{-1}$, $1\le i\le m$. Define a linear operator $S:\textrm{FVL}[E]\rightarrow X$ by $S g = \sum ^m_{i=1}g(x^*_i)\chi _{V_i}$. It is clear that S is a lattice homomorphism. By definition of $\rho$, if $\rho (g) \le 1$, then $\rho _{mr}(g(x^*_1),\dots , g(x^*_m)) \le 1$ and thus $\Vert Sg\Vert \le 1$. Let $T = S\phi _E:E \rightarrow X$. Then $\Vert T\Vert \le 1$ and hence

$$\begin{aligned} \rho _{\mathcal {C}}(f) \ge \Vert \widehat{T}f\Vert = \Vert \widehat{S\phi _E}f\Vert = \Vert G(S\delta _{x_1},\dots ,S\delta _{x_n})\Vert = \Vert Sf\Vert , \end{aligned}$$

where the last step holds since S is a lattice homomorphism. Thus, $\rho _{\mathcal {C}}(f) \ge \rho (f) -\varepsilon$ for any $\varepsilon >0$. Therefore, $\rho _{\mathcal {C}}(f) \ge \rho (f)$.

Conversely, given $\varepsilon > 0$, there exists a linear contraction $T:E\rightarrow X$ so that $\rho _{\mathcal {C}}(f) \le \Vert \widehat{T}f\Vert + \varepsilon$. Let $h_k = Tx_k$, $1\le k\le n$, and $h = \sum ^n_{k=1}|h_k|$. By Lemma 2.9, there is a sequence ${\mathcal {U}}= (U_i)$ of disjoint measurable subsets of $\Omega$ so that $0<\mu (U_i)<\infty$ for all i and $|h_k-{\mathcal {P}}_{\mathcal {U}}h_k| \le \varepsilon h$, $1\le k \le n$. There is a constant $C_G < \infty$, depending only on G, so that

$$\begin{aligned} |\widehat{T}f - G({\mathcal {P}}_{\mathcal {U}}h_1,\dots , {\mathcal {P}}_{\mathcal {U}}h_n)| = |G(h_1,\dots , h_n) - G({\mathcal {P}}_{\mathcal {U}}h_1,\dots , {\mathcal {P}}_{\mathcal {U}}h_n)| \le C_G\varepsilon h. \end{aligned}$$

Express ${\mathcal {P}}_{\mathcal {U}}h_k$ as $\sum _i a_{ki}\chi _{U_i}$, where $T^*\chi _{U_i} = y^*_i$ and

$$\begin{aligned} a_{ki} = \frac{1}{\mu (U_i)}\int _{U_i}h_k\,d\mu =\frac{y^*_i(x_k)}{\mu (U_i)}. \end{aligned}$$

Then

$$\begin{aligned} G({\mathcal {P}}_{\mathcal {U}}h_1,\dots , {\mathcal {P}}_{\mathcal {U}}h_n)&= G(\sum _ia_{1i}\chi _{U_i},\dots , \sum _ia_{ni}\chi _{U_i})\\&= \sum _i G(a_{1i},\dots ,a_{ni})\chi _{U_i} \\&= \sum _i G(\delta _{x_1},\dots , \delta _{x_n})(y^*_i)\,\frac{\chi _{U_i}}{\mu (U_i)} \\ &= \sum _i \frac{f(y^*_i)}{\mu (U_i)}\,\chi _{U_i}. \end{aligned}$$

Since X has the Fatou property, there exists $l\in {\mathbb {N}}$ so that

$$\begin{aligned} \left\| \sum ^l_{i=1} \frac{f(y^*_i)}{\mu (U_i)}\,\chi _{U_i}\right\| > \Vert G({\mathcal {P}}_{\mathcal {U}}h_1,\dots , {\mathcal {P}}_{\mathcal {U}}h_n)\Vert -\varepsilon \ge \Vert \widehat{T}f\Vert - (1+ C_G\Vert h\Vert )\varepsilon . \end{aligned}$$

Using the Fatou property once more, choose disjoint Lebesgue measurable sets $V_1,\dots , V_l$ so that $V_i \subseteq U_i$, $\mu (V_i) \in {\mathbb {Q}}$ and

$$\begin{aligned} \Vert \widehat{T}f\Vert - (1+ C_G\Vert h\Vert )\varepsilon < \left\| \sum ^l_{j=1}\frac{f(y^*_i)}{\mu (U_i)}\,\chi _{V_i}\right\| . \end{aligned}$$

Write $\mu (V_i) =\frac{j_i}{r}$, where $j_i, r\in {\mathbb {N}}$. Decompose each $V_i$ as a disjoint union $V_i = \bigcup ^{j_i}_{s=1}V_{is}$, with $\mu (V_{is}) = r^{-1}$. Take $j_0 = \sum ^l_{i=1}j_s$. It follows that $(j_0,r) \in I$. Consider the sequence where each $\frac{y^*_i}{\mu (U_i)}$ is repeated $j_i$ times. For any $x\in B_E$, let b be the element of ${\mathbb {R}}^{j_0}$ so that each $\frac{y^*_i(x)}{\mu (U_i)}$ occurs $j_i$ times. Then

$$\begin{aligned} \rho _{j_0r}(b) = \left\| \sum ^l_{i=1}\frac{y^*_i(x)}{\mu (U_i)}\,\chi _{V_i}\right\| \le \Vert {\mathcal {P}}_{\mathcal {U}}(Tx)\Vert \le \Vert T\Vert =1. \end{aligned}$$

Recall that elements of $\textrm{FVL}[E]$ have a homogeneity property given after Proposition 2.1. It follows from the definition of $\rho$ that, taking a to be the element of ${\mathbb {R}}^{j_0}$ where each $\frac{f(y^*_i)}{\mu (U_i)}$ occurs $j_i$ times,

$$\begin{aligned} \rho (f)&\ge \rho _{j_0r}(a) = \left\| \sum ^l_{i=1}\sum ^{j_i}_{s=1} \frac{f(y^*_i)}{\mu (U_i)}\,\chi _{V_{is}}\right\| = \left\| \sum ^l_{i=1}\frac{f(y^*_i)}{\mu (U_i)}\,\chi _{V_i}\right\| \\&> \Vert \widehat{T}f\Vert - (1+ C_G\Vert h\Vert )\varepsilon \ge \rho _{\mathcal {C}}(f) - (2+ C_G\Vert h\Vert )\varepsilon . \end{aligned}$$

Thus, $\rho (f) \ge \rho _{\mathcal {C}}(f)$.$\square$

The above construction of $\textrm{FBL}^\mathcal {C}[E]$ motivates the following question.

Question 2.11

It is well-known that the properties of ${\textrm{FBL}}^{(p)}[E]$ depend heavily on whether p is finite or infinite. Can the Banach lattices $\textrm{FBL}^\mathcal {C}[E]$ be used to interpolate such properties? For example, if $(e_k)$ is the unit vector basis of $\ell ^2,$ then by [31, Propositions 5.14 and 6.4] $(|\delta _{e_k}|)$ gives a copy of $\ell ^1$ in ${\textrm{FBL}}^{(p)}[\ell ^2]$ when $p<\infty$ and a copy of $\ell ^2$ when $p=\infty$. Can we find for each $r\in (1,2)$ a class $\mathcal {C}_r$ so that $(|\delta _{e_k}|)$ behaves like $\ell ^r$ in $\textrm{FBL}^{\mathcal {C}_r}[\ell ^2]$?

2.4 Injective homomorphisms need not extend to the norm completion

It is not obvious in general that a representation of a normed free object can be extended to its completion. In particular, in Sect. 2.2 we defined $\textrm{FBL}^\mathcal {C}[E]$ as the completion of $(\textrm{FVL}[E],\rho _\mathcal {C})$, for which there is a continuous lattice homomorphic injection $(\textrm{FVL}[E],\rho _\mathcal {C})\hookrightarrow C(B_{E^*})$. However, it is not clear in general whether the continuous extension to $\textrm{FBL}^\mathcal {C}[E]$ of this canonical inclusion will still be an injection. This is known to be true when $\textrm{FBL}^\mathcal {C}[E]={\textrm{FBL}}^{(p)}[E]$ with $1\le p\le \infty$, that is, when ${\mathcal {C}}$ is the class of all Banach lattices [8] or when it is the class of p-convex Banach lattices with p-convexity constant 1 [24] (see Theorem 3.8 for a complete characterization of the classes such that $\textrm{FBL}^\mathcal {C}[E]={\textrm{FBL}}^{(p)}[E]$). Theorem 2.10 also provides new cases where $\textrm{FBL}^\mathcal {C}[E]$ is fully contained in $C(B_{E^*})$, but in all of these situations the argument heavily relies on an explicit representation of the norm $\rho _{\mathcal {C}}$. This is the reason why the question of finding a function lattice representation for $\textrm{FBL}^{(p,\infty )}[E]$ asked in [24, Remark 6.2] has remained unanswered until the present. In [17] it is shown that under certain order continuity assumptions (which are not fulfilled in our situation) continuous lattice homomorphic injections retain their injectivity after norm completion. Moreover, it is wrongly claimed in [36, Lemma 14] that any continuous lattice homomorphic injection from a normed vector lattice to a Banach lattice lifts injectively to the completion. Note if this were the case, it would provide a straightforward solution to the question above. In this subsection, we give a complete characterization of when continuous injective lattice homomorphisms extend injectively to the completion, and, in particular, show that this need not always be the case. Afterwards, in Sect. 3, we will provide an explicit representation for the norm of $\textrm{FBL}^{(p,\infty )}[E]$, solving affirmatively the question raised in [24, Remark 6.2].

Throughout this subsection X will be a vector lattice equipped with a lattice norm and Y will be a Banach lattice. We denote the norm completion of X by $\widehat{X}$ and consider polar sets with respect to the duality $\langle \widehat{X},X^*\rangle$: for $A\subseteq \widehat{X}$, set $A^\circ =\{x^*\in X^*:\langle a,x^*\rangle \le 1,\,\forall a\in A\}.$

Theorem 2.12

Let X be a normed vector lattice equipped with a lattice norm and let $\widehat{X}$ be its norm completion. The following are equivalent.

• (1) For any Banach lattice Y, any bounded linear injective lattice homomorphism $T:X\rightarrow Y$ extends to a bounded linear injective operator $\widehat{T}:\widehat{X}\rightarrow Y$.

• (2) For any $0 < x \in \widehat{X}$, $I_x\cap X\ne \{0\}$, where $I_x$ is the closed ideal in $\widehat{X}$ generated by x.

Proof

(1)$\implies$(2): Suppose that there exists $0<x\in \widehat{X}$ so that $I_x\cap X = \{0\}$. By [35, Proposition II.4.7], $(I_x)^\circ$ is an ideal in $X^*$. Define $\rho :X\rightarrow {\mathbb {R}}$ by

$$\begin{aligned} \rho (u) = \sup _{x^*\in (I_x)^\circ \cap B_{X^*}}|\langle u, x^*\rangle |. \end{aligned}$$

Since $(I_x)^\circ \cap B_{X^*}$ is a solid subset of $B_{X^*}$, $\rho$ is a lattice semi-norm on X such that $\rho (\cdot ) \le \Vert \cdot \Vert$. If $0 < u\in X$, then $u\notin I_x$. Since $I_x$ is a closed subspace of $\widehat{X}$, there exists $x^*\in (I_x)^\circ \cap B_{X^*}$ such that $\langle u,x^*\rangle \ne 0$. Hence, $\rho (u) >0$. This shows that $\rho$ is a norm on X.

