## Free and projective Banach lattices.(English)Zbl 1325.46020

Let $$A$$ be a nonempty set. A free vector lattice over $$A$$ is a pair $$(F,i)$$, where $$F$$ is a vector lattice and $$i:A\rightarrow F$$ is a map with the property that, for any vector lattice $$E$$ and any map $$\phi:A\rightarrow E$$, there exists a unique vector lattice homomorphism $$T:F\rightarrow E$$ such that $$\phi = T\circ i$$. It follows that, if $$(F,i)$$ and $$(G,\kappa)$$ are free vector lattices over a nonempty set $$A$$, then there exists a unique vector lattice isomorphism $$T:F\rightarrow G$$ such that $$T(i(a)) = \kappa(a)$$ for $$a\in A$$. Because of this, a free vector lattice $$(F,i)$$ over $$A$$ is referred to as the free vector lattice generated by $$A$$ and is denoted by $$\operatorname{FVL}(A)$$. If $$A$$ and $$B$$ are sets of equal cardinality, then $$\operatorname{FVL}(A)$$ and $$\operatorname{FVL}(B)$$ are isomorphic vector lattices and so $$\operatorname{FVL}(A)$$ depends on the cardinality of the set $$A$$. Thus the notation $$\operatorname{FVL}(a)$$ for $$\operatorname{FVL}(A)$$, where $$a$$ is the cardinality of $$A$$.
If $$i:A\rightarrow \operatorname{FVL}(A)$$ is the embedding of $$A$$ into $$\operatorname{FVL}(A)$$, then $$i(a)$$ is denoted by $$\delta(a)$$ and $$\{\delta(a):a\in A\}$$ is called the free generator of $$\operatorname{FVL}(A)$$. Free vector lattices exist.
If $$A$$ is a nonempty set, then a free Banach lattice over $$A$$ is a pair $$(X,i)$$, where $$X$$ is a Banach lattice and $$i:A\rightarrow X$$ is a bounded map such that for any Banach lattice $$Y$$ and any bounded map $$\kappa:A\rightarrow Y$$ there exists a unique vector lattice homomorphism $$T:X\rightarrow Y$$ such that $$\kappa = T\circ i$$ and $$||T|| = \sup\{ ||i(a)||:a\in A\}$$. The authors show that free Banach lattices exist and denote it by $$\operatorname{FBL}(A)$$. For a nonempty set $$A$$ define a map from the Dedekind complete vector lattice $$\operatorname{FVL}(A)^\sim$$ into the extended non-negative reals by $||\phi||^{\dagger} = \sup\{|\phi|(|\delta_a|:a\in A\}.$
Let $$\operatorname{FVL}(A)^\dagger = \{\phi \in \operatorname{FVL}(A)^\sim: ||\phi||^\dagger <\infty\}$$, which is a vector lattice ideal in $$\operatorname{FVL}(A)^\sim$$. For $$f\in \operatorname{FVL}(A)$$, let $$||f||_F = \sup\{\phi(|f|):\phi\in \operatorname{FVL}(A)^\dagger_+,\;||\phi||^\dagger\leq 1\}$$. Then $$||.||_F$$ is a lattice norm in $$\operatorname{FVL}(A)$$. In fact, for any nonempty set $$A$$, the completion of $$\operatorname{FVL}(A)$$ with respect to the norm $$||.||_F$$ and the map $$i:a\rightarrow \delta_a$$ is the free Banach lattice over $$A$$.
The authors give a representation on a compact Hausdorff space and study the basic properties of free Banach lattices. It is well known that every Banach lattice is a quotient of a free Banach lattice. They make this statement quite precise and obtain that, if $$X$$ is a separable Banach lattice, then $$X$$ is isometrically order isomorphic to a Banach lattice quotient of $$\operatorname{FBL}(\aleph_0)$$ and show that the dual $$X^*$$ is isometrically order isomorphic to a weak*-closed band in $$\operatorname{FBL}(\aleph_0)^*$$.
$$\operatorname{FBL}(n)$$ is not an AM-space unless $$n= 1$$, but as the authors show $$\operatorname{FBL}(n)$$ has a lot of AM-space structure if $$n$$ is finite.
The authors then give some characterizations of free Banach lattices over, respectively, one, a finite number, or a countable number of generators amongst all free Banach lattices. They also study the question of when $$\operatorname{FBL}(A)$$ is a classical Banach lattice and consider various properties generally considered desirable. It turns out that properties that are considered to be good are only possessed by a free Banach lattice generated by a single generator.
They then investigate the question of when disjoint families in a quotient Banach lattice $$X/J$$ can be lifted to a disjoint family in $$X$$. They prove a positive result for countable families and a negative result for families having larger cardinality.
A Banach lattice $$P$$ is projective if, whenever $$X$$ is a Banach lattice, $$J$$ is a closed order ideal in $$X$$ and $$Q:X\rightarrow X/J$$ is the quotient map, then for every linear lattice homomorphism $$T:P\rightarrow X/J$$ and $$\epsilon > 0$$ there exists a linear lattice homomorphism $$\hat{T}:P\rightarrow X$$ such that (1) $$T = Q\circ\hat{T}$$ and (2) $$||\hat{T}|| \leq (1 +\epsilon) ||T||$$.
They show that a free Banach lattice is projective and give a characterization of projective Banach lattices. It is worth noting that for finite $$p$$, the Banach lattice $$L_p[0,1]$$ is not projective. Studying the problem which Banach lattices are projective, they show that every finite-dimensional Banach lattice is projective and study for which compact subset $$K\subset \mathbb R^n$$, $$C(K)$$ is projective with the supremum norm. They obtain that $$C[0,1]$$ is a projective Banach lattice, but the sequence spaces $$c$$, $$l_\infty$$ are not projective while $$l_1$$ is. It turns out that, if $$C(K)$$ is a projective Banach lattice under any norm, then $$K$$ has only finitely many components.
For more on this nicely written paper, we have to refer the reader to the paper. The paper ends with twenty-one open problems.

### MSC:

 46B42 Banach lattices 46A40 Ordered topological linear spaces, vector lattices
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