The problem of envelopes for Banach spaces

LetX be a Banach space. A Banach spaceY is an envelope ofX if (1)Y is finitely representable inX; (2) any Banach spaceZ finitely representable inX and of density character not exceeding that ofY is isometric to a subspace ofY. Lindenstrauss and Pelczynski have asked whether any separable Banach space has a separable envelope. We give a negative answer to this question by showing the existence of a Banach space isomorphic tol ₂, which has no separable envelope. A weaker positive result holds: any separable Banach space has an envelope of density character ≦ℵ₁ (assuming the continuum hypothesis).     

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.     

Bidual renormings of Banach spaces

Given a separable Banach space X with no isomorphic copies of ℓ₁ and a separable subspace Y of its bidual, we provide a sufficient condition on Y to ensure that X admits an equivalent norm such that the restriction to Y of the corresponding bidual norm is midpoint locally uniformly rotund. This result applies to the separable subspaces of the bidual of a Banach space with a shrinking unconditional Schauder basis and to the bidual of the James space.     

Banach spaces of universal disposition

In this paper we present a method to obtain Banach spaces of universal and almost-universal disposition with respect to a given class M of normed spaces. The method produces, among others, the only separable Banach space of almost-universal disposition with respect to the class F of finite-dimensional spaces (Gurari? space G); or the only, under CH, Banach space with density character the continuum which is of universal disposition with respect to the class S of separable spaces (Kubis space K). We moreover show that K is isomorphic to an ultrapower of the Gurari? space and that it is not isomorphic to a complemented subspace of any C(K)-space. Other properties of spaces of universal disposition are also studied: separable injectivity, partially automorphic character and uniqueness.     

Asymptotically isometric copies of c0 and l in certain Banach spaces

If X is a separable Banach space, then X∗ contains an asymptotically isometric copy of l¹ if and only if there exists a quotient space of X which is asymptotically isometric to c₀. If X is an infinite-dimensional normed linear space and Y is any Banach space containing an asymptotically isometric copy of c₀, then L(X,Y) contains an isometric copy of l^∞. If X and Y are two infinite-dimensional Banach spaces and Y contains an asymptotically isometric copy of c₀, then contains a complemented asymptotically isometric copy of c₀.     

Linearly ordered compacta and Banach spaces with a projectional resolution of the identity

We construct a compact linearly ordered space Kω₁ of weight ℵ₁, such that the space C(Kω₁) is not isomorphic to a Banach space with a projectional resolution of the identity, while on the other hand, Kω₁ is a continuous image of a Valdivia compact and every separable subspace of C(Kω₁) is contained in a 1-complemented separable subspace. This answers two questions due to O. Kalenda and V. Montesinos.     

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.     

Large sublattices in subsets of Banach lattices

A classical problem (initially studied by N. Kalton and A. Wilansky) concerns finding closed infinite dimensional subspaces of X / Y, where Y is a subspace of a Banach space X. We study the Banach lattice analogue of this question. For a Banach lattice X, we prove that X / Y contains a closed infinite dimensional sublattice under the following conditions: either (i) Y is a closed infinite codimensional subspace of X, and X is either order continuous or a C(K) space, where K is a compact subset of \({\mathbb {R}}^n\); or (ii) Y is the range of a compact operator.     

A non-archimedean Dugundji extension theorem

We prove a non-archimedean Dugundji extension theorem for the spaces C*(X, C* (X, K) of continuous bounded functions on an ultranormal space X with values in a non-archimedean non-trivially valued complete field K. Assuming that K is discretely valued and Y is a closed subspace of X we show that there exists an isometric linear extender T: C* (Y, K) → K* (X, K) if X is collectionwise normal or Y is Lindelöf or K is separable. We provide also a self contained proof of the known fact that any metrizable compact subspace Y of an ultraregular space X is a retract of X.     

Smooth extensions of functions on separable Banach spaces

Let X be a Banach space with a separable dual X*. Let ${Y\subset X}Let X be a Banach space with a separable dual X*. Let Y ì X{Y\subset X} be a closed subspace, and f:Y?\mathbbR{f:Y\rightarrow\mathbb{R}} a C ¹-smooth function. Then we show there is a C ¹ extension of f to X.     

Compactness and collective compactness in spaces of compact operators

This is a study of compactness in (a) spaces K_b(X, Y) of compact linear operators, (b) injective tensor products $\text{X}\text{}\text{?}\text{bo}{}_{\mathrm{?}}\text{Y}$, and (c) spaces L_c(X, Y) of continuous linear operators, and its various relationships with equicontinuity and collective compactness. Among the applications is a result on factoring compact sets of compact operators compactly and uniformly through one and the same reflexive Banach space.     

