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.     

On solutions to the Schroeder-Bernstein problem for Banach spaces

We investigate the geometry of the Banach spaces failing Schroeder-Bernstein Property (SBP). Initially we prove that every complex hereditarily indecomposable Banach space H is isomorphic to a complemented subspace of a Banach space S(H) that fails SBP in such a way that the only complemented hereditarily indecomposable subspaces of S(H) are those which are nearly isomorphic to H. Then we show that every Banach space having Mazur property is isomorphic to some complemented subspace of a Banach space which is not isomorphic to its square but isomorphic to its cube. Finally, we prove that if a Banach space X fails SBP then either it is not primary or the Grothendieck group K₀(L(X)) of the algebra of operators on X is not trivial.     

Remotality of Closed Bounded Convex Sets in Reflexive Spaces

Let X be a Banach space and E be a closed bounded subset of X. For x ? X, we define D(x, E) = sup{‖ x ? e‖:e ? E}. The set E is said to be remotal (in X) if, for every x ? X, there exists e ? E such that D(x, E) = ‖x ? e‖. The object of this paper is to characterize those reflexive Banach spaces in which every closed bounded convex set is remotal. Such a result enabled us to produce a convex closed and bounded set in a uniformly convex Banach space that is not remotal. Further, we characterize Banach spaces in which every bounded closed set is remotal.     

Banach空间上闭线性子空间强正交可补的充分必要条件

证明了闭的极大线性子空间是强正交可补的充分必要条件是,空间X是自反严格凸的.     

Algebra descent spectrum of operators

We say that a Banach space X satisfies the “descent spectrum equality” (in short, DSE) whenever, for every bounded linear operator T on X, the descent spectrum of T as an operator coincides with the descent spectrum of T as an element of the algebra of all bounded linear operators on X. We prove that the DSE is fulfilled by ℓ₁, all Hilbert spaces, and all Banach spaces which are not isomorphic to any of their proper quotients (so, in particular, by the hereditarily indecomposable Banach spaces [8]), but not by ℓ _p , for 1 < p ≤ ∞ with p ≠ 2. Actually, a Banach space is not isomorphic to any of its proper quotients if and only if it is not isomorphic to any of its proper complemented subspaces and satisfies the DSE.     

On Subspaces and Quotients of Banach Spaces C(K, X)

 We study the relation of to the subspaces and quotients of Banach spaces of continuous vector-valued functions , where K is an arbitrary dispersed compact set. More precisely, we prove that every infinite dimensional closed subspace of totally incomparable to X contains a copy of complemented in . This is a natural continuation of results of Cembranos-Freniche and Lotz-Peck-Porta. We also improve our result when K is homeomorphic to an interval of ordinals. Next we show that complemented subspaces (resp., quotients) of which contain no copy of are isomorphic to complemented subspaces (resp., quotients) of some finite sum of X. As a consequence, we prove that every infinite dimensional quotient of which is quotient incomparable to X, contains a complemented copy of . Finally we present some more geometric properties of spaces. Received 8 November 2000; in revised form 7 December 2001     

On Subspaces and Quotients of Banach Spaces C ( K , X )

 We study the relation of to the subspaces and quotients of Banach spaces of continuous vector-valued functions , where K is an arbitrary dispersed compact set. More precisely, we prove that every infinite dimensional closed subspace of totally incomparable to X contains a copy of complemented in . This is a natural continuation of results of Cembranos-Freniche and Lotz-Peck-Porta. We also improve our result when K is homeomorphic to an interval of ordinals. Next we show that complemented subspaces (resp., quotients) of which contain no copy of are isomorphic to complemented subspaces (resp., quotients) of some finite sum of X. As a consequence, we prove that every infinite dimensional quotient of which is quotient incomparable to X, contains a complemented copy of . Finally we present some more geometric properties of spaces.     

Relatively weakly open sets in closed balls of Banach spaces,and real <Emphasis Type="Italic">JB</Emphasis><Superscript>*</Superscript>-triples of finite rank

We prove that, given a real JB^*-triple X, there exists a nonempty relatively weakly open subset of the closed unit ball of X with diameter less than 2 (if and) only if the Banach space of X is isomorphic to a Hilbert space. Moreover we give the structure of real JB^*-triples whose Banach spaces are isomorphic to Hilbert spaces. Such real JB^*-triples are also characterized in two different purely algebraic ways.Mathematics Subject Classification (2000): 46B04, 46B22, 46L05, 46L70Partially supported by Junta de Andalucía grant FQM 0199.Revised version: 30 September 2003     

On the Nonseparable Subspaces of J(η) and C([1, η])

Let η be a regular cardinal. It is proved, among other things, that: (i) if J(η) is the corresponding long James space, then every closed subspace Y ⊆ J(η), with Dens (Y) = η, has a copy of 𝓁₂(η) complemented in J(η); (ii) if Y is a closed subspace of the space of continuous functions C([1, η]), with Dens (Y) = η, then Y has a copy of c₀(η) complemented in C([1, η]). In particular, every nonseparable closed subspace of J(ω₁) (resp. C([1, ω₁])) contains a complemented copy of 𝓁₂(ω₁) (resp.c₀(ω₁)). As consequence, we give examples (J(ω₁), C([1, ω₁]), C(V), V being the “long segment”) of Banach spaces X with the hereditary density property (HDP) (i.e., for every subspace Y ⊆ X we have that Dens (Y) = w*–Dens (Y*)), in spite of these spaces are not weakly Lindelof determined (WLD).     