Denote by $X_\rho$ the vector lattice X normed by $\rho$ and let $i:X\rightarrow X_\rho \subseteq \widehat{X_\rho }$ be the formal identity, where $\widehat{X_\rho }$ is the norm completion of $X_\rho$. By construction, i is a bounded linear injective lattice homomorphism. By (1), i extends to an injective bounded linear operator $\widehat{i}:\widehat{X}\rightarrow \widehat{X_\rho }$. Denote by $\widehat{\rho }$ the norm on $\widehat{X_\rho }$. By injectivity of $\widehat{i}$, $\widehat{i}x\ne 0$. Let $c = \widehat{\rho }(\widehat{i}x) > 0$. There exists $(x_n)$ in X that converges to x in $\widehat{X}$. By continuity, $(ix_n)$ converges in $\widehat{X_\rho }$ to $\widehat{i}x$. Choose N large enough so that $\widehat{\rho }(ix_N-\widehat{i}x) < \frac{c}{2}$. Then $\rho (ix_N) > \frac{c}{2}$. By definition of $\rho$, there exists $x^*\in (I_x)^\circ \cap B_{X^*}$ so that $|\langle ix_N, x^*\rangle | > \frac{c}{2}$. By definition of $\rho$ again, $x^*$ defines a bounded linear functional on $X_\rho$ with norm $\rho ^*(x^*)\le 1$. Hence,

$$\begin{aligned} |\langle x,x^*\rangle | = |\langle \widehat{i}x,x^*\rangle | \ge |\langle ix_N,x^*\rangle | - \widehat{\rho }(ix_N-\widehat{i}x)\cdot \rho ^*(x^*) > \frac{c}{2} - \frac{c}{2} = 0. \end{aligned}$$

This is impossible since $x\in I_x$ and $x^*\in (I_x)^\circ$.

(2)$\implies$(1): Let $T:X\rightarrow Y$ be as in (1), and assume that its extension $\widehat{T}$ is not injective. There exists some $x\in \widehat{X}$, $x\ne 0$, such that $\widehat{T}x=0$. Since $\widehat{T}$ is a lattice homomorphism, we can assume that $x>0$. Let $I_x$ be the closed ideal in $\widehat{X}$ generated by x. By (2), there is a non-zero element $z\in I_x\cap X$. Since $I_x$ is the closure of the ideal generated by x, there exists a sequence $(z_n)\subseteq \widehat{X}$ converging to z in $\widehat{X}$ and some positive scalars $(\lambda _n)$ such that $0\le |z_n|\le \lambda _n x$. Therefore,

$$\begin{aligned} 0\le |\widehat{T}z_n|\le \lambda _n \widehat{T}x=0 \end{aligned}$$

and

$$\begin{aligned} Tz=\widehat{T}z = \lim _n \widehat{T}z_n=0, \end{aligned}$$

so by the injectivity of T we conclude that z must be zero. This is a contradiction.$\square$

The following example taken from [12] illustrates the above situation.

Example 2.13

Let $X\subseteq c_0$ consist of all $(a_n)\in c_0$ such that there exists $m\in {\mathbb {N}}$ so that $a_n = \frac{a_1}{n}$ for all $n \ge m$. Clearly, X is a vector sublattice of $c_0$ with $e_n \in X$ if $n\ge 2$. Moreover, for any $n\ge 2$, $x_n: = e_1 + \sum ^\infty _{k=n}\frac{e_k}{k} \in X$ and $(x_n)$ converges to $e_1$ in $c_0$. Thus, $e_1\in \overline{X}$. It follows that $\overline{X} = c_0$, that is, the norm completion of X is $c_0$. Let $I\subseteq c_0$ be the span generated by $e_1$, which coincides with the ideal generated by $e_1$. Clearly $I\cap X=\{0\}$. On the other hand, if $P_1$ is the band projection onto I and T is the restriction of $id_{c_0}-P_1$ to the sublattice X, then $T:X\rightarrow c_0$ is an injective lattice homomorphism, but its extension $\widehat{T}=id_{c_0}-P_1: \widehat{X}=c_0 \rightarrow c_0$ vanishes on $e_1$, so it is not injective.

3 Applications to free p-convex and free $(p,\infty )$-convex Banach lattices

We now apply the results of Sects. 2.2 and 2.3 to the classes ${\mathcal {C}}_p$ and ${\mathcal {C}}_{p,\infty }$ consisting of all Banach lattices with p-convexity constant 1 and all Banach lattices with upper p-estimate constant 1, respectively. For this purpose, we recall the following factorization theorems.

Theorem 3.1

(Maurey’s Factorization Theorem) Let $1< p <\infty$, $A \subseteq L^1(\mu )$ and $0< M < \infty$. The following are equivalent.

• (1) For all finitely supported sequences $(\alpha _i)_{i\in I}$ of real numbers and $(f_i)_{i\in I}\subseteq A$,

  $$\begin{aligned} \left\| \left( \sum |\alpha _if_i|^p\right) ^{\frac{1}{p}}\right\| _{L^{1} (\mu )} \le M\left( \sum |\alpha _i|^p\right) ^{\frac{1}{p}}. \end{aligned}$$

• (2) There exists $g\in L^1(\mu )_+$, $\Vert g\Vert _{L^{1} (\mu )}= 1$, such that for any $f\in A$,

  $$\begin{aligned} \bigl \Vert \frac{f}{g}\bigr \Vert _{L^p(g\cdot \mu )} \le M. \end{aligned}$$

Theorem 3.2

(Pisier’s Factorization Theorem) Let $1< p <\infty$, $A \subseteq L^1(\mu )$, $0< M < \infty$ and $\gamma _p = (1-\frac{1}{p})^{\frac{1}{p}-1}$. Consider the following conditions.

• 1. For all finitely supported sequences $(\alpha _i)_{i\in I}$ of real numbers and $(f_i)_{i\in I}\subseteq A$,

  $$\begin{aligned} \bigl \Vert \bigvee |\alpha _if_i|\bigr \Vert _{L^{1} (\mu )} \le M\left( \sum |\alpha _i|^p\right) ^{\frac{1}{p}}. \end{aligned}$$

• 2. There exists $g\in L^1(\mu )_+$, $\Vert g\Vert _{L^{1} (\mu )}\le 1$, such that for any $f\in A$ and any $\mu$-measurable set U,

  $$\begin{aligned} \bigl \Vert f\chi _U\bigr \Vert _{L^{1} (\mu )} \le \gamma _p M\,\left( \int _Ug\,d\mu \right) ^{1-\frac{1}{p}}. \end{aligned}$$

Then (1)$\implies$(2).

Theorem 3.1 is very well-known. See, for example, [1, Chapter 7]. Theorem 3.2 is the implication (iii)$\implies$(i) in [33, Theorem 1.1]. Note that condition (2) in Theorem 3.2 is equivalent to

$$\begin{aligned} \bigl \Vert \frac{f}{g}\chi _U\Vert _{L^1 (g\cdot \mu )} \le \gamma _p M\bigl ((g\cdot \mu )(U)\bigr )^{1-\frac{1}{p}}. \end{aligned}$$

In particular, under the above conditions, $\bigl \Vert \frac{f}{g}\bigr \Vert _{L^{p,\infty } (g\cdot \mu )} \le \gamma _p M$ for all $f\in A$.

Recall that, given a measure space $(\Omega , \Sigma ,\mu )$, the space $L^{p,\infty }(\mu )$ is defined as the set of all measurable functions $h:\Omega \rightarrow {\mathbb {R}}$ such that the quasinorm

$$\begin{aligned} \sup _{t>0} t \mu (\{|h|>t\})^{\frac{1}{p}} \end{aligned}$$

is finite. Note, in particular, that every $h\in L^{p,\infty }(\mu )$ has $\sigma$-finite support. Unless otherwise specified, we will equip $L^{p,\infty }(\mu )$ with the norm

$$\begin{aligned} \Vert h\Vert _{L^{p,\infty }} = \sup \left\{ \mu (E)^{\frac{1}{p}-1}\int _E |h| d\mu : 0< \mu (E) <\infty \right\} . \end{aligned}$$

It is a standard fact (cf. [22, Exercise 1.1.12]) that the above norm is equivalent to the usual quasinorm on weak-$L^p$; one can also check that it is a lattice norm with upper p-estimate constant 1.

We will use the above factorization theorems to represent p-convex Banach lattices inside of infinity sums of $L^p$ spaces and Banach lattices with upper p-estimates inside of infinity sums of weak-$L^p$ spaces.

Proposition 3.3

• (1) Let X be a p-convex Banach lattice with constant $M\ge 1$. There is a family $\Gamma$ of probability measures and a lattice isomorphism

  $$\begin{aligned} J: X\rightarrow \bigl (\oplus _{\mu \in \Gamma }L^p(\mu )\bigr )_{\infty } \end{aligned}$$

  such that $\Vert x\Vert \le \Vert Jx\Vert \le M\Vert x\Vert$ for all $x\in X$.

• (2) Let X be a $(p,\infty )$-convex Banach lattice with constant $M\ge 1$. There is a family $\Gamma$ of probability measures and a lattice isomorphism

  $$\begin{aligned} J: X\rightarrow \bigl (\oplus _{\mu \in \Gamma }L^{p,\infty }(\mu )\bigr )_{\infty } \end{aligned}$$

  such that $\Vert x\Vert \le \Vert Jx\Vert \le \gamma _p M\Vert x\Vert$ for all $x\in X$.

Proof

Let $x\in S_X^+=\{z\in X_+:\Vert z\Vert =1\}$. Choose $x^*\in X^*_+$, $\Vert x^*\Vert =1,$ such that $x^*(x) = 1$. Define $\rho _x:X\rightarrow {\mathbb {R}}$ by $\rho _x(z) = x^*(|z|)$. Then $\rho _x$ is an L-norm on $X/\ker \rho _x$. Let $q_x: X\rightarrow X/\ker \rho _x$ be the quotient map. There exist a probability measure $\mu _x$ and a contractive lattice homomorphism $i_x: X/\ker \rho _x\rightarrow L^1(\mu _x)$ such that $i_xq_xx = 1$, the constant 1 function. Set $A_x = i_xq_x(B_X)\subseteq L^1(\mu _x)$. Let $(\alpha _{i})$ be a finitely supported real sequence and let $f_i = i_xq_xx_i$ for $x_i \in B_X$. Since $i_xq_x$ is a lattice homomorphism,

$$\begin{aligned} \left\| \left( \sum |\alpha _if_i|^p\right) ^{\frac{1}{p}}\right\| _{L^1(\mu _x)}&= \left\langle \left( \sum |\alpha _ix_i|^p\right) ^{\frac{1}{p}}, x^*\right\rangle \le \left\| \left( \sum |\alpha _ix_i|^p\right) ^{\frac{1}{p}}\right\| \\&\le M\left( \sum |\alpha _i|^p\right) ^{\frac{1}{p}} \text { if } X \text {is } p\text {-convex with constant } M,\\ \left\| \bigvee |\alpha _if_i|\right\| _{L^1(\mu _x)}&= \left\langle \bigvee |\alpha _ix_i|, x^*\right\rangle \le \left\| \bigvee |\alpha _ix_i|\right\| \\&\le M\left( \sum |\alpha _i|^p\right) ^{\frac{1}{p}} \text { if } X \text { is } (p,\infty )\text {-convex with constant } M. \end{aligned}$$

By the factorization theorems above, there exists $g_x\in L^1(\mu _x)_+$, $\Vert g_x\Vert _{L^1(\mu _x)}\le 1$, such that for any $f\in A_{x}$

$$\begin{aligned} \left\| \frac{f}{g_x}\right\| _{L^p(g_x\cdot \mu _x)} \le M,\ \text {respectively},\ \left\| \frac{f}{g_x} \right\| _{L^{p,\infty } (g_x\cdot \mu _x)} \le \gamma _p M. \end{aligned}$$

Define

$$\begin{aligned} J: X\rightarrow \bigl (\oplus _{x\in S_{X}^+}L^p(g_x\cdot \mu _x)\bigr )_{\infty }, \quad \text {respectively}\quad J: X\rightarrow \bigl (\oplus _{x\in S_{X}^+}L^{p,\infty }(g_x\cdot \mu _x)\bigr )_{\infty } \end{aligned}$$

by $Jz = (i_xq_x z/g_x)_x$. It is easy to check that J is a lattice homomorphism. If $z\in B_X$, we have $i_xq_xz\in A_x$. Hence,

$$\begin{aligned} \Vert i_xq_xz/g_x\Vert _{L^p(g_x\cdot \mu _x)} \le M,\ \text {respectively}\ \Vert i_xq_xz/g_x\Vert _{L^{p,\infty }(g_x\cdot \mu _x)} \le \gamma _p M. \end{aligned}$$

Thus, $\Vert Jz\Vert \le M\Vert z\Vert$, respectively $\Vert Jz\Vert \le \gamma _p M\Vert z\Vert$ for all $z\in X$.

