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Preprint Number 536

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536. Itaï Ben Yaacov, C. Ward Henson
Generic orbits and type isolation in the Gurarij space
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Submission date: 21 November 2012.

Abstract:

We study model-theoretic aspects of the separable Gurarij space G, in particular type isolation and the existence of prime models, without use of formal logic.
1. If E is a finite-dimensional Banach space, then the set of isolated types over E is dense, and there exists a prime Gurarij over E. This is the unique separable Gurarij space G extending E with the unique Hahn-Banach extension property (property U), and the orbit of id : E --> G under the action of Aut(G) is a dense G_\delta in the space of all linear isometric embeddings E --> G.
2. If E is infinite-dimensional then there are no non realised isolated types, and therefore no prime model over E (unless G \cong E), and all orbits of embeddings E --> G are meagre. On the other hand, there are Gurarij spaces extending E with property U.

We also point out that the class of Gurarij space is the class of models of an \aleph_0-categorical theory with quantifier elimination, and calculate the density character of the space of types over E, answering a question of Avilès et al.

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Full text arXiv 1211.4814: pdf, ps.


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