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Separable universal Banach lattices. (English) Zbl 1445.46019

The paper under review is a relevant contribution to the theory of Banach lattices. It is well known that every separable Banach space is linearly isometric to a subspace of \(C([0,1])\), the space of continuous functions on the unit interval. Since \(C([0,1])\) is also separable, it becomes a universal object in the class of separable Banach spaces. The purpose of this paper is to provide an analogue of this in the context of separable Banach lattices. More precisely, the space of \(L_1(0,1)\)-valued continuous functions on the Cantor space, \(C(\Delta, L_1)\) is shown to have the universal property that, for every separable Banach lattice \(X\), there is a lattice isometry from \(X\) into \(C(\Delta,L_1)\) (that is, a linear isometry which also preserves the lattice operations).
One should note that a standard application of Kakutani’s representation theorem can be used to show that every separable Banach lattice is lattice isometric to a sublattice of \((\bigoplus L_1)_{\ell_\infty}\) (see also the work of H. P. Lotz and N. T. Peck [Proc. Am. Math. Soc. 126, No. 1, 75–84 (1998; Zbl 0889.46018)] for further examples of non-separable Banach lattices with this property). The key difficulty so far, which is solved in this paper, has been to construct a universal Banach lattice which is also separable.
In addition, the authors study the dual problem, and they provide a separable Banach lattice \(E\) such that every separable Banach lattice is lattice isometric to a quotient of \(E\). Other examples of separable Banach lattices with the same property have been considered recently by B. de Pagter and A. Wickstead in [Proc. R. Soc. Edinb., Sect. A, Math. 145, No. 1, 105–143 (2015; Zbl 1325.46020)].

MSC:

46B42 Banach lattices
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