James’ quasi-reflexive space is not isomorphic to any subspace of its dual

We prove that each non-reflexive subspace ofJ ^* contains a subspace isomorphic toJ ^* and complmented inJ ^*. Consequences are thatJ is not isomorphic to any subspace ofJ ^*, and that every reflexive subspace ofJ is contained in a complemented reflexive subspace ofJ.     

DE LA VALLÉE POUSSIN'S THEOREM AND WEAKLY COMPACT SETS IN ORLICZ SPACES

Abstract

The classical theorem of Dunford and Pettis identifies the bounded, uniformly integrable subsets of L¹(μ) with the relatively weakly compact sets. Another characterization of uniform integrability is given in a theorem of De La Vallée Poussin which states that a subset K of L¹ (μ) is bounded and uniformly integrable if and only if there is an N-function F so that sup{f F(f)dμ: f ε K} < ∞. De La Vallée Poussin's theorem is the focal point of the fmt part of this paper as well as the driving force for the results in the second part. We refine and improve this theorem in several directions. The theorem of De La Vallée Poussin does not, for instance, specify just how well the function F can be chosen. It gives little additional information in case the set in question is relatively norm compact in L¹ (μ). Finally it gives no information on the structure of the set in the corresponding Band space of F-integrable functions. More specifically we establish the fact that a subset K of L¹ is relatively compact if and only if there is an N-function F ε δ' so that K is relatively compact in L*_F. Furthermore we prove that a subset K of L¹ is relatively weakly compact if and only if there is an N-function F ε δ' so that K is relatively weakly compact in L*_F. We then go on to show that a large class of non-reflexive Orlicz spaces has the weak Band-Saks property, by establishing a result for these spaces, very similar to the Dunford-Pettis theorem for L¹.     

On Orlicz sequence spaces. II

Given a separable Orlicz sequence spacel _F we investigate those Orlicz sequence spacesl _f which are isomorphic to subspaces (respectively complemented subspaces) ofl _F. We give in particular an example of a reflexive Orlicz sequence space which does not contain anyl _p, 1<p<∞, as a complemented subspace.     

Onl p-complemented copies in Orlicz spaces

Let 1<α≦β<∞ andF be an arbitrary closed subset of the interval [α,β]. An Orlicz sequence spacel ^φ (resp. an Orlicz function spaceL ^φ(μ)) with associated indices α and β is found in such a way that the set of valuesp for which thel ^p-space is isomorphic to a complemented subspace ofl ^φ (resp.L ^φ(μ)) is precisely the given setF (resp.F ∪ {2}). Also, a recent result of Hernández and Peirats [1] is extended showing that, even for the case in which the indices satisfy α_φ ^∞<2<β_φ ^∞, there exist minimal Orlicz function spacesL ^φ(μ) with no complemented copy ofl ^p for anyp ≠ 2. Supported in part by CAICYT grant 0338-84.     

Sous-espaces de L

A typical result of the paper isTheorem. LetE be a reflexive subspace ofL ¹ (Ω, A, P) [(Ω,A, P) a probability space]. IfE contains a subspace isomorphic to l^p then for every ε > 0,E contains a subspace (1 + ε) isomorphic to l^p. The technics are probability theory and ultraproducts.     

Banach spaces whose duals are isomorphic to l1(Γ)

Banach spaces X whose duals are isomorphic or isometric to l₁(Γ) are characterized by certain classes of operators on X. It is proved that a separable, conjugate space isomorphic to a complemented subspace of an L₁(S, Σ, μ) space is isomorphic to l₁; a $\text{L}$₁ space contained in a separable, conjugate space is isomorphic to a subspace of l₁.     

On the subspaces ofL p (p>2) spanned by sequences of independent random variables

Let 2<p<∞. The Banach space spanned by a sequence of independent random variables inL ^p , each of mean zero, is shown to be isomorphic tol ²,l ^p ,l ²⊕l ^p , or a new spaceX _p , and the linear topological properties ofX _p are investigated. It is proved thatX _p is isomorphic to a complemented subspace ofL ^p and another uncomplemented subspace ofL ^p , whence there exists an uncomplemented subspace ofl ^p isomorphic tol ^p . It is also proved thatX _p is not isomorphic to the previously knownℒ _p spaces. The work for this research was partially supported by the National Science Foundation GP-12997.     

Uniformly complementedl p n ’s in quasi-reflexive Banach spaces

It is shown that for every non-reflexive Banach spaceX withX ^**/X reflexive there exists a uniformly bounded sequence of projections {P _n } _n=1 ^∞ whose ranges are uniformly isomorphic to {l _p ⁿ } _{n = 1} ^∞ either forp=1, orp=2 or forp=∞. The proof uses knowledge of the transfinite dualX ^ω, ESA Schauder decompositions and proof of a similar statement for spaces with an unconditional basis due to Tzafriri.     

Reflexivity of bilattices

We study reflexivity of bilattices. Some examples of reflexive and non-reflexive bilattices are given. With a given subspace lattice ? we may associate a bilattice Σ_?. Similarly, having a bilattice Σ we may construct a subspace lattice Σ_?. Connections between reflexivity of subspace lattices and associated bilattices are investigated. It is also shown that the direct sum of any two bilattices is never reflexive.     

ℤ2-Codimensions of the Grassmann Algebra

Let E be the infinite-dimensional Grassmann algebra over a field F of characteristic zero, and consider L the F-vector space spanned by all generators of E. Let ? _l be any fixed automorphism of E of order 2 such that L is an homogeneous subspace.

