Ward Henson (UIUC)
Continuous model theory and Gurarij's universal homogeneous separable Banach space
Abstract: Gurarij’s Banach space was constructed in the 1960s using a metric version of a Fraïssé construction; it is universal isometrically (for separable Banach spaces) and homogeneous in an almost-isometric sense relative to its finite dimensional subspaces. It is the analogue (for Banach spaces) of such structures as the random graph and Urysohn’s metric space. General results in Banach space theory from the 1960s show that its dual space is of the form $L^1(\mu)$ for some measure $\mu$, so it falls into the important class of ``classical Banach spaces,’’ a fact that is far from obvious based on the original construction. Wolfgang Lusky showed in the 1970s that the Gurarij space is isometrically unique, a surprising result. He also indicated that the set of smooth points of norm 1 is an orbit of its automorphism group. In this talk it will be shown how these results can be seen and improved using continuous model theory. In particular, the class of separable Gurarij spaces can be realized as the class of separable models of a certain continuous theory $T$ (of unit balls of Banach spaces); this theory has quantifier elimination and is the model completion of the theory of all Banach spaces. An optimal amalgamation result due to the speaker yields a simple formula for the induced metric on the type spaces of $T$ over sets of parameters, which is the key to the applications that will be discussed in this talk. A highlight of recent research, proved in joint work with Itaï Ben Yaacov, is the following: let $X$ be Gurarij's space and let $E$ be a finite dimensional space whose unit ball is polyhedral (i.e., the convex hull of a finite set). There is an isometric linear embedding $S$ of $E$ into $X$ such that $S(E)$ has the unique Hahn-Banach extension property in $X$; moreover, the set of all such embeddings forms a full orbit under the action of the automorphism group of $X$. Model-theoretically this situation is equivalent to saying that $(X,a)_{a \in S(E)}$ is an atomic model of its theory. |
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