Injective Banach lattices: a survey. (English) Zbl 1463.46036
Summary: The work is aimed to survey recent results on injective Banach lattices, outline the Boolean-valued approach, and pose some open problems. The central idea of the investigation is the Boolean-valued transfer principle from \(AL\)-spaces to injective Banach lattices: Every injective Banach lattice is embedded into an appropriate Boolean-valued model, becoming an \(AL\)-space. To illustrate the method, a concrete description of injective Banach lattices similar to that of \(AL\)-spaces is presented and the isometric classification of injective Banach lattices is accomplished.
MSC:
| 46B42 | Banach lattices |
| 46B04 | Isometric theory of Banach spaces |
| 47B65 | Positive linear operators and order-bounded operators |
| 03C90 | Nonclassical models (Boolean-valued, sheaf, etc.) |
| 03C98 | Applications of model theory |
| 46-02 | Research exposition (monographs, survey articles) pertaining to functional analysis |
Keywords:
\(AL\)-space; \(AM\)-space; injective Banach lattice; Boolean-valued model; Boolean-valued transfer principle; tensor product; homogeneous Banach latticeReferences:
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