Generic orbits and type isolation in the Gurarij space

Itaï Ben Yaacov¹ ; C. Ward Henson²

¹ Université Claude Bernard – Lyon 1 Institut Camille Jordan CNRS UMR 5208 43 boulevard du 11 novembre 1918 69622 Villeurbanne Cedex, France URL: <a href="http://math.univ-lyon1.fr/~begnac/">http://math.univ-lyon1.fr/~begnac/</a>
² University of Ilinois at Urbana-Champaign Urbana, IL 61801, U.S.A. URL: <a href="http://www.math.uiuc.edu/~henson/">http://www.math.uiuc.edu/~henson/</a>

Fundamenta Mathematicae, Tome 237 (2017) no. 1, pp. 47-82.

Résumé

We study the question of when the space of embeddings of a separable Banach space $E$ into the separable Gurarij space $\mathbf {G}$ admits a generic orbit under the action of the linear isometry group of $\mathbf {G}$. The question is recast in model-theoretic terms, namely type isolation and the existence of prime models. We characterise isolated types over $E$ using tools from convex analysis. 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.

Détail

DOI : 10.4064/fm193-3-2016

Keywords: study question space embeddings separable banach space separable gurarij space mathbf admits generic orbit under action linear isometry group mathbf question recast model theoretic terms namely type isolation existence prime models characterise isolated types using tools convex analysis set isolated types dense dense delta orbit exists otherwise orbits meagre study families examples respect dichotomy point out class gurarij spaces class models aleph categorical theory quantifier elimination calculate density character space types answering question avil