A New Infinite Game in Banach Spaces with Applications

Let X be a separable infinite-dimensional Banach space, and let A be a set of normalized sequences in X. We can consider a two-player game in X each move of which consists of player S (subspace chooser) selecting some element Y from the set cof(X) of finite-codimensional subspaces of X, and P (point chooser) responding by selecting a vector y from the unit sphere SY of Y . The game, which we shall refer to as the A-game, consists of an infinite sequence of such moves generating a sequence X1, x1, X2, x2, . . . , where Xi ∈ cof(X) and xi ∈ SXi for all i∈N. S wins the A-game if (xi)i=1∈A. Of course this game, which has its roots in the game described by W. T. Gowers [5] and in the notion of asymptotic structure [9], has certain limitations. Unlike the theory of asymptotic structure (where, for each n ∈ N, a game is considered that consists of n moves, where each move is the same as above), there is generally no unique smallest class A (depending on X) for which S has a winning strategy. However, one can hypothesize certain specific classes A for which S has a winning strategy for a given X and deduce certain structural consequences. For example, if for some K>0 we let A be the class of sequences K-equivalent to the unit vector basis of `p (1<p<∞), then any reflexive space X in which S has a winning strategy for the A-game in X, embeds into an `p-sum of finite-dimensional spaces [10]. In fact, it was the problem of classifying subspaces of `p-sums of finite-dimensional spaces that motivated the study of this game. The general theme here is to take a coordinate-free property of a space X, recast it in terms of S having a winning strategy in the A-game for a suitable class A, and then to show that X embeds into a space with an FDD (finite-dimensional decomposition) which has the “coordinatized” version of the property we started with. In addition to the `p result in [10] cited above this general theme was followed in [11] and [12]. In [11] reflexive spaces X were studied for which S has a winning strategy for both games corresponding to the classes Ap of normalized basic sequences with an `p-lower estimate and A of normalized basic sequences with an `q-upper estimate (1<q≤p<∞). The end result was that X embeds into a reflexive space with an FDD such that every block sequence satisfies `p-lower and `q-upper estimates. A consequence of this is that one can construct a separable, reflexive space universal for the class of separable, uniformly convex spaces or, more generally, for the class Cω={X : X is separable, reflexive ,Sz(X)≤ω, Sz(X∗)≤ω}, where Sz(Y ) denotes the Szlenk index of a separable Banach space Y . Recently an alternative proof of the universal result was given [3] using powerful set-theoretical notions (although, the FDD structural results cannot be obtained in this way). We should also note that a set-theoretical study of A-games was given by C. Rosendal [14].