Banach sublattices of weak \(L_ 1\)
By definition, weak \(L_ 1\) is the space of all measurable functions f on (0,1) for which \[ p_ 1(f)=\sup_{t>0}\{t\lambda (| f| >t)\}<\infty \] (with \(\lambda\) the Lebesgue measure). The Banach envelope weak \(L_ 1^{\wedge}\) of weak \(L_ 1\) is the completion of the semi- normed space (weak \(L_ 1,p)\), where p is the gauge functional of the convex hull of the \(p_ 1\)-unit ball. The main results of the paper read as follows: 1) if X is a separable Banach lattice with order continuous norm, then X is lattice isometric to a sublattice of weak \(L_ 1^{\wedge}.\) This result is also shown to be true for certain non-separable Banach lattices. To be more specific: 2) let X be a Banach lattice with generalized basis \(\{x_ B:B\in {\mathcal B}\}\) (with \({\mathcal B}\) the set of all branches) such that: a) if \(x=\sum a_ Bx_ B\), then \(x\geq 0\) if and only if each \(a_ B\geq 0\) b) (spreading invariance condition) whenever n is a natural number, \(a_ 1,...,a_ n\) are real numbers and \(B_ 1<...<B_ n,B_ 1'<...<B_ n'\) are branches then \(\| \sum^{n}_{i=1}a_ ix_{B_ i}\| =\| \sum^{n}_{i=1}a_ ix_{B_ i'}\|.\) Then X is lattice isometric to a sublattice of weak \(L_ 1^{\wedge}\).
scientific article; zbMATH DE number 4046398
scientific article