On the other hand, if $x\in S_X^+$, then in the p-convex case,

$$\begin{aligned} \Vert Jx\Vert \ge \Vert i_xq_xx/g_x\Vert _{L^p(g_x\cdot \mu _x)} \ge \Vert i_xq_xx/g_x\Vert _{L^1(g_x\cdot \mu _x)} \end{aligned}$$

since $\Vert g_x\cdot \mu _x\Vert \le 1$. Therefore, $\Vert Jx\Vert \ge \Vert i_xq_xx\Vert _{L^1( \mu _x)}= x^*(x) =1$. In the $(p,\infty )$-convex case, we have

$$\begin{aligned} \int \frac{i_xq_xx}{g_x}\,d(g_x\cdot \mu _x) = \int {i_xq_xx}\,d\mu _x= x^*(x) =1 \ge \bigl ((g_x\cdot \mu _x)(\Omega _x)\bigr )^{1-\frac{1}{p}}. \end{aligned}$$

Hence, $\Vert Jx\Vert \ge \bigl \Vert \frac{i_xq_xx}{g_x}\bigr \Vert _{L^{p,\infty }(g_x\cdot \mu _x)} \ge 1$. It follows that $\Vert Jx\Vert \ge \Vert x\Vert$ for all $x\in X$ in both cases.$\square$

Remark 3.4

The isomorphism constant $\gamma _p$ in Proposition 3.3 cannot be chosen to be 1. In fact, if $X = {\mathbb {R}}^3$ so that the norm on $X^*$ is

$$\begin{aligned} \Vert (b_1,b_2,b_3)\Vert _{X^*} = \max \bigl \{\bigl (|b_i|^{p'} + (|b_j|+|b_k|)^{p'}\bigr )^{\frac{1}{p'}}: \{i,j,k\} = \{1,2,3\}\bigr \}, \end{aligned}$$

then it is clear that X is a Banach lattice with the coordinate-wise ordering. It can be proven that X satisfies an upper p-estimate with constant 1, but cannot be embedded lattice isometrically into any Banach lattice of the form $\bigl (\oplus _{\mu \in \Gamma }L^{p,\infty }(\mu )\bigr )_{\infty }$.

Proposition 3.3(1) has the following interpretation: Assume that $1< p< \infty$ and let ${\mathcal {C}}_p$ be the class of all p-convex Banach lattices with constant 1 and let ${\mathcal {X}}_p$ be the class of all $L^p(\mu )$ spaces with $\mu$ a probability measure. Then, by Proposition 3.3(1), ${\mathcal {C}}_p \subseteq \overline{{\mathcal {X}}}_p$ and therefore ${\mathcal {C}}_p = \overline{{\mathcal {X}}}_p$. This fact, together with the abstract framework developed in Sect. 2 provides an alternative proof of [24, Theorem 6.1]:

Theorem 3.5

Let E be a Banach space and let $1< p<\infty$. The norm $\rho _{{\mathcal {C}}_p}$ on $\textrm{FVL}[E]$ is given by

$$\begin{aligned} & \rho _{{\mathcal {C}}_p}(f) \\ & \quad = \sup \left\{ \Vert (f(x_k^*))_{k=1}^n \Vert _{\ell ^p(n)}: \, n\in \mathbb {N}, \, x_1^*,\dots ,x_n^*\in E^*, \, \sup _{x\in B_E}\Vert (x_k^*(x))_{k=1}^n\Vert _{\ell ^p(n)}\le 1\right\} \end{aligned}$$

for any $f\in \textrm{FVL}[E]$.

Proof

For short, let us denote the expression on the right-hand side of the statement by $\rho _p$. As observed above, by Proposition 3.3(1), ${\mathcal {C}}_p = \overline{{\mathcal {X}}}_p$, so by Corollary 2.6 we obtain that for any Banach space E, $\rho _{{\mathcal {C}}_p} = \rho _{{\mathcal {X}}_p}$ on $\textrm{FVL}[E]$. Now, if $\mu$ is a probability measure, there is always a non-atomic probability measure $\nu$ such that $L^p(\mu )$ embeds lattice isometrically into $L^p(\nu )$. Thus, $\rho _{\{L^p(\mu )\}} \le \rho _{\{L^p(\nu )\}}$, where the two norms here are the ones associated to the singleton classes $\{L^p(\mu )\}$ and $\{L^p(\nu )\}$, respectively. Moreover, by Theorem 2.10, the norm $\rho _{\{L^p(\nu )\}}$ coincides with $\rho _p$. This in particular yields that $\rho _p\le \rho _{{\mathcal {C}}_p}$, since $L^p(\nu )$ is in the class ${\mathcal {C}}_p$. Putting everything together we get

$$\begin{aligned} \rho _{{\mathcal {C}}_p}=\rho _{{\mathcal {X}}_p} =\sup _{\mu \text { finite}} \rho _{\{L^p(\mu )\}} \le \rho _p\le \rho _{{\mathcal {C}}_p}. \end{aligned}$$

Hence, $\rho _{{\mathcal {C}}_p} = \rho _p$, as claimed.$\square$

The situation presented in Proposition 3.3(2) is slightly different due to the isomorphism constant $\gamma _p$. Again, fix $1< p< \infty$ and let ${\mathcal {C}}_{p,\infty }$ be the class of all $(p,\infty )$-convex Banach lattices with constant 1 and let ${\mathcal {X}}_{p,\infty }$ be the class of all $L^{p,\infty }(\mu )$ spaces with $\mu$ a $\sigma$-finite measure. Clearly, $\overline{{\mathcal {X}}}_{p,\infty }\subseteq {\mathcal {C}}_{p,\infty }$. On the other hand, statement (2) in Proposition 3.3 establishes that any Banach lattice $X\in {\mathcal {C}}_{p,\infty }$ is $\gamma _p$-lattice isomorphic to an element of $\overline{{\mathcal {X}}}_{p,\infty }$. This correspondence allows us to find an explicit representation of an equivalent norm in $\textrm{FBL}^{{\mathcal {C}}_{p,\infty }}[E]$ for any Banach space E (equation (1.4)), solving the question raised in [24, Remark 6.2] in the upper p-estimates setting:

Theorem 3.6

Let E be a Banach space and let $1< p<\infty$. The norm $\rho _{{\mathcal {X}}_{p,\infty }}$ on $\textrm{FVL}[E]$ is given by

$$\begin{aligned} & \rho _{{\mathcal {X}}_{p,\infty }}(f) \\ & \quad = \sup \left\{ \Vert (f(x_k^*))_{k=1}^n \Vert _{\ell ^{p,\infty }(n)}: \, n\in \mathbb {N}, \, x_1^*,\dots ,x_n^*\in E^*,\, \sup _{x\in B_E}\Vert (x_k^*(x))_{k=1}^n\Vert _{\ell ^{p,\infty }(n)}\le 1\right\} \end{aligned}$$

for any $f\in \textrm{FVL}[E]$. Moreover, $\rho _{{\mathcal {X}}_{p,\infty }}\le \rho _{{\mathcal {C}}_{p,\infty }} \le \gamma _p\,\rho _{{\mathcal {X}}_{p,\infty }}$, where $\gamma _p = (1-\frac{1}{p})^{\frac{1}{p}-1}$.

Proof

The proof of the first part of the statement is similar to that of Theorem 3.5. Now, since ${\mathcal {X}}_{p,\infty } \subseteq {\mathcal {C}}_{p,\infty }$, $\rho _{{\mathcal {X}}_{p,\infty }} \le \rho _{{\mathcal {C}}_{p,\infty }}$. On the other hand, let $X\in {\mathcal {C}}_{p,\infty }$. By Proposition 3.3, there exists $Y\in \overline{{\mathcal {X}}}_{p,\infty }$ and a lattice isomorphism $J:X\rightarrow Y$ so that $\Vert x\Vert \le \Vert Jx\Vert \le \gamma _p\Vert x\Vert$ for all $x\in X$. Let $T:E\rightarrow X$ be a linear contraction. Then $\gamma _p^{-1}JT: E \rightarrow Y$ is a linear contraction. Since J is a lattice homomorphism, $\widehat{\gamma _p^{-1}JT} = \gamma _p^{-1}J\widehat{T}$. For any $f\in \textrm{FVL}[E]$,

$$\begin{aligned} \Vert \widehat{T}f\Vert \le \gamma _p\Vert \gamma _p^{-1}J\widehat{T}f\Vert = \gamma _p\Vert \widehat{\gamma _p^{-1}JT}f\Vert \le \gamma _p\rho _{{\mathcal {X}}_{p,\infty }}(f). \end{aligned}$$

Taking supremum over all linear contractions $T:E\rightarrow X$ for any $X\in {\mathcal {C}}_{p,\infty }$ shows that $\rho _{{\mathcal {C}}_{p,\infty }}\le \gamma _p \rho _{{\mathcal {X}}_{p,\infty }}$. $\square$

Note that $\textrm{FBL}^{{\mathcal {C}}_{p,\infty }}[E]$ endowed with the equivalent norm $\rho _{{\mathcal {X}}_{p,\infty }}$ satisfies the following universal property: given any Banach lattice X satisfying an upper p-estimate with constant 1 and a linear operator $T:E\rightarrow X$, there exists a unique lattice homomorphic extension $\widehat{T}:\textrm{FBL}^{{\mathcal {C}}_{p,\infty }}[E]\rightarrow X$ with $\Vert \widehat{T}\Vert \le \gamma _p \Vert T\Vert$. Moreover, if $X=L^{p,\infty }(\mu )$ for some $\sigma$-finite measure $\mu$, then actually $\Vert \widehat{T}\Vert =\Vert T\Vert$.

Finally, observe that Theorems 3.5 and 3.6 state that $\rho _{{\mathcal {C}}_p} =\rho _{\{\ell ^p\}}$ and that $\rho _{\{\ell ^{p,\infty }\}} \le \rho _{{\mathcal {C}}_{p,\infty }} \le \gamma _p\,\rho _{\{\ell ^{p,\infty }\}}$. As a result, $\textrm{FBL}^{{\mathcal {C}}_p}[E] = \textrm{FBL}^{\{\ell ^p\}}[E]$ and $\textrm{FBL}^{{\mathcal {C}}_{p,\infty }}[E] = \textrm{FBL}^{\{\ell ^{p,\infty }\}}[E]$ and it is clear that both of these spaces are vector sublattices of $C(B_{E^*})$. The proof works in exactly the same way if $p=1$, in which case we get that ${\mathcal {C}}_1$, the class of all (1-convex) Banach lattices, is equal to $\overline{{\mathcal {X}}}_1$, where ${\mathcal {X}}_1$ consists of all $L^1(\mu )$ spaces. The Banach lattice $\textrm{FBL}^{{\mathcal {C}}_1}[E]$ is the free Banach lattice generated by E, which is usually denoted by $\textrm{FBL}[E]$. From the above, we see that $\rho _{{\mathcal {C}}_1} = \rho _{{\mathcal {X}}_1} = \rho _{\{\ell ^1\}}$.

