On relations between weak approximation properties and their inheritances to subspaces

It is shown that for the separable dual X^∗ of a Banach space X, if X^∗ has the weak approximation property, then X^∗ has the metric weak approximation property. We introduce the properties W^∗D and MW^∗D for Banach spaces. Suppose that M is a closed subspace of a Banach space X such that M^⊥ is complemented in the dual space X^∗, where for all m∈M}. Then it is shown that if a Banach space X has the weak approximation property and W^∗D (respectively, metric weak approximation property and MW^∗D), then M has the weak approximation property (respectively, bounded weak approximation property).     

Weak and quasi approximation properties in Banach spaces

In this paper, we introduce weak versions (the weak approximation property, the bounded weak approximation property, and the quasi approximation property) of the approximation property and derive various characterizations of these properties. And we show that if the dual of a Banach space X has the weak approximation property (respectively the bounded weak approximation property), then X itself has the weak approximation property (respectively the bounded weak approximation property). Also we observe that the bounded weak approximation property is closely related to the quasi approximation property.     

On dual and three space problems for the compact approximation property

We introduce the properties W^∗D and BW^∗D for the dual space of a Banach space. And then solve the dual problem for the compact approximation property (CAP): if X^∗ has the CAP and the W^∗D, then X has the CAP. Also, we solve the three space problem for the CAP: for example, if M is a closed subspace of a Banach space such that M^⊥ is complemented in X^∗ and X^∗ has the W^∗D, then X has the CAP whenever X/M has the CAP and M has the bounded CAP. Corresponding problems for the bounded compact approximation property are also addressed.     

Compactness in

This paper is concerned with compactness for some topologies on the collection of bounded linear operators on Banach spaces. New versions of the Eberlein–Šmulian theorem and Day's lemma in the collection are established. Also we obtain a partial solution of the dual problem for the quasi approximation property, that is, it is shown that for a Banach space X if X^** is separable and X^* has the quasi approximation property, then X has the quasi approximation property.     

Lifting bounded approximation properties from Banach spaces to their dual spaces

Based on a new reformulation of the bounded approximation property, we develop a unified approach to the lifting of bounded approximation properties from a Banach space X to its dual X^*. This encompasses cases when X has the unique extension property or X is extendably locally reflexive. In particular, it is shown that the unique extension property of X permits to lift the metric A-approximation property from X to X^*, for any operator ideal A, and that there exists a Banach space X such that X,X**,… are extendably locally reflexive, but X^*,X***,… are not.     

The weak metric approximation property and ideals of operators

We study the weak metric approximation property introduced by Lima and Oja. We show that a Banach space X has the weak metric approximation property if and only if F(Y,X), the space of finite rank operators, is an ideal in W(Y,X∗∗), the space of weakly compact operators for all Banach spaces Y.     

The asymptotically commuting bounded approximation property of Banach spaces

We introduce and study the asymptotically commuting bounded approximation property of Banach spaces. This property is, e.g., enjoyed by any dual space with the bounded approximation property. The principal result is the following: if a Banach space X has the asymptotically λ-commuting bounded approximation property, then X is saturated with locally λ-complemented separable subspaces enjoying the λ-commuting bounded approximation property.     

A characterization of bounded convex approximation properties

The characterization of bounded approximation properties defined by arbitrary operator ideals due to Oja is extended to bounded convex approximation properties. As an application, it is shown that the unique extension property of a Banach space X enables to lift the metric convex approximation property from a Banach space X to its dual space X*.     

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.     

Genericity and amalgamation of classes of Banach spaces

We study universality problems in Banach space theory. We show that if A is an analytic class, in the Effros-Borel structure of subspaces of C([0,1]), of non-universal separable Banach spaces, then there exists a non-universal separable Banach space Y, with a Schauder basis, that contains isomorphs of each member of A with the bounded approximation property. The proof is based on the amalgamation technique of a class C of separable Banach spaces, introduced in the paper. We show, among others, that there exists a separable Banach space R not containing L¹(0,1) such that the indices β and rND are unbounded on the set of Baire-1 elements of the ball of the double dual R∗∗ of R. This answers two questions of H.P. Rosenthal.We also introduce the concept of a strongly bounded class of separable Banach spaces. A class C of separable Banach spaces is strongly bounded if for every analytic subset A of C there exists Y∈C that contains all members of A up to isomorphism. We show that several natural classes of separable Banach spaces are strongly bounded, among them the class of non-universal spaces with a Schauder basis, the class of reflexive spaces with a Schauder basis, the class of spaces with a shrinking Schauder basis and the class of spaces with Schauder basis not containing a minimal Banach space X.     

The approximation properties determined by operator ideals

We introduce the notion of the right approximation property with respect to an operator ideal A and solve the duality problem for the approximation property with respect to an operator ideal A, that is, a Banach space X has the approximation property with respect to A ^d whenever X* has the right approximation property with respect to an operator ideal A. The notions of the left bounded approximation property and the left weak bounded approximation property for a Banach operator ideal are introduced and new symmetric results are obtained. Finally, the notions of the p-compact sets and the p-approximation property are extended to arbitrary Banach operator ideals. Known results of the approximation property with respect to an operator ideal and the p-approximation property are generalized.     