Our goal is to finish the computation of the sequences of ? _l -codimensions, by finding its exact value for the unique open case, that is, when the subspace of L corresponding to the eigenvalue 1 is finite-dimensional. As a consequence we get the ?-codimensions for a large amount of arbitrary automorphisms ? of E of order 2.     

On the complemented subspaces problem

A Banach space is isomorphic to a Hilbert space provided every closed subspace is complemented. A conditionally σ-complete Banach lattice is isomorphic to anL _p -space (1≤p<∞) or toc ₀(Γ) if every closed sublattice is complemented.     

Best simultaneous approximation in L∞(μ, X)

Let Y be a reflexive subspace of the Banach space X, let (Ω, Σ, μ) be a finite measure space, and let L_∞(μ, X) be the Banach space of all essentially bounded μ ‐Bochner integrable functions on Ω with values in X, endowed with its usual norm. Let us suppose that Σ₀ is a sub‐σ ‐algebra of Σ, and let μ₀ be the restriction of μ to Σ₀. Given a natural number n, let N be a monotonous norm in ?ⁿ . We prove that L_∞(μ, Y) is N ‐simultaneously proximinal in L_∞(μ,X), and that if X is reflexive then L_∞(μ₀, X) is N ‐simultaneously proximinal in L_∞(μ, X) in the sense of Fathi, Hussein, and Khalil [3]. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

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 ofL ^p(E) (1≦p<∞) is stable. Our main result asserts that every infinite dimensional stable Banach space containsl ^p, for somep, 1≦p<∞. This is a generalization of a theorem due to D. Aldous: every infinite dimensional subspace ofL ¹ containsl ^p, for somep in the interval [1, 2].     

Extension of Operators Defined on Reflexive Subspaces of L 1 or L 1/H 1 0

Inperpolation theory is used to develop a general pattern for proving the extension theorems mentioned in the title. In the case where the range space G is a w ^*-closed subspace of L or H with reflexive annihilator F, a necessary and sufficient condition on G is found for such an extension to always be possible. Specifically, F must be Hilbertian and become complemented in L ^p (1 < p 2) after a suitable change of density. Bibliography: 17 titles.     

Jordan Derivations of Reflexive Algebras

Let ${\mathcal L}Let L{\mathcal L} be a subspace lattice on a Banach space X and suppose that ú{L ? L: L_- < X}=X{\vee\{L\in\mathcal L: L_- < X\}=X} or ${\land\{L_- : L \in \mathcal L, L>(0)\}=(0)}${\land\{L_- : L \in \mathcal L, L>(0)\}=(0)} . Then each Jordan derivation from AlgL{\mathcal L} into B(X) is a derivation. This result can apply to completely distributive subspace lattice algebras, J{\mathcal J} -subspace lattice algebras and reflexive algebras with the non-trivial largest or smallest invariant subspace.     

Über absolut repräsentierende Systeme aus Quasipolynomen in Räumen analytischer Funktionen

Criteria for Orlicz spaces containing an isomorphic as well as an almost isometric complemented copy of l¹ are given. The same is made for copies of l^∞ and c₀ in Orlicz spaces and in the subspace of order continuous elements in Orlicz space, respectively. The Hahn-Banach theorem is not used in the constructions of the projections.     

Weakly compact operators onH ∞

We prove that a weakly compact operator fromH or any of its even duals into an arbitrary Banach space is uniformly convexifying. By using this, we establish three dicothomies: (1) every operator defined onH or any of its even duals either fixes a copy ofl or factors through a Banach space having the Banach-Saks property; (2) every quotient ofH or any of its even duals either contains a copy ofl or is super-reflexive; (3) every subspace ofL ¹/H ₀ ¹ or any of its even duals either contains a complemented copy ofl ¹ or is super-reflexive.     

Large spaces of symmetric matrices of bounded rank are decomposable ∗

Let k and n be positive integers such that kn. Let S_n (F) denote the space of all n×n symmetric matrices over the field F with char F≠2. A subspace L of S_n (F) is said to be a k-subspace if rank Ak for every A?L.

Now suppose that k is even, and write k=2r. We say a k∥-subspace of S_n (F) is decomposable if there exists in Fⁿ a subspace W of dimension n?r such that x^tAx=0 for every x?W A?L.

We show here, under some mild assumptions on k n and F, that every k∥-subspace of S_n (F) of sufficiently large dimension must be decomposable. This is an analogue of a result obtained by Atkinson and Lloyd for corresponding subspaces of F^m,n .     

Lattice-universal Orlicz spaces on probability spaces

Lattice-universal Orlicz function spacesL ^F _α,β[0, 1] with prefixed Boyd indices are constructed. Namely, given 0<α<β<∞ arbitrary there exists Orlicz function spacesL ^F _α,β[0, 1] with indices α and β such that every Orlicz function spaceL ^G [0, 1] with indices between α and β is lattice-isomorphic to a sublattice ofL ^F _α,β[0, 1]. The existence of classes of universal Orlicz spacesl ^Fα,β(I) with uncountable symmetric basis and prefixed indices α and β is also proved in the uncountable discrete case. Partially supported by BFM2001-1284.     

Classes caractéristiques non triviales des feuilles de sous-feuilletages localement homogènes

We discuss the computation of the secondary characteristic classes and of the holonomy classes for the leaves of locally homogeneous subfoliations. Furthermore, we shall construct some examples of locally homogeneous subfoliations (F ₁, F ₂) so that a leaf (L ₁, L ₂) of (F ₁, F ₂) admits non-trivial secondary characteristic classes and non-trivial holonomy classes which cannot be obtained if we consider each leaf separately.