3.1 Classes of Banach lattices generating ${\textrm{FBL}}^{(p)}[E]$ and $\textrm{FBL}^{(p,\infty )}[E]$.

Let ${\mathcal {C}}_p$ and ${\mathcal {C}}_{p,\infty }$ be the class of p-convex Banach lattices with constant 1 and the class of Banach lattices satisfying an upper p-estimate with constant 1, respectively. We will characterize the classes of Banach lattices ${\mathcal {D}}$ so that $(\textrm{FBL}^{{\mathcal {C}}_p}[E],\rho _{{\mathcal {C}}_p})$ and $(\textrm{FBL}^{\mathcal {D}}[E], \rho _{\mathcal {D}})$ agree for any Banach space E and also the classes ${\mathcal {D}}'$ so that $(\textrm{FBL}^{{\mathcal {C}}_{p,\infty }}[E],\rho _{{\mathcal {C}}_{p,\infty }})$ and $(\textrm{FBL}^{{\mathcal {D}}'}[E], \rho _{{\mathcal {D}}'})$ are lattice isomorphic.

For the proofs of the above claims, we need an extension of the isometric lattice-lifting property which may be of independent interest. A Banach lattice X is said to have the isometric lattice-lifting property if there exists a lattice isometric embedding $\alpha$ from X to $\textrm{FBL}[X]$ such that $\widehat{id_X}\alpha =id_X$. This property was introduced in the article [4], inspired by previous works of Godefroy and Kalton [21] on Lipschitz-free spaces. Notably, Banach lattices ordered by a 1-unconditional basis have this property.

In [31, Theorem 8.3] an alternative proof of the lattice lifting property for spaces ordered by a 1-unconditional basis was given, which also worked with $\textrm{FBL}[E]$ replaced by ${\textrm{FBL}}^{(p)}[E]$. We now show how to generalize these results to $\textrm{FBL}^{\mathcal {C}}[E]$.

Proposition 3.7

Let $X\in \overline{{\mathcal {C}}}$ be a Banach space with a normalized 1-unconditional basis $(e_i)$, viewed as a Banach lattice in the pointwise order induced by the basis. Then $\textrm{FBL}^{\mathcal {C}}[X]$ contains a lattice isometric copy of X. Moreover, there exists a contractive lattice homomorphic projection onto this sublattice.

Proof

Using either [4, Theorem 4.1] or [31, Theorem 8.3] with $p=1$ we obtain a lattice isometric embedding $\alpha :X\rightarrow \textrm{FBL}[X]$. Since ${\mathcal {C}}\subseteq {\mathcal {C}}_1$, it follows that $\rho _{\mathcal {C}}(f)\le \rho _{{\mathcal {C}}_1}(f)$ for every $f\in \textrm{FVL}[X]$, so the formal identity extends to a norm one lattice homomorphism $j:\textrm{FBL}[X]\rightarrow \textrm{FBL}^{\mathcal {C}}[X]$. On the other hand, since $X\in \overline{{\mathcal {C}}}$, the identity operator on X extends to a contractive lattice homomorphism $\widehat{id_X}:\textrm{FBL}^{\mathcal {C}}[X]\rightarrow X$. It can be checked from the construction of $\alpha$ that $\widehat{id_X}j\alpha =id_X$, from which it follows that

$$\begin{aligned} \Vert x\Vert _X=\Vert \widehat{id_X}j\alpha x\Vert \le \rho _{{\mathcal {C}}}(j\alpha x)\le \rho _{{\mathcal {C}}_1}(\alpha x)\le \Vert x\Vert _X \end{aligned}$$

for every $x\in X$. $\square$

Theorem 3.8

The following are equivalent for a class of Banach lattices ${\mathcal {D}}$.

• (1) For any Banach space E, $\textrm{FBL}^{{\mathcal {C}}_p}[E] = \textrm{FBL}^{\mathcal {D}}[E]$ as sets and the norms $\rho _{{\mathcal {C}}_p}$ and $\rho _{\mathcal {D}}$ agree there.

• (2) ${\mathcal {D}}\subseteq {\mathcal {C}}_p$ and $\ell ^p \in \overline{{\mathcal {D}}}$.

Proof

(1)$\implies$(2): By Proposition 3.7, $\textrm{FBL}^{{\mathcal {D}}}[\ell ^p] = \textrm{FBL}^{{\mathcal {C}}_p}[\ell ^p]$ contains a lattice isometric copy of $\ell ^p$. By Proposition 2.7, $\textrm{FBL}^{\mathcal {D}}[\ell ^p]\in \overline{{\mathcal {D}}}$. Hence, $\ell ^p\in \overline{{\mathcal {D}}}$.

Suppose that $X\in {\mathcal {D}}$. Let $\phi _X:X\rightarrow \textrm{FBL}^{{\mathcal {D}}}[X]$ be the canonical embedding. The identity $i: X\rightarrow X$ induces a lattice homomorphic contraction $\widehat{i}:\textrm{FBL}^{\mathcal {D}}[X] \rightarrow X$ so that $\widehat{i}\phi _X = i$. Since $\textrm{FBL}^{\mathcal {D}}[X] = \textrm{FBL}^{{\mathcal {C}}_p}[X]$ lattice isometrically, this space belongs to $\overline{{\mathcal {C}}}_p = {\mathcal {C}}_p$ by Proposition 2.7. Hence, so does $\textrm{FBL}^{{\mathcal {D}}}[X]/\ker \widehat{i}$. Define $j: \textrm{FBL}^{\mathcal {D}}[X]/\ker \widehat{i} \rightarrow X$ by $j[f] = \widehat{i}f$, where [f] is the equivalence class of f. The map j is a lattice isomorphism so that $\Vert j\Vert \le 1$ and $x = j[\delta _x]$ for any $x\in X$. Hence, $B_X = j(B_{\textrm{FBL}^{\mathcal {D}}[X]/\ker \widehat{i}})$. This shows that j is an onto lattice isometry. Therefore, $X\in {\mathcal {C}}_p$.

(2)$\implies$(1): Since ${\mathcal {D}}\subseteq {\mathcal {C}}_p$, for any Banach space E, $\rho _{\mathcal {D}}\le \rho _{{\mathcal {C}}_p}$ on $\textrm{FVL}[E]$. Similarly, since $\ell ^p\in \overline{{\mathcal {D}}}$, $\rho _{\{\ell ^p\}}\le \rho _{\overline{{\mathcal {D}}}}$. However, $\rho _{{\mathcal {C}}_p} = \rho _{\{\ell ^p\}}$ and $\rho _{\overline{{\mathcal {D}}}} = \rho _{\mathcal {D}}$ by Corollary 2.6. Therefore, we have the reverse inequality $\rho _{{\mathcal {C}}_p}\le \rho _{\mathcal {D}}$. $\square$

For the case of ${\mathcal {C}}_{p,\infty }$, the following result can be proven in essentially the same way as Theorem 3.8. The only difference is that, since $\ell ^{p,\infty }$ does not have a basis, Proposition 3.7 cannot be applied directly. However, we can still use Proposition 3.7 to embed $\ell ^{p,\infty }$ lattice isometrically into $(\bigoplus _m\textrm{FBL}^{{\mathcal {C}}_{p,\infty }}[\ell ^{p,\infty }(m)])_\infty$ and continue with the rest of the proof with slight adaptations.

Theorem 3.9

The following are equivalent for a class of Banach lattices ${\mathcal {D}}$.

• (1) There is a constant $M_1 <\infty$ so that for any Banach space E, $\textrm{FBL}^{{\mathcal {C}}_{p,\infty }}[E] = \textrm{FBL}^{\mathcal {D}}[E]$ as sets and $\frac{1}{M_1}\rho _{\mathcal {D}}\le \rho _{{\mathcal {C}}_{p,\infty }}\le M_1\rho _{\mathcal {D}}$ there.

• (2) There is a constant $M_2 <\infty$ so that any $X\in {\mathcal {D}}$ satisfies an upper p-estimate with constant $M_2$, and $\ell ^{p,\infty }$ is $M_2$-lattice isomorphic to a space $Y \in \overline{{\mathcal {D}}}$.

4 Subspace problem

In this section we characterize when an embedding $\iota : F\hookrightarrow E$ induces a lattice embedding $\overline{\iota }: \textrm{FBL}^{(p,\infty )}[F]\hookrightarrow \textrm{FBL}^{(p,\infty )}[E]$. The analogous problem for ${\textrm{FBL}}^{(p)}$ was solved in [31, Theorem 3.7] by making use of an extension theorem for regular operators due to Pisier [34, Theorem 4]. This scheme of proof, however, does not seem to be applicable to $\textrm{FBL}^{(p,\infty )}$. For this reason, we develop in section 4.1 an entirely new approach to the subspace problem which is based on push-outs. Then, in section 4.2 we prove the injectivity of $\ell ^p$ in the class of p-convex Banach lattices, which shows that our solution to the subspace problem for ${\textrm{FBL}}^{(p)}$ is equivalent to the one in [31]. Finally, in Sect. 4.3 we prove that $\ell ^{p,\infty }$ is not injective in the class of Banach lattices with upper p-estimates.

We begin with some preliminaries. Let F be a closed subspace of a Banach space E and let $\iota :F\rightarrow E$ be the inclusion map. Let $q = \iota ^*:E^*\rightarrow F^*$ and note that q is weak$^*$-to-weak$^*$ continuous with $q(B_{E^*}) = B_{F^*}$. Hence, $f\circ q\in C(B_{E^*})$ if $f\in \textrm{FVL}[F]$.

Proposition 4.1

The map $\overline{\iota }: \textrm{FVL}[F]\rightarrow \textrm{FVL}[E]$, $\overline{\iota }f = f\circ q$, is a vector lattice isomorphism from $\textrm{FVL}[F]$ onto the sublattice ${\mathcal {L}}$ of $\textrm{FVL}[E]$ generated by $\{\delta _{\iota y}: y\in F\}$.

Proof

Clearly, $\overline{\iota }$ is a lattice homomorphism from $\textrm{FVL}[F]$ into $C(B_{E^*})$. Moreover, $(\overline{\iota }\delta _y)(x^*) =(\delta _y\circ q)(x^*) = \delta _{\iota y}(x^*)$ for all $y\in F$ and $x^*\in B_{E^*}$. Hence, $\overline{\iota }\delta _y = \delta _{\iota y}$. Thus, $\overline{\iota }(FVL[E])$ is a sublattice of $C(B_{E^*})$ that contains $\delta _{\iota y}$ for all $y\in F$, which implies that ${\mathcal {L}}\subseteq \overline{\iota }(FVL[E])$. On the other hand, $\overline{\iota }^{-1}{\mathcal {L}}$ is a sublattice of $\textrm{FVL}[F]$ containing $\delta _y$ for all $y\in F$. Hence, $\overline{\iota }^{-1}{\mathcal {L}}= \textrm{FVL}[F]$, which implies that $\overline{\iota }(FVL[E])={\mathcal {L}}$.$\square$

Note that $\phi _E\iota : F\rightarrow \textrm{FBL}^{\mathcal {C}}[E]$ is a contraction (in fact, an isometry) and $\textrm{FBL}^{\mathcal {C}}[E] \in \overline{{\mathcal {C}}}$. By Proposition 2.5, $\widehat{\phi _E\iota }: (\textrm{FVL}[F],\rho _{\mathcal {C}})\rightarrow \textrm{FBL}^{\mathcal {C}}[E]$ is a contraction.

Proposition 4.2

Let $\overline{\iota }$ be the map from Proposition 4.1. Then $\widehat{\phi _E\iota }f = \overline{\iota }f$ for any $f\in \textrm{FVL}[F]$.

