Event Detail

Fri Apr 5, 2013
60 Evans Hall, 4:10–6 PM
Logic Colloquium
C. Ward Henson (University of Illinois at Urbana-Champaign)
Continuous Model Theory and Gurarij’s Universal, Homogeneous, Separable Banach Space

Gurarij’s Banach space was constructed in the 1960s using a metric version of a Fraisse construction; it is universal isometrically (for separable Banach spaces) and is 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 L1(µ) for some measure µ, 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 extended 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 yields a simple formula for the induced metric on the type spaces of T over sets of parameters, which is a key to the applications that will be discussed in this talk. A highlight of recent research is the following: let X be Gurarij’s space and let E be any finite dimensional space. Then there is an isometric linear embedding J of E into X such that J(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 expanded by naming all elements of J(E) is an atomic model of its theory. This is joint work with Itai Ben Yaacov, and our paper is available at the arXiv. [Note: Earlier Gurarij’s name was transliterated “Gurarii” and in the west it is always pronounced “Ger-ar-eee”.]