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Isomorphic universality and the number of pairwise nonisomorphic models in the class of Banach spaces. (English) Zbl 1472.46003

Summary: We develop the framework of natural spaces to study isomorphic embeddings of Banach spaces. We then use it to show that a sufficient failure of the generalized continuum hypothesis implies that the universality number of Banach spaces of a given density under a certain kind of positive embedding (very positive embedding) is high. An example of a very positive embedding is a positive onto embedding between \(C(K)\) and \(C \left(L\right)\) for 0-dimensional \(K\) and \(L\) such that the following requirement holds for all \(h \neq 0\) and \(f \geq 0\) in \(C(K)\): if \(0 \leq T h \leq T f\), then there are constants \(a \neq 0\) and \(b\) with \(0 \leq a \cdot h + b \leq f\) and \(a \cdot h + b \neq 0\).

MSC:

46B03 Isomorphic theory (including renorming) of Banach spaces
46B26 Nonseparable Banach spaces
03C98 Applications of model theory
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