Generic orbits and type isolation in the Gurarij space

Itaï Ben Yaacov (1) , C. Ward Henson (2)

1 ICJ - Institut Camille Jordan
2 UIUC Department of Mathematics

We study the question of when the space of embeddings of a separable Banach space $E$ in the separable Gurarij space $\mathbf{G}$ admits a generic orbit under the action of linear isometry group of $\mathbf{G}$. The question is recast in model-theoretic terms, namely type isolation and the existence of prime models, albeit without use of formal logic. We show that if the set of isolated types over $E$ is dense then a dense $G_\delta$ orbit exists, and otherwise all orbits are meagre. We then study some (families of) examples with respect to this dichotomy. We also point out that the class of Gurarij spaces 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.

Mots clés: Gurarij space; Banach space; isolated type; atomic model; prime model; group action; generic orbit

Domaines: Analyse fonctionnelle [math.FA]; Logique [math.LO]

Itaï Ben Yaacov, C. Ward Henson. Generic orbits and type isolation in the Gurarij space. 2014. ⟨hal-00755022v4⟩