ICM 1998, Berlin, Aug. 18–27

Abstracts of Plenary and Invited Lectures

Section:

8. Analysis

1991 MS Classification: 46B07 46B20 46B09 Tomczak-Jaegermann, Nicole, Edmonton, Canada From finite- to infinite-dimensional phenomena in Geometric Functional Analysis, on local and asymptotic levels

We shall present several directions in which close connections can be found between the geometry and linear-metric structure of infinite-dimensional Banach spaces and finite-dimensional phenomena. The geometry of convex bodies in IR^{n} has a fundamental influence on structural phenomena in random quotients of the corresponding normed spaces; and this in turn can be used in infinite-dimensional constructions with various non-local properties. Other directions use, to various extents, a logical connection between the finite-dimensional (or local) structure of a Banach space and its infinitedimensional structure. Pairs of finite-dimensional spaces with unsually large norms of isomorphisms were discovered by Gluskin in the early 80’s (presented at the ICM ’86); these were random quotients of ℓ_{1}^{n} of dimension [n/2], say; and extensive investigations of this class of spaces by a group of researchers followed. We shall discuss analogous phenomenona for general normed spaces. Let X = (IR^{n}, K) be an n-dimensional normed space with unit ball K. A normed space E = (E, P_{E}(K)), where E is a “random” [n/2]-dimensional subspace of IR^{n} and P_{E}(K) is the image of K under the projection onto E, turns out to have very few well-bounded operators, depending on the volume of K. Precise quantitative formulations are asymptotic, as n → ∞. On the other hand, the volume of K also controls “closeness” of P_{E}(K) to an ellipsoid. This indicates a type of curious dichotomy for general random spaces. We shall show how the above phenomena can be used in the constructions of infinite-dimensional Banach spaces that do not admit continuous operators, or sequences of operators, with natural (algebraic) properties. In fact, such constructions require just a few canonical operations, starting from an arbitrary Banach space, non-isomorphic to ℓ_{2}. This in particular leads to characterizations of a Hilbert space. Another point of view that we shall discuss depends on stabilization of the structure of a given Banach space X over infinite-dimensional subspaces. For example, if X has a basis, the asymptotic structure of X comprises all finite-dimensional subspaces spanned by blocks of the basis living sufficiently far along the basis. So called asymptotic-ℓ_{p} spaces, for which the asymptotic structure is the simplest possible, generalize classical spaces ℓ_{p}, and they may still have an inherently different (and more interesting) geometry. As discovered in a series of deep results presented at the ICM ’94 in the talks by Gowers and by Odell and Schlumprecht, infinite-dimensional phenomena in general do not stabilize. Thus, a bridge between finite- and infinite-dimensional properties of Banach spaces is formed by localizing natural structural invariants to families of finite subsets of IN with increasing complexity. As well, Gowers’ Dichotomy Theorem may immediately reduce initially purely infinite-dimensional problems to local considerations.