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**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.

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.

Reviewer: Şafak Alpay (Ankara)