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.     

The fixed point property via dual space properties

A Banach space has the weak fixed point property if its dual space has a weak^∗ sequentially compact unit ball and the dual space satisfies the weak^∗ uniform Kadec-Klee property; and it has the fixed point property if there exists ε>0 such that, for every infinite subset A of the unit sphere of the dual space, A∪(−A) fails to be (2−ε)-separated. In particular, E-convex Banach spaces, a class of spaces that includes the uniformly nonsquare spaces, have the fixed point property.     

Lipschitz-type functions on metric spaces

In order to find metric spaces X for which the algebra Lip^∗(X) of bounded Lipschitz functions on X determines the Lipschitz structure of X, we introduce the class of small-determined spaces. We show that this class includes precompact and quasi-convex metric spaces. We obtain several metric characterizations of this property, as well as some other characterizations given in terms of the uniform approximation and the extension of uniformly continuous functions. In particular we show that X is small-determined if and only if every uniformly continuous real function on X can be uniformly approximated by Lipschitz functions.     

Compact adjoint operators and approximation properties

This paper is concerned with the space of all compact adjoint operators from dual spaces of Banach spaces into dual spaces of Banach spaces and approximation properties. For some topology on the space of all bounded linear operators from separable dual spaces of Banach spaces into dual spaces of Banach spaces, it is shown that if a bounded linear operator is approximated by a net of compact adjoint operators, then the operator can be approximated by a sequence of compact adjoint operators whose operator norms are less than or equal to the operator norm of the operator. Also we obtain applications of the theory and, in particular, apply the theory to approximation properties.     

Three-space problems for the bounded compact approximation property

In this paper, the notion of the bounded compact approximation property (BCAP) of a pair [Banach space and its subspace] is used to prove that if X is a closed subspace of L∞ with the BCAP, then L∞/X has the BCAP. We also show that X* has the λ-BCAP with conjugate operators if and only if the pair (X, Y) has the λ-BCAP for each finite codimensional subspace Y∈X. Let M be a closed subspace of X such that M⊥ is complemented in X*. If X has the (bounded) approximation property of order p, then M has the (bounded) approximation property of order p.     

The approximation property in terms of the approximability of weak∗-weak continuous operators

By a well-known result of Grothendieck, a Banach space X has the approximation property if and only if, for every Banach space Y, every weak∗-weak continuous compact operator T:X∗→Y can be uniformly approximated by finite rank operators from X⊗Y. We prove the following “metric” version of this criterion: X has the approximation property if and only if, for every Banach space Y, every weak∗-weak continuous weakly compact operator T:X∗→Y can be approximated in the strong operator topology by operators of norm ?‖T‖ from X⊗Y. As application, easier alternative proofs are given for recent criteria of approximation property due to Lima, Nygaard and Oja.     

Borel properties of linear operators

Given an injective bounded linear operator T:X→Y between Banach spaces, we study the Borel measurability of the inverse map T−1:TX→X. A remarkable result of Saint-Raymond (Ann. Inst. Fourier (Grenoble) 26 (1976) 211-256) states that if X is separable, then the Borel class of T⁻¹ is α if, and only if, X∗ is the αth iterated sequential weak∗-closure of T∗Y∗ for some countable ordinal α. We show that Saint-Raymond's result holds with minor changes for arbitrary Banach spaces if we assume that T has certain property named co-σ-discreteness after Hansell (Proc. London Math. Soc. 28 (1974) 683-699). As an application, we show that the Borel class of the inverse of a co-σ-discrete operator T can be estimated by the image of the unit ball or the restrictions of T to separable subspaces of X. Our results apply naturally when X is a WCD Banach space since in this case any injective bounded linear operator defined on X is automatically co-σ-discrete.     

Sigma-locally uniformly rotund and sigma-weak Kadets dual norms

The dual X^∗ of a Banach space X admits a dual σ-LUR norm if (and only if) X^∗ admits a σ-weak^∗ Kadets norm if and only if X^∗ admits a dual weak^∗ LUR norm and moreover X is σ-Asplund generated.     

Distortion and stability of the fixed point property for non-expansive mappings

Let X be a Banach space. We say that X satisfies the fixed point property (weak fixed point property) if every non-expansive mapping defined from a convex closed bounded (convex weakly compact) subset of X into itself has a fixed point. We say that X satisfies the stable fixed point property (stable weak fixed point property) if the same is true for every equivalent norm which is close enough to the original one. Denote by P(X) the set formed by all equivalent norms with the topology of the uniform convergence on the unit ball of X. We prove that the subset of P(X) formed by the norms failing the fixed point property is dense in P(X) when X is a non-distortable space which fails the fixed point property. In particular, no renorming of ?₁ can satisfy the stable fixed point property. Furthermore, we show some examples of distortable spaces failing the weak fixed point property, which can be renormed to satisfy the stable weak fixed point property. As a consequence we prove that every separable Banach space can be renormed to satisfy the stable weak fixed point property.