We prove three new dichotomies for Banach spaces à la W.T. Gowers' dichotomies. The three dichotomies characterise respectively the spaces having no minimal subspaces, having no subsequentially minimal basic sequences, and having no subspaces crudely finitely representable in all of their subspaces. We subsequently use these results to make progress on Gowers' program of classifying Banach spaces by finding characteristic spaces present in every space. Also, the results are used to embed any partial order of size ℵ 1 into the subspaces of any space without a minimal subspace ordered by isomorphic embeddability.

No abstract

We construct a uniformly convex hereditarily indecomposable Banach space, using similar methods as Gowers and Maurey in [GM] and the theory of complex interpolation for a family of Banach spaces of Coifman, Cwikel, Rochberg, Sagher and Weiss ( [5a]). n i=1 x i , then we say that X satisfies a lower f -estimate.Let q > 1 in R, q ′ such that 1/q + 1/q ′ = 1. Let θ ∈ ]0, 1[ , and p be the number defined by 1/p = 1 − θ + θ/q.Let S be the strip {z ∈ C/Re(z) ∈ [0, 1]}, δS its boundary, S 0 the line {z/Re(z) = 0}, S 1 the line {z/Re(z) = 1}. Let µ be the Poisson probability measure associated to the point θ for the strip S. We have µ(S 0 ) = 1 − θ. Let µ 0 be the probability measure on R defined by µ 0 (A) = µ(iA)/(1 − θ), µ 1 be the probability measure on R defined by µ 1 (A) = µ(1 + iA)/θ. Let A S be the set of analytic functions F on S, with values in c 00 , which are L 1 on δS for dµ and which satisfy the Poisson integral representation F (z 0 ) = δS F (z)dP z0 (z) on S ( this is well defined since such functions have finite ranges). If F is analytic and bounded on S, then F ∈ A S .We recall the definition of the interpolation space of a family of N -dimensional spaces from [5a]. Let . z for z in S be a family of norms on C N , equivalent with log-integrable constants, and such that z → x z is measurable for all x in C N . The interpolation space in θ is defined by the norm, where A N S denotes the image of the canonical projection from A S into the space of functions from S to C N .We generalize to the infinite-dimensional case as follows. Let {X z , z ∈ δS} be a family of Banach spaces in X , equipped with norms . z , such that for all x in c 00 , the function z → x z is measurable, and such that over vectors of finite range N , the norms . z are equivalent with log-integrable constants. Let X N z be E N X z , X N be the θ-interpolation space of the family X N z ; the interpolation space of the family in θ is completion(∪ N ∈N X N ). Now let {X t , t ∈ R} be a family of spaces in X , equipped with norms . t , such that for all t in R, X t satisfies a f -lower estimate and for all x in c 00 , the function t → x t is measurable. For vectors of range at most E N , we have f (N ) −1 x 1 ≤ x t ≤ x 1 , so that the norms . t are equivalent to . 1 with log-integrable constants. We are then allowed to define the θ-interpolation space of the family defined on δS as X t if z = it, l q if z = 1 + it. Let X θ be the class of spaces X obtained in that way.We shall sometimes use for z ∈ δS the notation . z , to mean . t if z = it, and . q if z = 1 + it. There will be no ambiguity from the context. We shall similarly use the notation . * z . The notation X N t stands for E N X t , and X N * t for E N X * t . Also, if not specified, the measure of a subset of R will be its measure for µ 0 .