Proof

Let $f\in \textrm{FVL}[F]$. Suppose that $f = G(\delta _{y_1},\dots , \delta _{y_n})$, where G is a lattice-linear expression and $y_1,\dots , y_n\in F$. For any $x^*\in B_{E^*}$,

$$\begin{aligned} (\overline{\iota }f)(x^*)&= f(qx^*) = G(qx^*(y_1),\dots , qx^*(y_n)) = G(x^*(\iota y_1),\dots , x^*(\iota y_n)) \\&= G(\delta _{ \iota y_1},\dots ,\delta _{ \iota y_n})(x^*). \end{aligned}$$

Thus, $\overline{\iota }f = G(\delta _{\iota y_1},\dots ,\delta _{ \iota y_n})$. On the other hand,

$$\begin{aligned} \widehat{\phi _E\iota }(f) = G(\phi _E\iota y_1,\dots , \phi _E \iota y_n) = G(\delta _{ \iota y_1},\dots , \delta _{\iota y_n}). \end{aligned}$$

Therefore, $\overline{\iota }= \widehat{\phi _E\iota }$. $\square$

The operator $\overline{\iota }$ extends to a contraction from $\textrm{FBL}^{\mathcal {C}}[F]$ into $\textrm{FBL}^{\mathcal {C}}[E]$, which we denote again by $\overline{\iota }$. The objective now is to determine when $\overline{\iota }$ is an embedding. We begin with the following elementary observation.

Proposition 4.3

Suppose that for any $X\in {\mathcal {C}}$, any contraction $T:F\rightarrow X$ extends to a contraction $S: E\rightarrow X$. Then $\overline{\iota }$ is a lattice isometric embedding.

Proof

In view of the preceding propositions and the fact that $\widehat{\phi _E\iota }$ is a contraction, it suffices to show that $\rho _{\mathcal {C}}(f) \le \rho _{\mathcal {C}}(\overline{\iota }f)$ for all $f\in \textrm{FVL}[F]$. Let $X\in {\mathcal {C}}$ and let $T:F\rightarrow X$ be a contraction. There exists a contraction $S:E\rightarrow X$ so that $S \iota = T$. Suppose that $f = G(\delta _{y_1},\dots , \delta _{y_n})$, where G is a lattice-linear expression and $y_1,\dots , y_n\in F$. We have

$$\begin{aligned} \widehat{S}(\overline{\iota }f) = \widehat{S}\widehat{\phi _E\iota }(f) = G(S\iota y_1,\dots ,S\iota y_n)= G(Ty_1,\dots , Ty_n) = \widehat{T}f. \end{aligned}$$

Thus, $\Vert \widehat{T}f\Vert \le \rho _{\mathcal {C}}(\overline{\iota }f)$. Taking supremum over all such T shows that $\rho _{\mathcal {C}}(f) \le \rho _{\mathcal {C}}(\overline{\iota }f)$. $\square$

Using the above results we may now characterize when $\textrm{FBL}^\mathcal {C}[F]$ embeds via $\overline{\iota }$ onto a complemented sublattice of $\textrm{FBL}^\mathcal {C}[E]$. This characterization should be contrasted with [31, Proposition 3.12] and our solution to the subspace problem in Theorem 4.8. Note, in particular, that if condition (1) in Theorem 4.4 holds for a class $\mathcal {C}$ then it also holds for any class $\mathcal {C}'$ such that $\mathcal {C}'\subseteq \mathcal {C}$.

Theorem 4.4

Let F be a closed subspace of a Banach space E and let $\iota : F\rightarrow E$ be the inclusion map. Set $\overline{\iota } = \widehat{\phi _E \iota }$. The following are equivalent.

• (1) For any $X\in \overline{{\mathcal {C}}}$, any bounded linear operator $T:F\rightarrow X$ has a bounded linear extension $\widetilde{T}:E\rightarrow X$ with $\Vert \widetilde{T}\Vert = \Vert T\Vert$.

• (2) $\overline{\iota }$ is a lattice isometric embedding and there is a contractive lattice homomorphic projection P from $\textrm{FBL}^{\mathcal {C}}[E]$ onto $\overline{\iota }(\textrm{FBL}^{\mathcal {C}}[F])$.

Proof

(1)$\implies$(2): By the above, $\overline{\iota }$ is a contraction. Furthermore, $\overline{\iota } =\widehat{\phi _E\iota }$ is a lattice homomorphism. Since $\textrm{FBL}^{\mathcal {C}}[F]\in \overline{{\mathcal {C}}}$, the isometric embedding $\phi _F: F\rightarrow \textrm{FBL}^{\mathcal {C}}[F]$ has a bounded linear extension $\widetilde{\phi _F}: E\rightarrow \textrm{FBL}^{\mathcal {C}}[F]$ such that $\Vert \widetilde{\phi _F}\Vert = 1$. Therefore, $\widehat{\widetilde{\phi _F}}:\textrm{FBL}^{\mathcal {C}}[E] \rightarrow \textrm{FBL}^{\mathcal {C}}[F]$ is a lattice homomorphism such that $\Vert \widehat{\widetilde{\phi _F}}\Vert = \Vert \widetilde{\phi _F}\Vert =1$ and $\widehat{\widetilde{\phi _F}}\phi _E = \widetilde{\phi _F}$. For $y\in F$, it is easy to see that

$$\begin{aligned} \widehat{\widetilde{\phi _F}}\overline{\iota } \delta _y = \widehat{\widetilde{\phi _F}}\delta _{\iota y} = \widetilde{\phi _F}(\iota y) = \phi _Fy = \delta _y. \end{aligned}$$

Since $\widehat{\widetilde{\phi _F}}\overline{\iota }$ is a lattice homomorphism, it follows that $\widehat{\widetilde{\phi _F}}\overline{\iota }$ is the identity map on $\textrm{FBL}^{\mathcal {C}}[F]$. In particular,

$$\begin{aligned} \rho _{\mathcal {C}}(f) = \rho _{\mathcal {C}}(\widehat{\widetilde{\phi _F}}\overline{\iota }f) \le \rho _{\mathcal {C}}(\overline{\iota }f) \le \rho _{\mathcal {C}}(f)\text { for any }f\in \textrm{FBL}^{\mathcal {C}}[F]. \end{aligned}$$

Hence, $\overline{\iota }$ is a lattice isometric embedding. Furthermore, $P:= \overline{\iota } \widehat{\widetilde{\phi _F}}$ is a lattice homomorphic projection on $\textrm{FBL}^{\mathcal {C}}[E]$ such that $P(FBL^\mathcal {C}[E]) = \overline{\iota }(FBL^\mathcal {C}[F])$ and $\Vert P\Vert =1$.

(2)$\implies$(1): Let $X\in \overline{{\mathcal {C}}}$ and let $T: F\rightarrow X$ be a bounded linear operator. Then $\widehat{T}:\textrm{FBL}^{\mathcal {C}}[F] \rightarrow X$ is a lattice homomorphism such that $\Vert \widehat{T}\Vert = \Vert T\Vert$ and $\widehat{T}\phi _F = T$. Let $\widetilde{T} = \widehat{T}\overline{\iota }^{-1}P\phi _E: E\rightarrow X$. For any $y\in F$,

$$\begin{aligned} \widetilde{T}\iota y = \widehat{T}\overline{\iota }^{-1}P\delta _{\iota y} = \widehat{T}\overline{\iota }^{-1}\delta _{\iota y} =\widehat{T}\delta _y = Ty. \end{aligned}$$

Hence, $\widetilde{T}$ is a bounded linear extension of T. Clearly, $\Vert \widetilde{T}\Vert \le \Vert \widehat{T}\Vert = \Vert T\Vert$. Hence, $\Vert \widetilde{T}\Vert = \Vert T\Vert$. $\square$

We conclude this preliminary subsection with another simple observation.

Proposition 4.5

Assume that $\overline{\iota }:\textrm{FBL}^{\mathcal {C}}[F]\rightarrow \textrm{FBL}^{\mathcal {C}}[E]$ is a lattice isometric embedding. Let $X\in {\mathcal {C}}$ and let $T:F\rightarrow X$ be a bounded linear operator. The following are equivalent.

• (1) There exists an operator $S:E\rightarrow X$ such that $S\iota = T$ and $\Vert S\Vert = \Vert T\Vert$.

• (2) There exists a lattice homomorphism $R: \textrm{FBL}^{\mathcal {C}}[E] \rightarrow X$ such that $R\overline{\iota } = \widehat{T}$ and $\Vert R\Vert = \Vert T\Vert$.

Proof

(1)$\implies$(2): Suppose that S is as given. Then $\widehat{S}:\textrm{FBL}^{\mathcal {C}}[E] \rightarrow X$ is a lattice homomorphism such that $\Vert \widehat{S}\Vert = \Vert S\Vert = \Vert T\Vert$. If $f= G(\delta _{y_1},\dots , \delta _{y_n})\in \textrm{FVL}[F]$, then from the proof of Proposition 4.2, $\overline{\iota }f = G(\delta _{\iota y_1},\dots , \delta _{\iota y_n})$. Thus,

$$\begin{aligned} \widehat{S}\overline{\iota }f = \widehat{S}G(\delta _{\iota y_1},\dots , \delta _{\iota y_n}) = G(S\iota y_1,\dots , S\iota y_n) = G(Ty_1,\dots , Ty_n) = \widehat{T}f, \end{aligned}$$

so (2) holds with $R= \widehat{S}$.

(2)$\implies$(1): Assume that R is as given. Let $S = R \phi _E: E\rightarrow X$. Then $\Vert S\Vert = \Vert R\Vert = \Vert T\Vert$. Moreover, if $y\in F$ then

$$\begin{aligned} S\iota y = R\phi _E\iota y = R \delta _{\iota y} = R\overline{\iota }\delta _y = \widehat{T}\delta _y = Ty. \end{aligned}$$

Thus, (1) holds. $\square$

4.1 Push-outs in $\mathcal {C}$

Recall that for objects $A_0,A_1,A_2$ and morphisms $\alpha _i:A_0\rightarrow A_i$, $i=1,2$, a push-out diagram is an object $PO=PO(\alpha _1,\alpha _2)$ together with morphisms $\beta _i:A_i\rightarrow PO$, $i=1,2$, making the following diagram commutative

and with the universal property that if $\beta '_i:A_i\rightarrow B$ are such that $\beta '_1\alpha _1=\beta '_2\alpha _2$, then there is a unique $\gamma :PO\rightarrow B$ such that $\gamma \beta _i=\beta '_i$ for $i=1,2,$ as follows:

In the category $\mathfrak {BL}$ of Banach lattices and lattice homomorphisms, isometric push-outs were shown to exist in [9] – these are push-outs satisfying the extra condition that $\max \{\Vert \beta _1\Vert ,\Vert \beta _2\Vert \}\le 1$ and which guarantee the inequality $\Vert \gamma \Vert \le \max \{ \Vert \beta '_1\Vert ,\Vert \beta '_2\Vert \}$ in the universal property. The construction of such push-outs can be adapted to Banach lattices within certain classes $\overline{\mathcal {C}}$, as follows.