Complementably universal Banach spaces, II

The two main results are:

A.

If a Banach space X is complementably universal for all subspaces of c₀ which have the bounded approximation property, then X^∗ is non-separable (and hence X does not embed into c₀).

B.

There is no separable Banach space X such that every compact operator (between Banach spaces) factors through X.

Theorem B solves a problem that dates from the 1970s.     

On a generalized Mazur-Ulam question: Extension of isometries between unit spheres of Banach spaces

We call a Banach space X admitting the Mazur-Ulam property (MUP) provided that for any Banach space Y, if f is an onto isometry between the two unit spheres of X and Y, then it is the restriction of a linear isometry between the two spaces. A generalized Mazur-Ulam question is whether every Banach space admits the MUP. In this paper, we show first that the question has an affirmative answer for a general class of Banach spaces, namely, somewhere-flat spaces. As their immediate consequences, we obtain on the one hand that the question has an approximately positive answer: Given ε>0, every Banach space X admits a (1+ε)-equivalent norm such that X has the MUP; on the other hand, polyhedral spaces, CL-spaces admitting a smooth point (in particular, separable CL-spaces) have the MUP.     

On spaces admitting no ℓ p or c 0 spreading model

It is shown that for each separable Banach space X not admitting ? ₁ as a spreading model there is a space Y having X as a quotient and not admitting any ? _p for 1 ≤ p < ∞ or c ₀ as a spreading model. We also include the solution to a question of Johnson and Rosenthal (Studia Math 43:77–92, 1972) on the existence of a separable space not admitting as a quotient any space with separable dual.     

The Bishop-Phelps-Bollobás theorem for operators

We prove the Bishop-Phelps-Bollobás theorem for operators from an arbitrary Banach space X into a Banach space Y whenever the range space has property β of Lindenstrauss. We also characterize those Banach spaces Y for which the Bishop-Phelps-Bollobás theorem holds for operators from ?₁ into Y. Several examples of classes of such spaces are provided. For instance, the Bishop-Phelps-Bollobás theorem holds when the range space is finite-dimensional, an L₁(μ)-space for a σ-finite measure μ, a C(K)-space for a compact Hausdorff space K, or a uniformly convex Banach space.     

C(α) preserving operators on separable Banach spaces

Let T be a bounded linear operator from a separable Banach space X to a Banach space Y. A necessary and sufficient condition on T for the existence of a subspace Z of X such that Z is isomorphic to C(α) and the restriction of T to Z is an isomorphism is given. The special case where X is the disc algebra is then considered and results similar to those previously obtained by the author for C(K) spaces are obtained for the disc algebra. Finally some additional results of the same type are proved for subspaces of C(K) with small annihilator.     

A note on Banach spaces containing c0 or C∞

It is shown that every separable Banach space X containing a subspace isomorphic to c₀ has a subspace Y with basis such that $\text{X}\text{Y}\text{~}\text{c}{}_0\text{C}{}_{\mathrm{\infty }}$ and the latter space has a shrinking basis and an unconditional FDD. Moreover, it is shown that X ⊕ C_∞ has a basis if X has the bounded approximation property.     

Operator spaces with prescribed sets of completely bounded maps

Suppose A is a dual Banach algebra, and a representation π:A→B(?₂) is unital, weak* continuous, and contractive. We use a “Hilbert-Schmidt version” of Arveson distance formula to construct an operator space X, isometric to ?₂⊗?₂, such that the space of completely bounded maps on X consists of Hilbert-Schmidt perturbations of π(A)⊗I_?2. This allows us to establish the existence of operator spaces with various interesting properties. For instance, we construct an operator space X for which the group K₁(CB(X)) contains Z₂ as a subgroup, and a completely indecomposable operator space containing an infinite dimensional homogeneous Hilbertian subspace.     

A renorming in some Banach spaces with applications to fixed point theory

We consider a Banach space X endowed with a linear topology τ and a family of seminorms {Rk(⋅)} which satisfy some special conditions. We define an equivalent norm ?⋅? on X such that if C is a convex bounded closed subset of (X,?⋅?) which is τ-relatively sequentially compact, then every nonexpansive mapping T:C→C has a fixed point. As a consequence, we prove that, if G is a separable compact group, its Fourier-Stieltjes algebra B(G) can be renormed to satisfy the FPP. In case that G=T, we recover P.K. Lin's renorming in the sequence space ?₁. Moreover, we give new norms in ?₁ with the FPP, we find new classes of nonreflexive Banach spaces with the FPP and we give a sufficient condition so that a nonreflexive subspace of L₁(μ) can be renormed to have the FPP.     

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.