Stable Banach Spaces Research Articles

Coarse embeddings into superstable spaces

Krivine and Maurey proved in 1981 that every stable Banach space contains almost isometric copies of lp, for some p ∊ [1, ∞). In 1983, Raynaud showed that if a Banach space uniformly embeds into a superstable Banach space, then X must contain an isomorphic copy of lp, for some p ∊ [1, ∞). In these notes, we show that if a Banach space coarsely embeds into a superstable Banach space, then X has a spreading model isomorphic to lp, for some p ∊ [1, ∞). In particular, we obtain that there exist reflexive Banach spaces which do not coarsely embed into any superstable Banach space.

Types on stable Banach spaces

We prove a geometric characterization of Banach space stability. We show that a Banach space X is stable if and only if the following condition holds. Whenever X is an ultrapower of X and B is a ball in X , the intersection B ∩ X can be uniformly approximated by finite unions and intersections of balls in X ; furthermore, the radius of these balls can be taken arbitrarily close to the radius of B , and the norm of their centers arbitrarily close to the norm of the center of B . The preceding condition can be rephrased without any reference to ultrapowers, in the language of types, as follows. Whenever τ is a type of X , the set τ−1[0, r] can be uniformly approximated by finite unions and intersections of balls in X ; furthermore, the radius of these balls can be taken arbitrarily close to r , and the norm of their centers arbitrarily close to τ (0). We also provide a geometric characterization of the real-valued functions which satisfy the above condition.

A representation of stable Banach spaces

We show that any separable stable Banach space can be represented as a group of isometries on a separable reflexive Banach space, which extends a result of S. Guerre and M. Levy. As a consequence, we can then represent homeomorphically its space of types.

Weakly stable Banach spaces and the Banach-Saks properties

In [9] J. L. Krivine and B. Maurey introduced the class of stable Banach spaces: a separable Banach space is called stable if for every pair of bounded sequences (xn)n, (yn)n and for every pair of ultrafilters on the natural numbers we have

Quotients and interpolation spaces of stable Banach spaces

Quotients and interpolation spaces of stable Banach spaces

Weakly stable Banach spaces

The class of stable Banach spaces, inspired by the stability theory in mathematical logic, was introduced by Krivine and Maurey and provided the proper context for the abstract formulation of Aldous’ result of subspaces ofL 1. In this paper we study the wider class of weakly stable Banach spaces, where the exchangeability of the iterated limits occurs only for sequences belonging to weakly compact subsets, introduced independently by Garling (in an earlier unpublished version of his expository paper on stable Banach spaces brought recently to our attention) and by the authors. Taking into account Rosenthal’s application of the study of pointwise compact sets of Baire-1 functions (Rosenthal compact spaces) in the study of Banach spaces (for whichl 1 does not embed isomorphically) and of the study of Rosenthal compact sets by Rosenthal and Bourgain-Fremlin-Talagrand, we prove the following analogue of the Krivine-Maurey theorem for weakly stable spaces:If X is infinite dimensional and weakly stable then either l p for some p≧1or co embeds isomorphically in X (§1). Garling (in the above reference) proved this result under the additional assumption thatX* is separable. We also construct an example of a Banach spaceX which is weakly stable, without an equivalent stable norm, and such thatl 2 embeds isomorphically in every infinite dimensional subspace ofX (§3).

Embedding unconditional stable banach spaces into symmetric stable banach spaces

In this paper we prove the following result which solves a question raised by A. Pelczynski: “Every stable Banach space with an unconditional basis is isomorphic to a complemented subspace of some stable Banach space with a symmetric basis.” Moreover, we show that all the interpolation spacesl p ,l q θ,X,1 1≦p, q<∞ andX stable, are stable.

L q -subspaces of stablep-Banach spaces, 0&lt;p?1

Each infinite dimensional subspace ofL p (0<p≦1) is shown to contain a copy of somel q p≦q<∞, using arguments similar to the ones that appearin Krivine and Maurey's paper concerning stable Banach spaces. Generally speaking, ifX is a stable infinite dimensionalp-Banach space, with 0<p≦1, then, there exists aq(p≦q<∞), such that,X contains (1+e)-isomorphic copies ofl q , for all e>0. Moreover, it is possible to prove that if a stablep-Banach space, 0<p≦1, contains an isomorphic copy ofl q,p≦q<∞, then, it also contains (1+e) -isomorphic copies ofl q , for all e>0.

Espaces de Banach superstables, distances stables et homeomorphismes uniformes

We introduce here the notion of superstable Banach space, as the superproperty associated with the stability property of J. L. Krivine and B. Maurey. IfE is superstable, so are theL p (E) for eachp∈[1, +∞[. If the Banach spaceX uniformly imbeds into a superstable Banach space, then there exists an equivalent invariant superstable distance onX; as a consequenceX contains subspaces isomorphic tol p spaces (for somep∈[1, ∞[). We give also a generalization of a result of P. Enflo: the unit ball ofc 0 does not uniformly imbed into any stable Banach space.

Espaces 𝑙^{𝑝} dans les sous-espaces de 𝐿¹

It is shown that every subspace E E of L 1 {L^1} contains a subspace isomorphic to l p ( E ) {l^{p(E)}} , where p ( E ) p(E) is the upper bound of the set of real p p ’s such that E E is of type p p -Rademacher. As p ( E ) p(E) is also the upper bound of the set of real p p ’s such that E E embeds into L p {L^p} , this result answers a question of H. P. Rosenthal. The proof uses the theory of stable Banach spaces developed by J. L. Krivine and B. Maurey.

Espaces de Banach stables

We define the notion of “stable Banach space” by a simple condition on the norm. We prove that ifE is a stable Banach space, then every subspace ofLp(E) (1≦p<∞) is stable. Our main result asserts that every infinite dimensional stable Banach space containslp, for somep, 1≦p<∞. This is a generalization of a theorem due to D. Aldous: every infinite dimensional subspace ofL1 containslp, for somep in the interval [1, 2].

Quelques proprietes des espaces de Banach stables

We prove that stable Banach spaces, introduced by J. L. Krivine and B. Maurey [7], are weakly sequentially complete and that every spreading model defined on a stable Banach space is stable.