Theorem 4.6

Let $\mathcal {C}$ be a class of Banach lattices such that $\overline{\mathcal {C}}$ is closed under lattice quotients. Given Banach lattices $X_0,X_1,X_2$ in $\overline{\mathcal {C}}$ and lattice homomorphisms $T_i:X_0\rightarrow X_i$ for $i=1,2$, there is a Banach lattice $PO^{\overline{\mathcal {C}}}$ in $\overline{\mathcal {C}}$ and lattice homomorphisms $S_1,S_2$ so that the following is an isometric push-out diagram in $\overline{\mathcal {C}}$:

Proof

Let $X_1\oplus _1 X_2$ denote the direct sum equipped with the norm $\Vert (x_1,x_2)\Vert =\Vert x_1\Vert +\Vert x_2\Vert$ and let $j_i:X_i\rightarrow X_1\oplus _1 X_2$ denote the canonical embedding for $i=1,2$. Let $\phi :X_1\oplus _1 X_2\rightarrow \textrm{FBL}^{\mathcal {C}}[X_1\oplus _1 X_2]$ be the canonical embedding and Z be the (closed) ideal in $\textrm{FBL}^{\mathcal {C}}[X_1\oplus _1 X_2]$ generated by the families $(\phi (j_1 |x|)-|\phi (j_1 x)|)_{x\in X_1}$, $(\phi (j_2 |y|)-|\phi (j_2 y)|)_{y\in X_2}$ and $(\phi j_1T_1z-\phi j_2T_2z)_{z\in X_0}$. Let

$$ PO^{\overline{\mathcal {C}}}=\textrm{FBL}^{\mathcal {C}}[X_1\oplus _1 X_2]/Z, $$

and let $S_i=q\phi j_i:X_i\rightarrow PO^{\overline{\mathcal {C}}}$, for $i=1,2$, where $q:\textrm{FBL}^{\mathcal {C}}[X_1\oplus _1 X_2]\rightarrow PO^{\overline{\mathcal {C}}}$ denotes the canonical quotient map.

Since $\overline{\mathcal {C}}$ is closed under lattice quotients and $\textrm{FBL}^{\mathcal {C}}[X_1\oplus _1 X_2]$ is in $\overline{\mathcal {C}}$, so is $PO^{\overline{\mathcal {C}}}$. The rest of the proof follows exactly as in [9, Theorem 4.3]. $\square$

Theorem 4.6 applies, in particular, when $\mathcal {C}$ is the class of p-convex Banach lattices or the class of Banach lattices with an upper p-estimate (with constant 1). Moreover, as in [9, Theorem 4.4], for these classes we have the following.

Theorem 4.7

Let $\mathcal {C}$ be the class of p-convex Banach lattices or Banach lattices with an upper p-estimate. Set $K_{\mathcal {C}}=2^{1-\frac{1}{p}}$ if ${\mathcal {C}}={\mathcal {C}}_p$ and $K_{\mathcal {C}}=2^{1-\frac{1}{p}}\gamma _p$ if ${\mathcal {C}}={\mathcal {C}}_{p,\infty }$, and let

be an isometric push-out diagram in $\overline{\mathcal {C}}$. If $\Vert T_1\Vert \le 1$ and the lower arrow $T_2$ is an isometric embedding then the upper arrow $\widetilde{T}_1$ is a $K_{\mathcal {C}}$-embedding.

Proof

Fix $x_1\in X_1$. We aim to show that $\Vert x_1\Vert \le K_{\mathcal {C}} \Vert \widetilde{T}_1(x_1)\Vert$. The reverse inequality is trivially true since $\Vert \widetilde{T}_1\Vert \le 1$. Since we are dealing with lattice norms and lattice homomorphisms, we can suppose that $x_1$ is positive. By the push-out universal property, as given in Theorem 4.6, it is enough to find a commutative diagram in $\overline{\mathcal {C}}$

such that $\Vert x_1\Vert \le \Vert \widehat{T}_1(x_1)\Vert$ and $\max \{ \Vert \widehat{T}_1\Vert ,\Vert \widehat{T}_2\Vert \} \le K_{\mathcal {C}}$.

To this end, by [9, Theorem 4.4] we have a Banach lattice isometric push-out diagram

such that $\widetilde{T}_1$ is an isometric embedding. Hence, by [9, Lemma 3.3], we can pick a positive $\sigma$-finite element $x^*\in PO^*$ of norm one such that $x^*(|\widetilde{T}_1 x_1|) = \Vert x_1\Vert$.

The functional $x^*$ induces a lattice semi-norm on PO given by $\Vert z\Vert _{x^*} = x^*(|z|)$. After making a quotient by the elements of norm 0, this becomes a norm that satisfies $\Vert |x|+|y|\Vert _{x^*} = \Vert x\Vert _{x^*} +\Vert y\Vert _{x^*}$ for all x, y. This identity extends to the completion of this normed lattice, which, by Kakutani’s representation theorem [26, Theorem 1.b.2], is lattice isometric to a Banach lattice of the form $L^1(\nu _{x^*})$. The formal identity induces a homomorphism $\psi _{x^*}:PO\rightarrow L^1(\nu _{x^*})$ of norm one.

For $i=1,2,$ let $A_i=\psi _{x^*}(\widetilde{T}_i(B_{X_i}))$ and set $A=A_1\cup A_2\subseteq L^1(\nu _{x^*})$. We will distinguish the following cases:

• (1) Let $\mathcal {C}={\mathcal {C}}_p$ denote the class of p-convex Banach lattices (with constant 1). Since the $X_i$ are p-convex and the $\psi _{x^*}\widetilde{T}_i$ are contractive lattice homomorphisms we have that for all finitely supported sequences $(\alpha _i)_{i\in I}$ of real numbers and $(f_i)_{i\in I}\subseteq A$,

  $$\begin{aligned} & \Big \Vert \left( \sum |\alpha _if_i|^p\right) ^{\frac{1}{p}}\Big \Vert _1 \le \Big \Vert \Bigg (\sum _{f_i\in A_1} |\alpha _if_i|^p\Bigg )^{\frac{1}{p}}\Big \Vert _1\\ & \qquad +\Big \Vert \Big (\sum _{f_i\in A_2} |\alpha _if_i|^p\Big )^{\frac{1}{p}}\Big \Vert _1\le 2^{1-\frac{1}{p}}\left( \sum |\alpha _i|^p\right) ^{\frac{1}{p}}. \end{aligned}$$

  Hence, by Theorem 3.1, there is $g\in L^1(\nu _{x^*})_+$ with $\Vert g\Vert _1\le 1$ and lattice homomorphisms $\widehat{T}_i:X_i\rightarrow L^p(g\nu _{x^*})$ that make the following diagram commutative

  

  is a commutative diagram in ${\mathcal {C}}_p$ and

  $$\begin{aligned} \Vert x_1\Vert =x^*|\widetilde{T}_1 x_1|=\Vert \psi _{x^*}\widetilde{T}_1 x_1\Vert _{L^1(\nu _{x^*})}\le \Vert \widehat{T}_1 x_1\Vert _{L^p(g\nu _{x^*})}. \end{aligned}$$

• (2) Let $\mathcal {C}={\mathcal {C}}_{p,\infty }$ denote the class of Banach lattices satisfying an upper p-estimate (with constant 1). Since the $X_i$ satisfy an upper p-estimate and the $\psi _{x^*}\widetilde{T}_i$ are contractive lattice homomorphisms we have that for all finitely supported sequences $(\alpha _i)_{i\in I}$ of real numbers and $(f_i)_{i\in I}\subseteq A$,

  $$\begin{aligned} \Big \Vert \bigvee |\alpha _if_i|\Big \Vert _1 \le \Big \Vert \bigvee _{f_i\in A_1} |\alpha _if_i|\Big \Vert _1+\Big \Vert \bigvee _{f_i\in A_2} |\alpha _if_i|\Big \Vert _1\le 2^{1-\frac{1}{p}}(\sum |\alpha _i|^p)^{\frac{1}{p}}. \end{aligned}$$

  Hence, by Theorem 3.2, there is $g\in L^1(\nu _{x^*})_+$ with $\Vert g\Vert _1\le 1$ and lattice homomorphisms $\widehat{T}_i:X_i\rightarrow L^{p,\infty }(g\nu _{x^*})$ that make the following diagram commutative

  

  is a commutative diagram in ${\mathcal {C}}_{p,\infty }$ and

  $$\begin{aligned} \Vert x_1\Vert =x^*|\widetilde{T}_1 x_1|=\Vert \psi _{x^*}\widetilde{T}_1 x_1\Vert _{L^1(\nu _{x^*})}\le \Vert \widehat{T}_1 x_1\Vert _{L^{p,\infty }(g\nu _{x^*})}. \end{aligned}$$

$\square$

As a consequence of the above, we get our desired solution to the subspace problem.

Theorem 4.8

Suppose that $\iota :F\rightarrow E$ is an isometric embedding and let $\mathcal {C}$ denote either the class of p-convex Banach lattices or the class of Banach lattices with upper p-estimates. The following are equivalent.

• (1) $\overline{\iota }:\textrm{FBL}^{\mathcal {C}}[F]\rightarrow \textrm{FBL}^{\mathcal {C}}[E]$ is a $c_1$-lattice embedding.

• (2) For every operator $T: F\rightarrow X$ with X in $\mathcal {C}$, there is Y in $\mathcal {C}$, a (norm one) $K_{\mathcal {C}}$-lattice embedding $j:X\rightarrow Y$ and $S:E\rightarrow Y$ so that $jT=S\iota$ and $\Vert S\Vert \le c_2 \Vert T\Vert$.

Here, $c_2\le c_1\le K_{\mathcal {C}}c_2$.

Proof

(1)$\implies$(2): Let $X\in \mathcal {C}$ and $T:F\rightarrow X$ an operator. By Theorem 4.6, let us consider the isometric push-out diagram in $\overline{\mathcal {C}}$ for $X_0=\overline{\iota }(\textrm{FBL}^{\mathcal {C}}(F))$, $X_1=X$, $X_2=\textrm{FBL}^{\mathcal {C}}(E)$, $T_1=\frac{\widehat{T}\overline{\iota }^{-1}}{\Vert \widehat{T}\overline{\iota }^{-1}\Vert }:X_0\rightarrow X_1$ and $T_2$ the formal identity from $X_0$ to $X_2$, where $\widehat{T}:\textrm{FBL}^{\mathcal {C}}[F]\rightarrow X$ denotes the lattice homomorphism extending T.

Let us write $Y=PO^{\overline{\mathcal {C}}}$ and $j=S_1$. Since $T_2$ is an isometric embedding, we can apply Theorem 4.7 to obtain that j is a norm one $K_{\mathcal {C}}$-embedding. On the other hand, let $S=\Vert \widehat{T}\overline{\iota }^{-1}\Vert S_2\phi _E:E\rightarrow Y$. It follows that

$$\begin{aligned} jT= j\widehat{T} \phi _F=\Vert \widehat{T}\overline{\iota }^{-1}\Vert j T_1 \overline{\iota }\phi _F=\Vert \widehat{T}\overline{\iota }^{-1}\Vert S_2 T_2 \overline{\iota }\phi _F=\Vert \widehat{T}\overline{\iota }^{-1}\Vert S_2 \phi _E \iota =S\iota , \end{aligned}$$

and $\Vert S\Vert \le \Vert \widehat{T}\Vert \Vert \overline{\iota }^{-1}\Vert \le c_1 \Vert T\Vert$, so $c_2\le c_1$.

