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The structure of lattice-subspaces. (English) Zbl 1047.46005

Given an ordered Banach space \(L,\) an ordered subspace \(X\) of \(L\) is said to be a lattice-subspace if it is a vector lattice (that is, for every pair of elements \(x,y\in X\) there exists a least upper bound in \(X)\). \(C[ 0,1] \) is a universal Banach lattice in the sense that each separable Banach lattice \(E\) is order isomorphic to a lattice-subspace \(X\) of \(C[ 0,1] .\) This isomorphism is realized by a bounded linear operator \(T:E\to X\) such that \(k\| x\| \leq\| Tx\| \leq\| x\| \) for every \(x\in E\) and \(\| Tx\| =\| x\| \) for \(x\in E_{+};\) here \(k\in [1/2,1].\) See I. A. Polyrakis [J. Math. Anal. Appl. 184, No.1, 1-18 (1994; Zbl 0802.46035)].
In the paper under review it is proved that the AM-spaces can be characterized by the existence of an order-isometry \(T\) for which \(k=1.\) For an AL-space, the constant \(k\) takes the lowest value, 1/2. The converse is open. Example 6 shows the interesting fact that the closure of a lattice-subspace is not necessarily a lattice-subspace. The main result is Theorem 7, which gives a necessary and sufficient condition that a Schauder basis \((b_{n})_{n}\) consisting of positive elements of a Banach lattice \(X\) should verify the condition \(X_{+}=\{ \sum_{n} \lambda_{n}b_{n}\mid \lambda_{n}\geq0\text{ for each }n\} .\) It is assumed that \(X\) is a lattice-subspace of the space of all real-valued functions defined on a nonempty set \(\Omega.\)

MSC:

46A22 Theorems of Hahn-Banach type; extension and lifting of functionals and operators
47B60 Linear operators on ordered spaces
47B65 Positive linear operators and order-bounded operators
46B42 Banach lattices

Citations:

Zbl 0802.46035
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