## 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

### Keywords:

Hahn-Banach theorem; ordered vector space; regular operator

Zbl 0802.46035
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