(2)$\implies$(1): First, note that $\Vert \overline{\iota }\Vert = 1$. Now, take $f\in \textrm{FVL}[F]$ and a contraction $T:F\rightarrow X$, X in $\mathcal {C}$. By the hypothesis, there exists Y in $\mathcal {C}$, a norm one $K_{\mathcal {C}}$-lattice embedding $j:X\rightarrow Y$ and $S:E\rightarrow Y$ so that $jT=S\iota$ and $\Vert S\Vert \le c_2$. It follows that $j\widehat{T}=\widehat{S}\overline{\iota }$, so

$$\begin{aligned} \Vert \widehat{T}f\Vert _X=\Vert j^{-1}\widehat{S}\overline{\iota }f\Vert _X\le \Vert j^{-1}\Vert \Vert S\Vert \rho _{\mathcal {C}}(\overline{\iota }f)\le K_{\mathcal {C}} c_2 \rho _{\mathcal {C}}(\overline{\iota }f) \end{aligned}$$

and $\overline{\iota }$ is a $K_{\mathcal {C}} c_2$-embedding. $\square$

4.2 Recovering the POE-p via injectivity

Recall that a Banach lattice X is injective if for every Banach lattice Z, every closed linear sublattice Y of Z and every positive linear operator $T:Y\rightarrow X$ there is a positive linear extension $\widetilde{T}: Z\rightarrow X$ with $\Vert \widetilde{T}\Vert =\Vert T\Vert .$ Equivalently [28, Proposition 3.5], X is injective if whenever X lattice isometrically embeds into a Banach lattice Y there is a positive contractive projection from Y onto X. Note that this notion of injectivity is closer to the category of Banach lattices with positive linear maps, which is considerably different from the category $\mathfrak {BL}$ of Banach lattices with lattice homomorphisms (where in fact there are no injective objects [9]). The study of injective Banach lattices is classical [16, 23, 25, 28, 29]; however, it seems that not much is known on injectivity (or projectivity) in subcategories of Banach lattices.

Here, we discuss the notion of injectivity in the class of p-convex Banach lattices with positive operators, with an aim towards providing an alternative argument to recover the POE-p (Property of operator extension into $L^p$, see [31, Definition 3.8]). The original proof of [31, Theorem 3.7] requires a result due to Pisier [34, Theorem 4] which implicitly implies the injectivity of $L^p$ spaces in the class of p-convex Banach lattices. This fact additionally allows one to establish an analogue of [28, Proposition 3.5] in this restricted context (see Proposition 4.11 below). However, it turns out that Theorem 4.8 together with Theorem 4.9 recover [31, Theorem 3.7] (up to constants) without making use of Pisier’s arguments. The first step in this proof is to show that $\ell ^p$ satisfies an a priori weaker property than the injectivity in the class of p-convex Banach lattices.

Theorem 4.9

Let X be a p-convex Banach lattice and suppose that $(x_n)$ is a disjoint positive sequence in X equivalent to the $\ell ^p$-basis. Then $[(x_n)]$ is the image of a positive projection on X.

Proof

We may assume that X is a closed sublattice of $Z=(\bigoplus _{\mu \in \Gamma }L^p(\mu ))_\infty$ by Proposition 3.3. It suffices to show that $[(x_n)]$ is the range of a positive projection on Z. Assume that $c\Vert (a_n)\Vert _p < \Vert \sum a_nx_n\Vert \le \Vert (a_n)\Vert _p$ for any non-zero $(a_n)\in \ell ^p$. Let $m\in {\mathbb {N}}$. There exists a finite set $\Gamma _m$, $k\in {\mathbb {N}}$, and a positive contraction $T:(\bigoplus _{\mu \in \Gamma }L^p(\mu ))_\infty \rightarrow (\ell ^p(k))^{\Gamma _m}_\infty$ so that $(Tx_n)^m_{n=1}$ is a positive disjoint sequence satisfying $c\Vert (a_n)^m_{n=1}\Vert _p < \Vert \sum ^m_{n=1} a_nTx_n\Vert \le \Vert (a_n)^m_{n=1}\Vert _p$ for any non-zero $(a_n)^m_{n=1}\in \ell ^p(m)$. Write $Tx_n = (y_n(\gamma ))_{\gamma \in \Gamma _m}$ with $y_n(\gamma )\in \ell ^p(k)$. Define $S: \ell ^1(m)\rightarrow \ell ^\infty (\Gamma _m)$ by

$$\begin{aligned} S(b_1,\dots , b_m) = \left( \sum ^m_{n=1}b_n\Vert y_n(\gamma )\Vert ^p\right) _{\gamma \in \Gamma _m}. \end{aligned}$$

Since $(y_n(\gamma ))^m_{n=1}$ is a disjoint sequence for each $\gamma$, if $b=(b_1,\dots , b_m)\ge 0$, then

$$\begin{aligned} \Vert Sb\Vert = \max _{\gamma \in \Gamma _m}\left\| \sum ^m_{n=1}b_n^{\frac{1}{p}}y_n(\gamma )\right\| _p^p = \left\| \sum ^m_{n=1}b_n^{\frac{1}{p}}Tx_n\right\| ^p \ge c^p\left\| (b_n^{\frac{1}{p}})^m_{n=1}\right\| _p^p = c^p\Vert b\Vert _1. \end{aligned}$$

We claim that $B^+_{\ell ^\infty (m)} \subseteq c^{-p}{\text {co}}{\text {so}}\{S^*e^*_\gamma :\gamma \in \Gamma _m\}$. Here, for a subset A of a Banach lattice X, ${\text {so}}A = \bigcup _{a\in A}[-|a|,|a|]$ denotes the solid hull of A and, as usual, ${\text {co}}(A)$ denotes the convex hull. To prove the claim, note that if $0\le y^* \notin c^{-p}{\text {co}}{\text {so}}\{S^*e^*_\gamma :\gamma \in \Gamma _m\}$ then by [32, Proposition 3.1] there exists $y\in \ell ^1(m)_+$ so that

$$\begin{aligned} \Vert y^*\Vert \,\Vert y\Vert \ge y^*(y) > c^{-p}\max _{\gamma \in \Gamma _m}(S^*e^*_\gamma )(y) = c^{-p}\Vert Sy\Vert \ge \Vert y\Vert . \end{aligned}$$

Hence, $\Vert y^*\Vert > 1$. This completes the proof of the claim.

By the claim we can find a convex combination $\sum _{\gamma \in \Gamma _m}\alpha _\gamma S^*e^*_\gamma \ge c^p\overbrace{(1,\dots , 1)}^m$. Choose $y^*_n(\gamma ) \in \ell ^{p'}(k)_+$ so that ${\text {supp}}y^*_n(\gamma ) = {\text {supp}}y_n(\gamma )=: I_n$, $\Vert y^*_n(\gamma )\Vert _{p'} = 1$ and $\langle y_n(\gamma ), y^*_n(\gamma )\rangle = \Vert y_n(\gamma )\Vert _p$. If $x \in (\bigoplus _{\mu \in \Gamma }L^p(\mu ))_\infty$ and $Tx = (y(\gamma ))_{\gamma \in \Gamma _m}$, let

$$\begin{aligned} z^*_n(x) = \sum _{\gamma \in \Gamma _m}\alpha _\gamma \Vert y_n(\gamma )\Vert _p^{p-1}\langle y(\gamma ), y^*_n(\gamma )\rangle . \end{aligned}$$

It is easy to see that $z^*_n$ is a positive linear functional and for any $x\in (\bigoplus _{\mu \in \Gamma }L^p(\mu ))_\infty$,

$$\begin{aligned} \Vert (z^*_n(x))^m_{n=1}\Vert _p&\le \sum _{\gamma \in \Gamma _m}\alpha _\gamma \bigl \Vert (\Vert y_n(\gamma )\Vert _p^{p-1}\Vert y(\gamma )\chi _{I_n}\Vert _p)^m_{n=1}\bigr \Vert _p\\&\le \sup _{\gamma \in \Gamma _m}\sup _n\Vert y_n(\gamma )\Vert ^{p-1}_p\bigl \Vert \sum _{n=1}^m y(\gamma )\chi _{I_n}\bigr \Vert _p \\&\le \sup _{\gamma \in \Gamma _m}\sup _n\Vert y_n(\gamma )\Vert ^{p-1}_p\Vert y(\gamma )\Vert _p\\&\le \sup _n\Vert Tx_n\Vert ^{p-1}\cdot \Vert Tx\Vert \le \Vert Tx\Vert \le \Vert x\Vert . \end{aligned}$$

On the other hand, $z^*_l(x_n) = 0$ if $l\ne n$ and

$$\begin{aligned} z^*_n(x_n) = \sum _{\gamma \in \Gamma _m}\alpha _\gamma \Vert y_n(\gamma )\Vert _p^{p-1}\langle y_n(\gamma ), y^*_n(\gamma )\rangle = \sum _{\gamma \in \Gamma _m}\alpha _\gamma \Vert y_n(\gamma )\Vert _p^p\ge c^p, \end{aligned}$$

since $\sum _{\gamma \in \Gamma _m}\alpha _\gamma \Vert y_n(\gamma )\Vert _p^p$ is the n-th coordinate of the sum $\sum _{\gamma \in \Gamma _m}\alpha _\gamma S^*e^*_\gamma$. Therefore, the map $P_m$ on $(\bigoplus _{\mu \in \Gamma }L^p(\mu ))_\infty$ given by

$$\begin{aligned} P_mx = \sum ^m_{n=1}\frac{z_n^*(x)}{z^*_n(x_n)}x_n \end{aligned}$$

is a positive projection from $(\bigoplus _{\mu \in \Gamma }L^p(\mu ))_\infty$ onto $[(x_n)^m_{n=1}]$ such that $\Vert P_m\Vert \le c^{-p}$. Finally, let ${\mathcal {U}}$ be a free ultrafilter on ${\mathbb {N}}$. Regard $[(x_n)^\infty _{n=1}]$ as a dual Banach lattice (either the dual of an isomorph of $\ell ^{p'}$ if $1< p \le \infty$ or the dual of an isomorph of $c_0$ if $p=1$). The operator $P:(\bigoplus _{\mu \in \Gamma }L^p(\mu ))_\infty \rightarrow [(x_n)]$ given by $Px = w^*$-$\lim _{m\rightarrow {\mathcal {U}}}P_mx$ is a positive projection onto $[(x_n)]$.$\square$

As a corollary of Theorems 4.8 and 4.9, we recover the solution to the subspace problem for ${\textrm{FBL}}^{(p)}$ given in [31, Theorem 3.7].

Corollary 4.10

Let $\iota :F\hookrightarrow E$ be an isometric embedding and $1\le p<\infty$. The following are equivalent.

• (1) $\overline{\iota }:\textrm{FBL}^{(p)}[F]\rightarrow \textrm{FBL}^{(p)}[E]$ is a $c_1$-lattice embedding.

• (2) For every operator $T: F\rightarrow \ell ^p$ there is $R:E\rightarrow \ell ^p$ so that $T=R\iota$ and $\Vert R\Vert \le c_2 \Vert T\Vert$.

Here, $c_1\le c_2\le 2^{p-\frac{1}{p}}c_1$.

Proof

We use the identification $\textrm{FBL}^{(p)}=\textrm{FBL}^{\mathcal {C}}$ where $\mathcal {C}$ denotes the class of p-convex Banach lattices and invoke Theorem 4.8. The implication $(2)\Rightarrow (1)$ is clear. For the converse, we get that there exist a p-convex Banach lattice Y, a $2^{1-\frac{1}{p}}$-lattice embedding $j:\ell ^p\rightarrow Y$ and $S:E\rightarrow Y$ such that $jT=S\iota$ with $\Vert S\Vert \le c_1\Vert T\Vert$. By Theorem 4.9, there is a (positive) projection $P:Y\rightarrow j(\ell ^p)\subseteq Y$ with $\Vert P\Vert \le (2^{1-\frac{1}{p}})^p$ (as j is a $2^{1-\frac{1}{p}}$-embedding). Thus, we can take $R=j^{-1}PS$. $\square$

Let us conclude this section with a note on how, in general, the property stated in Theorem 4.9 is actually equivalent to injectivity in the class of p-convex Banach lattices, provided we make use of Pisier’s result [34, Theorem 4].

Proposition 4.11

Let X be a p-convex Banach lattice with constant 1. The following are equivalent:

• (1) X is injective in the class of p-convex Banach lattices, i.e., given Z a p-convex Banach lattice, Y a closed sublattice of Z and $T:Y\rightarrow X$ a positive operator, there exists a positive extension $\widetilde{T}:Z\rightarrow X$.

• (2) If X lattice isomorphically embeds into a p-convex Banach lattice Z, there exists a positive projection from Z onto X.