Bases and quasi‐reflexivity in Fréchet spaces

A Fréchet space E is quasi‐reflexive if, either dim(E″/E) < ∞, or E″[β(E″,E′)]/E is isomorphic to ω. A Fréchet space E is totally quasi‐reflexive if every separated quotient is quasi‐reflexive. In this paper we show, using Schauder bases, that E is totally quasi‐reflexive if and only if it is isomorphic to a closed subspace of a countable product of quasi‐reflexive Banach spaces. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A characterization of the local dual spaces of a Banach space

A local dual of a Banach space X is a closed subspace of X^∗ that satisfies the properties that the principle of local reflexivity assigns to X as a subspace of X∗∗. Here we introduce a technical property which characterizes the local dual spaces of a Banach space and allows us to show new examples of local dual spaces.     

Banach spaces in various positions

We formulate a general theory of positions for subspaces of a Banach space: we define equivalent and isomorphic positions, study the automorphy index a(Y,X) that measures how many non-equivalent positions Y admits in X, and obtain estimates of a(Y,X) for X a classical Banach space such as ?p,Lp,L₁,C(ωω) or C[0,1]. Then, we study different aspects of the automorphic space problem posed by Lindenstrauss and Rosenthal; namely, does there exist a separable automorphic space different from c₀ or ?₂? Recall that a Banach space X is said to be automorphic if every subspace Y admits only one position in X; i.e., a(Y,X)=1 for every subspace Y of X. We study the notion of extensible space and uniformly finitely extensible space (UFO), which are relevant since every automorphic space is extensible and every extensible space is UFO. We obtain a dichotomy theorem: Every UFO must be either an L_∞-space or a weak type 2 near-Hilbert space with the Maurey projection property. We show that a Banach space all of whose subspaces are UFO (called hereditarily UFO spaces) must be asymptotically Hilbertian; while a Banach space for which both X and X^∗ are UFO must be weak Hilbert. We then refine the dichotomy theorem for Banach spaces with some additional structure. In particular, we show that an UFO with unconditional basis must be either c₀ or a superreflexive weak type 2 space; that a hereditarily UFO Köthe function space must be Hilbert; and that a rearrangement invariant space UFO must be either L_∞ or a superreflexive type 2 Banach lattice.     

On Banach spaces X for which Π2 (X,L 1) = N1(X,L 1)

We prove that every 2-summing operator from a Banach space X into an L ₁-space is nuclear if and only if X is isomorphic to a Hilbert space. Then we study the class of Banach spaces X for which Π₂(l ₂, X) = N ₁(l ₂, X).     

The Radon-Nikodým Property Does Not Imply the Separable Complementation Property

There is a Banach space X enjoying the Radon-Nikodým Property and a separable subspace Y which is not contained in any complemented separable subspace of X.     

Hilbert space structure and positive operators

Let X be a real Banach space. We prove that the existence of an injective, positive, symmetric and not strictly singular operator from X into its dual implies that either X admits an equivalent Hilbertian norm or it contains a nontrivially complemented subspace which is isomorphic to a Hilbert space. We also treat the nonsymmetric case.     

Yet on linear structures of norm-attaining functionals on Asplund spaces

In this paper, we show that if an Asplund space X is either a Banach lattice or a quotient space of C(K), then it can be equivalently renormed so that the set of norm-attaining functionals contains an infinite dimensional closed subspace of X^* if and only if X^* contains an infinite dimensional reflexive subspace, which gives a partial answer to a question of Bandyopadhyay and Godefroy.     

Uniform embeddings of metric spaces and of banach spaces into hilbert spaces

It is proved using positive definite functions that a normed spaceX is unifomly homeomorphic to a subset of a Hilbert space, if and only ifX is (linearly) isomorphic to a subspace of aL ₀(μ) space (=the space of the measurable functions on a probability space with convergence in probability). As a result we get thatl _p (respectivelyL _p (0, 1)), 2<p<∞, is not uniformly embedded in a bounded subset of itself. This answers negatively the question whether every infinite dimensional Banach space is uniformly homeomorphic to a bounded subset of itself. Positive definite functions are also used to characterize geometrical properties of Banach spaces. Partially supported by the National Science Foundation, Grant MCS-79-03322. Partially supported by the National Science Foundation, Grant MCS-80-06073.     

On spaces in which countably compact sets are closed, and hereditary properties

A space X is called C-closed if every countably compact subset of X is closed in X. We study the properties of C-closed spaces. Among other results, it is shown that countably compact C-closed spaces have countable tightness and under Martin's Axiom or 2^ω0<2^ω1, C-closed is equivalent to sequential for compact Hausdorff spaces. Furthermore, every hereditarily quasi-k Hausdorff space is Fréchet-Urysohn, which generalizes a theorem of Arhangel'sk in [4]. Also every hereditarily q-space is hereditarily of pointwise countable type and contains an open dense first countable subspace.     

Stampacchia’s Property,Self-duality and Orthogonality Relations

We show that if the conclusion of the well known Stampacchia Theorem on variational inequalities holds on a real Banach space X, then X is isomorphic to a Hilbert space. Motivated by this, we obtain a relevant result concerning self-dual Banach spaces and investigate some connections between properties of orthogonality relations, self-duality and Hilbert space structure. Moreover, we revisit the notion of the cosine of a linear operator and show that it can be used to characterize real Banach spaces that are isomorphic to a Hilbert space. Finally, we present some consequences of our results to quadratic forms and to evolution triples.     

Ergodic Banach spaces

We show that any Banach space contains a continuum of non-isomorphic subspaces or a minimal subspace. We define an ergodic Banach space X as a space such that E₀ Borel reduces to isomorphism on the set of subspaces of X, and show that every Banach space is either ergodic or contains a subspace with an unconditional basis which is complementably universal for the family of its block-subspaces. We also use our methods to get uniformity results. We show that an unconditional basis of a Banach space, of which every block-subspace is complemented, must be asymptotically c₀ or ?_p, and we deduce some new characterisations of the classical spaces c₀ and ?_p.