Generic orbits and type isolation in the Gurarij space
Résumé
We study model-theoretic aspects of the separable Gurarij space $\bG$, in particular type isolation and the existence of prime models, without use of formal logic. \begin{enumerate} \item 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 $\bG$ extending $E$ with the unique Hahn-Banach extension property (\emph{property $U$}), and the orbit of $\id\colon E \hookrightarrow \bG$ under the action of $\Aut(\bG)$ is a dense $G_\delta$ in the space of all linear isometric embeddings $E \hookrightarrow \bG$. \item If $E$ is infinite-dimensional then there are no non realised isolated types, and therefore no prime model over $E$ (unless $\bG \cong E$), and all orbits of embeddings $E \hookrightarrow \bG$ are meagre. On the other hand, there are Gurarij spaces extending $E$ with property $U$. \end{enumerate} 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 Avilés et al.
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