Proof

$(1)\Rightarrow (2)$: Let $J:X\rightarrow Z$ be a lattice isomorphic embedding. It suffices to apply the injectivity property of X to the sublattice $Y=J(X)\subseteq Z$ and the operator $J^{-1}:Y\rightarrow X$.

$(2)\Rightarrow (1)$: Let Z be a p-convex Banach lattice, Y a closed sublattice of Z and $T:Y\rightarrow X$ a positive operator. By Proposition 3.3, there is an isometric lattice embedding $J:X\rightarrow \bigl (\oplus _{\mu \in \Gamma }L^p(\mu )\bigr )_{\infty }$ for a certain family of measures $\Gamma$, and by the hypothesis, X is complemented inside this space by means of a positive projection $P:\bigl (\oplus _{\mu \in \Gamma }L^p(\mu )\bigr )_{\infty }\rightarrow X$. For every $\mu \in \Gamma$ let $P_\mu : \bigl (\oplus _{\mu \in \Gamma }L^p(\mu )\bigr )_{\infty } \rightarrow L^p(\mu )$ be the (positive) projection onto the factor indexed by $\mu$. Using [34, Theorem 4] we can extend each operator $P_\mu J T$ to a positive operator $S_\mu :Z\rightarrow L^p(\mu )$. Gluing them together we obtain $S:Z\rightarrow \bigl (\oplus _{\mu \in \Gamma }L^p(\mu )\bigr )_{\infty }$ given by $Sz=(S_\mu z)_{\mu \in \Gamma }$. The operator $\widetilde{T}=PS:Z\rightarrow X$ is the desired extension. $\square$

4.3 $\ell ^{p,\infty }$ is not injective for Banach lattices with upper p-estimates

We now show that for any $1< p < \infty$, $\ell ^{p,\infty }$ is not injective in the class of Banach lattices with upper p-estimates. As in the previous section, we say that a Banach lattice X satisifying an upper p-estimate with constant 1 is injective in the class of Banach lattices with upper p-estimates if given any Banach lattice with upper p-estimates Z, a closed sublattice Y of Z and $T:Y\rightarrow X$ a positive operator, there exists a positive extension $\widetilde{T}:Z\rightarrow X$.

Theorem 4.12

For any $1< p < \infty$ there is a Banach lattice X with an upper p-estimate and an uncomplemented closed sublattice Y of X that is lattice isomorphic to $\ell ^{p,\infty }$. In particular, $\ell ^{p,\infty }$ is not injective in the class of Banach lattices with upper p-estimates.

Proof

We begin with some notation. Let X be the Banach lattice $(\bigoplus _{\mathbb {N}}\ell ^{p,\infty })_\infty$. Clearly, X satisfies an upper p-estimate with constant 1. We may express any $x\in X$ as $x = (x(j))_{j=1}^\infty = (x(i,j))_{i,j=1}^\infty$, where (x(j)) is a bounded sequence in $\ell ^{p,\infty }$ and $x(j) = (x(i,j))^\infty _{i=1}$ for each j. Given $m\in {\mathbb {N}}$, define $P_m$ on X by

$$\begin{aligned} (P_mx)(i,j) = {\left\{ \begin{array}{ll} x(m,j)& \text {if }i =m,\\ 0 & \text {if }i\ne m. \end{array}\right. } \end{aligned}$$

Clearly, $P_m$ is a band projection on X and the projection band $P_mX$ is lattice isometric to $\ell ^\infty$ via the map

$$\begin{aligned} L_m:\ell ^\infty \rightarrow P_mX,\ (L_my)(i,j) = {\left\{ \begin{array}{ll} y(j) & \text {if }i = m, \text { where }y = (y(j))^\infty _{j=1},\\ 0& \text {otherwise}.\end{array}\right. } \end{aligned}$$

Let ${\mathcal {S}}$ be the Schreier family, i.e., all subsets I of ${\mathbb {N}}$ so that $|I| \le \min I$. Since ${\mathcal {S}}$ is countable, we can list its elements in a sequence $(S_j)^\infty _{j=1}$. Given $a= (a_i) \in \ell ^{p,\infty }$, let

$$\begin{aligned} x_a(i,j) = {\left\{ \begin{array}{ll} a_i & \text {if }i\in S_j,\\ 0& \text {otherwise.} \end{array}\right. } \end{aligned}$$

Denote the sequence of coordinate unit vectors in $\ell ^{p,\infty }$ by $(e_i)$. For any j,

$$\begin{aligned} \Vert (x_a(i,j))^\infty _{i=1}\Vert _{p,\infty } = \left\| \sum _{i\in S_j}a_ie_i\right\| _{p,\infty } \le \Vert (a_i)\Vert _{p,\infty }. \end{aligned}$$

Hence, $T: \ell ^{p,\infty }\rightarrow X$, $a \mapsto x_a$ is a bounded linear map which is clearly a lattice homomorphism. On the other hand, if $\Vert a\Vert _{p,\infty } >1$ then there exists a non-empty finite set L in ${\mathbb {N}}$ so that $|a_i| \ge (p'|L|^{\frac{1}{p}})^{-1}$ for all $i\in L$. Let $S_j\in {\mathcal {S}}$ be such that $S_j\subseteq L$ and $|S_j| \ge |L|/2$. Then,

$$\begin{aligned} \Vert x_a\Vert \ge \left\| \sum _{i\in S_j}a_ie_i\right\| _{p,\infty } \ge (p'|L|^{\frac{1}{p}})^{-1}\left\| \sum _{i\in S_j}e_i\right\| _{p,\infty } = \frac{1}{p'}\left( \frac{|S_j|}{|L|}\right) ^{\frac{1}{p'}} \ge \frac{1}{p'2^{\frac{1}{p'}}}. \end{aligned}$$

This shows that T is a lattice isomorphism from $\ell ^{p,\infty }$ into X.

Let $Y = T\ell ^{p,\infty }$. We will show that there is no bounded projection from X onto Y. Assume on the contrary that there is a projection P from X onto Y. Then there is a sequence $(x^*_i)$ in $X^*$ so that $(x^*_i(x))_i\in \ell ^{p,\infty }$ for any $x\in X$ and $x^*_i(Ta) = a_i$ for any i and $a = (a_k)\in \ell ^{p,\infty }$. It follows readily that $(x^*_i)_i\subseteq X^*$ is equivalent to the unit vector basis of $\ell ^{p',1}$. In particular, it is an unconditional basic sequence in a Banach lattice that is q-concave for some $q<\infty$. Since $(P_i^*)$ is a disjoint sequence of band projections on $X^*$, the sequence $(P^*_ix^*_i)$ is disjoint and hence also unconditional in $X^*$. By [26, Theorem 1.d.6(i)], for any finitely nonzero real sequence $(b_i)$,

$$\begin{aligned} \Vert (b_i)\Vert _{p',1} \sim \left\| \sum b_ix^*_i\right\| \sim \left\| \sqrt{\sum |b_ix^*_i|^2}\right\| \ge \left\| \sqrt{\sum |b_iP_i^*x^*_i|^2}\right\| \sim \left\| \sum b_iP^*_ix^*_i\right\| . \end{aligned}$$

Thus, the linear map $Q:X\rightarrow \ell ^{p,\infty }$ defined by $Qx = (P^*_ix^*_i(x))_i$ is bounded. On the other hand, observe that $Te_k \in P_kX$. Therefore,

$$\begin{aligned} \langle Te_k, P_i^*x^*_i\rangle = {\left\{ \begin{array}{ll} \langle T e_i, x^*_i\rangle = 1 & \text {if }k=i,\\ 0 & \text {otherwise}. \end{array}\right. } \end{aligned}$$

(4.1)

Hence, $QTe_i=e_i$ for any $i\in \mathbb {N}$.

Regard $L_i$ as a lattice isometric embedding from $\ell ^\infty$ into X. Define $y^*_i = L_i^*P_i^*x_i^*$ in $(\ell ^\infty )^*$. Let I be a finite subset of ${\mathbb {N}}$. Suppose that $(u_i)_{i\in I}$ is a sequence in $\ell ^\infty$ so that $\Vert (\sum _{i\in I}|u_i|^p)^{\frac{1}{p}}\Vert _\infty \le 1$. Set $u = \sum _{i\in I}L_iu_i\in X$.

Then $\Vert u\Vert \le \Vert (\sum _{i\in I}|u_i|^p)^{\frac{1}{p}}\Vert _\infty \le 1$. Thus,

$$\begin{aligned}\sum _{i\in I}y^*_i(u_i)&= \sum _{i\in I}\langle u_i, L^*_iP^*_ix^*_i\rangle = \sum _{i\in I}(P^*_ix^*_i)(u) \\ &\le \Vert (P^*_ix^*_i(u))\Vert _{p,\infty }\cdot |I|^{\frac{1}{p'}} = \Vert Qu\Vert \cdot |I|^{\frac{1}{p'}}\lesssim |I|^{\frac{1}{p'}}. \end{aligned}$$

Making use of the duality described on [26, p. 47], we conclude that there is an absolute constant $C<\infty$ so that

$$\begin{aligned} \left\| \left( \sum _{i\in I}|y^*_i|^{p'}\right) ^{\frac{1}{p'}}\right\| \le C|I|^{\frac{1}{p'}} \end{aligned}$$

(4.2)

for any finite subset I of ${\mathbb {N}}$.

Since $(\ell ^\infty )^*$ is an AL-space, the bounded sequence $(y^*_i)$ has a subsequence $(y^*_{i_k})$ that has a splitting

$$\begin{aligned} y^*_{i_k} = u^*_k + v^*_k,\ |u^*_k|\wedge |v^*_k| =0, (u^*_k)\, \text {is almost order bounded, }(v^*_k)\, \text {is disjoint}. \end{aligned}$$

See, e.g., [39].

Since $|v^*_k| \le |y^*_{i_k}|$ for all k, it follows from (4.2) that $\Vert v^*_k\Vert \rightarrow 0$. Thus, the whole sequence $(y^*_{i_k})$ is almost order bounded in $(\ell ^\infty )^*$ and hence it is relatively weakly compact.

Since $x_i:=Te_i\in P_iX$ and $L_i:\ell ^\infty \rightarrow P_iX$ is a surjective lattice isometry, we may find a sequence $(u_k)$ in $\ell ^\infty$ so that $L_{i_k}u_k = x_{i_k}$ for each k. If we write $u_k = (u_k(j))^\infty _{j=1}$ then $u_k(j) = 1$ if $k\in S_j$ and 0 otherwise.

Hence, for any finitely supported real sequence $(a_k)$,

$$\begin{aligned} \left\| \sum a_ku_k\right\| _\infty = \sup _j\left| \sum _{k\in S_j}a_k\right| . \end{aligned}$$

The expression on the right is the norm of the Schreier space.

The preceding equation says that $(u_k)$ is equivalent to the unit vector basis of the Schreier space.

Thus, $(u_k)$ is a weakly null sequence (in $\ell ^\infty$). Since $\ell ^\infty$ has the Dunford-Pettis property, the weakly null sequence $(u_k)$ converges uniformly to 0 on the relatively weakly compact set $(y^*_{i_k})$. In particular,

$$\begin{aligned} 0 = \lim y^*_{i_k}(u_k) = \lim \langle u_k, L^*_{i_k}P^*_{i_k}x^*_{i_k}\rangle = \lim \langle x_{i_k}, P^*_{i_k}x^*_{i_k}\rangle . \end{aligned}$$

However,

$$\begin{aligned} e_{i_k} = QTe_{i_k} = Qx_{i_k} = (P^*_jx^*_j(x_{i_k}))_j. \end{aligned}$$

Taking the $i_k$th component yields that $\langle x_{i_k}, P^*_{i_k}x^*_{i_k}\rangle =1$ for any k, which is the desired contradiction. $\square$